Objectivity with Objects. A Modal Explanation of the Infinite

INTERNATIONAL DOCTORAL SCHOOL OF THE USC Ismael Ordóñez Miguéns PhD Thesis OBJECTIVITY WITH OBJECTS. A MODAL EXPLANATION OF THE INFINITE Santiago de Compostela, 2024 Doctoral Programme in Logic and Philosophy of Science Ph.D. THESIS OBJECTIVITY WITH OBJECTS. A MODAL EXPLANATION OF THE INFINITE Ismael Ordóñez Miguéns Directora: Concha Martínez Vidal Director: José Ferreirós Domínguez Tutora: Concha Martínez Vidal ESCOLA DE DOUTORAMENTO INTERNACIONAL DA UNIVERSIDADE DE SANTIAGO DE COMPOSTELA PROGRAMA DE DOUTORAMENTO EN LÓXICA E FILOSOFÍA DA CIENCIA SANTIAGO DE COMPOSTELA 2024 Declaración do autor da tese D. Ismael Ordóñez Miguéns Título da tese: Objectivity with Objects. A Modal Explanation of the Infinite Presento a miña tese, seguindo o procedemento adecuado ao Regulamento, e declaro que: 1. A tese abarca os resultados da elaboración do meu traballo. 2. De ser o caso, na tese faise referencia ás colaboracións que tivo este traballo. 3. Confirmo que a tese non incorre en ningún tipo de plaxio doutros autores nin de traballos presentados por min para a obtención doutros títulos. 4. A tese é a versión definitiva presentada para a súa defensa e coincide a versión impresa coa presentada en formato electrónico. E comprométome a presentar o Compromiso Documental de Supervisión no caso de que o orixinal non estea na Escola. En Santiago de Compostela, 30 de Decembro de 2023 Asdo. Ismael Ordóñez Miguéns How the heart surges when it thinks of you, Infinite One! How it sinks when it gazes down upon itself! Lamenting, it sees but misery, night and death. Klopstock, To the Infinite One iv AGRADECEMENTOS O presente escrito é o resultado dun longo proceso. Durante estes anos varias persoas —en distintos momentos e de distintas maneiras— contribuíron a que sexa posible. Sen elas, non tería visto a luz. E a elas llo quero agradecer. Quero comezar dándolle as grazas aos meus directores, Concha Martínez e José Ferreirós, polo seu seguemento e atención todo este tempo. Especialmente, pola confianza depositada en min. En segundo lugar, quero expresar o meu agradecemento a Øystein Linnebo, pola súa hospitalidade durante a miña estadía na Universidade de Oslo, por múltiples e ricas conversas e, sobre todo, pola súa inestimable axuda co último capítulo da tese. A José Pedro Úbeda quero agradecerlle por ter espertado en min o interese na teoría de conxuntos tantos anos atrás, e por seguir acompañándome no estudo desta complexa teoría. Grazas, tamén, a Martín Pereira, polos seus bos consellos docentes, e pola axuda co soporte e presentación final da tese. Non é esaxerado dicir que, sen a súa axuda, este escrito non existiría. Tampouco podo esquecerme dos amigos atopados polo camiño, que fixeron máis levadeiro o que este conleva. Especialmente, grazas aos meus compañeiros de doutorado, Violeta e Alex, con quenes compartín tantas conversas, momentos e experiencias enriquecedores en Santiago de Compostela. Grazas a Oscar Parcero, Xavier de Donato e Laura Lojo. O seu apoio e bos consellos foron imprescindibles para que esta tese chege a bo porto. Grazas aos meus pais. E grazas a Cecilia, polo seu apoio e por acompañarme neste camiño. Finalmente, quero darlle as grazas, tamén, ás institucións que fixeron económicamente posible a presente investigación. En primeiro lugar, ao Ministerio de Educación, Formación Profesional e Deportes, pola financiación a través do Programa Nacional FPU (número de axuda: FPU18/02787). E tamén por financiar a miña estadía na Universidade de Oslo baixo o programa de axudas para estadías no estranxeiro (número de axuda: EST19/00949). En segundo lugar, ao grupo de investigación Episteme, que me financiou baixo os seguintes proxectos de investigación: “Abstract Objects: For and Against”, financiado por FEDER, o Ministerio de Ciencia, Innovación e Universidades, e a Axencia Estatal de Investigación (número de proxecto: FFI2017-82534-P); e “Deflationary views in ontology and metaontology”, financiado pola Axencia Estatal de Investigación” (número de proxecto: PID2020-115482GB-I00). 30 de Diciembre 2023 Abstract The objectivity of mathematics has been questioned for two reasons: Benacerraf’s challenge and the emergence of several limitation theorems in mathematics. This has led to the search for new explanations of mathematical knowledge and alternatives to universist platonism. Mainly, mathematical knowledge has been reduced to logical knowledge. Against this, this dissertation argues that mathematical knowledge cannot be reduced to logical knowledge. The two best proposals defending this reduction, fictionalism and multiversism, run into serious problems. Both will be challenged to explain how mathematical language acquires its content. Then, it will be argued that they do not satisfy this test. Faced with this problem, an alternative explanation of mathematical language and mathematical knowledge will be developed using abstraction principles. As a result, mathematical knowledge will be explained through linguistic competence. Finally, the proposal will be applied to ZF set theory. Keywords objectivity, fictionalism, multiversism, abstraction principles, potentialism, set theory Resumo A obxectividade das matemáticas foi cuestionada por dúas razóns: o problema de Benacerraf e a emerxecia de numerosos teoremas de limitación en matemáticas. Isto levou á busca de novas explicacións do coñecemento matemático e de alternativas ao platonismo universista. Principalmente, o coñecemento matemático tentouse reducir a coñecemento lóxico. En contra disto, esta tese argumenta que o coñecemento matemático non pode reducirse ao coñecemento lóxico. A dúas mellores propostas que defenden esta redución, o ficcionalismo e o multiversimo, incorren en serios problemas. As dúas serán desafiadas a explicar como a linguaxe matemática adquire o seu contido. Entón, argumentarase que ningunha satisfai esta proba. Ante este problema elabórase unha explicación alternativa da linguaxe e do coñecemento matemático usando principios de abstracción. Como resultado, a explicación do coñecemento matemático pasa pola explicación da competencia lingüística. Finalmente, a proposta aplicarase á teoría de conxuntos ZF. Palabras clave obxectividade, ficcionalismo, multiversismo, principios de abstracción, potencialismo, teoría de conxuntos vi Resumen La objetividad matemática ha sido cuestionada por dos razones: el problema de Benacerraf y la emergencia de numerosos teoremas de limitación en matemáticas. Esto ha conducido a la búsqueda de nuevas explicaciones del conocimiento matemático y de alternativas al platonismo unviersista. Principalmente, se ha intendado reducir el conocimiento matemático a conocimiento lógico. En contra de esto, esta tesis argumenta que el conocimiento matemático no se puede reducir al conocimiento lógico. Las dos mejores propuestas a favor de esta reducción, el ficcionalismo y el multiversismo, incurren en serios problemas. Principalmente, las dos serán desafiadas a explicar como el lenguaje matemático adquiere su contenido. Entonces, se argumentará que ninguna pasa esta prueba. Ante este problema se elaborará una explicación alternativa del lenguaje y el conocimiento matemático usando principios de abstracción. Como resultado, la explicación del conocimiento matemático pasa por la explicación de la competencia lingüística. Finalmente, la propuesta se aplicará a la teoría de conjuntos ZF. Palabras clave objetividad, ficcionalismo, multiversismo, principios de abstracción, potencialismo, teoría de conxuntos vii Contents Resumo 1 1 Introduction 9 1.1 Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Challenges to Mathematical Objectivity 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 A Realist Conception of Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Mathematical Platonism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4 What Mathematical Platonism is Not . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.5 Varieties of Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Appendix 2.A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Appendix 2.B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Appendix 2.C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Anti-realist Pluralism Fictionalism 59 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Fictionalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.3 The Reliability Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 Full Fictionalism and The Language Acquisition Challenge . . . . . . . . . . . . . . . 74 3.5 Algebraic Fictionalism and the Consistency Challenge . . . . . . . . . . . . . . . . . . 89 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4 Realist Pluralism Multiversism 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Knowledge in the Multiverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 matemática. Con respecto a este dilema, situareime entre Scila e Caribdis. Por unha pate, este texto é unha defensa dunha tese central e negativa: O coñecemento matemático non se reduce a coñecemento lóxico. Consecuentemente, a obxectividade matemática non se reduce a obxectividade lóxica. En contra das novas formas de platonismo pluralista e tamén en contra do fictionalismo, nego que o coñecemento matemático se poida explicar exhaustivamente apelando meramente ao coñecemento da consistencia e da consecuencia lóxica. Asumindo unha concepción eminentemente semántica da linguaxe, argumentarei que o coñecemento lóxico presupón a competencia lingüística. Polo tanto, calquera redución do coñecemento matemático ao coñecemento lóxico presupón dúas cousas: primeiro, que a linguaxe matemática é significativa, e segundo, que os matemáticas son capaces de comprendela. Porén, argumentarei que nin o multiversismo nin o fictionalismo son capaces de explicar isto. É dicir, estas posicións non son capces de proporcionar unha explicación metasemántica de como o vocabulario matemático adquire o seu significado, e de como os falantes adquiren a súa competencia lingüística. Así, opoñerei ás filosofías pluralistas das matemáticas unha nova versión do Desafío da Adquisición da Linguaxe usado (30) por en contra das matemáticas clásicas —porén, esta nova versión non está directamente relacionada co argumento de Dummett, nin coas súas preferencias intuicionistas. Por outra parte, o texto é unha defensa dunha tese positiva complementaria: O coñecemento matemático pode ser explicado mediante a competencia lingüística. Así, tratarei de complementar as concepcións pluralistas das matemáticas desenvolvendo unha explicación adecuada de como o vocabulario matemático adquire o seu contido, e de como os matemáticos adquiren a súa competencia lingüística e matemática. Para esto, usarei a recente aproximación abstraccionista aos obxectos abstractos deseñada por Linnebo (81; 82). O seu programa fai un forte uso dos principios de abstracción para explicar a referencia e o coñecemento matemático. A relevancia filosófica dos principios de abstracción foi orixinalmente sinalada por Frege, na súa Grundlagen der Arithmetik. Estes principios introducen unha perspectiva reducionista en metasemántica que os fai especialmente apropiados para elaborar unha explicación epistemolóxica dos obxectos abstractos: podemos explicar como os falantes adquiren unha linguaxe enteira comprometida con estes obxectos sin presupoñer pola súa parte ningún coñecemento dos mesmos. Ademais, a adquisición dunha linguaxe mediante o uso de principios de abstracción proporciona aos falantes un coñecemento non lingüístico. É dicir, a competencia lingüística ven acompañada do coñecemento dun punñado de verdades formuladas nesa linguaxe —non non soamente de verdades lóxicas. Desta forma, o logro cognitivo en matemáticas recibe unha explicación metasemántica seguindo as liñas de (79; 82) e (115; 116). Isto garantiza un mínimo de verdade matemática obxectiva sen abandonar os obxectos matemáticos. Neste sentido, a miña posición é unha modesta vindicación do platonismo matemático —a pesar das aparencias, inda hai un lugar para a combinación de ambos. No capítulo 2 analizo a explicación máis intuitiva da obxectividade matemática: o platonismo matemático. En primeiro lugar, caracterizo a obxectividade do coñecemento empírico como a conxunción 5 de dúas teses: Independencia e Reactividade. A primeira afirma que o contido das crenzas, xuízos ou afirmacións é independente das respostas epistémicas dos axentes. A segunda afirma que o coñecemento obxectivo implica unha relación forte e contrafáctica entre os axentes e o mundo. En segundo lugar, caracterizo o platonismo matemático como a tese filosófica de que os obxectos abstractos existen independentemente de nós. De acordo co platonismo, as matemáticas son a ciencia que estuda eses obxectos. En terceiro lugar, explico en que consiste o desafío de Benacerraf e argumento que este funciona só cando se acepta que o coñecemento matemático satisfai Reactividade. Finalmente, explico por que o platonismo non involucra nin unha teoría sobre a estructura do mundo ou sobre a verdade, e desenvolvo a idea de que os obxectos matemáticos son independentes de nos. No capítulo 3 analizo o ficcionalismo, a tese filosófica de que as teorías matemáticas son ficcións, e polo tanto son falsas. O ficcionalismo reduce o coñecemento matemático ao coñecemento lóxico: coñecemento sobre a consistencia e as consecuencias desas ficcións. Neste capítulo desafío o ficcionalismo por tres razóns. En primeiro lugar, argumento que baixo dous supostos razoables non pode explicar a fiabilidade matemática. Entón, intentarei reforzar esta obxección, argumentando que o ficcionalismo non pode explicar a fiabilidade dos matemáticos inda se os devanditas supostos non son aceptados. En segundo lugar, distingo dous tipos de ficcionalismo: o ficcionalismo pleno e o ficcionalismo alxébrico. O primeiro afirma que, a pesar de que as teorías matemáticas son falsas, a linguaxe matemática é significativa e trata sobre obxectos abstractos. Pola súa parte, o segundo non atribúe ningún contido á linguaxe matemática. Usando esta distinción, defendo que se o ficcionalismo pleno pode explicar como o vocabulario matemático adquire o seu contido e como os matemáticos adquiren a súa competencia lingüística, entón o platonismo tamén pode. Pero, dado que o mellor argumento para o ficcionalismo consiste no fracaso do platonismo, isto é fatal para a primeira posición filosófica. Finalmente, defendo que o ficcionalismo alxébrico non está nunha posición máis vantaxosa. A redución do coñecemento matemático ao coñecemento lóxico presupón a competencia lingüística dos matemáticos. O ficcionalismo alxébrico non pode explicar isto a menos que, á súa vez, sexa reducido ao ficcionalismo pleno. En conclusión, o ficcionalismo debe ser complementado con unha explicación alternativa da competencia lingüística dos matemáticos. No capítulo 4 analizo o multiversismo. O multiversismo é unha nova versión pluralista do platonismo matemático que defende a existencia de múltiples universos matemáticos. En lugar dun universo único e privilexiado habería un multiverso. Neste, todos os universos son equivalentes dende o punto de vista da semántica e da metafísica. Coma o ficcionalismo, o multiversismo reduce o coñecemento matemático ao coñecemento lóxico. En primeiro lugar, explico que hai moitos tipos diferentes de multiversismo. En segundo lugar, explico os dous principais argumentos a favor do multiversismo: o argumento dende a independencia lóxica e o argumento dende a categoricidade. En terceiro lugar, argumento que as dúas versións máis radicais do multiversismo non son boas explicacións das matemáticas. Ningunha delas impón límites á extensión do multiverso. Por este motivo ambas poden reducir o coñecemento matemático ao coñecemento lóxico. O multiversismo hamkinsiano non pode facer fronte ao desafío de explicar como o vocabulario matemático adquire o seu significado. Máis concretamente, a súa explicación da referencia matemática é circular. Á súa vez, o platonismo full-blooded carece de unha noción adecuada de consistencia para explicar o coñecemento matemático. O uso desta noción presupón unha explicación 6 da competencia lingüística dos falantes. O problema é que o platonismo full-blooded tampouco pode proporcionar a devandita explicación. No capítulo 5 utilizo o enfoque abstraccionista dos obxectos matemáticos desenvolto por Linnebo (81; 82) para explicar a competencia lingüística dos matemáticos e o coñecemento matemático. En primeiro lugar, explico que son os principios de abstracción. Estes introducen un enfoque metasemántico reducionista permitindo explicar como unha comunidade de falantes é capaz de adquirir a linguaxe matemática sen presupoñer ningunha relación contrafáctica ou lóxica entre os seus membros e os feitos matemáticos. En segundo lugar, explico en que consisten os dous principais problemas para as perspectivas abstracionistas: o problema do Xulio César e o problema da mala compañía. Despois, mostro como estes problemas poden ser resoltos. Unha vez que se demostre que os princios de abstracción están en boas condicións, mostrarei como tamén poden ser empregados para explicar o coñecemento matemático. O proceso de abstracción mediante o cal os falantes extenden a súa linguaxe tamén lles proporciona coñecemento sobre unha colección de verdades matemáticas—e non soamente verdades lóxicas. Así, tal e como Linnebo (79; 82) e Rayo (115; 116) afirman, o coñecemento matemático ven da man da comprensión lingüística. Fronte á perspectiva pluralista do ficcionalismo e do multiversismo, o enfoque abstraccionista permítenos reintroducir un mínimo de obxectividade matemática. No capítulo 6 explico como os principios de abstracción son suficientes para recuperar a teoría de conxuntos de Zermelo-Fraenkel (81, Ch 3,12). En primeiro lugar, explico como a iteración dos intentos de abstracción xera unha serie de expansións do dominio. Así mesmo, explico como se poden expresar as expansións do dominio a nivel da linguaxe de obxectos usando operadores modais. En segundo lugar, contrapoño as concepcións potencialista e actualista da teoría de conxuntos. O actualismo afirma que o universo conxuntista forma unha pluralidade de obxectos. Á súa vez, o potencialismo nega isto. Entón, argumento que o segundo está en mellores condicións que o primeiro. En terceiro lugar, introduzo a concepción iterativa dos conxuntos. Unha vez feito isto, explico como a aplicación repetida dos principios de abstracción proporciona unha explicación das súas operacións primitivas: a operación ‘conxunto de’ e o conxunto potencia. Esta explicación implica o potencialismo en teoría de conxuntos. Deste xeito, expoño, por unha parte, os principios modais que describen o proceso iterativo e, por outra, aqueles principios que caracterizan a natureza específica dos conxuntos. Combinados, todos estes principios forman a teoría de conxuntos potencialista. Finalmente, mostro como esta teoría interpreta a teoría de conxuntos de Zermelo-Fraenkel sen os axiomas de Infinitude e Reemprazo. Para recuperar ambos axiomas, a teoría de conxuntos potencialista debe complementarse con principios adicionais máis fortes. Finalmente, no capítulo 7 desenvolvo un novo tipo de principios de abstracción modal. A través deles, explico como os falantes adquiren a capacidade de conceptualizar o infinito en acto mediante un proceso de abstracción. O principal beneficio da abstracción modal é que se trata dunha forma de abstracción carente de especificacións. É dicir, o proceso de abstracción non ten lugar no mundo presente, sobre un dominio de obxectos relacionados mediante a relación de unidade. En cambio, ten lugar sobre o que é posible relativamente a esta relación. Así, a mera posibilidade de que dous obxectos estean relacionados por ela é suficiente para que a abstracción introduza o correspondente obxecto abstracto especificado por eles. En primeiro lugar, descarto algunhas formas insatisfactorias de enunciar estes principios. Despois, estendo a linguaxe da teoría de conxuntos potencialista, introducindo unha serie de operadores modais. 7 Unha vez feito isto, os principios de abstracción modal pódense formalizar apropiadamente. Finalmente, mostro como os principios de abstracción modal interpretan o axioma do Infinito cando se engade á teoría de conxuntos potencialista. Deste xeito, o abstraccionismo pode explicar como os falantes captan a noción de infinito. O obxectivo principal desta investigación é o de recuperar un mínimo de obxectividade non lóxica para o coñecemento matemático ao mesmo tempo que se defende a súa condición de ciencia de obxectos abstractos, independentes de nós. O coñecemento matemático implica unha forma moi particular de logro cognitivo: a correcta comprensión da linguaxe matemática. As mellores explicacións pluralistas das matemáticas, o ficcionalismo e o pluralismo radical, non son capaces de proporcionar unha explicación satisfactoria da competencia lingüística dos matemáticos. Fronte a estas, a defensa feita do programa abstraccionista pode entenderse coma unha defensa do platonismo universista. Porén, isto non debe ser entendido coma unha refutación do ficcionalismo ou do multiversismo. Unha vez introducido un mínimo de obxectividade matemática, estas concepcións pódense defender para outras teorías matemáticas adicionais. O debate entre as dúas posicións filosóficas mencionadas está fóra do alcance do presente texto e, de ser posible a súa solución, terá que ser resolto mediante o uso de outros argumentos filosóficos. A introducción dun mínimo de obxectividade matemática só constitúe o punto de partida na exploración da multiplicidade de ficcións ou universos alternativos. 8 1 Introduction The objectivity of mathematics has been called into question because of two main reasons: Benacerraf’s challenge and limitation theorems. Let’s start with the former. According to a simple and informal characterization, mathematics is the science of numbers, functions, sets, geometrical figures and many other objects inhabiting the mathematical realm. All these objects have something in common: they are very different from those objects that surround us in our daily lives. More specifically, what makes them so unique is their non-spatiotemporal nature. Mathematical objects —whether they exist or not— are abstract objects. They are “out” of space and time. For example, people read books, people chop wood, or they climb mountains. On the contrary, nobody has seen the number 3 —and nobody will see it ever. Similarly, nobody has touched ever a function —even if real-valued functions and vectors are used by physicists to describe the world. In general, mathematical objects cannot be perceived. We cannot interact with them. The situation is very different from that of distant galaxies or objects too small to be seen. It is not simply that they are causally isolated from us because they are remote or inaccessible. They are causally ineffective and, by their very nature, do not exemplify physical properties. The peculiar metaphysical status of mathematical objects raises some suspicions about the epistemological condition of mathematics. At first glance, they are so different that they might seem to be unknowable. This is in stark contrast to empirical knowledge. We have good theories that explain how people acquire knowledge about empirical matters. People interact with medium-sized objects under stable perceptual conditions for long periods. They observe the movement of celestial bodies, the free fall of objects, the passing of seasons, and many other phenomena. They elaborate theories to explain this complex web of events. And, with a lot of effort, people manage to acquire information about the world beyond that provided by physiological channels. Of course, explanations of empirical knowledge are much more sophisticated than these superficial phrases. But the point is that there is agreement that we at least have some empirical knowledge of the world. Tuning into the world in an epistemologically relevant way is not easy. Nor is it impossible. Most explanations of empirical knowledge —at least of empirically justified belief— contend that people are causally or, generally, counterfactually connected to empirical events. And, these strong connections guarantee their epistemic reliability and the objectivity of natural science. Objectivity is —in part (23; 24; 25)— the correct knowledge of the world. In turn, mathematics resists these explanations. The abstract nature of mathematical objects makes it difficult to think what kind of counterfactual relation could exist between them and us. They are causally ineffective. Therefore, causal relations are ruled out. We cannot perceive them. No experiment will provide us with new information about them. This leaves us with few alternatives. It is much more difficult to find a channel that allows people to tune in to the mathematical realm. What kind of counterfactual relation could exist between mathematical facts and people that justifies our beliefs about them and makes mathematicians reliable professionals? Benacerraf’s challenge (7) uses the difficulty of providing an answer to this question against the objectivity of mathematics: if mathematics is a science of abstract objects, then we cannot explain mathematical knowledge and mathematical reliability. Abstract objects would be epistemically untraceable. Specifically, the challenge is based on two points. First, it concludes that there is no counterfactual connection between mathematical facts and people. Note that counterfactual relations are not necessarily causal. All that is needed for there to be a counterfactual relation between two types of events is a strong —in a sense to be specified— coordination between the variation of one and the other. In this case, the variation of the mathematical facts should affect the epistemic responses of the mathematicians. Virtually no one accepts that this is possible. Indeed, if mathematical facts are necessary —as many authors claim (74)—, then we could not make sense of possible variations of these facts. Instead, we would need to make sense of impossible variations —(123; 124; 45; 67). But, this is a controversial and difficult idea to pin down. Second, most advocates of the challenge (7; 40) contend that if there is no counterfactual relation, no suitable explanation of mathematical language and mathematical knowledge can be elaborated. We could not explain how reference is fixed, how mathematical vocabulary acquires its meaning, and how mathematicians become reliable professionals. Consequently, since mathematicians speak a meaningful language and provide us with mathematical knowledge —at least with justified mathematical beliefs—, we must rethink our view of mathematics. Unlike the natural sciences, which describe an empirical world populated by many kinds of objects, mathematics would not describe a parallel world populated by all the aforementioned inhabitants —numbers, sets, vectors, etc. Mathematical knowledge and mathematical objectivity would not be a function of the mathematical world. This leaves us with two alternatives. Either mathematical objectivity is achieved by other means —leaving strong relations to abstract objects aside—, or mathematical objectivity is abandoned. Mathematical objects are not indispensable for mathematical objectivity. At this point, we can follow Kreisel’s dictum: what matters is mathematical objectivity, not mathematical objects. Now, the best option to save the objectivity of mathematics —while reading it at face value, and keeping the image of a science of objects— is to resort to weaker linguistic and logical relations. It is a matter of turning the question around. Instead of starting from the world, we would start from language. The idea is to impose enough logical and theoretical constraints to select a mathematical theory, fix mathematical reference, and acquire a meaningful language. In this way, we could explain mathematical knowledge and the reliability of mathematicians without using strong, counterfactual relations. For example, these constraints would include the consistency of the theory, that it be powerful enough to include basic mathematical reasoning —such as arithmetic—, or that it respects the metrizations used in the natural sciences. These constraints would serve a dual function. On one side, they would select a collection of sentences: the axioms of the desired mathematical theory —the objectively correct mathematical theory. On the other side, they would select a collection of objects: the objects that satisfy the properties coached in the theory. Once reference is fixed in this domain of objects, the axioms of the theory turn out to be true. As a result, the Leibnizian ideal could finally become reality: mathematical knowledge reduces to consequence inside this ideal theory. Indeed, this picture fits well with the algebraic nature of contemporary mathematics (56; 35; 76). 10 Mathematicians select a set of axioms. These characterize the concepts they wish to study. As a system of equations with several unknowns selects its solutions, these axioms would select the objects that make them true. Then, mathematicians study what follows from these axioms. Here is where the second problem for mathematical objectivity bursts in: limitation theorems. As their name suggests, these theorems show the deductive limitations of mathematical theories as well as the expressive limitations of mathematical languages. We find the first examples of these theorems in the 19th century when Bolyai and Lobachevsky proved the logical independence of the fifth postulate with respect to the other axioms of Euclidean geometry (133, Ch. 6-7). However, the most famous independence results are Gödel’s incompleteness theorems. He proved that any recursively axiomatizable theory —i.e. a theory whose set of axioms is recursive— that includes Peano-Dedekind arithmetic and is consistent, is incomplete. There is a sentence such that the theory proves neither it nor its negation. But, what is the relevance of independence results for mathematical objectivity? Remember, we were trying to recover mathematical objectivity by laying down a collection of logical and theoretical constraints that select our desired theory —the objectively correct mathematical theory. Let’s see in detail what role these restrictions play. First, they must select a theory that is in principle manageable for people. This means a recursively axiomatizable theory. On the contrary, we would not be able to know which sentences are part of it! People would not be able to recognise the theory. Second, the theory must recover enough mathematics. At least, it must recover Peano-Dedekind arithmetic —this is a minimal theory involved in most mathematical reasonings. Third, the theory must be consistent —we are dealing with classical mathematics. Consequently, such a theory will be incomplete. Any theory selected by a set of logical and theoretical constraints of the above-mentioned kind will be incomplete. Consequently, mathematical knowledge and mathematical objectivity do not reduce to logical entailment inside an ideal theory. Indeed, we know that most of our best mathematical theories, such Zermelo-Fraenkel set theory, are highly incomplete: there are infinitely many sentences that these theories or natural extensions do not decide. Independence results are a serious drawback for mathematical objectivity. However, somebody could argue that not everything is lost. This kind of results simply show that our mathematical knowledge is far from being perfect. There are many mathematical questions for which we have no answer. Perhaps we never will. But this does not mean that people do not have mathematical knowledge. Indeed, these sort of results should be expected. No parcel of human knowledge is complete. Natural science also harbours a large number of unanswered questions. What was there before the Big Bang? How did life on earth originate? Is the universe finite or infinite? But, the lack of an answer to all these questions does not lead us to conclude that natural science is not objective —at least up to some point. Therefore, if a collection of logical and theoretical constraints selects a mathematical theory —even if an incomplete theory— and fixes mathematical reference, this would be enough to explain the objectivity of mathematicians’ limited knowledge. These constraints select the interpretation of the mathematical language. In turn, the interpretation selects a complete theory —the objectively correct mathematical theory. As a result, the theory decides all mathematical questions. And, this is so even if there are sentences that mathematicians do not know whether or not they belong to the theory. This response to independence results is simple and effective. 11 Nevertheless, there is a second kind of limitation theorem affecting the expressive power of mathematical language that tells against it: no uncontroversial collection of logical and theoretical constraints is enough to fix mathematical reference (128; 111). Those first-order theories which are powerful enough to recover Peano-Dedekind arithmetic can be interpreted in domains of objects which differ structurally from each other —first-order mathematical theories are not categorical. In turn, while second-order logic is strong enough to do the job, this is highly controversial (114; 54; 36). Consequently, we will not be able to isolate one of these interpretations just by listing a collection of logical and theoretical constraints on theories: these will only select a set of sentences from the language, which in turn will have multiple interpretations. But, if we cannot select a privileged interpretation in this way, nothing else will select a complete mathematical theory. We are just left with an incomplete set of axioms. And all theories extending them will be on a par. If mathematics is a science of abstract objects, mathematical knowledge and mathematical objectivity are not a function of our theoretical desiderata either. Benacerraf’s challenge questions the use of strong, counterfactual relations to explain mathematical knowledge and mathematical objectivity. In turn, limitation theorems have been used to question the use of weaker relations based on the imposition of logical and theoretical constraints on our theories. These are not enough to select an objectively correct mathematical theory either. Thus, both kinds of arguments have been used to question the informal and amenable characterization of mathematics as a science of abstract objects independent from us. If a universe of abstract objects exists independently of us, then it is plausible that there is no semantic or epistemic relation between it and people. In this way, both arguments have been used to question mathematical platonism, the thesis that mathematical, abstract objects exist independently of us and that mathematicians aim to study them. Consequently, we are faced again with a tightened version of the above dilemma: either we rethink our view of mathematics —i.e., platonism—, or we give up mathematical objectivity. All philosophical accounts of mathematics that have been proposed so far have embraced at least one horn of this dilemma. Those philosophies of mathematics that seek to preserve a minimal notion of mathematical objectivity —i.e., mathematical objective truth and knowledge— have chosen the first option, rethinking what mathematics is about. Some of them deny that mathematics is a science of objects. Instead, mathematicians would be involved with complex, modal facts (106; 55; 8) or with structures (119; 125; 56; 65). Others, while accepting that mathematics is a science of objects, have reconsidered their metaphysical nature. For example, some authors understand mathematical objects as symbolic objects, explaining mathematical objectivity in terms of intersubjectivity within a community (34; 35; 37; 38). Alternatively, mathematical platonism has not disappeared. Nevertheless, it has changed its form, becoming a plural metaphysical thesis (2; 54). Faced with the difficulties of justifying the existence of a single correct theory of mathematical objects —the objectively correct mathematical theory— its advocates have accepted that there are a multiplicity of correct theories. All of them are equivalent from the point of view of semantics and metaphysics. Instead of a mathematical universe, platonism contends the existence of a mathematical multiverse. This metamorphosis of platonism comes with a new understanding of mathematical knowledge. The explanation of mathematical knowledge no longer needs a relation between mathematical facts and speakers. On the contrary, this reduces to logical knowledge —knowledge of consistency and entailment— and mathematical objectivity reduces to logical objectiv- 12 ity. Finally, though not exhaustively, other accounts of mathematics embrace both horns of the dilemma. This is the case of fictionalism (76; 16; 5). Fictionalism accepts that mathematical theories are about mathematical, abstract objects. However, it denies that such objects exist. According to it, mathematical theories are fiction. And mathematics, rather than a science of abstract objects, is the study of what happens in those fictions. Since all fictions are semantically and metaphysically on a par, fictionalists also deny the existence of the objectively correct mathematical fiction. The single privilege of mathematical theories is to be studied historically by mathematicians. In this way, fictionalists, as the new platonists, reduce mathematical knowledge to logical knowledge —knowledge about consistency and entailment in mathematical theories. In this dilemma, I will place myself between Scylla and Charybdis. On one side, this text is a defence of a central, and negative thesis: Mathematical knowledge does not reduce to logical knowledge. Consequently, mathematical objectivity does not reduce to logical objectivity. Against the new pluralist versions of platonism as well as against fictionalism, I deny that mathematical knowledge can be explained exclusively by appealing to knowledge of consistency and logical consequence. Assuming an eminently semantic conception of language, I will argue that logical knowledge presupposes linguistic competence. Therefore, any reduction of mathematical knowledge to logical knowledge presupposes two things: first, that mathematical language is meaningful, and second, that mathematicians understand it. Nevertheless, I will argue that neither multiversism, nor fictionalism can explain this. That is, they cannot provide a metasemantic explanation of how mathematical vocabulary acquires its meaning, and how speakers acquire their linguistic competence. Thus, I will oppose to pluralist philosophies of mathematics a new version of the Language Acquisition Challenge used by (30) against classical mathematics —however, this new version is not directly related to Dummett’s argument, nor with his intuitionist preferences. On the other side, the text is a defence of a complementary, positive thesis: Mathematical knowledge can be explained through linguistic competence. Thus, I will try to complement those pluralist accounts of mathematics by developing an adequate explanation of how mathematical vocabulary acquires its content, and how mathematicians acquire their linguistic and mathematical competence. For this, I will use Linnebo’s (81; 82) recent abstractionist approach to abstract objects. The program makes strong use of abstraction principles to account for mathematical reference and mathematical knowledge. The philosophical relevance of abstraction principles was originally brought out by Frege, in his Grundlagen der Arithmetik. These introduce a reductionist perspective in metasemantics that makes them especially suited to elaborate an epistemological account of abstract objects: we can explain how speakers acquire a whole language committed to abstract objects without presupposing that they have any prior knowledge of these objects. Moreover, the acquisition of a language through abstraction provides speakers with some non-linguistic knowledge. That is, linguistic competence comes with knowledge of a handful of truths formulated in that language —and not just logical truths. As a result, cognitive achievement in mathematics receives a metasemantic explanation 13 following the lines of (79; 82) and (115; 116). This guarantees a minimum of objective mathematical truth and knowledge without abandoning mathematical objects. In this sense, my position is a modest vindication of objectivity and mathematical platonism —despite appearances, there is still a place for the combination of both. However, this must not be understood as a refutation of fictionalism or multiversism. Once a minimum of mathematical objectivity has been introduced, these conceptions can be defended for other mathematical theories. The debate between them is beyond the scope of the present text, and —if possible— will have to be solved by other philosophical arguments. The minimum of mathematical objectivity introduced here only constitutes the starting point in the exploration of the multiplicity of alternative mathematical fictions or universes. 1.1 HYPOTHESIS The reduction of mathematical knowledge to logical knowledge proposed by fictionalism and the most extreme forms of multiversism entails a plural conception of mathematics and the abandonment of a proper notion of mathematical objectivity: all consistent mathematical theories are correct. However, this fails as an explanation of mathematical knowledge. Logical knowledge —i.e. knowledge of consistency and logical entailment— presupposes the linguistic competence of speakers. But, neither fictionalism nor multiversism can explain how mathematical vocabulary acquires its content, and how speakers understand it. An alternative explanation is required. This explanation can be developed using abstraction principles. In addition, the use of abstraction principles will show how mathematical knowledge is acquired to some extent at the same time as linguistic competence is acquired. 1.2 OBJECTIVES The main objective of this research is to explain how a minimum of non-logical objectivity can be recovered for mathematics. In particular, the recent abstractions approach of (81; 82) will be used to show how cognitive achievement in mathematics can be explained as a form of linguistic knowledge. The achievement of this objective will be articulated through the following sub-objectives: 1. To elaborate an analysis of the most intuitive explanation of mathematical objectivity: mathematical platonism. Also, identify the main characteristics of objectivity in the natural sciences and show that the main argument against platonism, Benacerraf’s challenge, is biased by attributing them to mathematical knowledge. 2. To analyse the explanation of mathematical knowledge proposed by fictionalism. The analysis will show how the reduction of mathematical knowledge to logical knowledge presupposes the linguistic understanding of speakers. 3. To analyse the explanation of mathematical knowledge proposed by Hamkinsian multiversism and full-blooded platonism. As in the previous sub-objective, the analysis will show how the reduc- 14 serting and judging.2Under some necessary or sufficient conditions, statements and judgements satisfy the required standards. In the best scenario, correct statements and judgements about the world are constituted as knowledge-bearers.3 So far nobody has been able to offer a reasonable characterization of the norms involved in knowledge. Gettier cases have been extremely resistant to philosophical treatment, and nobody expects a successful reductionist analysis soon. All it is agreed is that knowledge is factive: if it is known that p, then p. Consequently, knowledge is usually taken as a primitive concept. And, this transitively applies to objective knowledge. A reductionist analysis of this concept seems unreasonable.4Nevertheless, objectivity hides an appealing and intuitive idea. This consists in the fact that the content of beliefs, judgements, or statements bearing knowledge is —partially at least— non-epistemic. That is, they are not about agents’ epistemic responses —for example, judging, believing, asserting, reporting or answering. Instead, they are about the world. Again, consider the former example. The sailor was tired and anxious to reach the mainland. Furthermore, evidence thresholds did not support the opposite judgment that the tiny, brownish spot was a cloud. As a result, she erroneously “concluded” that the crew was arriving mainland. A few minutes later they knew that the boat was facing an unexpected storm. The looker’s report was incorrect. This example shows that the content of her report exceeded her epistemic responses. Indeed, the error was caused because she included her responses —her belief that the spot was not a cloud, caused by her desires and expectations— among appropriate evidence required by the situation. As the usual slogan says: objective knowledge is independent —in some sense of “independence”— from agents (23; 24; 25; 118). I will not attempt any definition or reductionist analysis of “objectivity”. Nevertheless, I will attribute two necessary theses to the norms ruling objective knowledge. Independence The content of beliefs, judgements, or statements involved in objective knowledge does not depend on epistemic responses as believing, judging, asserting, reporting, answering, etc. Responsiveness Objective knowledge partially depends on the content of beliefs and theories involved in knowledge. The idea behind the conjunction of these theses is the following: against anti-realist conceptions of knowledge, knowledge about the world exceeds knowledge about agents’ states. Independence makes this explicit. Moreover, it is reasonable to expect the content involved in knowledge —the world— to be before knowledge itself. The priority direction is the opposite of the direction involved in anti-realist conceptions of knowledge. This is what Responsiveness expresses. Because the world is such that the law mistake to conclude that she is no longer playing football. At most, what is indispensable is that agents understand the norms governing their activity. See (147). 2An assertion is a communicative action that consists of sanctioning the content of some sentence as true. A judgment is a psychological action that consists of sanctioning some propositional content as true. 3Knowledge qualifies actions and entities relating to propositional content, such as statements, judgements and beliefs. 4See (118) for several attempts of analysis and why they have failed. A further challenge arises from the complex etymology of “objectivity”. The word has been used in a variety of different ways, acquiring a complex entanglement of meanings. Sometimes it applies to theories, while others it applies to methods or attitudes. It is employed as a moral value too. See (23; 24; 25). 21 of gravitation5holds, agents can gather knowledge about it. Compare this with knowledge about fashion or comedy. Some clothing is fashionable because, in the first place, people find it so. Also, jokes are funny, or disgusting, because of people’s sense of humour. In both cases, truths are grounded in people’s responses. 2.2.1 Independence Both Independence and Responsiveness include a notion of dependency in their formulation. This notion is used very often in descriptions of what objective knowledge is. But, how should it be understood? In the case of Independence, (151) has proposed a characterization in terms of analyticity. In his book Truth and Objectivity, Wright develops four criteria for assessing the objectivity of the theory formed by the true sentences of a language —thus, for the objectivity of languages. Two of these criteria — the projectivist position concerning the Euthyphro contrast and Cognitive Command— explain how a language is epistemically loaded by appealing to the notion of analyticity. Simplifying things a bit, the former criterion holds that a language is not objective when the Knowability Principle (KP) KP φ →K φ —where “ φ ” is a metavariable representing sentences of the language and “K” is a sentential operator expressing knowledge.6— is analytic of the content of the language. Consequently, KP would imply —modulo factivity— that φ and K φ are analytically equivalent. In turn, the second criterion holds that a language is objective when the fact that KP does not hold is analytic of the language. Leaving aside the challenges of analyticity, the proposal was criticized by (148) on the basis that it imposes unreasonable constraints on metasemantic explanations7of reference. Wright’s criteria entail that both non-objective and objective languages fix reference by analytic factors alone. However, this is rarely the case. Remember the moral of Putnam’s (108) Twin Earth thought experiment. For example, even in the case of colour —which Wright considers a case of an anti-realist discourse— empirical discoveries determine the reference of colour words and which truths about colours speakers accept. Physics has something to say about that. Consequently, Wright’s criteria for objectivity or non-objectivity fail systematically. I will not endorse the required notion of dependency in Independence with any commitment to analyticity. On the contrary, I will assume that what is required is an appropriate notion of explanation 5Newton’s gravitation law states that a mass point is attracted by another mass point with a force proportional to the product of the masses and inversely proportional to their distance. The law is taken as an approximation to the solutions of relativistic field equations under low-speed conditions. 6Usually, KP is formulated as hidden a modal operator: φ →♢K φ . Nevertheless, Fitch’s paradox (44; 14) shows that this formulation collapses in the former under two modest assumptions: that Kdistributes over conjunction and factivity. Because of this, and Wright’s exposition of the issue, it does not matter which formulation is chosen. 7Metasemantics explains how words acquire its meaning —I am using here a vague notion of meaning. The fact that a word has a particular meaning is not a metaphysical primitive. In turn, semantics describes what each word means and how their meanings relate to each other. 22 —maybe, a metaphysical notion of explanation. If the content of the language depends on speakers’ epistemic responses this means that it is explained at least in part by speakers’ epistemic responses. This is consistent with the fact that metasemantic explanations do not include exclusively analytical factors. Even if the content of language involves speakers’ epistemic responses, those facts explaining how the language acquires its meaning can involve empirical matters. According to this, Independence says that the content of languages is not explained by the epistemic responses of speakers. Using the symbol “⊩” to represent the appropriate notion of explanation, this can be expressed as follows: Independence:C φ ⊮ φ , where “C” represents speakers’ cognitive responses regarding the sentence φ .8,9 There is a reasonable doubt as to whether this formulation implies that knowledge is not factive. For, suppose that conditions C φ are entangled with knowledge of φ . Then, Independence would entail K φ ⊮ φ . Fortunately, even if this were the case this is not equivalent to the negation of factivity. Factivity is the conditional K φ → φ . If this were read as expressing something stronger than a material conditional, surely this would not be the same notion of explanation as the represented by “⊩”. Factivity is analytically true of the notion of knowledge —i.e. A(K φ → φ ). On the contrary, the idea of dependency involves a notion of explanation that must be weaker than analyticity. Therefore, factivity and Independence are mutually consistent. Conversely, ⊩must be stronger than a plain strict conditional —i.e. □( φ → ψ ). The idea of dependency entails that some facts10 are more fundamental than others. Thus, φ ⊩ ψ says that φ is more fundamental than ψ — ψ is the case because of φ . As every explanatory relation, dependency is asymmetric.11 On 8As relations of entailment, the notion of explanation could be generalized as holding between sets of facts and facts: Γ⊩ φ , where Γis a set of facts. 9Rosen (121, p. 127) proposes a similar analysis of Idealism in terms of metaphysical grounding. 10I will use a minimal notion of fact. This ensures that facts are simply the metaphysical correlate of the concept of truth. Thus, given a sentence φ , I will speak interchangeably of “the truth of φ ” or “of the fact that φ ”. See 2.6 for a proof that a minimal notion of fact is conservative over a minimal notion of truth. For those who distrust metaphysical concepts, any mention of fact can be replaced by speaking of a sentence being true. 11I will leave open which one is the precise relation expressed by “⊩”. A candidate is the relation of metaphysical grounding (120). Either way, it should satisfy the following properties —where ∆and Γrepresent sets of sentences expressing facts or propositions: Transitive If Γ, φ ⊩ ψ and ∆⊩ φ , then Γ,∆⊩ ψ Irreflexive φ ⊮ φ Asymmetric If φ ⊩ ψ , then ψ ⊮ φ . Not connected It is not the case that: Γ, φ ⊩ ψ or Γ, ψ ⊩ φ Non-monotonic It is not the case that if Γ⊩ φ , then Γ, ψ ⊩ φ Explanation must be non-monotonic. On the contrary, every fact would partially explain every other fact. This is highly unpleasant, almost trivializing the explanatory role. For instance, assume that mental states are grounded —or supervene— on the brain’s physiology. Then, by no means they would be explained —or supervene— by the 23 the contrary, the paradoxes of the strict conditional show that it is not asymmetric at all. The explanation hidden in the idioms of dependency is in between analyticity and the strict conditional. In turn, ifC φ is entangled with K φ , Independence is equivalent to the negation of certain formulation of KP. This thesis can be understood as something stronger than a material conditional. According to KP, facts are intrinsically epistemic: there is no gap between them and agents’ knowledge of them. Therefore, KP can be understood as saying that φ depends on or is explained by K φ :K φ ⊩ φ . A realist conception of knowledge strongly rejects this assumption. This is precisely what Independence says. Consequently, ⊩could help to identify what is the core of the division between realist and anti-realist conceptions of knowledge. Note that Independence does not imply that the language is referentially successful. That is, it does not imply that the language is geared with terms and generalized quantifiers, as well as committed to the existence of a particular realm of objects. For example, suppose that you accept a de re conception of modality. Terms and quantifiers nested inside intensional contexts would be referentially innocuous. However, according to such a conception of modality, it does not reduce to analyticity or other representational concepts. De re modality is realist in spirit, regarding modal laws as independent from agents as natural laws. Also, see (12) for the possibility of languages with no referential apparatus at all. Despite its intuitive appeal, it might seem that Independence faces some counterexamples. For example, does psychology contradict this thesis? After all, psychology is intended as an objective science, but its theories are precisely about human cognitive responses. This seems to imply a contradiction: psychology would not be an objective science after all. Fortunately, this is not the case. What Independence requires is that the content of those statements endorsed by any speaker does not depend on her epistemic responses. This is completely satisfied by psychology as well as any other human science. The knowledge gathered by psychologists is not determined by their epistemic responses. Their object of study is human beings —generally, cognitive organisms—, but how these subjects are psychologically constituted does not depend on the researchers’ epistemic responses. Thus, that C-fibers are responsible for pain is by no means the case because some researchers judged it so. Also, that a particular subject faces some psychological problem is by no means the case because her therapist judged it so. Truthfully, because researchers correctly judge that pis the case, they obtain knowledge about p. But pitself —the content of their judgement— is not the case because they judge it so. But, is not Independence too strong? After all, all attempts to analyse objectivity have failed so far (118). The efforts made to isolate those factors tied to the researchers’ judgement from knowledgegathering processes have shown hopeless. As a result, a substantial amount of studies —especially those focused on understanding how social and pragmatic factors affect science— have criticized the very idea brain’s physiology and the fact that the Earth and the Moon satisfy —approximately— Newton’s gravitational law —leaving aside marvellous surprises in physics! As a result, classical semantic entailment is not a sub-relation of the required notion of explanation: ⊨⊈⊩. Otherwise, I will not commit myself to a Lewisian conception of a fundamental metaphysical domain over which everything else pends. Thus, I will leave it open whether ⊩is well-founded. That is, I will leave open the question concerning the existence of infinite descendent chains of fundamentality between facts —... φα +1⊩ φα ⊩... ⊩ φ 3⊩ φ 2⊩ φ 1. It is natural to extrapolate humans’ usual experience about composites, such as sets or mereological fusions, concluding that there should be a fundamental domain of grounding facts which depend on nothing. However, humans’ experience is fallible. There is nothing incoherent about atomless worlds. 24 contained in Independence. As a human activity, science and, more generally, knowledge —including mathematics— would involve an irreducible remainder of the researchers’ dispositions. For instance, mathematical axioms, as Power Set12, would be justified not because of mathematicians’ privileged access to the set-theoretic universe, but because of pragmatic reasons involved in the practice of 18th and 19th-century that lead to contemporary set theory (35, Ch. 8). These kinds of proposals tend to reject a precise distinction between the context of discovery and the context of justification. According to them, epistemology would hopelessly involve the study of agents’ responses and it would be impossible even in principle to disentangle where the frontier between the world and their responses is. Therefore, is not Independence overlooking all this? I believe the answer is no. The previous considerations are orthogonal to Independence. This is a characterization of a realist conception of knowledge. Such a characterization is consistent with any challenge to humans’ cognitive abilities, even if the latter entails the unattainability of the former. Humans are far from being perfect when it comes to knowledge about the world. Their justifications of empirical statements as well as statements of formal science involve mistakes and inaccuracies —for instance, physical magnitudes can only be approached to a rational value— and several sorts of bias —gender, social or economical (31; 118). But, this is consistent with a meaningful notion of objectivity. Moreover, Independence is a thesis about the content of agents’ epistemic actions, not about the justification of those actions. Thus, think again about the first example and reverse it. Imagine that indeed there was land in front of the boat but the looker confounded it with a storm. Consequently, the crew would change direction getting lost in the ocean. Even if they remained convinced forever that there was a storm far away, this would not be the case. To think the opposite would be almost to believe in magic. Hence, it is important to distinguish between the content of language and the evidence thresholds attained to theories. 2.2.2 Responsiveness The notion of objective knowledge as correct knowledge about the world is overtly elaborated with empirical science as a reference. Thus, the notion of dependency that features in Responsiveness is usually taken as hiding a counterfactual relation between speakers and facts. As Benacerraf puts it: We think that X could not know that p. What reasons can we offer in support of our view? If we are satisfied that X has normal inferential powers, that pis indeed true, etc., we are often thrown back on arguing that X could not have come into possession of the relevant evidence or reasons: that X’s four-dimensional space-time worm does not make the necessary (causal) contact with the grounds of the truth of the proposition for X to be in possession of evidence adequate to support the inference (if an inference was relevant). The proposition pplaces restrictions on what the world can be like. Our knowledge of the world, combined with our understanding of the restrictions placed by p, given by the truth conditions of p, will often tell us that a given individual could not have come into possession of 12A set bis a subset of another set aif and only if every element of bis also an element of a. The powerset of a —in symbols P(a)— is the set of all its subsets. The powerset axiom asserts that the powerset of every set exists. 25 evidence sufficient to come to know p, and we will thus deny his claim to the knowledge. (7, 671-672) According to this understanding of Responsiveness, this should be expressed as follows using the notion of explanation introduced in the previous section: Responsiveness C φ + φ +X⊩K φ , where C φ expresses some counterfactual relation between agents and the facts depicted by φ , and “X” is a variable standing for some appropriate facts —this unknown prevents any simplistic definition of knowledge. Xcan be non-constructively thought of as the required epistemic conditions under which φ is known. Thus, C φ + φ partially explains K φ . Responsiveness says that when agents are appropriately related to the world by C, φ is true and X, then they know that φ . In the case of empirical knowledge, the counterfactual relation Cis a causal relation between agents and the world. For example, knowledge about Newton’s law of inertia partially depends on physical objects satisfying it. Also, knowledge about continental drift partially depends on a complex geological history of changes in Earth’s lithosphere. As can be seen, Responsiveness involves an explanation of knowledge. Therefore, the following worry will arise: does it entail some kind of solution to Gettier’s challenge? After all, this challenge calls for necessary and sufficient conditions under which agents can know that φ . If the antecedent of Responsiveness collapses into these conditions, requiring a viable solution to the challenge, there would be little chance of satisfying this criterion. Fortunately, I believe this is not the case. At best, Responsiveness entails a local solution to the challenge. On the contrary, it does not entail a global solution. But, this is precisely what the challenge asks for. Let Lbe the speakers’ language. A global solution to the challenge involves an explanation of the form For all φ ∈L:Γ⊩K φ , where Γis a set of sentences. Indeed, this would be a definition of knowledge. On the contrary, a local solution to the challenge only affects a restricted collection of sentences ∆of the language: For all φ ∈∆⊆L:Γ⊩K φ . Therefore, even if “X” is solved in many cases, Responsiveness does not collapse into a solution to Gettier’s challenge. While a global solution to Gettier’s challenge —especially, an explanation of perceptual knowledge— is an excessive requirement, a local solution is precisely what the epistemology of special sciences calls for. For instance, it is completely coherent to explain mathematical knowledge, without elaborating a solution to the epistemic difficulties involved in perception. A realist epistemology of arithmetic calls at least for a solution for X in the following expression For all φ ∈ { ψ /PA ⊢ ψ }:C φ + PA + X ⊩K φ . And a realist epistemology of sets at least requires a solution for X in 26 For all φ ∈ { ψ /ZF ⊢ ψ }:C φ + ZF + X ⊩K φ . Nevertheless, it is this approach that creates the biggest problems for mathematical knowledge. As I will argue, Responsiveness is a bad characterization of mathematical knowledge. It does not provide a de facto global solution, but because it cannot do so in principle. While it is appropriate as a characterization of empirical knowledge, it fails as a global explanation of knowledge. 2.3 MATHEMATICAL PLATONISM The previous characterization of a realist conception of knowledge provides a helpful explanation of the most prominent challenge that mathematical epistemology faces: Benacerraf’s challenge (7). In this section, I will explain what mathematical platonism is. Then, I will explain which are the main epistemological challenges it faces by using Independence and Responsiveness. 2.3.1 What is Mathematical Platonism? Mathematical platonism is the philosophical position resulting from the conjunction of the following three theses: Existence, Abstractness and Independence (138, Ch. 3-4), (2, p. 9), (116, pp. 9, 74),(85). Existence Mathematical entities exist. Abstractness Mathematical entities are abstract. Independence Mathematical entities are independent from agents. In what follows, I will comment on each one of these theses. 2.3.1.1 Existence Existence is a metaphysical thesis. —i.e., a thesis about what there is. Metaphysical theses are opposed to hermeneutic theses concerning the content of language.13 While the latter aims to explain what mathematical language, as employed by mathematicians when doing mathematics, is about, the former does not. Consequently, it is coherent to endorse Existence and at the same time to claim that mathematicians were not historically committed to any special realm of entities. For instance, their theories and methods could have been highly constructive, supplying their languages with anti-realistic semantics, such as intuitionistic or procedural semantics —as proposed by (41; 43). In parallel, using sophisticated metaphysical arguments, philosophers could have concluded that a weird domain of objects obeying mathematicians’ theories does exist. Even more, these objects could support classical logic, and the theories about them 13I take the “hermeneutic” label from (2) and (102). The distinction between these two kinds of thesis resembles the Quinean distinction between explanatory and hermeneutic philosophies of mathematics. An explanatory philosophy of mathematics is prescriptive. It aims to settle what mathematics ought to be about. Take set-theoretic reductions of arithmetic. According to explanatory proposals of arithmetical language, arithmetical terms should be understood as referring to a particular collection of sets —for example, von Neumann ordinals. Explanations prescribe what language should be about. On the contrary, hermeneutic philosophies of mathematics aim to describe what the languages used by mathematicians are about. 27 would be proper extensions of mathematicians’ old constructive ones. Although this picture is highly implausible, it is completely coherent with Existence as stated above. Nonetheless, is natural to expect some connection between metaphysical and hermeneutic accounts of language. After all, the relevant entities are qualified as “mathematical” because they obey mathematicians’ theories. Consequently, Existence must be understood as a statement about mathematical language: if read at face value, this language is about a distinctive realm of entities, mathematical entities. There is a meta-ontological delicate point concerning this thesis: what does it mean “to exist”? From now on I will assume Frege’s and Russell’s monochronism (137, p. 9) about meta-ontological notions. Therefore, I will reduce existence to the concept expressed by the existential quantifier: a second-order concept loaded with ontological commitments. This is the relevant notion of existence for mathematical platonism. Otherwise, note that the above three theses speak about entities, not about objects. I consciously employ this notion to be neutral about the ontological status of the mathematical realm. More precisely, it should not be presupposed that mathematical platonism is about objects. For instance, ante rem structuralism claims that mathematics is about structures. Moreover, ante rem structuralism is usually regarded as a platonic philosophy of mathematics. But, are structures objects? A negative answer calls for fine-grained classifications of different types of platonism. For instance, ante rem structuralism, object-platonism, etc. To simplify things, I will adopt a Fregean concept of object.14 This is generous enough to encompass the notions of entity, thing, individual, and the like. As a result, the considerations in this text apply both to ante rem structuralism and to object-platonism. Finally, Existence can be relativized to particular mathematical theories. ExistencePA Natural numbers exist. ExistenceZF Sets exist Note that if mathematical theories T are framed in first-order logic and they are read at face value, the rejection of ExistenceTimplies that such theories are at least partially false. 2.3.1.2 Abstractness Abstractness makes explicit the most remarkable metaphysical feature of mathematical objects: they are abstract. Despite the complexity of this expression and the doubts expressed by some people (2, Ch. 8), there is a simple way to characterise it. An entity is abstract when it does not exemplify spatio-temporal properties. Thus, provided that every physical entity is spatiotemporally located, abstract entities are not among the physics inventory. For instance, think about real numbers, real-valued functions, or manifolds. Such entities are essential for empirical sciences, as physical magnitudes reveal. Physical magnitudes, such as temperature, speed, intensity, or mass, are understood as total, real-valued functions defined on the domain of concrete, spatiotemporally located objects. However, neither real numbers nor real-valued 14According to this, objects are characterized semantically as possible referents of terms. The result is a broad notion of object. For instance, as soon as theoreticians refer to concepts, concepts are objects too. Nonetheless, the characterization allows for objects of different order and different kinds. This suffices to do justice to Frege’s maxim: “never to lose sight of the distinction between concept and object.” (46, p. x) 28 functions are part of this domain. Abstract objects are, loosely speaking, “out” of space and time. These objects are not just spatiotemporally isolated from us —like distant galaxies and other remote events. On the contrary, they are —so to say— causally ineffective. Abstractness can be understood as a thesis about the semantic status of physical attributions in two different ways. First, it can be understood as claiming that statements endorsing abstract objects with physical properties are false. For example, suppose that somebody says “the number 2 is blue”. Furthermore, assume that the context disambiguates her utterance, making it clear that she is referring to an abstract object —i.e. the number two. Then, if her utterance is an assertion, this will at least be false — numbers have no colour. Alternatively, abstractness can be understood as claiming that such statements are meaningless. That is, physical attributions involving abstract objects would be ill-formed expressions breaking the rules of grammar (137, pp. 93-94), or at best, they would bear a third truth value —neither the truth nor the false. According to the first, the sentence “the number 2 is blue” is ungrammatical. Grammatical rules forbid “to be blue” to be predicated on the term “the number 2”. Although predicates are allowed to apply to terms, this type of construction could be an exception. Maybe constructions combining terms that refer to some kind of entities —abstract objects— and predicates that express a property about other, very different kinds —spatio-temporal objects— hide a categorical mistake. Thus, compositionality would obey a finer classification of terms and predicates. On the other side, these attributions may lack a determined truth value. Even if they are grammatically correct, they may be insufficient to select well-determined semantic content. Or, if they are, the world may be insufficient to make them true or false. Which one is the correct reading is not especially important for my purposes. 2.3.1.3 Independence This is the thesis explained in section 2.2.1. Independence C φ ⊮ φ , where C φ represents speakers’ cognitive responses regarding the sentence φ . Independence constitutes the real core of mathematical platonism. Mathematical entities and mathematical facts are epistemically independent from agents. As some counterexamples show, Existence and Abstractness are insufficient to provide a satisfactory characterization. Let’s start with Existence, and think about mathematical psychologism. Simplifying things a bit, this is the anti-realist thesis that mathematics is about mental events. According to psychologists, numbers exist as mental constructions of some type. Consequently, they would be at least temporarily located objects. This example shows that Existence is consistent with anti-realism. Thus, it is not enough to identify the realist bedrock of mathematical platonism. Also, Abstractness is not enough either. Think about fictionalism, the thesis that mathematical objects and mathematical facts are fictitious —see Chapter 3. According to this, the number 3 would be as Hamlet or Oliver Twist: they do not exist. However, the fictionalist believes there is a clear difference between these two kind of fictitious objects. While Hamlet and Oliver Twist are spatio-temporal objects, the number 3 is abstract. It does not exist. But, if it existed, it would not be a spatiotemporal object. Therefore, Abstractness is also consistent with anti-realism and is not enough to identify platonism. 29 Mental constructions and fictitious objects are not epistemically independent from agents. This is what constitutes both psychologism and fictionalism as anti-realist philosophies of mathematics. None of these satisfy Independence. When applied to mathematics, this says that mathematical facts, for instance, arithmetical facts, are independent of agents: For all φ ∈ { ψ /PA ⊢ ψ }, then C φ ⊮ φ , where Cis the corresponding epistemic context. For example, the arithmetical fact that 5+7=12 is independent from agents. Independence is about facts. However, as stated in section 2.3.1, it is about objects: mathematical objects are independent of agents. Therefore, the form of the latter statement may give rise to some concern. Are the two formulations of Independence equivalent? Fortunately, the answer is “yes”. The independence of objects can be retrieved as the independence of identity facts. More specifically, an object α is independent of agents when the fact that this object is self-identical — α = α — satisfies Independence. α is independent from agents ≡d f C( α = α )⊮ α = α As it was explained in section 2.2.1, the notion of dependency is analyzed as an appropriate notion of explanation (⊩) between facts —recall that this is a minimal notion of fact, see 2.6. However, the literature on the philosophy of mathematics contains alternative ways of understanding the notion of dependence. For instance, (82, Ch. 11) argues that mathematical objects satisfy Independence if this is understood as expressing counterfactual independence. This form of independence claims that mathematical objects may exist when agents do not —i.e. mathematical worlds uninhabited by agents are possible. If mathematical objects exist necessarily, they will be independent in this sense. However, counterfactual independence is a relatively weak notion and it was criticized by (38). Moreover, if an anti-realist concept of modality is accepted, the counterfactual independence of mathematical facts could be compatible with their inherently epistemic nature. To say that they are counterfactually independent of agents would have no metaphysical import if the space of possibilities is just an epistemic device and not a metaphysical category. Stronger notions of independence elaborate on comparisons between mathematical and physical objects. Such comparisons demand that numbers, functions and other inhabitants of the mathematical realm must have the same status as mountains, tables or animals —see (82, Ch. 11)Linnebo3 and (93). However, I believe that the best way of understanding such analogies is not by charging mathematical platonism with Independence, but with Responsiveness. Responsiveness C φ + φ +X⊩K φ , where C φ expresses some counterfactual relation between agents and the facts depicted by φ . Numbers and mountains are alike not only because they are epistemically independent of agents. According to the analogy, they are similar in the way they relate to agents. In the case of mountains, C φ would express a causal relation between them and agents. In the case of mathematical objects, it would express other, non-causal kind of relation —mathematical objects are causally ineffective. However, as mathematical 30 challenges. It does not immediately provide us with such an answer. Indeed, what remains to be done is the most difficult: to provide an acceptable explanation of mathematical knowledge compatible with platonism. Surely, this requires the rejection of Responsiveness —the main feature of empirical knowledge. Surprisingly, when this has been done, the result has involved substantial revisions of traditional platonism. As fictionalists, advocates of mathematical realism argue that the negation of Responsiveness entails the abandonment of a strong form of mathematical objectivity. Platonism becomes a pluralistic philosophy of mathematics, adopting the form of Balaguer’s full-blooded platonism (FBP) (2) or Hamkins’ multiversism (54). Consequently, both anti-realist and realist positions that have survived the epistemological challenges share the same characteristic: essentially, they are pluralist philosophies that deny a strong notion of mathematical objectivity. In chapters 3 and 4 I will analyse the main representatives of these two major positions. In chapter 3 I will analyse fictionalism. Then, in chapter 4 I will deal with FBP and multiversism. Using an eminently semantic conception of language, I will argue that none of these provide us with a satisfactory explanation of mathematical knowledge. Indeed, I will argue that their very formulation requires that speakers have a minimum of prior mathematical knowledge. Consequently, contrary to what they say, a minimum “amount” of mathematical objectivity must be compatible with the rejection of Responsiveness. Finally, in chapters 5, 6 and 7 I will explain how this is the case and how human beings acquire enough mathematical knowledge to elaborate serious mathematical theories. 2.4 WHAT MATHEMATICAL PLATONISM IS NOT In the previous section, I explained what mathematical platonism is. Mathematical platonism is the conjunction of three theses: Existence, Abstractness and Independence. In this section, I will explain what mathematical platonism is not. Due to its non-sceptical status, platonism has been associated with more substantial metaphysical or semantic theses on several occasions. I will expose the two main groups of ideas that were charged to platonism: metaphysicalism and the semantic theory of truth. I will analyze which assumptions they rely on and which of those assumptions are doubtful. Then, I will argue that none of these assumptions are among the defining theses of platonism. As a result, it will be clear that it is innocent of many of the accusations made against it. 2.4.1 Metaphysicalism First, mathematical platonism must not be confounded with metaphysicalism.18 Metaphysicalism is the conjunction of the following three theses: Fundamentality The world’s metaphysical structure uniquely carves it up into its constituent parts: objects, properties and relations. Structure The world’s metaphysical structure is independent of epistemic factors. 18I take this label from (116, p. 5). The core ideas of Metaphysicalism were exposed and criticized by several philosophers under different headings. For example, (112), (94), (60), (82, p. 31). 37 Correspondence It analytically follows from the concept of truth that an atomic sentence P τ 1... τ n is true if and only if: (i) the term τ 1refers to the object a1, and..., and the term τ nrefers to the object an, (ii) the predicate Pexpresses the relation R, and (iii) it is the case that Fa1...an. I have restricted Correspondence to atomic formulas for the sake of simplicity. This thesis can be extended to complex formulas in an obvious way by employing Tarski’s recursive clauses. Alternatively, it can be postulated inside it that each complex formula corresponds to a unique complex fact.19 Metaphysicalism is a cousin of Russell’s and Wittgestein’s Logical Atomism. It provides us with an appealing and simplistic explanation of how language’s content, truth and beliefs relate to each other: the truth of sentences and the truth of our beliefs would be a function of the world’s metaphysical structure. This explanation, simplistic as it is, was qualified as a metaphysical fantasy. The metaphysical fantasy is that there is a totality of Forms, or Universals, or “properties”, fixed once and for all, and that every possible meaning of a word corresponds to one of these Forms or Universals or properties. The structure of all possible thoughts is fixed in advance —fixed by the Forms. (113, p. 6) Metaphysicalism is a dubious philosophical conception. It involves very strong metaphysical and semantic concepts. For example, the metaphysical concept of metaphysical structure. Such a structure would articulate the world in a series of basic components. Or the semantic concept of correspondence. According to this, the meaning and truth of sentences would consist in conforming to this structure. Sentences that conform correctly would be true, and those that do not would be false. None of these concepts is part of platonism. In the following I will discuss what each of the three theses composing Metaphysicalism consists of, showing that they are orthogonal to the latter. Consequently, Metaphysicalism can be abandoned as bad metaphysics and bad philosophy of language, while platonism is preserved. 2.4.1.1 Fundamentality and Structure Fundamentality involves the idea of a metaphysical structure that carves the world up in a unique, specific way. According to this, the world would be composed of simple items —commonly, objects, properties, 19Otherwise, I am assuming that logical form can be read directly from surface syntax. This is not a trivial assumption. Logical forms are theoretical entities. They are extremely useful in formal semantics —for instance, enabling a precise definition of logical entailment and related concepts, such as validity, consistency, completeness and the like. However, it is doubtful that logical forms play any substantive role in communication or in truth semantic clauses for natural language (60), (115, pp. 7-8). For the sake of expository simplicity, I will accept such entities, as well as the fact that they correspond to the surface syntax. 38 relations, categories, etc.— arranged in complexes which compose this structure.20 For example, according to Logical Atomism, objects and properties as well as objects and relations combine to form atomic facts. And, atomic facts combine between them to form complex facts. The exact components of the world and the operations between them vary depending on the particular metaphysical proposal. The important point is that platonism is not committed to any metaphysical conception of this kind. First, metaphysical structures are not mathematical structures. They are not mathematical objects. Especially, they are not ordered tuples, as the “structures” used in model-theoretic definitions of truth. Sometimes, platonism is said to be committed to the existence of a mathematical structure. Loosely speaking, the true mathematical structure. What mathematicians standardly mean when they say that a sentence is true is that it is true in the standard model, or the intended structure —or as we’ll see, the class of intended structures— for the given branch of mathematics. [Italics in the original] (2, p. 60) For example, according to this view, set theory would be a description of the universe of sets. The structure where all sets live. Similarly, Peano-Dedekind arithmetic would be a description of the sequence of natural numbers. However, this is just an elaborate way of saying something simpler: that platonism accepts a robust form of objectivity. Among all the theories stated in set-theoretical language, there is an exclusive, maximally consistent set of sentences that provides us with knowledge about these objects: the true set theory, or the true number theory. Generally, when someone speaks about the standard or intended structure of a given mathematical theory, this is only an indirect form of referring to the set of true sentences of its language. No metaphysical concept is involved here. Platonism is not immediately committed to metaphysical structures —only set theoretic platonism is committed to mathematical structures, in the sense of ordered tuples.21 Indeed, the status of metaphysical structures is unclear. What does it mean that the world —whatever this is— has a structure? Presumably, metaphysical structures are not architectural entities, such as buildings or bridges —perhaps metaphysicians can be considered architects of the world? For example, if we speak about “the existence” of such structures, we will be quantifying over them. Consequently, it is reasonable to think that structures are objects. Even if quantification is devoid of ontological commitments —i.e. it does not entail existence—, what is quantified acquires the status of an object —see (15; 17) for a detailed argument. But, if this is the case, it is doubtful that metaphysical structures can perform the task they are supposed to do. Allegedly, the world’s metaphysical structure carves it up into its constituent parts. Among them are the objects inhabiting the world. Now, suppose that the metaphysical structure of the world is also an object. If it is an object, it must be part of the world too. Therefore, we are faced with a dilemma. Either the structure explains itself, or we need a second structure to explain the ’position’ that the first structure occupies in the world. Now, the second structure will be an object too. Therefore, the second option leads us to an infinite regress. As for the first, it implies that the structure explains itself. 20I will put in brackets here any criticism on the already troublesome expression “the world”, as well as the gloss that it comes with parts. 21It could be retorted that ante rem structuralism is a form of platonism committed to metaphysical structures. However, this is not synonymous with mathematical platonism. An rem structuralism arises when the commitment to structures is added to the latter. 39 Thus, the two positions imply a form of circularity. And, although not every form of circularity is pernicious, it is pernicious if the relation is explanatory. And, in this case, it is. Consequently, it is plausible that structures are not objects —and cannot be quantified— or they do not perform the explanatory role desired by the metaphysician. But, if they are not objects, what are they? And, if they do not play an indispensable explanatory role, why should they be accepted? As Isaacson puts it: What are particular mathematical structures? Despite asserting e.g. that the natural numbers constitute a particular mathematical structure, I also say that the structure of the natural numbers (in the sense under discussion) is not a mathematical object nor indeed an object of any sort. In that sense talk of ‘the structure of the natural numbers’ is a façon de parler. In discussion Stewart Shapiro offered the suggestion that this construal of particular mathematical structures is a form of if-then-ism. [Italics in the original] (65, p. 31) Second, none of the three theses composing mathematical platonism —Existence, Abstractness and Independence— entails Fundamentality. Fundamentality entails the existence of those objects that are part of the structure of the world. Even if existence is thought of as a second-order concept, not as a property carved up into the world, this concept takes into account the objects and properties that are part of the world’s structure. Indeed, it is plausible that Metaphysicalism supports the following connection between existence and structure: an object exists if and only if it is a constituent of the world. Consequently, if mathematical objects are part of the word, Fundamentality —modulo the previous, reasonable assumption— entails Existence. However, the opposite is not the case. Existence does not entail Fundamentality. That some kinds of objects, such as mathematical objects, exist, by no means entails a position concerning the metaphysical structure of the world. In particular, the description of the existing objects remains the same with and without metaphysical structure. Again, structures should be taken as a façon de parler, that is, as an abbreviation of a more fundamental idea. This idea would be that there is a unique correct description of objects and how they relate to each other. Finally, the epistemological independence of the world’s structure —i.e. Structure— does not follow from platonism either. As has just been seen, the commitment to this metaphysical concept does not follow from platonism. So, neither does any particular claim involving it. The thesis Structure is similar to Independence —which is part of platonism. Both assert that some kind of entity and facts involving such entities are epistemically independent of agents. Besides, as in the case of Fundamentality and Existence, Structure entails the epistemological independence of the objects, properties, relations, etc. which compose the world’s structure. If this is epistemological independent from agents, so will be the former. Consequently, if mathematical objects are part of this structure, Structure entails Independence. But, the opposite —once more— is false. Platonism is about objects, not about metaphysical structures. 2.4.1.2 Correspondence Correspondence constitutes a substantial account of meaning and truth. Once the world’s structure has done its job and objects, properties and the like are on the table, agents will use adequate representational devices to depict them —or at least the metaphysicalist thinks so. As a result, Correspondence comes as an elaboration of the metasemantic “picture theory” advocated by Wittgenstein in the Tractatus. Linguistic 40 components are correlated with components of the structure. Terms are correlated with objects, and predicates are correlated with properties and relations. Then, the semantic content of every expression is explained by using this correlation. Therefore, Correspondence is intended as a metaphysical elaboration of semantic definitions of truth, such as Tarski’s (140) model-theoretic definition. According to semantic definitions, truth is a substantive concept which is defined using additional semantic concepts. In the case of Tarski’s definition, the concepts of reference and satisfaction. Instead of satisfaction, Correspondence relies on the metaphysical notion of the world’s carving. Thus, according to it, P τ 1... τ nis true if and only if the world’s metaphysical structure carves it up in such a way that Fa1...anis the case. Semantic definitions of truth and other substantive definitions —for example, causal conceptions of truth—, have been criticized on several occasions by those who defend a minimal or deflationist theory of truth (151; 64). Every theory of truth considers that the T-scheme φ is true iff φ follows analytically from this concept. According to the proponents of the minimal theory, this would be the only thing that follows analytically from the concept of truth. Consequently, a complete definition of truth reduces to the T-scheme. Truth is extensionally specified using a recursive and contextual definition —see 2.6 for some comments on minimalism. Therefore, if platonism entails Correspondence or a semantic theory of truth, the criticism posed by minimalists on the latter would transitively apply to the former. However, mathematical platonism does not entail any particular theory of truth. As I will argue in the next section, it is consistent with the minimal theory truth, with Tarski’s definition, or with some version of Correspondence. After all, platonism is a position about what kinds of objects there are — mathematical abstract objects—, not about how truth must be understood. All this may seem obvious, but Platonism has often been assimilated into semantic theories of truth. The reason is that it accepts the following two claims: mathematical objects exist and mathematical terms refer to these objects. Now, if the claim that mathematical terms refer to mathematical objects is loaded into Correspondence, this entails that mathematical theories are true. But, once more, the opposite is false. If mathematical objects exist, mathematical terms refer to these objects, and mathematical theories are true —i.e. if platonism is true—, Correspondence does not follow. The assumptions say nothing about what human beings mean when they speak about truth. Even if mathematical terms are referentially successful, it is not immediate that truth should be defined through the notion of reference. Platonism does not entail Correspondence. The present discussion shows the following. Assume that mathematical objects exist and that mathematical terms refer to them. Then, Metaphysicalism entails platonism: Existence + Reference + Metaphysicalism →platonism, where “Reference” abbreviates the fact that mathematical terms refer to mathematical objects. This may suggest a close connection between both positions. However, the connection is exhausted by the previous conditional. Against appearances, the opposite conditional is false. Platonism does non entail Metaphysicalism: 41 platonism ↛Metaphysicalism. Indeed, it does entail none of its three theses —Fundamentality, Structure, and Correspondence. These three theses are highly controversial. They were criticized on several occasions. Fortunately, platonism is not committed to any of them. Consequently, mathematical platonism can be accepted while these abandoned. 2.4.2 Theories of Truth As I said in the previous section, platonism was identified with semantic theories of truth on several occasions. Therefore, I will insist on this point here: platonism does say nothing about the content of the truth predicate or about what truth is. One of the clearest examples is Benacerraf, who endorses platonism with a commitment to Tarski’s model-theoretic definition of truth. I call the “platonistic” account that analyzes (2) [“There are at least three perfect numbers greater than 17”] as being of the form (3) [“There are at least three FG’s that bear R to a”] “the standard view”. [...] One of its primary advantages is that the truth definitions for individual mathematical theories thus construed will have the same recursion clauses as those employed for their less lofty empirical cousins. (7, 668-668) Moreover, he adds: “I take it that we have only one such account: Tarski’s, and that its essential feature is to define truth in terms of reference (or satisfaction)” (7, 667). Another example of this tendency is Dummett, who identifies platonism with classical mathematics, and both with the semantic conception: On a platonistic interpretation of a mathematical theory, the central notion is that of truth: a grasp of the meaning of a sentence of the language of the theory consists in a knowledge of what it is for that sentence to be true. [...] On the theory of meaning which underlies platonism, an individual’s grasp of the meaning of such a sentence consists in his knowledge of what the condition is which has to obtain for the sentence to be true, even though the condition is one which he cannot, in general, recognise as obtaining when it does obtain. (30, 14-15) Dummett uses this identification to build up two arguments against Tarski’s definition and, transitively, against Platonism. 2.4.2.1 The Infinite Regress Challenge The first argument is the Infinite Regress Challenge (IRC). Let’s assume platonism. Also, assume that Tarski’s semantic definition is also an explanation of speakers’ linguistic competence —i.e. an explanation of their understanding of language. Although Dummett does not do so, this is tantamount to accepting Davidson’s (26; 27) truth-conditional semantics. (4.i) Accept Davidson’s truth-conditional semantics. 42 According to this, knowledge of a sentence’ meaning consists of the speakers’ ability to state its truth condition.22 (4.ii) Understanding a sentence is tantamount to understanding its truth condition. Then, he notes that the truth condition of a given sentence involves the same expressive resources as the sentence. (4.iii) A sentence and its truth condition involve the same expressive resources. At this point, he notes that (4.ii) and (4.iii) entail an infinite regress. The explanation of how speakers understand the meaning of a sentence reduces to their understanding of its truth condition. But, this truth condition involves the same expressive resources as the sentence. Therefore, the explanandum features in the explanans —i.e. we use what we want to explain in the explanation. Thus, speakers’ understanding of a sentence’s truth conditions needs explanation too. But, (4.ii) reduces this explanation to the speakers’ understanding of the truth condition of this truth condition. And, (4.iii) requires to repeat this movement again. Consequently, an infinite regress arises (4.iv) An infinite regress arises. Dummett uses this to accuse Tarski’s semantic definition —i.e. Davidson’s truth-conditional semantics— and platonism of being unable to explain the linguistic competence of speakers. Since speakers understand mathematical language, platonism must be false. 2.4.2.2 The Language Acquisition Challenge The second argument is Dummett’s famous Language Acquisition Challenge (LAC) (66).He opposes it against Tarski’s definition —i.e. Davidson’s truth-conditional semantics— and platonism’s ability to explain speakers’ linguistic competence. Namely, he claims they do not meet the following maxim, which every such explanation must satisfy: Dummett’s Maxim Use exhaustively determines meaning. (30, p. 14) The argument goes as follows. Assume platonism. Thus —modulo his identification— assume Tarski’s definition of truth —i.e. Davidson’s truth-conditional semantics. However, Tarski’s semantics equips mathematical language with evident-transcendent truth conditions. There are sentences such that speakers do not know if their truth condition is satisfied or not. (4.a) Mathematical language has evident-transcendent truth conditions. To take a classic example, nobody knows if CH is true or false. Now, take one of these sentences φ and —without loss of generality— assume it is true. The assertion of a sentence at least involves the ability to recognize that it is true. Therefore, this sentence will not be assertible. 22For very different reasons, this is true of the minimalist theory of truth, which considers that “ φ is true” is analytically equivalent to φ . 43 (4.b) φ is true and not assertible. In turn, Dummett’s maxim entails that truth is revealed in use. Therefore, if a sentence is true, it must be assertible. That is, all the instances of the following scheme are true: (4.c) If sentence ψ is true, then ψ is assertible. However, (4.b) provides a counterexample to (4.c). Consequently, Dummett’s maxim and Tarski’s semantics are inconsistent. From this Dummett concludes the falsity of the latter. Thus, platonism is false. On the contrary, he takes as characteristic of the practice of assertion a version of the KK-principle: If φ is assertible, then it is assertible that φ is assertible. This means that assertion satisfies his maxim: if a sentence is assertible, speakers recognize it is assertible. Instead of truth conditions, Dummett proposes that meaning should be explained by providing the language with assertibility conditions. Verificationism-friendly philosophers tend to dismiss truthconditional semantics because it does not pass LAC. 2.4.2.3 Avoiding the Challenges To insist once more, Platonism does not imply Tarski’s definition of truth or any other definition. It is a metaphysical thesis concerning mathematical objects, not a semantic thesis about truth or any other related concepts. Therefore, this simple fact safeguards platonism from criticism posed by IRC and LAC. Nevertheless, I will show why this criticism does not work against Tarski’s semantic definition of truth either. As a result, there will be no doubt that platonism is safe. First, both challenges conclude that semantic conceptions of truth override speakers’ linguistic competence. Here is the first problem. Despite the intuitive connection between truth and meaning, Tarski’s truth definition does not entail a theory of speakers’ linguistic competence. The reduction of the theory of meaning —an intensional notion— to the theory of truth was obtained by Davidson after imposing a collection of specific constraints on any satisfactory theory of meaning. The concept of truth played no ostensible role in stating our original problem. That problem, upon refinement, led to the view that an adequate theory of meaning must characterize a predicate meeting certain conditions. It was in the nature of a discovery that such a predicate would apply exactly to the true sentences. (26, 310) Therefore, I take Dummett’s arguments as arguments against Davidson’s truth-conditional semantics — an extensional approach to meaning—, instead as arguments strictly against Tarski’s definition of truth. Now, I accept Dummett’s conclusion: Davidson’s truth-conditional semantics —i.e. Tarski’s semantic definition of truth— provides a partial explanation of speakers’ linguistic competence. The point is that this is not evidence against Davidson’s theory. The task of this theory is not to perform the explanatory functions demanded by Dummett. So, if it fails on this point, this is something that cannot be held against it. 44 Let’s start with IRC. The regress affects a particular aspect of speakers’ linguistic competence: how speakers acquire their competence. There is a difference between explaining what understanding consists of and the requirement of explaining how understanding is achieved. Therefore, both IRC and LAC lead to the same problem: speakers’ acquisition of their linguistic competence. When I say that Davidson’s truth-conditional semantics is a partial explanation of speakers’ linguistic competence is because this theory does not explain how speakers acquire it23 However, this is so because the theory is not designed to explain how speakers learn to speak. Davidson’s semantics states that to know the meaning of a sentence reduces to knowing its truth condition. This is a partial —but perfectly adequate— explanation of speakers’ competence. And the theory seeks nothing more than this. Furthermore, it can be supplemented with additional explanations. For instance, by adding a use theory of meaning able to deal with language learning. Truth-conditional semantics does not forbid this kind of extension. Let’s turn to LAC. This argument relies on Dummett’s maxim: use exhaustively determines meaning. This imposes the condition that truth does not exceed assertion —i.e. it entails (4.c). But, this is —modulo the T-schema— to accept the following version of KP: φ →A φ , where the operator “K” —expressing knowledge— is substituted by the operator “A”, that expresses assertion. This says that whenever a sentence is true it is assertible, and therefore recognisably true. From this, Dummett concludes that the only way of explaining speakers’ linguistic competence is to accept this sort of connection between meaning and epistemic factors. Once this connection is broken —i.e. truth and assertion take different ways—, language learning is not explainable. However, there are a variety of problems with this line of thought. First, there is a difference between learning a language and knowing the collection of its true sentences —the extension of the truth predicate. Davidson’s semantics does not collapse this difference. Truth conditional semantics explains meaning by using truth, but it does not reduce knowledge of the former —i.e. linguistic competence— to knowledge of the latter. What it prescribes is that to understand a sentence is to understand its truth condition: speakers must know which is the truth condition associated with each sentence. But, this is tantamount to requiring knowledge about the consequences of the T- schema. At most, this semantics accepts the previous version of KP to this theory. Now, as far as the T-schema generates a recursively, axiomatized theory, there is no problem in meeting this requirement. On the contrary, Dummett’s maxim does collapse the aforementioned difference between learning a language and knowing the extension of the truth predicate. However, this is an unreasonable demand. Knowledge of the consequences of the T-schema is enough to explain language learning when it is combined —for example— with a use theory of meaning. Consider the following toy model —it is not a serious attempt to explain how speakers acquire their linguistic competence. Let’s assume the division between primitive and non-primitive vocabulary for simplicity. At the beginning, a use theory of meaning could explain how speakers learn to use the primitive vocabulary, and how some words —as proper names— are linked to some items in the environment. Think about kids naming some objects. Doing this, the theory will explain speakers’ understanding of some simple sentences —i.e. how these sentences 23Horwich (64) offers the same criticism. 45 become associated with a particular truth condition— and their understanding of the sentences’ components as simultaneous events. Since Frege and Quine it is clear that sentences are the minimal empirical unity of semantics. At this point, Davidson’s semantics enters the picture. The compositional nature of truth conditions explains how more complex sentences are linked to their corresponding truth conditions. Thus, this rather simplistic —and unrealistic— model shows how Davidson’s semantics is consistent with a plausible theory that explains language learning. Furthermore, since this semantics entails the negation of Dummett’s thesis, the model shows that this thesis must not be imposed as an axiom on explanations of linguistic competence. Truth-conditional semantics has room for answering LAC. This is enough to dismiss Dummett’s criticism of Davidson’s semantics. However, one question remains to be answered. To what extent is Dummett’s maxim reasonable? To the point that it entails some version of KP, it is at odds with those discourses satisfying Independence. Indeed, if KP is analyzed as the negation of the latter, it would be falsified in those languages. Consequently, the question reduces to this dilemma: does a language satisfy Independence or not? I believe plenty of discourses do —the language of physics, perception, or “pure” mathematics are just a few candidates.24 2.5 VARIETIES OF INDEPENDENCE In this section, I will say a bit more about Independence. This is the thesis that the content of sentences is not metaphysically explained by appealing to epistemic factors Independence C φ ⊮ φ , where Care speakers’ cognitive responses. According to it, the language is not covertly about agents’ epistemological responses. For example, suppose that perceptual discourse has this feature. Then, when somebody says “the table is blue”, her statement is about an object which is not metaphysically explained by her epistemic responses —the table. The statement is not about her sense impressions or her judgements. Consequently, a statement such as “the table is blue” is not synonymous with “it seems that the table is blue” or “I judge that the table is blue”. Epistemic operators come as an extra component when the discourse satisfies Independence. As I have argued in section 2.3.2, Responsiveness causes Benacerraf’s challenge. The platonist’s most immediate response is to reject this thesis. Mathematical knowledge does not have the same standards as empirical knowledge. Therefore, to demand it is unreasonable. However, the rejection of Responsiveness does not solve the challenge. At most, it opens the door for platonism to find a solution, i.e., a reasonable epistemology of mathematics. But, this task is difficult because platonism entails Independence. Independence opens a gap between between agent’s knowledge-gathering activities and the world —i.e. language’s content. According to Independence, mathematical facts are not explained by the 24For criticism on Dummett’s thesis and KP based on a simple condition see (145; 146). The condition is that predicates of language have a fadeout. This is a numerically indexed series of epistemic contexts such that at nit is known that not p, at m>n+r, it is known that pand at ksuch that n<k<mit is not known that por not p. According to Williamson’s arguments, this feature entails the failure of some instances of KP, and thus the failure of Dummett’s maxim. Moreover, according to him, almost all discourses have fadeouts. Therefore, Dummett’s maxim turns out to be almost globally implausible for linguistic phenomena. 46 Proof. First of all, lets define a translation τ from L+into L∗. Let h(⌜ φ ⌝) = φ be a function from Γ into φ /SenL( φ )—i.e. a function from the set of names of formulas into the set of formulas. Axiom vi ensures that, for all terms pthere is a φ ,SenL( φ )such that C(⌜ φ ⌝,p). τ is defined recursively as follows: 1 For SenL( φ ): τ ( φ ) = φ 2 τ (⌜ φ ⌝) = ⌜ φ ⌝ 3 τ (p) = ⌜ φ ⌝(such that C(⌜ φ ⌝,p)) 4 τ (x) = x 5 τ (t1=t2) = τ (t1) = τ (t2) 6 τ (T(⌜ φ ⌝)) = T(⌜ φ ⌝) 7 τ (RL(p)) = τ (p) = τ (p) 8 τ (FL(p)) = T( τ (p)) 9 τ (C(⌜ φ ⌝,p)) = φ ↔h◦ τ (p) 10 τ is extended over connectives and quantifiers as expected. In particular: τ (∃x ψ (x)) = ∃x τ ( ψ (x)) The axioms i and vi guarantee —modulo ZF— the existence of a one-to-one mapping from RLonto Sen(L). Therefore, 9 could be simplified as follows: τ (C(⌜ φ ⌝,p)) = φ = τ (p). Now, I will prove TFL⊢ γ ↔ τ ( γ ), for SenL+( γ ), by induction on the complexity of formulas. I Atomic γ . Cases 5, 6 and 7 are trivial. Case 8. Assume TFL⊢FL(p). By Axiom vi, TFL⊢there is some φ ∈Sen(L)such that C( φ ,p). Thus, τ (p) = ⌜ φ ⌝. By Axiom v, TFL⊢T(⌜ φ ⌝). Assume TFL⊢T(⌜ φ ⌝), such that τ (p) = ⌜ φ ⌝. By Axiom v, TFL⊢F(q), for some q,C(⌜ φ ⌝,q). By Axiom ii, TFL⊢p=q. Therefore, TFL⊢FL(p). Case 9. Assume TFL⊢C(⌜ φ ⌝,p). Then: h(⌜ φ ⌝) = h◦ τ (p) = φ . So, trivially: TFL⊢ h(⌜ φ ⌝)↔h◦ τ (p). Now, assume TFL⊢h(⌜ φ ⌝)↔h◦ τ (p). By the definition of τ : TFL⊢ C(⌜ φ ⌝,p). II Case 10. The proof for complex formulas proceeds as expected through induction. In particular, assume TFL⊢ ∃x ψ (x). Instantiating: TFL⊢ ψ (t). By the induction step, TFL⊢ τ ( ψ (t)). Introducing the quantifier: TFL⊢ ∃x τ ( ψ (x)). The converse case is similar. Corollary 1. For all γ ,SenL+( γ )there is a ψ ,SenL∗( ψ )such that: if TFL⊢ γ , then MTL⊢ ψ . 53 Proof. Follows from Theorems 1 and 2. Corollary 2. TFLis consistent relatively to MTL. TFLand MTLare coextensive in the strong sense of Theorem 2. Despite talking about facts, everything that TFLimplies is equivalent to something told by MTL. TFLdoes show nothing new about truth. APPENDIX 2.B. Objectivity is related to substantive disagreements and speakers’ convergence on truth. The issue has been addressed originally in (144), who proposed that speakers’ convergence in their assertions is a distinctive feature of objective discourses. For example, physicians tend to agree on their assertions as well as on which theories are correct about the world. Moreover, when two physicians disagree over some issue, at least they agree that one and only one of them is right. On the contrary, speakers’ opinions about fashion or comedy tend to diverge to a much greater extent. Usually, what some person finds fashionable or funny, another find disgusting or offensive. Speakers do not converge on comedy or fashion. Moreover, speakers do not try to convince those who have different opinions about such matters by appealing to correct theories. Nevertheless, things are not so simple. Even though the picture is naively correct, it is far from clear that physicians have the required resources to always determine which of two disputants is right — especially since quantum mechanics appeared. In turn, speakers who belong to the same population share largely the same ideas concerning fashion and comedy. To avoid these problems, (151) shifted the attention from speakers’ empirical behaviour to the rules underlying disagreements in objective contexts. According to him, speakers converge on truth because of a deeper reason. Namely, because the rule governing their assertions is representational: speakers aim to depict a portion of the world. Consequently, where a discourse aims to depict some facts, disagreements cannot be solved at random. “Where we deal in a purely cognitive way with objective matters, the opinions which we form are in no sense optional or variable as a function of permissible idiosyncrasy, but are commanded to us” (p. 146). His analysis of the phenomena supplies us with an explanation in terms of language’s analytic features. The resulting criteria for objectivity are too weak and were criticized by (148). In turn, I will explain that cognitive failures in objective disagreements as well as speakers’ convergence on truth in objective contexts are explained by the conjunction of three simple items: (i) The K-rule, (ii) Factivity of knowledge, and (iii) The Law of non-contradiction. To deal with several speakers I will treat “K” not as a sentential operator, but as a binary predicate: thus, “K(x, φ )” means that xknows that φ is the case. Also, I will employ a binary predicate “A” for 54 assertion: “A(x, φ )” means that xasserts “ φ ”. Now, assume that the rule governing assertion in objective contexts is the K-rule —see (148) for a defence of the K-rule:29 K-Rule ∀x■(A(x, φ )→K(x, φ )), where the black square “■” expresses deontic necessity. The K-rule says that all speakers ought to assert a sentence only when they know that it is true. Also, assume that knowledge is factive: Factivity ∀x(K(x, φ )→ φ ). If speakers know that φ , then φ is the case. Adisagreement between two speakers xand yis as a situation where xasserts φ and yasserts ¬ φ . This case can be generalized to situations where xasserts that φ ,yasserts that ψ , but φ and ψ together entail a contradiction. For the sake of simplicity, I will restrict the exposition to the previous case. Disagreement xand ydisagree on φ if and only if A(x, φ )and ¬A(y, φ ), or ¬A(x, φ )and A(y, φ ). A speaker xexemplifies a cognitive shortcoming just in case it asserts φ , but she does not know that φ . Cognitive Shortcoming xexemplifies a cognitive shortcoming if and only if A(x, φ ), but not K(x, φ ). Thus, those speakers who do not obey the K-rule are cognitively faulty. Convergence on truth can be explained as a consequence of the fact that if φ is the case, no speaker is sanctioned to assert “not φ ” Convergence φ → ∀x¬K(x,¬ φ ) Convergence entails that, in the idealized situation where all the speakers obey the K-rule, all of them assert the same sentences, and only true sentences. Now, I will prove that the K-rule, the factivity of knowledge and the Law of non-contradiction (NC) entail Convergence. Theorem 3. K-Rule + Factivity + NC ⊢Convergence Proof. Assume the K-rule, Factivity and NC. Suppose that φ is the case. For reductio, suppose that for some speaker a, it happens that K(a,¬ φ ). By Factivity, ¬ φ is the case. So, φ and ¬ φ is the case, which contradicts NC. Consequently, if the language of a community of speakers satisfies the K-rule, Factivity, and NC, and they obey this rule, then they will converge on the sentences asserted —which will be true sentences. The language of empirical sciences as well as the language of perception are candidates to satisfy the requirements. Consequently, this would explain why speakers exemplify a high degree of convergence in these areas. 29Any rule involving factive epistemic conditions Cwould do. 55 These three factors also entail that disagreements are objective. When a discourse obeys the K-rule, the Law of non-contradiction, and knowledge is factive, then one of two disputants must be faulty when a disagreement arises. Theorem 4. Assume that speakers aand bdisagree on φ . Then: L-Rule + Factivity + NC ⊢aexemplifies a cognitive shortcoming or bdoes. Proof. Assume that speakers aand bdisagree on φ . By definition, A(a, φ )and A(b,¬ φ ). Also, assume the K-Rule, Factivity and NC. Suppose —without loss of generality— that φ is the case. For reductio suppose that K(b,¬ φ ). Factivity entails that ¬ φ is the case. Therefore, φ and ¬ φ , what contradicts NC. Consequently, A(b,¬ φ ), but not K(b,¬ φ ). By definition, bexemplifies a cognitive shortcoming. The proof is analogous if φ is not the case. When a discourse obeys the K-rule, NC, and knowledge is factive, then no contradictory assertions are sanctioned by the rules of the language. The fundamental reason is that the factivity of knowledge and the Law of non-contradiction entail that knowledge is consistent, in the sense that Consistency-K ∀x(K(x, φ )→ ∀y¬K(y,¬ φ )). This is the main feature of those discourses we usually tend to qualify as objective. The languages of physics, mathematics, logic, or perception are ruled by consistent epistemic conditions, such as knowledge. These command us which one of the horns in a dilemma is the correct one —if any. Does this mean that anti-realistic discourses are ruled by epistemic conditions allowing massive inconsistencies as well as choosing at will what is true and what is false? Not necessarily. Anti-realist discourses are consistent with the Law of non-contradiction. What happens is that, usually, anti-realist discourses allow —even if not massively— at least some contradictions. This is the case of fashion. Nothing forbids that some clothing is fashionable and disgusting at the same time. Because for some clothing to be fashionable all that is required is that some people find it fashionable. Equally, if it is disgusting. As a result, anti-realist discourses of this kind should be ruled by paraconsistent logic. On the contrary, it is harder to endorse languages holding cases of extensional or counterfactual independence with the failure of the Law of non-contradiction. This would be tantamount to endorsing the world with metaphysical or physical contradictions. I do not claim that metaphysical contradictions are senseless —for instance, quantum mechanics is a candidate for hosting contradictory situations. However, they come with a cost and we need to be extremely careful when assessing if it fits the metaphysical budget. To claim that mathematics or empirical science is contradictory would entail massive revisions in the whole body of human knowledge. Finally, note that the rejection of the Law of non-contradiction does not entail that individual speakers are systematically inconsistent. In particular, the negation of Consistency-K is consistent with the following one: Consistency-Individuals ∀x(K(x, φ )→ ¬K(x,¬ φ )). 56 Therefore, while two speakers are allowed by the failure of the Law of non-contradiction to sustain contradictory judgements about fashion or comedy, each one would not be allowed to make inconsistent claims. Nobody could be K-related both to φ and ¬ φ . For instance, while K(x, φ )and K(y,¬ φ ), this does not mean that both speakers xand yare allowed to assert that φ and ¬ φ . APPENDIX 2.C. Margin for error principles are used by epistemicism to capture the characteristic features of knowledge in vague contexts. According to advocates of epistemicism, they describe the epistemic situation underlying the sorites paradox. This puzzle affects predicates such as “heap”, “bald”, “red”, or “tall”, which have unclear or fuzzy boundaries of application. For all these predicates, there is no sharp boundary between the objects which satisfy them and those which do not. For example, one grain of sand does not make a heap. In turn, ten thousand grains will do. The problem arises because there are numbers ngreater than one, but smaller than ten thousand such that is not clear if ngrains of sand make a heap or not. The epistemicist solution claims that there is a definite answer for all these cases. What happens is that we do not or cannot know what the answer is. Thus, for all numbers n,ngrains of sand are a heap, or ngrains of sand are not a heap, even if nobody can give a definite answer. Margin for error principles are conditionals of the following form: if xand ydiffer incrementally with respect to φ , and K(Px), then Py, where “K” is an operator expressing knowledge, Pis a vague predicate, and φ is a feature whose variation affects the satisfaction of P. For example, if xis formed by ngrains of sand, and yis formed by n+1grains, and it is known that xis a heap, then yis also a heap. Epistemicists regard these principles as a characterization of inexact knowledge. They claim that knowledge in vague contexts must be “surrounded” by the satisfaction of the predicate in question —on the contrary, it would not be knowledge. If speakers know that an object satisfies the vague predicate this is because the closer cases also satisfy it. Therefore, they provide knowledge with a safety area surrounding it —this is why it is knowledge. Williamson (145; 146) has shown that margin for error principles are inconsistent with the knowability principle KP — φ →K φ . This result can be reconstructed in modal logic. First, let’s formalize KP as the following strict conditional KP □( φ →K φ ). Indeed, it is natural to read KP as entailing a strict conditional of this form. Its standard formulation only involves the material conditional. However, KP is usually taken as the characteristic feature of anti-realist conceptions of knowledge. Consequently, it is reasonable for advocates of these positions to read KP as an analytical truth of anti-realist knowledge or as a metaphysical principle of anti-realism. In both cases, KP will contain an operator strong enough to imply the previous strict conditional. Similarly, margin for error principles can be formalized as a strict conditional too: ME □(♢K φ → φ ). 57 This says that, if in some proximate —possible— situation φ is known, then φ must be the case. As the classical case, ME establish a safety area for knowledge. Now, it can be proved that KP and ME are mutually inconsistent in normal modal logic. Theorem 5. Let ⊢be probability in any normal modal logic. Assume ♢ φ and ♢¬ φ . Then: ME ⊢ ¬KP. Proof. The proof can be recovered in tableaux form. Let Rbe the accessibility relation between worlds. Assume ♢ φ and ♢¬ φ (in w0). Assume ME: ⊢□(♢K ψ → ψ ), (in w0). Now, for reductio assume KP: ⊢□( ψ →K ψ ), (in w0). By elimination of the diamond, suppose that φ , (in w1,such that w0Rw1). By KP, we have that φ →K φ , (w1). Thus, K φ , (in w1), and ♢K φ , (in w0). But, by ME, φ , (in w0). Now, by elimination of the diamond again, assume ¬ φ , (in w2,such that w0Rw2). By KP, ¬ φ →K(¬ φ ), (in w2). Thus, K(¬ φ ), (in w2), and ♢K(¬ φ ), (in w0). But, again by ME, we have ¬ φ , (in w0). Hence, φ ∧ ¬ φ , (in w0): a contradiction. This little result tells us something important about Independence. As in the case of counterfactual independence, suppose that modality in KP and ME is metaphysical modality —see section 2.5.3. Independence —both in the form of contextual or counterfactual independence— establishes a gap between the world and knowledge. Then, in the case of contingent sentences, ME shows that a certain margin of safety is necessary to know the world. The open gap introduces an epistemic risk: the risk of not knowing how things are. Therefore, bridging that gap comes at a cost. And, the margin established by ME assures us that it is worth paying it. However, what happens if the sentences are necessarily true? For instance, this could be the case of mathematical sentences. In this case, the theorem does not follow. There is no risk of failure! If one thing is always true, we cannot miss. We will always hit the target, and the margin set by ME is irrelevant. At this point, the following question arises: if mathematics is necessary, does Independence imply no epistemic difficulties? If this were so, then Benacerraf’s challenge would not really be a problem. Mathematical knowledge would be free as soon as mathematical language is understood. For example, if the axioms of arithmetic are necessarily true, and speakers understand what “5 + 7 = 12” means, then they will know that this is necessarily true —therefore, true. However, even if this picture is correct, it does not eliminate the explanatory pressure put on platonism by the epistemological challenges. It only shows that the gap open by Independence must be a hyperintensional gap —i.e. the concept of explanation contained in this thesis is a hyperintensional concept. Some instances of the gap can be analyzed by the modal idioms —for example, cases of extensional and counterfactual independence. Nevertheless, this may not be all that it contains. The explanation of mathematical knowledge introduced in Chapter 5 will be hyperintensional in nature. 58 3 Anti-realist Pluralism Fictionalism 3.1 INTRODUCTION In this chapter, I will analyze fictionalism. Fictionalism accepts mathematical language at face value: mathematical theories are about mathematical, abstract objects. However, it denies the existence of these objects. According to fictionalism, mathematical theories are fictions developed by mathematicians throughout history for various reasons. Consequently, mathematics would not be a science of objects. No more than other fictions, such as Hamlet,Oliver Twist, or War and Peace. Instead of truthfully describing the mathematical realm, mathematicians create stories and study what happens in them. Thus, fictionalism reduces mathematical knowledge to logical knowledge: knowledge about the consistency and consequences of those fictions. I will challenge fictionalism for three reasons. First, I will argue that under two reasonable assumptions, fictionalism cannot explain mathematical reliability. Then, I will try to do better, arguing that even if these assumptions fail, fictionalism cannot explain mathematical reliability. In consequence, it cannot solve Benacerraf’s reinforced challenge. Second, I will distinguish between two types of fictionalism: full fictionalism and algebraic fictionalism. The former claims that mathematical language is a meaningful language about abstract objects. In turn, the latter does not attribute any content to mathematical language. I will argue that if full fictionalism can explain how mathematical vocabulary acquires its content and how mathematicians acquire their linguistic competence, then so can platonism. Since the failure of platonism is the best argument for fictionalism, this is fatal for it. Finally, I will argue that algebraic fictionalism is in no better position. The reduction of mathematical knowledge to logical knowledge presupposes the linguistic competence of mathematicians. But, if algebraic fictionalism tries to explain this, then it reduces to full fictionalism. 3.2 FICTIONALISM Fictionalism is the philosophical account of mathematics that results from the rejection of Existence. According to fictionalists, mathematical objects do not exist. Consequently, fictionalism is a form of nominalism and a sceptic philosophical position. In turn, fictionalists agree with platonists in taking mathematics at face value. For them, the sentence “2 is prime” aims to be about an object, the number two. It asserts that this object has the property of being prime. As a result, fictionalism entails that mathematical theories are not true —they are false, or neither true nor false. Even weak mathematical theories, such as Peano-Dedekind arithmetic, entail existential claims —for example, that there is a prime number between 1 and 3. Thus, mathematics would be similar to fiction. Think about the famous play Hamlet. Hamlet —the main character of the tragedy, not Shakespeare’s son— did not exist. Therefore, claims about him are not true —they are false, or neither true nor false. For example, it is not true that Hamlet killed Laertes. Nor is that Ophelia jumped into the river. None of these events have taken place. Nevertheless, the analogy only goes this far. Fictionalism does not entail any further connection between fiction and mathematical theories. In particular, it does not deny important disanalogies between them. Fictionalism offers a promising account of mathematics. If it is correct, it would be in a position to explain mathematical knowledge. This would simply be explained using the best theories of knowledge about fiction at our disposal. More specifically, fictionalism denies that mathematical knowledge satisfies Responsiveness Responsiveness C φ + φ +X⊩K φ , where C φ expresses some counterfactual relation between agents and the facts depicted by φ , and “X” is a variable standing for some appropriate facts —this unknown prevents any simplistic definition of knowledge. This is the thesis that knowledge of some facts is —partially— explained by being C-related to these facts. For example, to know that the Sun is a star, Responsiveness requires that: (i) the Sun is a star, (ii) we are causally related to this fact, and maybe (iii) some unknown factors X. Usually, empirical knowledge is regarded as a paradigmatic case of Responsiveness. On the contrary, knowledge about the plot of a novel or a play does not obey the same demands. If these are fictions many of the sentences they contain will be not true —i.e. what the plot depicts is not the case. This immediately invalidates Responsiveness. In turn, fictionalism solves Xin the following statement X⊩K φ with the best explanation of knowledge about mathematical fiction. In Chapter 2 I argued that Responsiveness is the main cause of Benacerraf’s challenge and its reinforced version. By failing to find satisfactory instances of this scheme, platonism gets into trouble. Fictionalism avoids the same challenge by rejecting Responsiveness. However, this does not constitute the final response. Like all accounts of mathematics, fictionalism faces an important explanatory pressure. Even if mathematical theories are fictions, fictionalism must explain the knowledge provided by them. At this point, it comes with a rather simple idea. Knowledge about a mathematical, fictitious sentence φ reduces to two things: (i) knowing that φ follows from a given mathematical, fictitious theory Γ, and (ii) knowing that Γis consistent: K(Γ⊢ φ +Γis consistent) =d f KF φ , where KFis “fictionalist knowledge.” In this way, fictionalism reduces mathematical knowledge to logical knowledge —knowledge about logical consequence and consistency. As a result, as long as mathematicians are good at knowing logical facts, then the fictionalist can explain mathematical knowledge. And, fictionalists claim that mathematicians are very good at logic. 60 Benacerraf’s first problem is avoided on this account since the idea that a good mathematical theory must successfully refer to any objects, let alone a unique system of objects, is abandoned. Asking which object, precisely, is the number 2 is analogous to asking who, precisely, is Hamlet. On a plausible view of fictionalist discourse, 7 the name “Hamlet” really refers to no one, and so it is wrongheaded to ask, outside of the context of the fiction, who Hamlet is. (76, p. 125) Mathematical knowledge does not concern bare statements such as “there are no naturals x,y,z,nsuch that n>2and xn+yn=zn”. On the contrary, mathematical practice is about generating fictions. In the case of arithmetic, a fiction about numbers. In the case of set theory, a fiction about sets. Thus, mathematical knowledge is knowledge about entailment, i.e., about what follows and what does not from the corresponding generative assumptions of these “fictions” —the axioms of the theory. Strictly speaking, fictionalism denies Abstractness and Independence too. Abstractness Mathematical objects are abstract. Independence C φ ⊬ φ , where Care agents’ epistemic responses concerning φ . Abstractness claims that mathematical objects are abstract. In turn, independence claims that mathematical facts are not explained by epistemic factors. Thus, both sentences involve reference to mathematical objects, which do not exist. Consequently, they are not true. However, fictionalism does not deny the weaker version resulting from prefixing them with the condition that mathematical objects exist. It accepts that, if mathematical objects existed, they would be abstract objects. Similarly, if mathematical objects existed, they would be independent of agents’ epistemic responses. The failure of these two theses shows the real strength of fictionalism. If there are no abstract objects, the need to explain how speakers relate to them —a mysterious relation— disappears. Even more, if there are no mathematical objects —of whatever kind— the gap between the world and mathematicians vanishes. The explanation of mathematical knowledge does not consist in explaining how this gulf is bridged. Thus, the task becomes simpler. As a result of its scepticism, fictionalism rejects a strong notion of mathematical objectivity. There is no such thing as the true mathematical theory. The only relevant objectivity for mathematicians is logical objectivity. This dictates that there are many consistent mathematical theories, all equally acceptable from a metaphysical and semantic point of view. Fictionalism is inherently plural. Sometimes, this has been used against it —after all, mathematical practice harbours a certain degree of objectivity. However, fictionalists have a chance of getting back a weaker form of objectivity. They introduce the predicate “to be true in the fiction”, such that a sentence is true in a given fiction just in case it follows from the axioms of the theory. For example, “there are infinitely many primes” is true in the fiction PA. Similarly, “there are inaccessible cardinals” is true in the fiction ZF+Con(ZF). Once this is done, the fictionalist can understand plain truth as truth in the canonical story of mathematics, i.e., the story developed through centuries by mathematicians. This form of objectivity will be much weaker than the objectivity accepted by platonism. For example, assume that the canonical story about sets is ZF and is consistent. Then, this 61 story is highly incomplete. It does not prove nor disprove CH or V = L. None of these sentences will be true in the canonical story of sets. Hence, no one of them will be true simpliciter.1 Fictionalism has been criticized on several occasions —see (2, p. 98) and (5, pp. 8-15)for a survey of the criticism posed on fictionalism. The Indispensability Challenge and the accusation that inadvertently reintroduces abstract objects —sentence types— (2, pp. 99) and (5, p. 15) are among the most famous criticisms. Here, I will leave aside all these worries. Instead, I will pursue a different route of attack, opposing two challenges to it. First, I will argue that fictionalism does not meet the reinforced version of Benacerraf’s challenge. Fictionalism offers a very effective solution to its original version. By treating mathematical theories as fictions, it achieves a nominalistically acceptable epistemology —if logical knowledge is nominalistically acceptable. However, this is not enough to explain the reliability of mathematicians. Under extremely weak assumptions, this picture entails that mathematicians make systematic mistakes. This makes them highly unreliable. Second, using an eminently semantic conception of language, I will argue that fictionalism cannot explain how mathematical language acquires its meaning, nor how mathematicians understand it. The core idea of the argument is that language is not a syntactic or grammatical category. Indeed, grammatical categories are at the same time semantic categories and vice-versa. For example, consider proper names. An object is a proper name because it attempts to perform a semantic function: pointing to an object, among other things. The name “Aristotle”, among other things, points to the ancient Greek philosopher Aristotle. If it did not perform this function, it would not be a name. It would just be a sound, a spot on the screen, or some other object. The same applies to predicates. An object is a predicate because it attempts to perform a semantic function: to apply with truth to other objects, among other things. Thus, using the semantic nature of language, I will argue that fictionalism either (1) Cannot explain how the content of mathematical language is fixed, nor how mathematicians understand it, or (2) Its explanation allows platonism to solve Benacerraf’s challenge. (2) is fatal for fictionalism, since platonism’s inability to respond to Benacerraf’s challenge is the main argument in favour of it. In turn, (1) is the Language Aquisition Challenge (LAC) of Chapter 2 —see Chapter 2, 2.4.2.2. Now, this challenge invalidates the very formulation of fictionalism. This reduces mathematical knowledge to logical knowledge. However, logic is also a linguistic phenomenon. Consequently, if fictionalism fails to explain how mathematicians understand mathematical language, its explanation of mathematical language will fail too. In some sense, this argument tells us that fictionalism is in no better position than platonism concerning mathematical knowledge —it does not make it less mysterious. From this I will conclude that an alternative explanation of the understanding of mathematical language is required. At the same time, this must be an explanation of mathematical knowledge. Therefore, what we need is an explanation of linguistic competence in a language that comes equipped with certain 1See (3; 4) for an attempt to solve this problem. 62 speech acts are assertions. Instead of asserting, their practice would be one of acceptance. The distinction between asserting and accepting would allow mathematicians to accept sentences without believing or asserting them. For example, suppose that mathematical sentences are acceptable if and only if they follow from the canonical story of mathematics —(5, p. 14). Then, mathematicians will accept that 2 + 2 = 4. And they accept that 2 + 3 =4. But in no case will they assert these sentences at the textual level. Thus, they could accept untrue sentences without asserting untrue sentences. Their speech acts would only be assertions in appearance —they would be “acceptances”. This move would allow the revolutionary to reconcile two things. On one side, they can accept the K-rule. On the other, they prevent mathematicians from systematically making mistakes. These would rarely assert anything at the textual level of the mathematical fiction. This solution is problematic for two reasons. First, the fictionalist must explain what the linguistic or epistemic practice of acceptance consists of. It is not enough to say that this practice exists. She must describe what its rules are and explain how it differs from other normative practices, such as asserting or judging. This is a theoretical task that no one has done so far. Until then, this will not be an acceptable solution. Second, in this context “to accept” seems to be synonymous to “to believe” or “to judge positively” (62; 18). Agents accept that φ when they believe or judge positively that φ is the case. For example, mathematicians accept that 2 + 2 = 4 when they believe that this is the case. Also, they do not accept that 2 + 2 = 5 because they do not believe this is the case. But, if this is so, accepting would not constitute a different practice —contrary to the practice of giving and accepting things. Moreover, by the K-rule, mathematicians must assert φ only if they know that φ . Therefore, they must assert φ only if they believe that φ —knowledge entails believing. And, if the equivalence between believing and accepting is correct, mathematicians assert φ only if they accept that φ . Consequently, if believing and accepting were really synonymous, assertion would be the linguistic correlate of acceptance. No difference would be found between them. Thus, there are serious reasons to believe that acceptance is not a normative practice in itself. The fictionalist could reply that the expression “to accept” is used by the theoretician to describe a convention. In this case, the convention of point out a few mathematical sentences among the myriad of them —those sentences that follow from the canonical story of mathematics. Conventions are different from constitutive rules, as the rules governing speech acts or games (147). The latter are necessary to perform the acts ruled by them. More precisely, the rule that one must φ to perform an act is constitutive of the act if and only if it is necessary that one must φ it in order to perform the act.6Take the case of chess. Necessarily, the rule that the knight moves in L must be observed to be playing chess. If the knight moved four squares at a time, it would be a different game. On the contrary, conventions are arbitrary. They are not “essential” to any activity, and can be replaced by others —more or less— at will. Using the notion of convention, revolutionary fictionalism can argue that mathematicians “accept by convention” certain mathematical sentences. Those sentences belonging to the canonical story of mathematics. Thus, they could accept by convention untrue sentences, without believing or asserting at the textual level these sentences. 6Demanding that φ is necessarily satisfied will be excessive. This would imply that playing a game is incompatible with breaking the rules. 69 This solution is as problematic as the previous one. The main problem is the collapse of revolutionary fictionalism into other forms of anti-realism. For example, it could collapse into conventionalism — which is very problematic (110). In general, it will collapse into a position very similar to hermeneutic fictionalism. This is because the appeal to conventions reintroduces a level distinction very similar to the one made by the latter. Mathematicians will not assert sentences accepted by convention. That is, they will not make assertions at the textual level of the convention. Thus, they will not make untrue assertions, solving the challenge posed by the reliability of mathematicians. Instead, they will assert that this or that sentence is accepted by convention. But, this is similar to claim that mathematicians move at the metatextual level of convention. However, as I argued in the previous section, this level distinction does not allow to explain the reliability of mathematics. The real problem of appealing to different constitutive norms or conventions is that the second assumption of the challenge is inadvertently denied: that mathematicians do assert some simple arithmetical sentences on a systematic basis. For these kind of solutions to work, mathematicians cannot assert any mathematical sentence. They are allowed to do something else with them: what is commanded by the alternative rules or by the conventions. But, they cannot make mathematical assertions on a systematic basis. On the contrary, both solutions would fail to solve the challenge posed by mathematical reliability, because they say nothing about mathematical assertion. Consequently, they do not prevent mathematicians from making untrue assertions on a systematic basis if they make mathematical assertions. As I said in the previous section, this is more plausible than the opposite, and the revolutionary fictionalist should accept it. Consequently, no one of these solutions is helpful. 3.3.4 Rethinking Assertion The only solution left is to deny the first assumption of the challenge, i.e., to deny the K-Rule ■(A φ →K φ ). If assertion obeys a non-factive, weaker condition than knowledge —which, by factivity entails truth—, it would be safe to assert untrue sentences as well as to assert sentences without knowing if they are true. There are two options: to deny that the K-Rule is a rule governing mathematical discourse or to deny that the K-Rule is the unique rule governing assertion. Let’s start with the former case. Assume the K-Rule is not among the rules governing mathematical discourse. Moreover, remember that the K-Rule is a constitutive rule —as any rule governing speech acts (147). By assumption, it is the unique constitutive rule of assertion. Now, there are two options again: or assertion is governed by the K-Rule in other discourses, or no discourse is. Take the first option, and suppose —without loss of generality— that assertions about ordinary objects, such as trees, houses or mountains, are ruled by the K-Rule, as it seems to be the case. Then, because of the constitutive nature of the rules governing speech acts, mathematicians simply would assert nothing. The rules governing mathematical discourse are constitutive. But, the K-Rule —the unique rule governing assertion— is not among them. Consequently, mathematical discourse will not be endowed with the practice of making 70 assertions. Mathematicians would not make untrue assertions because they would not make assertions at all. This is simply false. Conditionalization and Modus Ponens are sufficient conditions for the language to allow assertion. Thus, to deny that mathematical discourse is not one of assertion is to deny mathematics these two phenomena. But, they are essential to mathematical practice. Just think of the use of Modus Ponens in classical and non-classical mathematical proofs. This is more than enough to reject the first option. Even worst, think about the practice of making questions. No one will deny that mathematicians, as well as usual speakers, ask questions about mathematical matters. Often, such questions are what guide their research. Is V = L true? Does every polynomial with complex coefficients and higher degree than zero has complex roots? Any discourse holding the practice of asking questions, also holds the practice of answering them: this is the practice of making assertions. Even if mathematicians are unreliable, this is so precisely because they make false assertions. Alternatively, the revolutionary fictionalist can claim that no discourse is governed by the K-Rule. Thus, assertion would be governed by other rule. But, which one? An in-depth discussion of this issue would take me too far away from my objectives. Let me point out that it is sufficient that the K-Rule governs assertion in some discourse for it to govern assertion in all discourses. Again, this follows from the constitutive nature of rules governing speech acts. Furthermore, it is highly plausible that the K-Rule governs assertion in some discourse. For example, assertion concerning ordinary objects. When people claim —sincerely— things such as “the tree is taller than the house” or “the table is made of wood”, this is because they at least consider that they know that this is the way things are —i.e. they consider that they know that the tree is taller than the house or that the table is made of good. Note that this is consistent with the fact that these people do not really know how the things are. All that K-Rule requires is that they consider that they do know. Conversely, if people is sincere and do not know these things they would output rather different reports. For instance, they will say something as “I do not know how tall the tree is” or “this table seems to be made of wood, but I am not sure”. When people do not know something and act sincerely —i.e. in accordance with the rules—, they make it explicit in their statements. Therefore, a weaker rule than the K-Rule seems implausible. This rule would involve a non-factive condition weaker than knowledge —for example, belief. I.e., it would be of the form ■(A φ →C φ ), where Cis some non-factive condition of assertion: it is not the case that C φ → φ . In turn, a stronger rule —resulting from a stronger condition Cthan knowledge— will be as fatal as the K-Rule for revolutionary fictionalism. For, in this case Cwill be factive. Therefore, it will forbid the assertibility of untrue sentences. Moreover, the rule will govern assertibility in all discourses —including mathematics. Consequently, if the second assumption is also accepted —mathematicians assert simple arithmetical sentences—, the reliability challenge arises again. See (147) for a detailed defence of the K-Rule against weaker and stronger alternatives. This is enough to put some pressure against this option. It is highly implausible that no discourse is governed by the K-Rule. Consequently, is highly plausible that assertion is governed by the K-Rule in any discourse whatsoever —including mathematics. 71 Finally, there is one option left —the second horn of the first dilemma. Revolutionary fictionalism could defend that assertion, as most games, is governed by several rules. When talking about ordinary objects, speakers would obey one of those rules. When talking about sets, functions and numbers, mathematicians would follow another, different rule. Furthermore, the latter should involve non-factive conditions Cfor warranted assertibility —see the previous paragraph. Such a rule would make safe the assertion of falsities and sentences lacking a definite truth value. For instance, “2 + 2 = 4” could be untrue while C(2 + 2 = 4). The condition of assertion Cdoes not entail truth —in this case it does not entail that 2 + 2 = 4. Thus, untrue sentences could satisfy it without contradiction. Again, I cannot stop here to provide a careful analysis of this option. Nevertheless, I also find it highly implausible. Let me briefly state two reasons why I think so. First, non-factive rules of assertion for mathematics meet the same criticism exposed in the previous paragraph. The comments made on the discourse of concrete objects apply in this case too. Mathematicians assert that 2 + 2 = 4 because —at least— the consider that they know that 2 + 2 = 4. Similarly for other arithmetical sentences. When they consider that they do not know something, mathematicians do not assert that this is the case. For example, take CH. Some set theorists believe this is a false statement. Moreover, some of them claim that they want to prove that this is so. However, nobody dares to assert that CH is true or is false. Thus, belief does not seem enough for mathematical assertion. This speaks against rules weaker than the K-Rule —i.e. a rule resulting from replacing Cby a non-factive condition. On the other side, stronger, factive conditions are useless to solve the challenge posed by mathematical assertion. Consequently, it is highly plausible that mathematical assertion is governed by a rule at least as strong as the K-Rule. But, if this is so, revolutionary fictionalism cannot explain mathematical assertion. Second, if assertion is governed by several rules, all of them will apply in all discourses allowing for assertion. For example, they will rule assertion about ordinary objects as well as about mathematical objects. For, suppose that an activity is governed by several rules. Also, suppose this activity is present in different discourses. Then these rules will have an effect on all of these discourses. On the contrary, it is doubtful whether this activity is the same activity across discourses. After all, its identity depends on the constitutive rules. So, suppose that in one discourse no activity is governed by the same rules as the activities present in another discourse. In this case we are forced to conclude that the former is not present in the latter. From this it follows that if mathematical assertion involves a non factive condition, this will also apply to the discourse about ordinary objects. However, as I argued in the previous paragraph, this is implausible. In conclusion, the rejection of the K-Rule as the unique rule governing assertion is highly problematic. The consequences of this rejection spread beyond mathematical language, reaching other discourses that fictionalism wishes to remain untouched. If mathematical language is as any other language which allows for assertion, any change made in the former will affect the latter. 3.3.5 The Unreliability of Fictionalist Mathematicians It follows from all the previous arguments that fictionalism must plausibly accept the two assumptions of the challenge: that the K-Rule is the unique rule governing assertion and that mathematicians make simple, arithmetical assertions. Moreover, if these two assumptions are true, then fictionalism entails that 72 mathematicians make untrue, forbidden assertions on a systematic basis. Consequently, if fictionalism is true it is highly plausible that mathematicians are highly unreliable professionals. This is enough to argue against it. Any position that plausibly makes mathematicians unreliable professionals —concerning mathematics— is a bad explanation of mathematics. In this sense, fictionalism is not so different from platonism. Like this, it cannot explain Benacerraf’s —reinforced— challenge. However, this result can be improved: fictionalism cannot explain the reliability of mathematicians in principle. The solution of Benacerraf’s reinforced challenge requires solving Xin the following template.7 (3.i) Fictionalism can explain why mathematicians are highly reliable when gathering knowledge about X (3.ii) If (3.i) is the case, then fictionalism can explain why mathematicians accept a mathematical sentences φ only if φ conforms to X. Then, (3.i) and (3.ii) entail that (3.iii) Fictionalism can explain why mathematicians accept mathematical sentence φ only if φ conforms to X. Moreover, (3.iv) According to fictionalism, all mathematical sentences φ conforming to Xare true Therefore, from (3.iii) and (3.iv), (3.v) Fictionalism can explain why mathematicians accept mathematical sentences φ only if φ is true. Here “to accept a sentence” means “to be disposed to assert a sentence.” Now, by definition, fictionalism does entail no true instances of (3.iv). According to fictionalism, any solution of Xthat satisfies (3.i) is not truth-conductive. It says that mathematicians are good at two things: knowing which fictions are consistent and which sentences follow from them. Thus, X=Consistency + Logical entailment. However, consistency and logical entailment are not criteria for the truth of sentences —the fact that a sentence is consistent, or the fact that it follows from a consistent theory, is not enough for its truth. Consequently, the following instance of (3.iv) is false: (3.iv∗) According to fictionalism, all sentences φ that are consistent or follow from a consistent theory are true. 7I take the form of this argument from (2, pp. 51-52). 73 Furthermore, fictionalism does not entail any other solution for X. Consequently, it does not solve Benacerraf’s reinforced challenge. And, this is so even if the K-Rule is denied. For the latter, just note that the K-Rule is not necessary for the challenge. The fact that mathematicians are reliable is not equivalent to the fact that they obey a rule of assertion based on a factive condition —such as the K-Rule. Even if this is the case, they may break the rule when they realise that it does not lead them to the truth. In this situation the rules of language are at odds with the rules of epistemology —breaking the former is an epistemological virtue. But, this is a coherent situation. For example, imagine that for asserting something it is enough to believe it. Believe is not factive. Thus, mathematicians would have to be vigilant to know when their beliefs are not truth-conductive. As a result, in this situation it makes even more sense to require an explanation of mathematical reliability. Concerning the second assumption, its status is more dubious. If mathematicians do not make assertions at face value at the textual level of mathematics, the challenge runs the risk of being trivialized. Consider again the above template. The absence of mathematical assertions makes the material conditional in (3.iii), (3.iv) and (3.v) vacuously true. Mathematicians make assertions only if they conform to Xbecause —in this situation— they do not make assertions at all. Then, any solution for Xmakes the material conditionals true. Nevertheless, (3.iii) and (3.v) ask for an explanation of the conditionals. If this requires intensional or hyperintensional resources, then making the conditionals true will not be enough. An explanation of mathematicians silence still will be required. Therefore, the absence of mathematical assertions does not trivialize the challenge either. In sum, the failure of the K-Rule does not trivialize Benacerraf’s challenge. Moreover, if explanations require something more than making material conditionals true, then the absence of mathematical assertions does not trivialize the challenge either. In any event, fictionalism cannot provide in principle a satisfactory solution for X. Consequently, fictionalism does not solve in principle Benacerraf’s reinforced challenge. 3.4 FULL FICTIONALISM AND THE LANGUAGE ACQUISITION CHALLENGE Fictionalism denies the existence of mathematical objects. At the same time, it agrees with platonism in taking mathematical language at face value. According to fictionalism, sentences such as “2 is prime” or “the rationals are dense” aim to be about some objects, the number two and the rational numbers. Specifically, mathematical terms such as ”2” aim to refer to mathematical, abstract objects. Consequently, according to it most mathematical theories will contain untrue sentences. Even the language of weak theories, such as Peano-Dedekind arithmetic, is endowed with a referential apparatus involving quantifiers and terms. However, if mathematical objects do not exist, mathematical sentences aiming to quantify or refer to them will be untrue —they will be false, or neither true nor false. Even worst, most mathematical theories entail existential claims falsified by fictionalism. For instance, the claim that the number two exists. Supposedly, the failure of the referential apparatus of mathematical language does not imply that it lacks content. After all, fictions use terms that refer to no object. But, people manage to read, understand and enjoy them. For instance, take —once more— the case of the term “Hamlet” as it is used in Hamlet. 74 This term does not refer to a danish person who discovered that his father was murdered with ear’s poison. Hamlet did not exist. Despite this, people have read and enjoyed this play many times. They understand untrue sentences made at the textual or metatextual levels of fiction. For example, they understand the sentence “Hamlet slayed his aunt Claudius”, or the sentence “Horatio has seen the ghost of Hamlet’s father”. Verbs such as “to slay” or “to see” are part of common English and express well-known actions. Thus, they can be used to attribute actions to fictional characters —even if we do fail in doing this. The language of fiction is not an empty frame. Fictionalism claims that the same is true of mathematical language. Mathematical vocabulary expresses operations and relations —such as addition, exponentiation, or set formation— which are familiar to us. And, mathematicians try to attribute them —unsuccessfully— to some objects. Usually, this is taken as an advantage of fictionalism over alternative versions of anti-realist anti-platonism. Moreover, it is worth pointing out here that even if it were the case that mathematical practice ran counter to fictionalism as well as the other versions of anti-realism, we would still have reason to favour fictionalism over these other views, because it would still be true that fictionalism provides a standard semantics for the language of mathematics. In other words, whereas non-fictionalistic versions of anti-realism provide non-standard views of the truth conditions of mathematical sentences, fictionalism provides a standard view here. Thus, even if it provides a non-standard view of whether or not these truth conditions are actually satisfied, fictionalism jibes with mathematical practice more than other versions of anti-realism do. (2, pp. 103-104) In this section, I will challenge this assertion. Despite appearances, if mathematical language is taken at face value, fictionalism cannot explain how it acquires its content, nor how speakers understand it. Call the version of fictionalism endorsing mathematical language with standard truth conditions —i.e. platonistic truth conditions— full fictionalism. I will argue that full fictionalism is an untenable account of mathematics. It fails to meet the Language Acquisition Challenge (LAC) exposed in Chapter 2, 2.4.2.2. The argument relies on the similarity between full fictionalism and platonism concerning the content of mathematical language. If mathematical language is taken at face value, i.e., as speaking about abstract objects, then both are faced with the same explanatory pressure: they have to explain how the language acquires its content, and how speakers acquire their competence. Moreover, any fictionalist explanation will be acceptable to platonism, which will use it to dispel Benacerraf’s challenges. Thus, if full fictionalism is true, then platonism dispels Benacerraf’s challenge. This is fatal for fictionalism, because platonism’s epistemological problems are the main argument in its favour. To understand the situation better, think of Charles Dickens’ Oliver Twist. This the story of an orphan growing up on the tough streets of 19th century London. The novel is a stark depiction of the harsh living conditions of the city’s inhabitants. Also, it is a fiction. The events it is about did not really happen. Strictly speaking, most of the plot is untrue. Nevertheless, this is not a problem for readers. People, houses, animals, etc, are mentioned in story, but these words are part of English. That is, Dickens wrote in aperfectly meaningful language. There are perfectly good empirical explanations of how people acquire concepts such as person,house or animal. However, the case of mathematics is “a bit” different. Most of 75 the vocabulary of Oliver Twist is empirically grounded. It acquires its content through complex physical relations with the environment —people, houses, animals, etc. Conversely, people is not physically related to mathematical facts. Then, the fictionalist owes us an explanation of how mathematese acquires its content. To say that mathematics is a fiction does solve the LAC. As a result, full fictionalism is not in a better position than platonism. 3.4.1 The Challenge Let Pbe an n-ary predicate. Also, let Dbe a collection of objects.8Let’s say that Pis a meaningful predicate on the collection Dif and only if it expresses some intensional entity —a concept, property, relation, category, function, etc.— that can hold on D, or Pcan be used to describe the objects in D. Pis meaningful on D≡d f (a) Pexpresses some intensional entity Iwhich can hold on Dor (b) P can be used to describe the objects in D. The side (a) of the disjunction means that I’s extension is a subcollection of D, if the objects in D exist. Concerning the side (b), if a predicate Pcan be used to describe the objects in D, then Pfeature in sentences whose truth condition involve the objects in D—this fact will be important below. If a predicate Pcan be used to describe the objects in D, then Pfeature in sentences whose truth condition involve the objects in D. This is enough to assign Pthe desired semantic function without appealing to intensional entities. I believe that (a) and (b) taken together cover the semantic function attributed to predicates by the main semantic theories on the table.9If this is correct, (a) and (b) are at least a good approximation to an uncommitted gloss of what it means to be meaningful. Now, assume full fictionalism. I.e., the idea that mathematical language has platonistic content. Full fictionalism entails that mathematical predicates are meaningful on some collection of mathematical objects. For example, the arithmetical binary predicate “to be less than or equal to” (≤) is meaningful on the natural numbers. It expresses a relation —here, an intensional entity— that can hold on the natural numbers or it can be used to describe the naturals. Let Pbe one of those predicates. In what follows I will argue for the plausibility of the following conditional: (*) If Pis a mathematical predicate meaningful on D, then: or platonism about the objects in Dsolves Benacerraf’s challenge or some objects in Dexist (or have existed). Full fictionalism entails that mathematical vocabulary —including predicates such as P— is meaningful on some collection of mathematical objects D. From this and (*) it follows that if full fictionalism 8This way of expressing things implies that Dis a plurality of objects. This could be avoided by speaking in a slightly more complicated way. We could say that Dis a collection such that any element in Dis an object. For simplicity, I will use the plural vocabulary. However, note that this is avoidable. The argument is not restricted to pluralities of objects. 9See (130) for a survey of the main semantic and metasemantic theories available. 76 concerning the objects in Dis true, then platonism —about the objects in D— solves Benacerraf’s challenge or some mathematical objects in Dexist (or have existed). By definition of fictionalism, the second option is no available. Therefore If full fictionalism about the objects in Dis true, then platonism about the objects in Dsolves Benacerraf’s challenge. This is enough to cancel the main argument in favour of full fictionalism: the failure of platonism to solve the epistemological challenges. The argument for (*) will be not deductive. Hence, I will only claim plausibility for this conclusion. Two comments are in order. First, the fact that a predicate Pis meaningful on Dis compatible with the fact that no object in Dexists (or has existed). This is guaranteed by the “can” in its definition. This only requires that Pexpresses an intensional entity which can apply to the objects in D. It does not require that Papplies to them as a matter of fact. Similarly, it only requires that Pcan be used to describe these objects. This ensures that to be meaningful on Ddoes not rule out fictionalism concerning the objects in Dby fiat. For instance, some predicates in The Neverending Story are meaningful on the collection of dragons. But, dragons never existed. Second, (*) only involves predicates. However, the language of mathematics contains other kinds of expressions, such as operators expressing functions.10 Therefore, this conditional could seem unreasonably limited. If the argument does not work for other types of expression, the fictionalist might object that his position still works for them. Could fictionalism be the correct position concerning the objects affected by the other types of expression? The answer is negative for two reasons. First, the collection Don which a mathematical predicate Pis meaningful is the same collection of objects where the other vocabulary is interpreted. For instance, the arithmetical operators “+” and “×” express total functions defined on the collection of natural numbers —or they are used to describe these objects. But, this is the same collection of objects on which the predicate “to be less than or equal to” is meaningful —according to the above definition. Consequently, if (*) works for some mathematical predicate Pmeaningful on D, then fictionalism about the objects in Dwill be in trouble no matter what happens with other kinds of expression used to talk about these objects. Second, the conditional (*) can be formulated in terms of other expressions. Again, take the case of operators. Let fbe an n-ary operator. We start by defining the notion of an operator being meaningful on some collection of objects: fis meaningful on D≡d f (a) fexpresses some intensional entity Fwhich can hold on Dor (b) f can be used describe the objects in D. This definition has exactly the same form as before. Then, we state (*) using operations: (*)’ If fis a mathematical operator meaningful on D, then: or platonism about the objects in Dsolves Benacerraf’s challenge or some objects in Dexist (or have existed). 10By operator I mean a symbol expressing a mathematical operation. For example, the arithmetic operators “+” or “×”. To test the argument, let’s grant that these symbols not only form complex terms, but also express or describe functions. 77 Consequently, as soon as full fictionalism claims that some portion of mathematical language has standard content, this fact will turn against it. Moreover, this is so no matter the vocabulary used by the language. 3.4.2 Structure of the Argument The argument has the following form. Let Pbe a mathematical predicate meaningful on some collection Dof mathematical objects. First, I will argue that: (4.i) If (b) Pcan be used to describe the objects in D, then It is plausible that extensional approaches to semantics entail that platonism about the objects in Dsolves Benacerraf’s challenge, or that some objects in Dexist (or have existed). Then, I will argue for its complementary: (4.ii) If (a) Pexpresses some intensional entity Iwhich can hold on D, then it is plausible that intensional approaches to semantics entail that platonism about the objects in Dsolves Benacerraf’s challenge. This is enough for concluding that (*) is plausible. For, Pexpress an intensional entity or not. In the second case, Pwill still be used to describe objects, but we are left only with extensional explanations of this phenomena. Thus, if this is the case, (4.i) entails that: (4.iii) If (b) Pcan be used to describe the objects in D, then it is plausible that some objects in Dexist (or have existed). In turn, if Pexpresses an intensional entity, this will be explained by means of intensional semantics. Therefore, (4.ii) entails that: (4.iv) If (a) Pexpresses some intensional entity Iwhich can hold on D, then it is plausible that platonism about the objects in Dsolves Benacerraf’s challenge. Finally, (4.iii) and (4.iv) entail that: (4.v) If (a) or (b), then it is plausible that platonism about the objects in Dsolves Benacerraf’s challenge, or some objects in Dexist (or have existed). But, by definition of “being meaningful on a collection D”, (4.v) is equivalent to (*). And, by distribution of plausibility over the material conditional, we have proved that (*) is plausible. Clearly, the argument must not be taken as a deductive argument. Specifically, it only offers plausible support to (4.iii) and (4.v). One reason is that labels such as “extensional semantics” or “intensional semantics” are informal and too vague to be taken as part of a deductive argument. They constitute a clumsy classification of the main approaches in contemporary semantics. Below, I will stop and consider the main representatives of each of these two strands. Yet, there is no precise definition or description 78 any metaphysical or semantic thesis concerning non-abstract objects compatible with fictionalism, will also be compatible with platonism. The former differs from the latter only in that it rejects Existence. Therefore, if full fictionalism can use linguistic resources to meet the explanatory pressure posed by LAC, then platonism can meet this explanatory pressure too. As a result, it is highly plausible that it can solve Benacerraf’s epistemological challenge. If intensional theories of meaning are rejected, then the direct —or causal— theory of meaning, the use theory of meaning, and the descriptive theory of reference are the main candidates to dispense with LAC: i.e., to explain how language acquires its meaning, and how speakers acquire their competence. But, given a mathematical predicate Pthat can be used to describe the mathematical objects in D, I have shown how the descriptive theory of reference allows platonism concerning the objects in Dto solve Benacerraf’s challenge. Also, I have shown how the use theory of meaning entails that some objects in Dexist (or have existed). Moreover, the direct theory of meaning is plausibly useless for dealing with abstract objects. All this is enough to conclude (4.i): (4.i) If (b) Pcan be used to describe the objects in D, then it is plausible that extensional approaches to semantics entail that platonism about the objects in Dsolves Benacerraf’s challenge, or that some objects in Dexist (or have existed). Consequently, the first step of the argument is achieved. 3.4.4 Intensional Semantics In this section I will argue for (4.ii): (4.ii) If (a) Pexpresses some intensional entity Iwhich can hold on D, then it is plausible that intensional approaches to semantics entail that platonism about the objects in Dsolves Benacerraf’s challenge. Intensional theories of meaning explain meaning in terms of intensional objects, such as properties, relations, concepts, or propositions. The mark of identity of intensionality is that it does not allow for substitution salva veritate. In turn, this is the main feature of extensionality. Extensional contexts preserve the extension under substitution of expressions sharing the same extension. For example, the connectives of n-order classical logic are paradigmatic cases of extensional operators. Let Hbe an n-ary operator applying to sequences of expressions e1,...,enof the language. Extensional operator His an extensional operator if and only if for all expressions e11,...,e1nand e21,...,e2nsuch that each pair e1i,e2i,0<i≤n,has the same extension, then H(e11...e1n)and H(e21...e2n)also has the same extension. For instance, if two sentences φ and ψ have the same truth value, then ¬ φ and ¬ ψ —where “¬” is the negation of operator of first-order logic— also have the same truth value. On the contrary, intensional operators do not satisfy these conditions. That is, even if the e1i,e2ihave the same extension, the extension 85 of H(e11...e1n)can be different to the extension of H(e21...e2n). For instance, consider contexts generated by believing.15 The terms “Brutus” and “Julius Caesar’s murderer” have the same extension, both refer to the same person: Brutus. Thus, the sentences “Brutus murdered Julius Caesar” and “Julius Caesar’s murderer murdered Julius Caesar” also have the same truth value —both are true. It is reasonable to expect a person who has only been told that Julius Caesar was murdered, but does not know about Brutus, to believe that Julius Caesar’s murdered Julius Caesar. However, this same person could not believe that Brutus murdered Julius Caesar —she lacks this piece of information. However, if this is so, how can the difference be explained? If there is no semantic difference between the expressions e1i,e2i—they have the same extension—, how can H(e11...e1n)and H(e21...e2n)differ in extension? Intensional approaches to meaning explain this phenomenon by endorsing fine-grained semantic distinctions. While extensional approaches take the expressions e1i,e2ias being semantically equal —they have the same extension—, intensional approaches believe that the extensional ones are missing part of the whole story about the meaning of expressions. Specifically, they attribute additional semantic differences to the expressions e1i,e2i. Now, these share the same extension. Hence, itensional approaches have to expand semantics by adding additional semantic categories —intensional categories, such as properties, relations, concepts or propositions. Then, they associate entities I1...Inbelonging to some of these categories with each one of the ei. As a result, they are able to explain why H(e11...e1n)differs in meaning from H(e21...e2n): they differ because I1i=I2ifor some 0<i≤n. Intensional operators have into account intensions —the objects falling under the new semantic categories. Intensional operator His an intensional operator if and only if for all expressions e11,...,e1nand e21,...,e2nsuch that each pair e1i,e2i,0<i≤n,has the same intension, then H(e11...e1n)and H(e21...e2n)also has the same intension. Again, take the case of Brutus and Julius Caesar. Speakers can believe that Brutus =Julius Caesar’s murderer because the intension associated with the name “Brutus” is different from the intension associated with the name “Julius Caesar’s murderer”. This difference transmits to the intensions attributed to the sentences φ ≡“Brutus murdered Julius Cesar” and ψ ≡“Julius Caesar’s murdered murdered Julius Caesar”, explaining the difference between believing that φ and believing that ψ . Given the limitations of extensional semantics, it is worth asking whether intensional entities can be of help to full fictionalism. Can full fictionalism resort to the use of intentional entities to deal with LAC? I believe it cannot. Despite its limitations, I will use possible worlds semantics through the argument for simplicity. Possible world semantics analyzes intensions in terms of possible worlds and the extensions corresponding to them. According to this approach, the intension associated with an expression is identified by the extension of at all relevant possible worlds. Let Ibe an intensional entity and fa binary function assigning the extension of Iat every relevant possible world w. Then, I1=I2if and only if for all possible worlds w:f(I1,w) = f(I2,w). 15Many of the phenomena related to belief can be treated as intentional phenomena. However, many others cannot. Beliefs are now considered to be hyperintentional phenomena. 86 Of course, possible world semantics owes us a metaphysical explanation of what possible worlds are. However, I will use possible worlds just as an heuristic device to clarify the argument on some points. Nothing of what I will say depends on the use of this formal tool. 3.4.4.1 Fictionalism and Intensions Before arguing for the negative result, I want to say a few words about the fit between fictionalism and intensions. An immediate worry arises: intensions are abstract entities. Then, the use of intensional semantics is in tension with fictionalism. For, if the latter rejects the existence of mathematical objects to avoid the problems posed by their abstract nature, it is to be expected that fictionalism rejects the existence of all abstract objects. As a result, intensional theories of meaning would be as false as mathematical theories. For the sake of the argument, I will consider that intensions are available to the fictionalist. Moreover, it is perfectly coherent to be fictionalist about one kind of object, but not about others. For example, about mathematical objects, but not about concepts or propositions. At last, I am in position to argue for (4.ii). Assume that Pis a mathematical predicate expressing some intensional entity Iwhich can hold on D. LAC requires an explanation of how speakers relate to intensions. There are two options. First, to elaborate an explanation of how speakers relate to Iwithout involving other objects —let’s say, an explanation of the “direct” relation between speakers and I. Second, to elaborate an explanation that involves other objects than I. In what follows I will consider both options. 3.4.4.2 Direct Grasp Suppose full fictionalism about the objects in Dhas at its disposal an explanation of the direct relation that speakers maintain with I—i.e., an explanation that only involves the speakers and this intension. An extremely simple —and bad— explanation is that speakers have a sixth sense that allows them to grasp these entities. Obviously, if this were so, the epistemology of modal concepts would not be a major philosophical problem. Unfortunately, a direct explanation is not of help for full fictionalism. If this can solve the LAC using such an explanation, so can the platonist. Then, platonism will explain mathematical reference in the same way as it would do it using a descriptive theory of reference. First, it will explain how speakers grasp the intensions associated with the other expressions of mathematical language. Plausibly, if speakers directly grasp I, they will do the same with the other intensions expressed by mathematical vocabulary. Second, the platonist will stipulate that mathematical terms refer to the objects which satisfy the descriptions coached in the language. Finally, if platonism can explain mathematical reference, it is highly plausible that it can dispense with Benacerraf’s problem too. 3.4.4.3 Indirect Grasp Let’s see what happens with the second option. Assume that objects other than Iare required to explain how speakers grasp the latter. We have to consider three cases: (i) the grasp of Iinvolves the objects it can describe —i.e., objects in its extension f(I,w), for some possible world w; (ii) the grasp of Idoes 87 not involve objects in the extension of I; (iii) Iis grasped through analytic relations to other intensional entities. First, suppose the grasp of Iis explained using some objects it can describe. Here, there are two options: these objects exist (or have existed) or they never existed. Suppose they never existed. If these objects never existed, then no counterfactual connection could have been established between them and speakers. Consequently, they can hardly be of help to indirectly grasp the intension I. What explanatory role can be played by entities with which the speakers have had no relationship? Simply, they play no role in agents’ epistemological and linguistic actions. At this point, the notion of non-existent objects could come to the rescue. But, this would be in vain. Even if such entities are accepted in our metaphysical inventory, this does not change their status concerning agents: if some objects never existed, then they are not counterfactually related to agents and should not play any explanatory role concerning epistemology and language. On the contrary, if agents somehow were related to objects that never existed, this must be an epistemological or linguistic non-counterfactual relation. Therefore, the explanatory relation is the other way around. Agents would understand what non-existent objects are because they grasp enough concepts to describe them. For example, we can understand what Pegasus is because we have been told how Pegasus looks like. Similarly, we know what dragons are because there are tales and movies where dragons are depicted in great detail. Agents are related to Pegasus or dragons —if they are accepted as non-existent objects— because they grasp the concepts employed to describe them. Moreover, the category of non-existent object is metaphysically controversial. And, those explanations that use it will also be highly controversial —alternative explanations which dispense with them should be given preference. This is enough to discard non-existent objects. Now, suppose the grasp of Iis explained using some objects it can describe —i.e., some objects in f(I,w), for some possible world w. Moreover, suppose these objects exist or have existed. The problem with this option is the same problem I opposed to the direct grasp of intensions. If fictionalism can explain the indirect grasp of Ithrough some objects that exist of have existed, then so can platonism. Platonism accepts all the existential assumptions accepted by fictionalism. Unlike fictionalism, Platonism accepts all its existential commitments. Thus, if an explanation is acceptable to the former, it will be acceptable to the latter. Besides, using the indirect grasp of intensional objects, platonism can explain mathematical reference —it stipulates reference to mathematical objects using the appropriate definite descriptions— and dispense with Benacerraf’s epistemological challenge. Consequently, the first option (i) leads to the desired conclusion. The second option (ii) faces the same problem. What about the third option (iii)? Can agents’ grasp of Ibe explained by means of some analytical entailment from other intensions I1...In? Suppose so. Then, agents’ grasp of some Ii,0<i≤n, must be explained through non-analytic relations. On the contrary, speakers’ understanding of Iwill be explained by means of an infinite descending chain of intensional entities. But this kind of situation is unacceptable in explanations of linguistic competence. It is highly implausible that in order to explain how agents grasp —for example— a concept, they need to make use of infinitely many concepts.16 Now, agents’ grasp of Iiis direct or indirect. And this leads us back to the problem of previous explanations: if fictionalism can explain how agents —directly or indirectly— grasp Ii, then so can platonism. 16See Chapter 4, 4.5.1 for an argument against this situation. 88 The use of intentional entities puts the fictionalist in a bind, and not only because of their metaphysical status. However, this should not be a surprise. If full fictionalism about such entities explains the content of mathematical language and meets the challenge posed by LAC, then a stronger metaphysical position may also be able to do so. For example, platonism. This is sufficient to conclude (4.ii): (4.ii) If (a) Pexpresses some intensional entity Iwhich can hold on D, then it is plausible that intensional approaches to semantics entail that platonism about the objects in Dsolves Benacerraf’s challenge. Finally, (4.i) and (4.ii) are enough to prove the desired conclusion: (*) If Pis a mathematical predicate meaningful on D, then: or platonism about the objects in Dsolves Benacerraf’s challenge or some objects in Dexist (or have existed). This is enough to pose full fictionalism in serious trouble. Full fictionalism entails that mathematical predicates Pare meaningful on some collection of mathematical objects D. But, this and (*) entail that or platonism about the objects in Dsolves Benacerraf’s challenge or some objects in Dexist (or have existed). By assumption, the second case is forbidden. Therefore, If full fictionalism about the objects in Dis true, then platonism about the objects in Dsolves Benacerraf’s challenge. Nevertheless, the main arguments for fictionalism are the epistemological challenges opposed to platonism. Consequently, full fictionalism entails its own misfortune. If it is accepted, the main argument in its favour vanishes. 3.5 ALGEBRAIC FICTIONALISM AND THE CONSISTENCY CHALLENGE The failure of full fictionalism does not immediately entail the failure of the fictionalist program. Fictionalism is not committed with any particular semantics. It just claims that mathematical objects do not exist and mathematical mathematical theories incorporate untrue sentences. It does not need to endow mathematical language with a standard semantics, nor with a platonic content. All it says is that mathematicians write untrue stories. These stories are summarized in the axioms of different canonical theories. Then, they devote all their efforts to find out what follows from these stories as well as to continue writing them. Fictionalists view the basic assumptions of a pure mathematical theory (its axioms, or perhaps if we wish to follow Balaguer, its “full conception” of its objects) not as truths, but as generative of a fiction. Mathematical practice then becomes the practice of developing these mathematical fictions, by working out what does and does not follow from their generative assumptions. (76, 125) Call this version of fictionalism algebraic fictionalism.17 17I take this label from (76) and (5), who distinguishes between formalist and non-formalist versions of platonism. 89 Fictionalism abandons the idea of strong objectivity. Unlike platonism, it denies the existence of a unique, true mathematical theory. And, mathematical theories need not be complete —indeed, incompleteness does not pose any pressure on fictionalism. Mathematics does not tell a closed story. Fictionalism poses a unique constraint on classical theories: consistency. It reduces knowledge about classical mathematics to knowledge about consistent fictions. Since (classically, at least) inconsistent assumptions imply everything, only consistent fictions are mathematically interesting. So, as with other algebraic views, for fictionalism what makes a theory mathematically acceptable is consistency, and what makes a mathematical utterance appropriate in the context of mathematical theorizing is that it is a consequence of the assumptions of the theory in which it is put forward. (76, 125) Therefore, knowledge about a mathematical sentence φ reduces to two things: (i) knowing that φ follows from a given mathematical, fictitious theory Γ, and (ii) knowing that Γis consistent: K(Γ⊢ φ +Γis consistent) =d f KF φ . In this way, fictionalism exchanges abstract objects for consistency. The question is: is this a good exchange for algebraic fictionalism? In what follows I will argue that the answer is negative. (**) Algebraic fictionalism lacks an adequate concept of consistency to explain mathematical knowledge. The argument goes as follows. There are two main concepts of consistency. On one side, the concept of consistency commonly used in model theory and proof theory is a mathematical concept. Although this is perfectly useful, for fictionalism it is only part of an untrue fiction. As a result, it cannot be used for the benefit of its epistemological explanations. On the other side, algebraic fictionalism resorts to a primitive concept of consistency. In this case, the problem is that this is a semantic concept. However, algebraic fictionalism does not attribute any content to mathematical language —this is the main difference between it and full fictionalism. The use of a primitive concept of consistency turns algebraic fictionalism into full fictionalism. But, as I argued in the previous section, this is an implausible account of mathematics. Therefore, algebraic fictionalism cannot make use of any concept of consistency. As a result, its explanation of mathematical knowledge fails. Furthermore, full fictionalism and algebraic fictionalism exhaust fictionalism. Therefore, if both are in trouble, fictionalism as such is also in trouble. The arguments contained in the previous section and in this section are fatal to this philosophical position. 3.5.1 Mathematical Notions of Consistency Consistency poses a problem on the fictionalist’s reduction of mathematical knowledge (2, p. 99), (76, 126). Let’s start with platonism. Platonism has no problem in dealing with consistency and other modal notions, such as possibility. Following usual model-theoretic definitions it can postulate a reductive account according to which for a collection of sentences to be consistent just is for it to have a model 90 —a model in classical first order logic is a non-empty set embedded with a collection of relations and operations. Model-theoretic Consistency A set of sentences Γis model-theoretically consistent if and only if there is a model M such that for all φ ∈Γ, M is a model of φ (M |= φ ), where “M |= φ ” is defined as usual in terms of Tarski’s satisfaction relation. Alternatively, the platonist can use a proof-theoretical definition. According to this, a set of sentences is consistent just in case it does not entail a contradiction. Indeed, in classical logic a contradiction entails every sentence. Therefore, it is enough to require that the theory does not prove some sentence. Proof-theoretical Consistency A set of formulas Γis proof-theoretically consistent if and only if there is a sentence ψ such that Γ⊬ ψ . By correctness and completeness, both definitions are equivalent in classical first-order logic. The concepts of model and proof in a formal system are mathematical concepts. A model is a — non-empty— set embedded with a collection of relations and operations. In turn, a —finitary— proof is a number-theoretic recursive function. Therefore, the platonist can employ them to provide a perfectly good analysis of consistency. Consequently, knowledge about consistency as well as logical knowledge become entangled with mathematical knowledge. On the contrary, no reductive analysis of this kind is available for the sceptic. Fictionalism denies the existence of models and functions. So, any of the previous definitions would be false by fiat. Take the model-theoretic definition. According to the fictionalist, no set of sentences Γhas a model because no model exists. Generally, no set of sentences Γwould be consistent because all attributions of consistency would involve claims about non-existent objects. This is a highly unpleasant conclusion. We have strong intuitions about the consistency of —at least— some theories. For instance theories with a finite domain. Theories with finite domains are decidable, and PA proves their consistency. If the definition of consistency entails that no theory is consistent, this will be a bad definition. Besides, fictionalism reduces mathematical knowledge to knowledge about consistency. Then, if fictionalism is true, the model-theoretic and the proof-theoretic definitions of consistency entail that there is no mathematical knowledge. Certainly, algebraic fictionalism needs an alternative to explain consistency attributions. 3.5.2 Consistency as a Fiction One option is to maintain the model-theoretic and proof theoretic definitions of consistency, while understanding consistency attributions as a case of applied mathematics. According to this, consistency would be just one piece of a cognitively useful fiction. It would be an important device to describe some empirical facts: the behaviour of mathematicians concerning a collection of statements. After all, there is no difference between this and how mathematical theories are used when doing physics or economy. Consistency attributions are untrue, let’s grant this. Also, they are valuable cognitive resources that —allegedly— do not add anything metaphysically relevant to the world. Therefore, to explain these attributions all we need is to explain how the mathematical theories explaining the concept of consistency 91 —model theory and arithmetic— are applied to describe and explain the mathematicians’ behaviour. Specifically, the process by means of which they accept and study a theory. Once this is done —either if mathematics are dispensable or they are nominalistically indispensable—, it will become clear what mathematicians are exactly doing when they recognize a theory as consistent. This simple idea accords with fictionalism’s spirit: to deny the existence of mathematical objects, but not of mathematics. Mathematics can be used for many things, among them to explain mathematicians labour. Then, the fictionalist can resort on this particular application of mathematics to state her philosophical position concerning mathematical knowledge. There is nothing circular here. Furthermore, this should not be objected by saying that the fictionalist owes us —on pain of pragmatic incoherence— a non-mathematical depiction of her philosophical position. That is, a depiction that does not involve any mathematical vocabulary —in this case, the mathematical expression “consistency”. Indispensable applications of mathematics are compatible with nominalism (2, pp. 96-97). However, this simple solution indeed leads to circularity. According to fictionalism, consistency is intended both as a criterion for theory choice and mathematical knowledge. Mathematicians know which theories are consistent, which theories are not, and decide to study the former. Then, according to the previous proposal, mathematicians’ choices and mathematical knowledge are explained by applying a particular theory Tin which consistency is defined —model-theory or arithmetic. Consequently, the explanation of theory choice and mathematical knowledge is relative to the previous choice of T—which already involves consistency. Now, the problem is: how do we explain the choice and knowledge concerning T? First, it is not self-explainable. No mathematical theory including Peano-Dedekind arithmetic as a subtheory proves its own consistency —if it is consistent. But, Tincludes arithmetic or model theory as a subtheory. Consequently, Tdoes not prove its own consistency —if it is consistent. It will not be possible to explain the choice and knowledge provided by Tby means of Titself. Thus, another theory T∗ is required to explain the mathematical knowledge about T. To speak about consistency, T∗must include Peano-Dedekind arithmetic. Consequently, mathematical knowledge concerning T∗cannot be explained in T∗. This situation generates an infinite regress. In this case, the infinite regress is unacceptable. It is highly implausible that knowledge of the consistency of one theory Tis equivalent to the knowledge of the consistency of infinitely many theories. Moreover, the model-theoretic and the proof-theoretic definitions of consistency entail that consistency is a relative concept. Different mathematical theories entail different facts about which theories are consistent. For example, ZF + Mod(ZF) —the sentence claiming that ZF has a model— entails the consistency of ZF, while ZF remains silent on this. However, facts about mathematical knowledge should not be relative to any mathematical theory. Consequently, the idea that consistency is a mathematical fiction is unworkable for fictionalism. 3.5.3 Primitive Consistency and the Semantic Nature of Language Fictionalism cannot explain logical knowledge by means of mathematical theories. Indeed, the explanatory direction goes in the opposite way: using an appropriate notion of consistency, the fictionalist explains mathematical knowledge —i.e., knowledge of consistent fictions. Thus, the model-theoretic and 92 proof-theoretic definitions of consistency are useless for fictionalism’s epistemological purposes. To explain mathematical knowledge through consistency, fictionalism needs a more robust and realist conception of the latter. Similarly, note that it cannot define consistency through modality either. For instance, it cannot stipulate that a theory is consistent just in case it is possible that all its sentences are true at the same time. There are two reasons for this. First, possibility, as well as consistency, is in need of explanation. Second, the fictionalist will be suspicious about some kinds of modality. More specifically, she will oppose any conception of de re modality because it would commit her with necessary properties or with possibilia —such entities do not fit very well in the nominalist’s metaphysical inventory. Nevertheless, given the relation between consistency and possibility, the fictionalist could use the latter as long as it does not imply too strong metaphysical commitments. In sum, the fictionalist needs a non-reductive account of consistency and possibility. Many authors (71; 39; 55; 2; 75), have advocated intuitive and primitive notions of consistency and logical possibility. Neither of these two concepts would be a mathematical concept —precisely because they are primitive, that is, they do not admit definition. To accept them is tantamount to accepting brute facts concerning consistency and modality. So long as the logical notions are not defined in model-theoretic (or proof-theoretic) terms, but are instead considered as modal primitives, there is no principled Benacerrafian objection to our having knowledge of logical consistency, consequence, and validity. I have suggested that there are reasons even for the platonist to resist attempts to reduce these modal notions to non-modal alternatives. (75) Fictionalists such as Field (39) and Leng (75) have used this kind of primitive concepts to secure fictionalism —i.e., to explain mathematical knowledge in terms of consistency or possibility. Suppose that speakers —including mathematicians— understand these primitive concepts and are able to handle brute facts involving consistency and modality. This is tantamount to attributing to them a sui generis logical knowledge. Mathematicians would have the ability to acquire knowledge about the logical properties of their theories. They will know which theories are consistent, which ones are not, and will devote their efforts to study the logical properties of the former —i.e., which sentences follow from them. Thus, the fictionalist can employ a primitive notion of consistency to state her philosophical proposal. This solution seems promising. It would ensure that fictionalism has an appropriate concept of consistency to explain mathematical knowledge. However, here comes the problem. Despite its appeal, the solution is not available for all types of fictionalism. In particular, it is not available for algebraic fictionalism: Algebraic fictionalism cannot use a primitive notion of consistency or logical possibility. The point is that the primitive concepts of consistency and possibility are eminently semantic. Logical knowledge, including knowledge about these two phenomena, is knowledge concerning a relation of entailment. And, this is a linguistic relation. A set of sentences entails other sentences —alternatively, a collection of propositions entails other propositions. Now, language is a semantic category. Grammatical categories, such as the categories of names, predicates, or sentences, are already semantic categories. 93 Take the case of proper names. An object is a proper name because it performs an specific function: among other things, it attempts to refer to another object. Even if it does not succeed, its function is essential to its identity and its status as a proper name. For example, consider the name “Aristotle”. The spot on the screen is just that, a spot. The sound emissions of this name are just that, sound emissions. What makes “Aristotle” a proper name is its semantic and communicative functions: it attempts to denote a very specific object, Aristotle. The same applies to predicates. Consider the predicate “to be a Greek philosopher”. As in the previous case, this predicate has a number of complex semantic and communicative functions. Among them, it applies to those objects which are Greek philosophers. If it did not perform this function, then it would not be a predicate —it would be another object, such as a stain or a sound. Although not all grammatical distinctions are logically relevant, they introduce semantic or communicative distinctions. Thus, the point is: it makes no sense to vary the semantic content of a name, a predicate or a sentence without varying its identity. If the linguistic expressions e1and e2perform different semantic functions, then e1=e2. Thus, let us imagine that there were two expressions e1and e2with the same spelling, “Aristotle”. However, while e1would refer to an object —Aristotle—, the other would perform a very different function: the same function that the predicate “to be a Greek philosopher” has in English. In this situation, while the first would be a name, the second would be a predicate. This is where the problem lies. The semantic nature of language puts the algebraic fictionalist in a bind. Algebraic fictionalism does not attribute any content to mathematical language. This is precisely where it differs from full fictionalism. Therefore, if the former denies the semantic nature of mathematical language, then it is denying that mathematicians speak any language at all. In principle, this need not be a problem —although it would complicate things for fictionalism. It’s could be the case that mathematical practice is linguistic only in appearance. Mathematical practice may be linguistic in appearance only. At first glance, mathematicians appear to be talking, asking questions and making assertions. In reality, however, they would be doing something quite different. Although this “alien” view of mathematics is coherent, it is hardly plausible: no one will accept it. In addition, the real problem for algebraic fictionalism is other. As I said, primitive notions of consistency and possibility are eminently semantic. Logical knowledge is knowledge about the entailment relation, thus about a linguistic relation between sentences —i.e., between semantic entities belonging to a particular language. This leads to the following conclusion: if mathematicians do not speak any mathematical language, then they cannot have logical knowledge about the consistency —in a primitive sense— of theories stated in such a language, nor about what is mathematically possible. Consequently, algebraic fictionalism cannot explain mathematical knowledge by using a primitive notion of consistency or possibility. Can algebraic fictionalism object to this challenge in any way? Let me briefly explore two options. One option is to attribute some kind of semantic content to mathematical language. As a result, it will be endorsing a semantic concept of language and will be able to attribute logical knowledge to mathematicians. But, what kind of content can it attribute to mathematical language? No kind of fictionalism can resort to paraphrase functions to explain mathematical language. Otherwise, it would lose its identity. For instance, if fictionalists accept Hellman’s (55) modal-structuralist translation of mathematics, they 94 The dual position of height multiversism is width or horizontal multiversism. According to this, the concept —or better concepts— of set fixes the height of its models, but not their width. Width Multiversism There is a multiplicity of metaphysically equivalent universes of sets varying in width, but not in height, each one of them providing an appropriate interpretation for set-theoretic terms and quantifiers. Accordingly, there are several concepts of set corresponding to the variety of universes and set theorists, instead of depicting one, privileged universe, seek to understand what is true in different universes varying in width and how they relate to other universes. Unlike height multiversism, for width multiversism there are universes which verify CH and universes that verify its negation. Additional versions of multiversism arise from the combination of the two previous types —i.e., height and width multiversism. For instance, the multiverse view advocated by Hamkins (54) combines both height and width multiversism. His rejection of a fixed width is based on Quine’s (114, 66) criticism of second-order logic. Second-order logic would be part of set theory, and is affected by the relativism inherent to the concept of set. The point is that a second-order categoricity argument, even just for the natural numbers, requires one to operate in a context with a background concept of set. And so although it may seem that saying “1, 2, 3,... and so on”, has to do only with a highly absolute concept of finite number, the fact that the structure of the finite numbers is uniquely determined depends on our much murkier understanding of which subsets of the natural numbers exist. (54, p. 428) Consequently, the quasi-categoricity result obtained by Zermelo (see note 3) does not explain how the concept of set fixes the width of the universe. On the contrary, it only tells us what holds on a particular subclass of universes inhabiting the bigger multiverse. A more fine-grained classification of multiversist positions arises when the ontology underlying them is also taken into account. As I said at the beginning, I will limit the exposition to realist versions of multiversism. According to these, the objects inhabiting the universes and the universes themselves, if they can be understood as objects, are abstract objects that exist independently of agents —in the sense of Independence Chapter 2, 2.2. Realist Multiversism The objects inhabiting the universes and the universes —if “universe” expresses a concept characterizing a particular kind of object—, are abstract objects that exist independently of agents. Realist versions of multiversism constitute a special form of platonism. They accept the three theses composing this philosophical account of mathematics: Existence, Abstractness, and Independence. But, at the same time, realist multiversism rejects the traditional belief in a single realm of mathematical objects. As a result, multiversism introduces a new division inside platonism: while traditional forms of platonism are monist, the former is a form of pluralist platonism. Indeed, the seminal works which gave birth to multiversism (2; 54) overtly declare it as a new form of mathematical platonism. 101 [Multiversism] holds that there are diverse distinct universes [...]. Each such universe exists independently in the same Platonic sense that proponents of the universe view regard their universe to exist [...] The multiverse view is one of higher-order realism-Platonism about universes —and I defend it as a realist position asserting the actual existence of the alternative set-theoretic universes into which our mathematical tools have allowed us to glimpse. (54, 416-417) The plural spirit of multiversism allows it to combine the lightweight approach to mathematical knowledge advocated by fictionalism, and a realist ontology. To understand the force of this, consider any incomplete mathematical theory —for instance ZF. We know that if ZF is consistent, ZF+CH and ZF+¬CH are consistent too —CH is independent of ZF. Now, let us return to the template schema needed to explain mathematical reliability —Chapter 2, 2.3.2.1. As fictionalism, multiversism solves Xwith the property of being consistent —or similar properties. Thus, ZF+CH and ZF+¬CH accommodate to X. But now, unlike in the case of fictionalism, (2.iv) turns out to be true, ensuring that both theories are true relative to some universe of sets. Multiversism exchanges fiction for universes. As in the case of the former, there is no single metaphysically or semantically correct universe. However, they are not fictional entities. This exchange allows multiversism to preserve truth, assertion, and to solve many of the problems faced by fictionalism. In turn, anti-realist versions of multiversism deny —or at least refrain from asserting— that sets and universes exist in any metaphysically loaded notion of existence. Anti-realist Multiversism The objects inhabiting the universes and the universes —if “universe” expresses a concept characterizing a particular kind of object—, do not exist. The supporter of this view regards the multiverse discourse as a useful tool to describe the practice of contemporary set theorists. They spend a lot of time investigating which theories are consistent and which are not, and what follows from them and what does not. And all the resulting theories are perfectly acceptable from the point of view of set theorists. Indeed, they also make use of the multiverse —entering the region of meta-practice— to guide their research about consistency and to produce new proofs of independence. According to this view, a universe is no more than the correlate of a theory proved to be consistent by some well-established model-theoretic method. Some of these universes —i.e., consistent theories— fit better with our informal understanding of sets as well as with additional criteria for theory choice —aesthetics, simplicity, or the broadness of the resulting concept of set (126). Otherwise, “all the universes” are metaphysically at the same level: they do not exist. Some antecedents of this view are Hilbert’s coherentism (57) or Carnap’s pluralism (20), who advocated a deflationary reading of existence.4Recently, Shelah (126) has explicitly advocated an anti-realist 4In “On the Infinite” Hilbert differentiates between real or finitely verifiable statements and ideal elements. The former are the statements corresponding to speakers’ informal understanding of arithmetic. Hilbert took them to be provable employing only finitist resources, in some unspecified sense of “finitism” —see (138, Ch.1-3) for an attempt to precise the concept of finitism. Similarly to observational sentences in empirical science, Hilbert took these as “verifiable” in the previous sense. In turn, ideal elements are sentences that are not finitely verifiable, usually quantified sentences —Π0 nand Σ0 ksentences for n≤1and k≤2— purporting to make claims about in- 102 conception of the multiverse. According to him, multiversism is a false theory, and consistent extensions of ZF are not true in some domains of set-like objects. Since multiversism and fictionalism elaborate a very similar explanation of mathematical knowledge, I believe that anti-realist multiversism can be assimilated into the latter. While realist multiversism rejects the sceptical ontology of fictionalism, antirealist versions of multiversism accept it. Thus, universes can be given the same treatment as fiction. The only difference between Shelah, on one side, and Hilbert and Carnap, on the other, is that the latter are non-factual. They deflate existence inside of mathematical discourse, remaining neutral regarding the existence —in an ontologically loaded sense— of mathematical objects. On the contrary, Shelah, like fictionalists, positively asserts that sets do not exist. I will focus on realist versions of multiversism —specifically, on Balaguer’s (2) and Hamkins’ (54) proposals. Thus, I will focus also on those versions which combine height and width multiversism. These proposals use minimal, epistemologically treatable resources to explain mathematical knowledge. Remember, mathematical knowledge is reduced to logical knowledge —knowledge about consistency and entailment: K(Γ⊢ φ +Γis consistent) =d f K φ M, where —informally— φ Mmeans that φ is true in some universe M. This minimisation simultaneously leads to a maximisation of mathematically acceptable theories: all consistent theories —or the theories proven to be consistent by model-theoretic methods— will be acceptable. Further limitations on acceptable theories, or on the extent of the multiverse, will be the result of additional constraints imposed on the explanation of mathematical knowledge. That is, logical knowledge will not be sufficient to explain it. Consequently, radical pluralism will be limited, reintroducing a concept of mathematical objectivity —although a minimal one— stronger than logical objectivity. This is precisely what I will argue for in Chapters 5-7. The exact relation between the proposal advocated there and height and width-multiversism remains open. In this chapter, I will argue in favour of this conclusion: that is, I will show why the extreme pluralism of Hamkins’ and Balaguer’s positions is problematic. 4.4 MOTIVATING THE MULTIVERSE Before going into the critique, I will explain which are the main reasons in favour of multiversism. There are two broad families of arguments. First, the promised solution to Benacerraf’s epistemological problem —including its strengthened version. The rejection of responsiveness cancels the challenge. Then, multiversism positively explains mathematical knowledge through logic. The second family of arguments consists of arguments against universism. Since these are opposed positions, showing that universism has a problem is equivalent to showing that multiversism has an advantage. Otherwise, realist multiversism finitely many objects. He understood them analogously to the instrumentalist’s understanding of non-observational sentences: as ontologically neutral devices employed to move from real statements to real statements. Thus, ideal elements are just a convenient tool for satisfying mathematicians’ pragmatic objectives —for instance, to simplify reasoning in mathematics. Aside from its usefulness, all that is required is that ideal elements are consistent with the theories composed of finitely verifiable sentences. 103 faces the metaphysical debate against anti-realist accounts of mathematics, such as fictionalism. The advocate of multiversism must provide additional reasons to support her favourite against the latter. Some authors —(2, Ch. 8), (94, Ch. 4)— have argued that this is an unsolvable dispute: there is and could be no facts to determine which is the metaphysically correct conception of mathematics. However, the arguments exposed in Chapter 3 suggest that this is too hasty a conclusion. For example, multiversism accounts for mathematical reliability, while fictionalism does not. Thus, the former scores better on this point. Nevertheless, I will not enter into the metaphysical debate here. Instead, I will expose the main arguments opposed to universism. According to multiversists, universism is false because it cannot explain how mathematical terms and quantifiers become interpreted in the metaphysically privileged universe of mathematics. Equivalently, this can be put in terms of intensional entities —if wording about these entities is allowed— saying that universism can not explain how mathematical predicates and functors express unique, well-identified concepts, relations and operations which are about the objects in the universe. More specifically, the multiversist endorses (4.i) The logical and linguistic features of mathematical theories do not fix —up to isomorphism— a unique universe of objects where terms and quantifiers are interpreted, —in terms of concepts: the logical and linguistic features of mathematical theories do not fix unique concepts, relations or operations holding among the objects in the universe. In addition, the multiversist advocates (4.ii) Mathematical reference can only be explained using descriptive resources, specifically using the logical and linguistic properties of axioms coached in mathematical language. The reasons for (4.ii) are both metaphysical and methodological. First, Abstractness and Independence claim that mathematical objects are abstract objects which exist independently of agents. All counterfactual connections with facts involving them are forbidden. Therefore, no connection of this kind plays a relevant role in explaining reference and meaning acquisition. The only relevant connections left to be used in metasemantic explanations are purely descriptive —i.e., non-causal connections based on the semantic properties of the mathematical language. These properties, together with the target objects, should impose sufficient constraints to achieve reference to them. Moreover, contemporary mathematical practice is eminently algebraic —(56),(76, 120),(35, Ch. 6) According to this, mathematicians identify a set of axioms and study what follows from them. The axioms are intended as a description of the objects under consideration: any consistent system of axioms contextually defines its subject matter. Thus, the role of axioms can be compared to the role of a system of equations with several unknowns: together they determine the collection of acceptable solutions to the system —the interpretation of mathematical vocabulary. In consequence, the descriptive resources relevant to mathematicians are those used in the axioms of their theories. From (4.i) and (4.ii) immediately follows that: 104 (4.iii) If (4.i) and (4.ii), then it is not explainable how mathematical terms refer to a privileged universe —or how concepts, relations and operations holding among the objects in this universe become uniquely expressed by the mathematical vocabulary. If reference and semantic content are not uniquely determined by logical properties of axiom systems —according to (4.i)— and no other resources are available to explain reference —as (4.ii) demands—, then it is simply not explainable how mathematics is only about a privileged universe of mathematical objects. Now, we have that: (4.iv) If it is not explainable how mathematical terms refer to a privileged universe —or how concepts, relations and operations holding among the objects in this universe become uniquely expressed by the mathematical vocabulary—, then universism is false. But, from (4.i), (4.ii) and (4.iii) it follows that: (4.v) It is not explainable how mathematical terms refer to a privileged universe —or how concepts, relations and operations holding between the objects in this universe become uniquely expressed by the mathematical vocabulary. Consequently, from (4.iv) and (4.v), the multiversist concludes that: (4.vi) Universism is false. The unique premise of the argument that remains to be justified is (4.i). There are two main arguments supporting it —both are based on the methodological procedures used in meta-mathematics: the argument from logical independence and the argument from categoricity. 4.4.1 The Argument from Logical Independence The argument from logical independence builds on independence results —see 4.7 for a brief survey of the incompleteness affecting set theory. Zermelo-Fraenkel set theory (ZF) is highly incomplete. Indeed, it is incomplete at very low levels of the hierarchy. There are several Π0 1-sentences —as its Gödel sentence GZF or Con(ZF)— that the theory does not decide. All these are examples of a benign kind of incompleteness: once we are justified in accepting Peano-Dedekind arithmetic (PA) the independence of Π0 1-sentences is enough for their truth. Moreover, the combination of intrinsic and extrinsic sources of evidence makes a strong case for completing ZF with new axioms. In his seminal (49), Gödel proposed to extend the theory by adding principles inductively supported by their fruitfulness in consequences —as non-observable statements in the case of empirical science. More precisely, if a candidate for extending ZF (i) affords us proofs of new theorems and (ii) it also simplifies to a large extent the proofs of old ones —Gödel takes these as verifiable consequences—, then it can be regarded as sufficiently justified. Thus, fruitfulness becomes a criterion for the objective truth of set-theoretic statements in the universe.5The 5See (94, Ch. 3) for a recent defence of fruitfulness as a criterion of set-theoretic objectivity. 105 hope was to press inductive justifications of this kind far enough to trigger a considerable reduction of the plural space of ZF’s extensions, obtaining a complete theory —at least complete for natural statements—: the correct theory of the universe of sets. Gödel proposed statements asserting the existence of large cardinals as the most natural candidates for this task. As a result, he gave birth to the program for large cardinal axioms, one of the most ambitious attempts to complete ZF. While Π0 1-sentences are incomplete relatively to particular, recursively axiomatized systems —such as Robinson arithmetic (Q), PA or ZF—, the notion of a large cardinal axiom is informal in nature, and it is not subject to a precise, recursive definition. Therefore, these present themselves as a perfect background to test candidates for absolute undecidability —sentences undecidable from any natural, justified extension of ZF. Gödel envisioned an informal notion of absolute provability —i.e., provability from true large cardinal axioms— and conjectured a generalized completeness theorem: “every proposition expressible in set theory is decidable from the present axioms plus some true assertion about the largeness of the universe of all sets” (1946, 151). Large cardinal axioms are a resounding success below CH. They yield a completeness result for the theory of L(R)in the sense that this theory is preserved by set forcing constructions.6 However, large cardinal hypotheses present several limitations that seem hard to overcome. First, this type of model-theoretic construction does not alter the truth of Σ2 1sentences as CH. Hence, large cardinal axioms cannot settle the status of such complex open problems. Some philosophers and mathematicians (69; 70) have argued that this is not the last word on the debate. According to them, the weakness of large cardinal hypotheses does not entail the existence of an absolute undecidable sentence and, thus, a negative answer to Gödel’s conjecture —if it is appropriately strengthened. The door is open to supplement the program with additional, stronger and natural principles. While this is true, and hope can be sustained, the contrary is also true: there is no reason to believe that all natural open problems in set theory will be solved one day or another. After several years of research, no final decision has been taken over any of the available candidates. Moreover, inductive justifications are not truth-conductive, and the criterion of fruitfulness as well as the notion of naturalness employed to characterize stronger set-theoretic principles are pragmatic. Therefore, it is far from clear that these attempts to complete the hierarchy succeed in recovering a high degree of objectivity. The multiversist denies that this is the case, and claims that set theory does not capture a single, well-identified concept of set —thus, (4.i). 4.4.2 The Argument from Categoricity The second argument builds on the fact that first-order theories with infinite domains are not categorical. The concept of categoricity is model-theoretic —thus, a mathematical concept. It involves the —also mathematical— concepts of language, theory, and model. A theory T is categorical if and only if all 6This is the theory of definable sets of reals. A definable set of reals is a set of reals characterized by a Σ2 nformula —this can be seen as a formula of second-order arithmetic. L(R)is the hierarchy that results from starting with the reals and iterating the definable powerset operation —the operation of taking definable subsets— into the transfinite. Those sets of reals in L(R)are the definable sets of reals. 106 models M1and M2of T are isomorphic.7Results about categoricity are proved in mathematics. Then, it is reasonable to expect some explanation of what these results and the very notion of categoricity have to do with semantics, a philosophical discipline. The answer is that formal semantics, that is, the mathematical study of semantic concepts such as reference or truth, provides an analytically tractable environment for obtaining knowledge about these concepts — similar to computational models applied in meteorology, which supply us with affordable information about such a complex system as it is the weather. Also, the notion of independence from a recursive collection of axioms is a mathematical concept. Nevertheless, it is known that sufficiently strong first-order theories are far from being categorical. The Downward Lowënheim-Skolem theorem entails that any first-order theory T that has an uncountable model M1also has a countable model M2—thus, M1and M2cannot be isomorphic.8Furthermore, completeness does not fix categoricity. There are first-order complete theories with non-isomorphic models. For instance, arithmetic has non-standard models. As Skolem showed, the theory of the structure ⟨N,<⟩—which, by definition, is a complete arithmetical theory containing PA— has non-standard models ⟨N∪ {a},<⟩where ais a non-standard number such that for all n∈N,n<a. Subsequently, Robinson developed nonstandard analysis as the result of making precise the notion of infinitesimals —reals smaller than all standard reals (96). The downward side of the Löwenheim-Skolem theorems can be applied to ZF —(128),(111)— supplying us with a countable, transitive model of sets. For, assume that ZF has a model M. By the axiom of Pairs, Mmust be infinite —countable or uncountable. Therefore, by the Downward Löwenheim-Skolem theorem, ZF has a countable model N. At the same time, by Cantor’s theorem, ZF entails that ℘ (N)is uncountable. Moreover, it will be uncountable in N—this satisfies the Powerset axiom. How is this possible? This result, which has the air of a contradiction, is famously known as Skolem’s Paradox. Yet, this curious result does not show any hidden incoherence in our understanding of sets. What happens is that the open formula “xis uncountable” is satisfied by countable sets inside N—informally, we could say that Nhas no resources to capture the intended notion of being uncountable, that is, it cannot see which sets are uncountable and which are not. In Nthere is no one-to-one map from ℘ (N)N—the set denoted by “ ℘ (N)” in N— onto N, even though such a map exists in V. Indeed, this is how Skolem originally understood the result, as a failure of formal resources to uniquely fix the concept of set. Thus, axiomatizing set theory leads to a relativity of set-theoretic notions, and this relativity is inseparably bound up with every thoroughgoing axiomatization. The relativity is due to the fact that to be an object in B means something different and far more restricted than merely to be in some way definable. That this relativity must be inseparably bound up with every thoroughgoing axiomatization is clear; for it rests upon the general theorems of mathematical logic mentioned above [Italics in the original] (128, p. 296) 7A model Mof classical first-order language is a n-tuple ⟨D,⟨Ri⟩i< α ,⟨fj⟩j< αβ , where Dis a non-empty set of objects, ⟨Ri⟩i< α is a sequence of relations of length α , and ⟨fj⟩j< α is a sequence of operations of length β . Given two models M1and M2, an isomorphism from M1to M2, is a bijection from D1to D2preserving relations and operations. 8Besides, the companion Upward Lowënheim-Skolem theorem entails that if T has a countable model M3it also has an uncountable model M4. 107 A purely algebraic approach to reference —in accordance with (4.ii)— entails an extreme relativism of set-theoretic concepts. This relativism of set-theoretic concepts has been welcomed by (54), who has strengthened the paradox including among the axioms of his theory of the multiverse the Countability Principle —this says that every universe V is countable from the perspective of another universe V∗— as well as the Well-foundedness Mirage —it says that every universe V is well-founded from the perspective of another universe V∗.9 The universist might object that categoricity is retrieved in second-order logic with standard semantics. Thus, second-order PA (PA2) is categoric. Also, (154) proved that second-order ZF (ZF2) is quasicategoric, in the sense that for any two models M1and M2of ZF2, there is an embedding from M1to M2 or vice-versa. Nevertheless, the standard semantics of second-order logic is highly controversial. According to the standard semantics, n-adic predicates are interpreted in ℘ (Dn)—the power set of the n-ary Cartesian product of the domain. Thus, standard models are stated in terms of the set-theoretic notion of powerset. This is the reason why Quine qualified second-order logic as “set theory in sheep’s clothing.” (114, p. 66) This opinion is shared by many philosophers and mathematicians, who believe that it hides strong mathematical notions in the metatheory (36). Alternatively, if weaker semantic systems are used to interpret the second-order language, such as Henkin’s frames or plural logic, categoricity vanishes again as these systems are similar in expressive power to first-order logic. To sum up, if logical resources are kept at the safe level of first-order logic, most mathematical theories —from the elementary PA to the powerful ZF— are not categorical. Again, this has been used to argue that these resources fail to fix unique, well-defined mathematical concept. Multiversists such as (54) only accept first-order logic. 4.4.3 Morals from the Limits of Logic According to the argument from logical independence, ZF is highly incomplete and, as a result, its axioms are not enough to single a unique, metaphysically privileged universe where set-theoretic terms and quantifiers are interpreted —similarly, a unique, privileged set-theoretic concepts about the objects in this universe. Furthermore, the argument from categoricity tells us that even if ZF is improved up to a complete theory ZF+, this will not be categorical. The logical resources afforded by the language where ZF+is formulated —if these resources are not controversial— are again too weak to fix reference and quantification into a unique, privileged universe. This is precisely what (4.i) states. Consequently, the multiversist claims that her argument against universism is solid. However, if multiversists want to be in a better position than universism, they must show that multiversism does not face an analogous objection to that posed by (4.v), i.e., they must show that the following is false: 9The relativity of the concepts finite and infinite had already been pointed out by Skolem: “Even the notions “finite”, “infinite”, “simply infinite sequence”, and so forth turn out to be merely relative within axiomatic set theory. A set M is finite, according to Dedekind’s definition, if it is not equivalent to any of its proper subsets. But the fact that the axioms hold does not rule out the possibility that we can define, first, parts of M that are not subsets or, second, correspondences that are not mappings, that is, “sets” of pairs. It is therefore quite possible that, within a domain B in which Zermelo’s axioms hold, there exist sets that are “finite” in the sense of Dedekind and for which there are one-to-one mappings onto some of their proper parts; but these “mappings” are not sets of the domain.” (128, p. 295) 108 (4.v∗) It is not explainable how mathematical terms refer into different universes inhabiting the multiverse —or how concepts, relations and operations holding between the objects in the multiverse become expressed by the mathematical vocabulary. Once more, we meet the Language Acquisition Challenge (LAC). The multiversist must explain how the mathematical language acquires its plural meaning —or how mathematicians become equipped with the appropriate resources to navigate the multiverse. As I said in 4.2, multiversists claim that once Responsiveness is rejected they can explain mathematical reference, mathematical knowledge and the reliability of mathematicians all at once —unlike fictionalism. Instead of causal or similar strong connections between speakers and mathematical facts, all multiversism requires are weaker connections between speakers and the logical properties of mathematical theories. Specifically, it only requires theories to be consistent — mathematical knowledge reduces to logical knowledge. According to the multiversist story, mathematical terms and quantifiers featuring in consistent theories are interpreted into some universe, and these mathematical theories are true relative to this universe. In the following sections, I will argue that this is not the case. Neither Hamkins’ nor Balaguer’s form of multiversism can cope with the explanatory pressure posed by LAC. Against appearances, I will argue that such extreme forms of realist pluralism cannot solve Xin the template argument of 4.2. As in the case of fictionalism, any solution provided by them fails to make (2.iv) true. They fail to explain how mathematical theories are true in different universes because, and they fail to explain how mathematical terms and quantifiers become interpreted into these universes. Furthermore, Balaguer’s full-blooded platonism uses the notion of consistency to explain how the mathematical language acquires its meaning, and how mathematicians acquire their linguistic competence. Thus, I will argue that this fails as an account of mathematics for the very same reasons as fictionalism does. Despite this negative critique, note that the door remains open to less extreme forms of pluralism —for example, versions of multiversism that impose some limits to the extent of the multiverse. 4.5 THE SET-THEORETIC MULTIVERSE Contemporary set theorists have devoted huge efforts to understand the phenomenon of independence. As a result of these efforts a variety of model-theoretic techniques —such as inner models, ultrapowers, set forcing, or class forcing— have been developed to show which extensions of ZF are consistent — relative to ZF’s consistency— and, thus, which sentences are independent of it. Astonishingly, set theory has been populated with an unexpected variety of set-theoretic possibilities, and research about models of ZF has become a central part of its agenda. The fundamental objects of study in set theory have become the models of set theory, and set theorists move with agility from one model to another. While group theorists study groups, ring theorists study rings and topologists study topological spaces, set theorists study the models of set theory. (54, p. 418) Hamkins (54) takes this phenomenon as strong evidence that our concept of set is inherently relativistic and, thus, as evidence against universism —he takes the argument from independence and the argument 109 from categoricity as supporting (4.i). Furthermore, Hamkins does not accept second-order logic as “true logic.” This has two immediate consequences. First, the concept of set framed in ZF cannot be completed by employing logical resources alone. Second, set theorists study models of first-order ZF and its extensions. As a result, he endorses an extreme form of pluralism. Hamkins advocates multiversism as a satisfactory account of set-theoretic practices. The multiverse is navigated through the variety of model-theoretic procedures used to build ZF models. This navigation always takes place against a background concept of set —i.e., it requires a departure point. The reason is that most model-theoretic techniques are relative construction methods: they are performed against a background, previously given notion of set. Once the set theoretician is given this background, different model-theoretic techniques tell her how to build different models —the journey starts. Hamkinsian Multiversism There are several universes of sets manifested through the application of model-theoretic techniques. Each universe elucidates a particular concept of set. From the perspective of the set theorists, each of these universes is a model of first-order ZF —i.e. a set. Moreover, each model of ZF exerts a partition of the language in true and false sentences, and thus elucidates a particular, well-defined concept of set —indeed, Hamkins identifies concepts and models. But, which kind of entity are models? From the perspective of the mathematician, models are just sets —i.e., mathematical objects. Note that this is a mathematical statement made against the background of an —at least partially specified— concept of set S. According to universism, this is the unique concept of set we have. As a result, given an object, Sdetermines whether this is or is not a set in an absolute sense. Conversely, the very notion of set turns out to be relative on the face of multiversism. As I said, multiversism denies that there is a single concept of set. As a result, an object is a set —or not— depending on the background concept of set that we are taking into account. For example, take two models V α and V κ , such that α < κ . Then, ifV α is a substructure ofV κ , what in the former is a proper class, in the second is a set. In the multiverse, everything is a set —or not— depending on where you look from. This reveals a surprising —but entirely coherent with its extreme relativism— feature of Hamkins’ multiversism: it cannot be characterised from within the multiverse, but only from outside it. The assertion that there are diverse concepts of set is a metamathematical as opposed to a mathematical claim, and one does not expect the properties of the multiverse to be available when undertaking an internal construction within a universe. That is, we do not expect to see the whole multiverse from within any particular universe. (54, p. 417) This kind of pluralism implies an extreme relativism about mathematical concepts. This situation entails certain limitations on what can be expressed with them. What the universes are or what the multiverse looks like cannot be stated from the point of view of the mathematician, who is always working inside the multiverse. Hamkins defends multiversism not only as an account of set theorists’ practice, but also as a suitable account of the metaphysics of mathematics, and as a suitable account of mathematical reference and knowledge. The idea is that model-theoretic techniques have semantic consequences for the language: they explain how mathematical reference is fixed inside the multiverse. The explanation involves two 110 conclusion, the multiversist cannot provide evidence that reference in the multiverse is achieved by only employing finite chains of concepts. Finally, the multiversist is left only with one alternative: to accept a privileged concept of set, imposing a limit on the extent of the multiverse. Reference to the objects inhabiting this universe would be explained by other means than the performance of some model-theoretic construction. Hence, this constitutes a modest vindication of universism: the universe of sets would be this familiar and initial universe from where all constructions and incursions into the multiverse start.13 The multiverse needs a starting point. Nevertheless, this answer by no means entails that all versions of multiversism are false —only the radical Hamkinsian version. Once a background concept of set V is fixed and reference to such sets is secured, then model-theoretic methods can be interpreted as shifting reference from V into other, different universes gaining new, previously unknown concepts of set. Set theorists become explorers of the logical space of sets and their journey starts at home. In particular, V is not metaphysically different from the other inhabitants populating such a vast space. All it does is to provide set theorists with a point of departure in their journey. In any philosophy of mathematics that tries to make sense of a multiverse picture of set theory as concerned with reference, there is likely to be an element of relativism of certain concepts. However, there is a limit to how far this relativism can go. In particular we need to be precise about which universes (off the bat) are to count as privileged, and in doing so will have to specify a list of concepts that we are simply taking to have absolute significance to restrict our models. Moreover, this stock of concepts must be sufficiently rich to allow the relevant model-theoretic techniques to be absolutely specified, and multiverse given precise limit. The acceptance of ‘any particular’ first-order model as a legitimate universe results in a relativism so strong that it cripples our ability to refer at all. (6, p. 204) If such a departure point is rejected, I believe that multiversism fails at explaining mathematical reference and knowledge. Then, multiversism fails to solve LAC and Benacerraf’s challenge. Consequently, it fails as an account of mathematics —in particular, of set theory. Indeed, the view collapses into an algebraic, anti-realist position similar to algebraic fictionalism — see Chapter 3, 3.5. Without a minimal starting point, Hamkinsian multiversism turns out to be another algebraic approach to mathematics according to which set theorists do not deal with a variety of concepts of set —least of all with abstract objects, such as sets and universes. Instead, they would lay down some —consistent— axioms and study what follows and what does not from them. As a result, their activity should be understood not as asserting something about a domain of existing objects, but as studying what will happen if some axioms were true. Unfortunately, as I argued in Chapter 3, 3.5 this position is also full of problems. 13The question of whether the multiverse has a core has been widely debated. See (132) for a positive development of this position. 117 4.6 FULL-BLOODED PLATONISM The other version of multiversism I am going to discuss is full-blooded platonism (FBP). This was elaborated in (2) explicitly as a solution to Benacerraf’s Challenge. He proposes to solve the challenge by abandoning Responsiveness and denying that counterfactual relations between mathematical facts and mathematicians are needed to explain mathematical knowledge. As fictionalists and advocates of other versions of multiversism, Balaguer explains mathematical knowledge as a case of logical knowledge. Therefore, the rejection of Responsiveness leads, again, to a plural account of mathematics: FBP is a form of pluralistic platonism. There is no universe of metaphysically privileged mathematical facts. In the absence of such a universe, there are either many or none, and Balaguer chooses the former option. FBP is a simple thesis: all possible mathematical universes exist.14,15 Full-blooded Platonism All logically possible universes exist. FBP is the result of supplementing multiversism with a maximal claim. Not only are there a variety of different —many of them strange— universes. Moreover, none of them is missing. Its mere logical possibility suffices for its existence. Hence, FBP envisions a full, plenitudinous multiverse of mathematical objects. According to Balaguer, this is what allows for a weak, non-counterfactual explanation of mathematical knowledge. If FBP is correct, then all consistent purely mathematical theories truly describe some collection of abstract mathematical objects. Thus, to acquire knowledge of mathematical objects, all we need to do is acquire knowledge that some purely mathematical theory is consistent. (It doesn’t matter how we come up with the theory; some creative mathematician 14Unlike Hamkinsian multiversism, full-blooded platonism is a general proposal for all mathematical theories. Balaguer does not focus on set theory. However, for the sake of continuity, I will present the proposal in set-theoretic terms. 15See (2, pp. 5-7) for a characterization of FPB. Balaguer spends some time explaining how FBP can be stated formally. “I do think it might help clarify FBP to say a few words about how one might go about trying to state it in a formal language. Before I do this, however, I want to emphasize that my sole aim here is to help the reader get clear about what FBP says; nothing important depends on finding an adequate formalization of FBP.” (2, p. 6) Unlike Hamkins, he employs second-order logic to make himself available resources to talk about universe-like entities —despite he does not talk about universes. Then, FBP is formulated as follows: ∃xMx ∧∀V(♢∃z(Mz ∧Vz)→ ∃z(Mz ∧Vz)), where “M” is a predicate expressing the property of being a mathematical object and “V” is a second-order variable ranging over universe-like entities. Of course, one wonders whether this does not condemn Balaguer’s proposal from the outset. If second-order resources are available to talk about universes, what prevents us from accepting the categoricity of mathematical theories? The answer is that, for the full-blooded platonist, second-order variables do not range over subclasses of a well-determined universe. On the contrary, they encompass a multiplicity of such universes. The metaphysical status of the universes, as well as the linguistic resources used to talk about them, is especially pressing. The answer to these questions will determine the best formalisation of the proposal. Here I will keep the exposition on an informal level, assuming that the multiversist discourse is in good shape —in particular, that talking about the existence of universes is safe. 118 might simply “dream it up”.) But knowledge of the consistency of a mathematical theory —or any other kind of theory, for that matter— does not require any sort of contact with, or access to, the objects that the theory is about. Thus, the Benacerrafian objection has been answered: we can acquire knowledge of abstract mathematical objects without the aid of any sort of contact with such objects. (2, pp. 48-49) Like fictionalists, Balaguer reduces mathematical knowledge to logical knowledge. Knowledge about a mathematical sentence φ reduces to two things: (i) knowing that φ follows from a given mathematical theory Γ, and (ii) knowing that Γis consistent. Thus, he solves Xin the argument-template of 4.2 with the property of being consistent. Now, as in the case of Hamkinsian multiversism, (2.iv) turns out to be true: (2.iv) According to FBP, all consistent mathematical sentences φ are true —at least φ is true relatively to some universe of objects. Indeed, this is tantamount to the very definition of FBP. Its plural realism guarantees an intimate correlation between consistency and truth. As a result, Balaguer explains mathematicians’ reliability exactly in the same way as the Hamkinsian multiversist does —replacing the property of having a model, with the property of being consistent.16 Again, the rejection of Responsiveness prevents any counterfactual connection between mathematicians and mathematical facts. However, there is a counterfactual connection between agents and the process that leads them to acquire their linguistic competence, as well as with the procedures that allow them to acquire knowledge about consistency. Agents have to learn to speak and to determine which theories are consistent and which are not. This is something that does not come without effort. Additionally, the correlation between consistency and the multiverse guarantees that knowledge of the former is in tune with the truth of sentences — relative to some universe. As a result, the FBP-ist endorses (2.v) (2.v) FBP can explain why mathematicians accept mathematical sentences φ only if φ is true —at least relatively to some universe of objects, with an explanatory connection between mathematicians’ acceptance of sentences and their truth. Unlike in the case of fictionalism —see Chapter 3, 3.5.3—, not only are all the lines of the argument true. The argument is also explanatory. Knowledge about consistency —i.e. logical knowledge— ensures mathematicians’ reliability.17 If this is correct, FBP solves the major epistemological challenges of platonism: Benacerraf’s epistemological challenge and the challenge posed by mathematicians’ reliability. As a result, it would be a refined and philosophically sound form of platonism. However, as in the case of fictionalism and Hamkinsian multiversism, I doubt that its explanation of mathematical knowledge works. Indeed, because FBP reduces mathematical knowledge to logical knowledge, I will argue it faces similar problems 16These properties are not equivalent in higher-order logic. However, because of the argument from categoricity, multiversists tend to accept first-order languages. But, completeness and correctness entail that both properties are equivalent at this level. In this case, if the notion of consistency used by the FBP-ist coincides with the notion of consistency at stake in set theory, the explanations of the Hamkinsian multiversist and the FBP-ist are the same. 17See[p. 51] (2) for some comments on mathematical reliability from the point of view of FBP. 119 to fictionalism. First, like algebraic fictionalism —see Chapter 3, 3.5.3, FBP lacks an appropriate notion of consistency to perform the desired explanatory function. Again, the two candidates are a defined notion of consistency or a primitive one. The former presupposes mathematical knowledge, leading unavoidably to circularity. In turn, the second presupposes an understanding of mathematical language. Let us compare its situation with fictionalism. In Chapter 3, 3.5.3, I argued that a primitive notion of consistency must be a linguistic notion, thus a semantic notion. This puts algebraic fictionalism in serious trouble, since the latter either attributes no content to mathematical language or collapses into full fictionalism —see Chapter 3, 3.5.3. Both options are dead ends. Now, the situation is similar for FBP. This only has to deal with the second horn of this dilemma —due to its underlying realism, it attributes semantic content to language. However, FBP fails to explain how mathematical language acquires its meaning, and how speakers understand it. That is, we meet once more the Language Acquisition Challenge (LAC). As it happens to full fictionalism and Hamkinsian multiversism, FBP can not solve this challenge. In this case, it attempts to explain reference through consistency. But, this is circular. Second, the realist metaphysics supported by FBP puts it in additional trouble. Assuming a minimum of mathematical complexity, the criterion of mathematical knowledge —consistency— becomes stronger than the theories themselves. There is nothing suspicious in acquiring knowledge about the facts depicted by some theory from other, stronger theory. Indeed, this is quite usual. For example, the premises of a deductive argument are —often— stronger than the conclusion. But they allow us to know the latter. The problem is to establish a criterion of knowledge that is stronger than what is known as a general rule. Both criticisms are exposed below. 4.6.1 The Consistency Challenge Again FBP turns out to be in a similar situation to algebraic fictionalism. While it attempts to reduce mathematical knowledge to logical knowledge —knowledge about consistency and entailment— it lacks an appropriate notion of consistency. Remember, there are two candidates for the job. First, the mathematical concept of consistency that is used in model theory and proof theory. Second, a primitive notion of consistency. However, the FPB-ist can use none of them. Consistency receives a smooth mathematical treatment in model theory and proof theory. Take the former. Using the notion of model, the consistency of a collection of sentences Γreduces to the existence of a model Mfor the theory. Model-theoretic Consistency A set of formulas Γis model-theoretically consistent if and only if there is a model Msuch that for all φ ∈Γ,M|= φ , where “M|= φ ” is defined as usual in terms of Tarski’s satisfaction clauses. Alternatively, consistency can be analyzed using a formal notion of proof —expressed by “⊢”. The consistency of a collection of sentences Γreduces to the fact that it does not prove a contradiction. Proof-theoretical Consistency A set of formulas Γis proof-theoretically consistent if and only if there is no sentence φ such that Γ⊢ φ and Γ⊢ ¬ φ . 120 Both the model-theoretic and the proof-theoretic notions of consistency reduce consistency to a mathematical concept. As a result, both are unacceptable for FBP’s purposes. FBP attempts to explain mathematical knowledge in terms of consistency. But, if consistency is already a mathematical concept, it cannot play this explanatory role. If this were so, the explanation of mathematical knowledge would already presuppose that agents possess mathematical knowledge —i.e., knowledge about consistency, a particular mathematical concept. However, this is a bad explanation. Explanations are not reflexive (120) —facts are not self-explanatory. For instance, if someone asks why the moon takes 27 days to orbit the Earth, it would be absurd to answer “because it takes 27 days to orbit the Earth.” Consequently, FBP needs a different, non-mathematical concept of consistency. At this point, (2, 69-75) appeals to a primitive and intuitive notion of consistency —this is the same as the one I exposed in Chapter 3, 3.5.3. I believe this notion is in perfectly good condition. It is highly plausible that linguistic competence comes with an intuitive understanding of such matters as consistency or possibility. But, even if this was not so, assume for the sake of the argument that the FBP-ist has some understanding of an intuitive notion of consistency. Balaguer claims that such a notion of consistency does not involve any “contact” with those facts depicted by mathematical language. In general, knowledge of the consistency of a set of sentences —whether the sentences are purely mathematical, purely physical, mixed, or whatever —does not require any sort of epistemic access to, or contact with, the objects that the sentences are about. (2, p. 73) This is true when “contact” is understood as involving some strong, metaphysical relation. For instance, no counterfactual relation with El Capitan —a rock formation 914 metres high located in Yosemite National Park, California— is required to know that those sentences claiming that it is a monolith and that it is 500 metres high are consistent with each other. Similarly, no counterfactual relation is required to know that the sentences “El Capitan is 500 metres high” and “El Capitan is not 500 metres high” are inconsistent with each other —these are the kind of examples Balaguer uses in (2, p. 73). Indeed, one may not even know what El Capitan is. All that is required is some understanding of the expressions composing the sentences —this is the crux of the matter. What is the problem? As I argued in Chapter 3, 3.5.3, primitive notions of consistency and possibility are eminently semantic. Logical knowledge is knowledge concerning a relation of entailment. But, this is a linguistic relation. That is, a relation between sentences, propositions, or other similar entities. And these categories, as well as the category of language are semantic. Grammatical categories, such as the category of names, of predicates, or sentences, are already semantic categories. For example, a name is a name because it performs a series of semantic and communicative functions. Among them, it attempts to refer to an object. Thus, the name “Aristotle” is a name because speakers —among other things— attempt to refer to an object using it, the Greek philosopher Aristotle. Similarly, the name “El Capitan” is a name because speakers attempt to refer to El Capitan using it. The semantic function of linguistic expressions is indissolubly tied to their identity. If the linguistic expressions e1and e2perform different semantic functions, then e1=e2. The same happens with other expressions, as predicates. Consider the predicate “to be 500 metres high”. As in the case of names, this predicate has several complex semantic and communicative functions. For 121 example, to convey information about those objects which are 500 metres high. If it did not perform this function, then it would not be a predicate —at least not this particular predicate. It would be another object, such as a stain or a sound. Although not all grammatical distinctions are logically relevant, they introduce semantic or communicative distinctions. The point is: it makes no sense to vary the semantic content or communicative function of a name, a predicate, a sentence, an adverb, etc. without varying its identity. Therefore, speakers must understand the language of a given theory or a collection of sentences to claim that they are consistent. The theory is a linguistic entity coached in a meaningful language. Therefore, to use an intuitive notion of consistency, the FBP-ist must solve LAC: she must explain how the mathematical language acquires its meaning, and how mathematicians are competent in it. Indeed, Balaguer accepts the challenge (2, pp. 49, 51-52): “After all, platonists need to explain not just how we could acquire knowledge of mathematical objects, but also how we could do things like have beliefs about mathematical objects and refer to mathematical objects.” (2, p. 49) Not only that. In addition, he claims that FBP solves the challenge: according to him, the FBP-ist can explain how mathematicians formulate meaningful mathematical theories —this is the first premise in his argument for FBP (2, p. 51).18 However, what is his explanation? The truth is that Balaguer does not elaborate any. He takes the explanatory pressure off himself by saying that mathematical language is about mathematical objects in athin sense: The argument for (i) has already been given: this premise is trivial because it is not making any strong claim to the effect that our purely mathematical theories have unique domains of mathematical objects that they are “about” in some metaphysically thick sense of the term. (2, p. 52) However, Balaguer does not explain what “to be thinly about” consists of. He simply points to some alleged examples of beliefs which are thinly about a domain of objects —for instance, people’s beliefs about mathematical objects.19 This is not enough to meet the explanatory pressure put by LAC. The challenge asks for a full explanation of mathematicians’ linguistic competence, as well as for a metasemantic explanation of how mathematical expressions acquire their meaning. To claim that mathematical language is about —in a thin sense— mathematical objects, or that people believe —in a thin sense— in these objects is to beg the question. It is far from clear how the notion of “to be thinly about” alone helps to understand any of these matters. That is, how it helps to develop a suitable account of mathematical language. At best, merely pointing to this notion is too weak to perform any explanation. Therefore, the notion of “to be thinly 18“FBP-ists can account for the fact that human beings can —without coming into contact with the mathematical realm— formulate purely mathematical theories.” (2, p. 51) 19“Now, no one doubts that human beings could have beliefs that are thinly about mathematical objects. Indeed, the existence of a single mathematical platonist establishes this, for in such a person, we have someone who wholeheartedly believes that there are abstract mathematical objects (e.g., the number 3) and that these objects have certain properties (e.g., primeness). Now, of course, it may be that there are no such things as mathematical objects and, hence, that there is nothing ”out there in the world” corresponding to our platonist beliefs, but this does not change the fact human beings are capable of arriving at beliefs that are thinly about mathematical objects.” (2, p. 39) 122 about” must be properly developed and explained in order to elaborate a complete theory that meets the challenge posed by LAC. Crucially, what FBP needs is an explanation that: (1) assigns to each sentence its truth condition and (2) according to the denial of Responsiveness, does so without involving counterfactual connections between speakers and mathematical facts. In this way, it can avoid at the same time the challenge imposed by LAC and Benacerraf’s challenge. Moreover, this would entail an explanation of what “to be thinly about” consists in. Now, what kind of relation can FBP use that fulfils this double function? The only relation available is, precisely, the one that follows from his definition. FBP is the thesis that every possible universe of mathematical objects exists. Then, given the relation between the primitive concepts of possibility and consistency, FBP entails that every consistent theory can be interpreted in a universe —i.e. every consistent theory is true in some universe. In consequence, consistency should be the vehicle that connects sentences with their truth condition —in general, consistency should connect each mathematical expression with its semantic value. Thus, the FBP-ist could elaborate the following three-step explanation —recall the similarity with fictionalism. First, classical mathematicians are interested in consistent theories. Thus, they understand which theories are consistent and which are not. Second, FBP equates consistency and truth in the multiverse. If mathematicians have logical knowledge about consistency, this connection guarantees that the language is meaningful and that mathematicians understand the language. As a result, a suitable metasemantic explanation can be developed to meet the challenge posed by LAC. Finally, this picture would be a perfect explanation of what “to be thinly about” means too: to be thinly about some objects in the multiverse is nothing more than to be a consistent sentence, belief or theory. How successful is this explanation? Certainly, despite appearances, it is not a good explanation. It reduces linguistic competence to logical knowledge. Now, remember that the notion of consistency at use is a primitive one. And, a primitive notion of consistency must be semantic. That is, to know which theories are consistent and which ones are not, the mathematician must understand the mathematical vocabulary and the sentences composing those theories. The former cannot occur without the latter: Logical knowledge takes place only if agents understand the language. Nevertheless, here we are going the other way around. We are trying to explain language understanding through logical knowledge. Consequently, the explanation is circular. Furthermore, there is no other connection between sentences and their truth conditions in the multiverse to which the FBP-ist can appeal. As a result, she fails to meet the explanatory pressure put by LAC. The FBP-ist cannot explain either how mathematical vocabulary acquires its meaning, or how mathematicians speak about mathematical objects. The unique option left for FBP is to resort to formal languages. There are perfectly good formal notions of language, formula, and theory used in formal logic. All of these are purely syntactic categories —no one presupposes meaning or semantic content. Moreover, they allow to speak about consistency. However, as I argued in Chapter 3, 3.5.3, formal languages are mathematical devices employed in model theory, proof theory, and other branches of formal logic. As a result, when the logician claims that a collection of formulas is consistent she employs the model-theoretic or the proof-theoretic notion of consistency. But, as we have seen, these are not available to the FBP-ist. 123 In conclusion, FBP is not allowed to use primitive notions of consistency or possibility to explain mathematical knowledge. Both demand linguistic competence on the part of the mathematician: that is, they have logical knowledge concerning the language of mathematics only if they understand this language. However, FPB fails to provide a suitable explanation of the latter. Thus, as fictionalism and Hamkinsian multiversism, it cannot solve the challenge posed by LAC. They do not explain how vocabulary acquires its meaning or how speakers become linguistically competent. Moreover, FBP cannot use mathematical notions of consistency either. These concepts lead FBP into an irremediable circularity. But, the mathematical and primitive notions of consistency —or possibility— are the unique notions available for FBP. Consequently, FBP fails to explain mathematical knowledge. The latter does not reduce to logical knowledge. Something more is needed. 4.6.2 The False Weakness of Logic Finally, I will expose an additional challenge to the reduction of mathematical knowledge to logical knowledge. Assuming a minimum of mathematical complexity, knowledge of the consistency of a theory exceeds the knowledge of the theory. Thus, knowledge of consistency is too strong an epistemological criterion. I will oppose the challenge both to intuitive and mathematical concepts of consistency. For this, note that there is a close connection between intuitive notions of consistency and the model-theoretic or proof-theoretic definitions. Roughly speaking, from a platonist point of view the latter offers valuable information to determine the extension of the former. In the case of first-order logic, the three concepts become coextensive for the platonist. (a) if a theory T is semantically consistent, then it is intuitively consistent; and (b) if T is syntactically inconsistent, then it is intuitively inconsistent. Moreover, if we combine these two points with the completeness theorem [...] we arrive at the result that (among firstorder theories) the intuitive notion of consistency is coextensive with both formal notions of consistency. (2, p. 70) Because completeness breaks down when we move up to higher-order logic, this result does not hold in stronger systems. However, I believe that it is plausible to accept a weaker relation between primitive and syntactic consistency even at these levels. Specifically:20 (*) A theory Γis consistent according to our intuitive standards if and only if Γis syntactically consistent. 20Fictionalism avoids the challenge exposed below because it forbids this equivalence. Technically speaking, the arithmetical sentence expressing consistency will be false: this sentence quantifies over numbers, but numbers do not exist. According to fictionalism, mathematical theories are fictions. Therefore, mathematical knowledge of arithmetical sentences taken at face value simply vanishes. Thus, it is trivial that logical knowledge is stronger than mathematical knowledge for all mathematical theories. Nevertheless, I believe the challenge could be reintroduced in a more subtle way. 124 Otherwise, we would take as intuitively consistent some theory Γthat syntactically implies a contradiction. This would hold against the different logical calculus, including the first-order calculus, a conclusion that is hard to bite: mathematical knowledge about logic would turn out to be false. Otherwise, note that the analogous equivalence concerning semantic consistency is far more dubious. There are no extramathematical reasons to think that in higher-order logic the consistency of a theory suffices for the theory to have a model —which is a set. In short, given any n-order system, it is reasonable for the platonist to accept that the intuitive consistency of a set of sentences Γis equivalent to its syntactic consistency. This is enough to develop the challenge. Let’s abbreviate “ φ is consistent” as Con( φ ). As Gödel showed, Con( φ ) is a number-theoretic statement —modulo some appropriate coding of the vocabulary of the language. Now, suppose that a mathematician knows (*) —I believe this is highly plausible. Suppose also that knowledge is preserved by Modus Ponens —this is almost universally accepted. Then, if this mathematician knows that Γis intuitively consistent, by (*) she also knows that Con(Γ). Knowledge about a primitive concept of consistency is tantamount to arithmetical knowledge. I am now in a position to pose the following difficulty. According to FBP, assume that mathematical knowledge reduces to knowledge of consistency and entailment. Thus, knowledge about a mathematical sentence φ reduces to knowing its consistency. That is, knowing that φ follows from knowing Con( φ ). Similarly, for a set of sentences Γ, knowing the theory Γis achieved through knowledge of Con(Γ): K(Con(Γ)) →K(Γ). The problem is that if PA ⊆Γ, then Con(Γ) is a stronger statement than any of those sentences in Γ. Gödel second incompleteness theorem says that if PA is consistent, then PA ⊬Con(PA) and PA ⊬¬Con(PA). Con(PA) →PA ⊬Con(PA) ∧PA ⊬¬Con(PA). Assume, then, that a mathematician has arithmetical knowledge —i.e., she knows PA.21 According to FBP, she obtains this knowledge because she knows that PA is consistent. Hence, by factivity of knowledge, PA is indeed consistent —we have Con(PA). And, by Gödel’s second incompleteness theorem, PA ⊬Con(PA). Consequently, knowledge of PA does not transmit through logical consequence to knowledge of Con(PA) —the mathematical sentence stating that PA is consistent. Moreover, given the connection between syntactic consistency and intuitive consistency, knowledge of PA does not transmit through logical consequence to knowledge of the intuitive consistency of PA. As a result, FBP entails that for any mathematical theory T ⊇PA, the unique criterion of knowledge for T is stronger than T. For platonism, logical knowledge is stronger than mathematical knowledge in some cases. This situation is challenging for FBP for two reasons. If logical knowledge is stronger than mathematical knowledge, and mathematical knowledge is regarded as problematic, is the problem not transmitted to logical knowledge? As far as arithmetical knowledge is concerned, logical knowledge —even knowledge about a primitive notion of consistency— makes more demands on the world than mathematical knowledge. Thus, if the latter is problematic, the former will be even more so. Second, it is controversial that knowledge about a sentence or a theory requires as a general rule knowledge of something 21Knowledge about Robinson Arithmetic (Q) is enough. This is PA without induction. 125 independent from them. Indeed this is often the case. The clearest example is —precisely— inferential knowledge. If we know that the glass is not on the table and that the glass is either on the table or under the table, then we can know that the glass is under the table. This is so even if the disjunction “the glass is either on the table or under the table” does not follow from the conclusion. Nevertheless, what is controversial is to establish that the unique criterion of knowledge of a sentence or of a theory must be always epistemologically stronger than them. The FBP-ist can object that knowing that a sentence or a theory is consistent is not stronger than knowing them. Thus, she can endorse the following conditional: K(Γ)→K(Con(Γ)). As a result, knowledge of theories becomes equivalent to knowledge of their —intuitive and mathematical— consistency. Thus, the previous criticism simply vanishes. Knowing a theory T we already know that it is consistent. But, to what extent is this plausible? At least, I believe this is highly controversial. It is not obvious that speakers’ knowledge of a theory already conveys knowledge of its consistency. On the contrary, it seems that knowledge of consistency requires a cognitive achievement beyond the efforts made to know the facts depicted by T. For instance, suppose that current physics affords us knowledge about how things are at an empirical level. Thanks to it, we know the laws governing elastic collision, how entropy characterizes the evolution of thermodynamic systems, or how elementary particles behave. However, this situation does not seem to involve knowledge about the state of the theory itself. That is, whether it is a good description of the world or not, whether it is consistent, etc. Something more is required to know this thing. Knowing that a theory is consistent —or what it depicts is possible— requires an extra cognitive achievement —even if the notions of consistency and possibility used are primitive. Indeed, if knowledge of a theory already entails knowledge about its consistency, the FBP-ist must accept the following situation. Let’s define recursively the following sequence of theories as follows: PA0= PA and PAn+1= PAn∪Con(PAn). Then, knowledge of PA would be equivalent to knowledge of PAn, for arbitrary n. Again, this is at least highly controversial. To sum up, the reduction of mathematical knowledge to logical knowledge faces two challenges. First, FBP lacks an appropriate notion of consistency to perform this reduction. On one side, mathematical concepts of consistency are useless for this purpose. In turn, FBP can not make use of a primitive notion of consistency —or possibility— until the challenge posed by LAC is solved. However, it is not enough to meet this explanatory pressure —i.e., FBP cannot explain how the mathematical vocabulary acquires its meaning, and how mathematicians understand this vocabulary. Second, given a minimum of mathematical complexity —arithmetic—, knowledge of consistency turns out to be stronger than mathematical knowledge. As a result, it is highly controversial that it can be established as a general criterion of mathematical knowledge. Consequently, the reduction of mathematical knowledge to logical knowledge is highly controversial too. Nevertheless, as in the case of Hamkinsian multiversism, this does not necessarily entail a total abandonment of FBP. Specifically, what is at issue here is its explanation of mathematical knowledge. This requires something more than logic. Conversely, FBP is fundamentally a metaphysical thesis: the thesis that all possible mathematical universes exist. But, this metaphysical thesis and the epistemology envisioned by Balaguer are independent. Thus, it is perfectly consistent to 126 for this case, but to use the concept of identity, taken as already known, as a means for arriving at that which is to be regarded as being identical. Admittedly, this seems to be a very odd kind of definition, to which logicians have not yet paid enough attention; but that it is not altogether unheard of, may be shown by a few examples. (46, §63, p. 73) Another example are directions. The identity of directions is given by the following equivalence Dir d(l1) = d(l2)↔l1∥l2, where “d(li)” abbreviates “the direction of line li” and ∥is the relation of parallelism. Thus, two lines specify the same direction if and only if they are parallel. According to the first step of the abstractionist explanation, the semantic content of the identity “d(l1) = d(l2)” is stipulated to be the same as the righthand side of the equivalence. It will be true just in case the line l1is parallel to the line l2. Then, according to the second step, directions are selected as referents of “d(l1)” and “d(l2)” subject to the constraint that they compositionally determine the semantic value of the identity. This is why the explanation is endowed with a top-down flavour. The semantic value of a range of sentences is stipulated to accord with some previous standards. Once this is done, reference is explained in terms of these semantic values and compositionality. Similarly, rational numbers can be given in terms of equivalences. Let “r(n,m)” abbreviate “the rational number corresponding to the pair ⟨n,m⟩of integers nand m”.Then, rationals can be introduced using the equivalence relation n1·m2=n2·m1defined on pairs of integers. RTL r(n1,m1) = r(n2,m2)↔n1·m2=n2·m1. That is, the rational corresponding to the pair of integers ⟨n1,m1⟩is the same as the rational corresponding to the pair ⟨n2,m2⟩if and only if the product of n1and m2is the same as the product of m1and n2. Indeed, this accords with the usual definition of rationals. As Frege said, the history of mathematics is full of examples.5 An abstraction principle is an equivalence of the form AP f( α ) = f( β )↔ α ∼ β , where α and β are variables ranging over objects6of some kind, ∼is an equivalence relation7defined on these objects and “f” is a singular-term-forming-operator. Let’s call the complex terms of the form 5For a detailed exposition of the use of abstraction principles in the history of mathematics see (95). Also, Hale (51) has shown how Dedekind’s definition of reals (28) can be explained in terms of abstraction principles. 6Here, the notion of object must be understood in a broad sense. It includes first-order entities (individuals), as well as higher-order ones (concepts, properties, relations, etc.). Most of the time the context will suffice to determine the order of variables. I will use Greek symbols “ α ”, “ β ”, both as metavariables standing for terms of a particular n-order language and as n-order variables standing for the objects this language speaks about. Instead of quotes, the context will suffice to disambiguate both uses. This greatly simplifies the exposition. 7An equivalence relation is a relation which is reflexive, symmetric and transitive. Equinumerosity, parallelism or the relation holding between pairs ⟨n1,m1⟩and ⟨n2,m2⟩of integers when n1·m2=n2·m1are cases of equivalence relation. 133 “f( α )”abstract terms and their referents —if there were any— f-abstracts or simply abstracts. Also, let’s call the entities over which the variables α and β range specifications. The reason is that these objects specify the abstracts for which AP holds. Finally, let’s call the equivalence ∼aunity relation. AP relates the identity of abstracts f( α )to the unity relation ∼holding between the specifications.8Abstractionism makes strong use of abstraction principles to explain how mathematical terms successfully refer and, a fortitori, to answer the aforementioned question of how numbers, as well as other inhabitants of the mathematical realm, are given to us. 5.2.0.1 The Structure of Abstraction Attempts Consider an idealized community of speakers who initially have a language L1. Let’s call the language L1the base language of abstraction. Let’s assume that L1does not contain the singular-term-forming- operator frequired to form abstract terms, as well as other similar expressive resources. Thus, the language L1is free from ontological commitments concerning the abstracts satisfying AP. At most, it contains enough expressive resources to deal with the equivalence ∼and other linguistic resources required to speak about the specifications on which this is defined. Then, speakers will use this resources to express a variety of facts concerning the equivalence. Specially, they will be able to state the right-hand side of AP. On the contrary, at this initial state the left-hand side remains unnoticed by them. An attempt to extend L1by means of AP takes three steps —for similar presentations of abstractionism in terms of communities of speakers see (82, pp. 148-151) and (135; 136). (i) First, the community extends the base language L1by adding an operator fwhich applied to norder variables α , β , etc. of the base language form expressions f( α ),f( β ), etc. —at this point of the explanation, it must not be assumed that these expressions are terms. Let’s call the resulting language the extended language L2. Additionally, it also contains a symbol “=” —at this point, it must not be assumed that this is the identity predicate. Then, the community uses the sentences of the extended language in accordance with some set of assertibility conditions stated in the base 8Abstraction principles provide a definition of abstracts. As Frege (46, §63) claims, “this seems to be a very odd kind of definition”. However, this sort of definition can be understood as a contextual definition, similar to the definition of truth afforded by the T-schema φ is true iff φ . This definition offers a criterion —a truth condition— which allows us to identify when a sentence φ satisfies the concept of truth and when it does not —that is, when φ is true and when it is not. Thus, the concept of truth is explained in terms of its extension. Similarly, AP provides a criterion stated in terms of a unity relation that identifies those elements —the abstracts f( α )— satisfying the concept being defined. Indeed, if we had a predicate Cexpressing the concept being defined, AP would be equivalent to a principle identical in form to the T-schema C(f( α )) ↔ α ∼ α . The concept Csatisfied by the abstracts is explained in terms of its extension. This reading of abstraction omits further comments on how the semantic content of sentences expressing facts about the equivalence relates to the content of identities between abstract terms. In particular, it omits Frege’s (46, §65) and Rayo’s (116) attributions of sameness of content, or Linnebo’s (81; 82) metaontological minimalism. This is equivalent to saying that such a reading does say nothing about how to interpret the equivalence ↔. More on this on 5.3. 134 language. In particular, the use of apparent identities of the form f( α ) = f( β )is governed by the assertibility condition α ∼ β —see 5.6 for a precise statement of the assertibility conditions. Thus, the rule governing the use of such apparent identities is given by the corresponding instance of AP. By obeying this rule, members of the community come to speak as if there were abstracts f( α ),f( β ), etc. (ii) Second, the content of new L2-sentences φ is stipulated by φ ’s assertibility condition. Specifically, the content of identities f( α ) = f( β )is stipulated by the corresponding L1-sentence α ∼ β about the unity relation ∼and specifications α and β . Thus, f( α ) = f( β )is true if and only if α ∼ β is also true. The assertibility conditions turn out to be truth conditions. Importantly, this specification does not rely on the ability of speakers to translate L1expressions into L2. (iii) Third, semantic values for subexpressions of new L2-sentences φ are selected to meet the constraint that these compositionally determine the semantic value of φ . Thus, if the abstraction attempt is successful, expressions f( α ),f( β ), etc. are assigned abstracts as their semantic values. Thus, f( α )and similar expressions are terms referring to a particular object. Speakers, as a result of a successful abstraction attempt, are furnished with the ability to speak about these abstracts. This three-step account summarizes the main features of abstractionist accounts of reference. The sketch can be specified in slightly different ways attending to how stipulation, sentential content, the relation between sentential content and reference, and so on are understood. Either way, I will assume this suffices to understand the kernel of the proposal. Before going on, let me address a reasonable worry concerning the previous picture of abstractionism. The first step involves talking about how the lexicon of L1is expanded. In particular, the operator f —and, presumably, other sorts of expressions— is added to L1. However, semantics only enters the explanation in the second step. Furthermore, the second step is only concerned with sentential content. The third step is the only one which mentions terms and reference. However, the arguments in Chapters 3 and 4 rely on the maxim that syntactic categories, such as the category of terms, are already semantic categories. So, does not the first step of the explanation lead back to the same problems concerning syntax faced by fictionalism and multiversism? For, when talking about the operator fis involved, this seems to qualify it as a purely syntactical element to which speakers accord some meaning. But, if this is the case, the concept of syntax and language at use must be a formal and mathematical one. Consequently, a vicious circle arises: the abstractionist explanation presupposes an understanding of mathematical facts and will be useless to explain mathematical reference. The answer to this worry is to observe that the above three steps are explanatory. Any talk about syntactic and semantic categories featuring in them is performed by the theoretician who is trying to explain the speakers’ linguistic practice. Conversely, speakers are not required to understand what they are doing and, thus, they are not required to understand this explanation. Therefore, the three steps composing the abstractionist explanation do not correspond to a temporal sequence of events underlying speakers’ practice. In particular, speakers do not extend L1by employing a formal language to which they attach some entities as semantic value —nor are they required to understand what a formal language 135 is. The unique stipulation they are required to do is a stipulation concerning the language they end up speaking. But, this is radically different from model-theoretic interpretations of formal expressions. So, abstractionism is free from the problems affecting pluralist philosophies of mathematics. 5.2.1 Language Expansions Now, I will show that the following conditional is true: If abstractionism can explain mathematical reference, it can also explain how speakers acquire a whole new language. Again, take the case of directions. A variety of properties and relations between directions are introduced from properties and relations on lines. For instance, orthogonality. Two directions d(l1)and d(l2)are regarded as orthogonal if and only if the lines l1and l2are orthogonal. Let ⊥and ⊥∗be orthogonality between lines and directions respectively. Then, the previous claim can be symbolized as follows: d(l1)⊥∗d(l2)↔l1⊥l2. Similarly, a variety of properties and relations on rationals are also “inherited” from properties and relations of pairs of integers. This is the case of the ordering between rationals. Let <and <∗be the usual orderings on integers and rationals respectively. Then r(n1,m1)<∗r(n2,m2)↔n1·m2<n2·m1, —if miis negative, then other nkand mksuch that ni·mk=nk·mimust be found. These examples show that the relations of orthogonality between lines and the ordering relation between rationals are “inherited” from analogous relations on specifications. Indeed, this phenomenon takes place when the latter meets the appropriate condition. Namely, that the property or relation does not discriminate between specifications related by the unity relation ∼. Thus, assume that lines l1and l2are parallel, and that l2is orthogonal to l3. Then, l1and l3will be also orthogonal —parallelism is a congruence with respect to the relation of orthogonality. Generally, for any sentence φ which does not discriminate between parallel lines —any sentence such that ∥is a congruence with respect to φ —, a predicate φ ∗can be introduced such that it holds of the directions of some lines if and only if φ holds of the very lines.9A whole language about directions can be acquired through abstraction from a language only committed to lines. This explanation can be generalized to other languages as follows. Let’s say that the unity relation ∼ is a congruence with respect to a L1-formula φ ( α n... β n)if and only if it satisfies the following condition10 α 1∼ β 1∧...∧ α n∼ β n→( φ ( α 1... α n)↔ φ ( β 1... β n)). 9“The meaning of any other type of assertion about directions would have first of all to be defined, and in defining it we can make it a rule always to see that it must remain possible to substitute for the direction of any line the direction of any line parallel to it.” (46, §65) 10Again, the formula must be read as the corresponding universal closure. 136 The formula φ ( α 1... α n)does not discriminate between equivalent specifications. Moreover, for simplicity, consider formulas which only hold in the field of ∼: φ ( α 1... α n)→ α 1∼ α 1∧...∧ α n∼ α n. That is, if φ ( α 1... α n)holds of some specifications α 1,..., α n, then these are related by ∼to other specifications, and —because ∼is reflexive— they are related by ∼to themselves. Then, for any formula satisfying these two conditions, abstraction explains how an n-ary predicate about abstracts satisfying the following inheritance principle is introduced Inheritance φ ∗(f( α 1)... f( α n)) ↔ φ ( α 1... α n). In consequence, the abstractionist three-step explanation can be enhanced by adding the assertibility conditions corresponding to the each predicate φ ∗—see 5.6. A whole new set of useful expressions is available to speak about the abstracts. 5.2.2 Reductionist Metasemantics Abstractionism offers a workable metasemantics which looks promising to deal with the case of abstract objects. The key feature of this metasemantic account is its reductionist nature. To explain how directionterms dsuccessfully refer, the abstractionist only makes use of lines land an equivalence relation ∼with which speakers are familiar. Also, the explanation of how a relation of congruence ⊥∗holds on directions d(l1)and d(l2)only involves the lines l1and l2and, once more, ∼. Directions themselves are not used to explain these facts. While abstracts feature in the explanandum —what is explained is how terms of the form f( α )refer to these abstracts—, they do not feature in the explanans. This is why abstraction and inheritance principles are so appropriate to deal with cases of reference to abstract objects. Together, they impose a minimal requirement for reference: to assign truth-conditions to identities and other sentences which respect the logic of identity11.(81; 82; 84) A minimal requirement is that truth-conditions have been assigned to all identity statements and other predications that involve the new singular terms to be introduced. Moreover, this assignment must be done in a way that respects the laws of logic. The most important part of logic in this connection is what we may call the logic of identity, which describes the identity relation and its interaction with predication. (82, p. 30) Furthermore, abstraction principles are reasonably strong. For instance, HP is enough to interpret in second-order logic PA’s axioms —this result, originally suggested by (99) and proved by (150), is known as Frege’s Theorem. All this makes a strong case for abstraction principles to be considered as serious candidates for the design of an account of mathematics that avoids the challenges exposed in Chapters 3 and 4. 11Frege’s (46, §62, p. 73) words: “we shall be giving a general criterion for the identity of numbers. When we have thus acquired a means of arriving at a determinate number and of recognizing it again as the same, we can assign it a number word as its proper name.” 137 5.3 METAONTOLOGICAL MINIMALISM Originally, Frege’s interest in abstraction principles was motivated by his desire to reduce arithmetic to logic. Simplifying things a bit, he regarded HP as a definition of the concept of cardinal number, and thus as a logical principle. Therefore, platonism, abstractionism and logicism —the thesis that arithmetic can be reduced to logic— are mixed together in his philosophical project. Similarly, neo-logicism (150) regards HP as an implicit definition —thus, as an analytic truth— of cardinal numbers. But, unlike Frege, they do not consider it a logical law. Neo-logicism relies on Frege’s Theorem and the analytic nature of definitions to claim that arithmetic is also analytic. Both logicism and neo-logicism read the biconditional ↔in HP as an analytic operator ⇔ #(X) = #(Y)⇔X≈Y, expressing some kind of analytic relation between its two sides.12 Modulo this reading of abstraction principles, both projects aim to reap the benefits of what seems a promising explanation of mathematical knowledge. If arithmetical truths are logical truths, or at least they are analytically true, it is possible to gain knowledge about them by laying down the right definitions —definitions of the form of AP. However, the connection between abstraction and analyticity is not a matter of necessity. As Linnebo (81), (82, p. 3) —by appealing to Kant and Boolos— points out, it is controversial that analytic claims entail existence. Logic and conceptual analysis are not enough to settle the debate about the existence of mathematical objects. This is why he suggests an alternative reading of abstraction as a form of metaphysical explanation, which retains the epistemological benefits promised by analyticity. According to his reading, abstraction principles provide us with an explanation of what it takes, that is, of what is metaphysically sufficient for some objects —such as cardinal numbers, rationals, or directions— to exist. Similarly, inheritance principles explain what is metaphysically sufficient for some facts —those facts depicted by atomic sentences of the form φ ∗(f( α 1)... f( α n))— to be the case. Consequently, instead of wondering whether there are analytic existence claims, this reading shifts attention to another question: are there thin objects “in the sense that their existence does not [..] amount to very much? Presumably, an analytic truth does not make a substantial demand on the world. But perhaps being analytic is not the only way to avoid imposing a substantial demand” (Linnebo, 2018, p. 3). A metaphysical reading of abstraction and inheritance principles suggests that the answer to this question is a resounding “yes” and the metaphysical picture arising from such a reading is known as metaontological minimalism. (81; 82) Ontology is the study of what there is. Metaontology is the study of those concepts involved in ontology, such as the concepts of object, existence or fact. Then, metaontological minimalism is the thesis that such concepts are minimal or thin. And, a thin concept of object allows for thin objects: objects whose existence “does not impose substantive demands on the world.” Minimalists do not claim 12On several occasions ⇔was read as expressing some kind of “sameness” in semantic content or meaning. This is usually referred as Frege’s recarving thesis. The thesis claims that the identity #(X) = #(Y)and X≈Y share the same content, which is carved up in different ways by each sentence. While the right side of HP is not committed to cardinal numbers, the identity recarves its content yielding commitment to such entities. However, this idea of “recarving” as well as its relevance in Frege’s philosophy is highly controversial. See (46, §64), (150) and (52) for uses of the thesis. Recently, Rayo (116) has advocated the idea of sameness in content and recarving by means of his just-is statements. See (82, Ch 7) for a careful analysis of the role of recarving in the Grundlagen and Grundgesetze, and its relation to the context principle. 138 that all objects are thin in this sense. Trees, mountains, or —token— books are quite thick objects. Their existence requires very specific physical and contingent circumstances. Thus, for a —token— book to exist it is required a complex chain of events leading to some agents developing language, creating paper and finally creating the book by printing some ink marks on the paper and arranging this in the form of a book. However, minimalists claim that thickness comes from the kind of objects trees, mountains or —token— books are. Thickness should not be loaded on the concept of object itself. Historically, there have been a variety of versions of this idea. Each one of them provides an analysis of what “to be thin means” —that is, an analysis of what it means not to impose substantive demands on the world in order to exist. Coherentism is a well-known example. According to coherentism, the consistency of a theory suffices for the existence of the objects the theory describes to exist.13 For example, the existence of natural numbers only demands from the world that arithmetic is a consistent theory. More recent examples of minimalism are Eklund’s (32) priority-based minimalism, Thomasson’s (141) easy ontology, or Ferreirós’ (38) practice-based account of mathematical objects. Linnebo’s proposal is that abstraction principles provide an alternative explanation of the metaphysical idiom of sufficiency. To the question raised by Koellner (70, p. 113, n. 48) How cheap is existence in mathematics? Does consistency suffice? Consider ZFC+CH versus ZFC+¬CH. There is reason to believe that both are consistent. The trouble is that CH and ¬CH are existential claims and, on a straightforward reading, the objects that they assert to exist cannot coexist. I am inclined to think that existence in mathematics is “Consistency +X” but I do not know how to solve for X” [italics in the original], abstractionist minimalism suggests solving Xvia abstraction and inheritance principles. The result is an abstractionist version of metaontological minimalism. In consequence, abstraction is not only a metasemantic mechanism for fixing reference. Furthermore, it delivers an explanation of what it takes for some objects to exist. Again, consider material bodies — such as trees, mountains or books. Assume talk about parcels of matter is in order. Surely, a necessary condition for the existence of material bodies is that some parcels of matter —the paper and ink stains— are spatiotemporally coordinated in a very specific way. First, these parcels must be coordinated under motion in such a way that any two parcels change their spatio-temporal location if and only if their relative positions are preserved by the movement. More formally, two parcels uand vmove from a1 and a2to b1and b2respectively, if and only if ρ (a1) = a2and ρ (b1) = b2—where ρ is a permutation preserving the relative position of uand vto each other. Second, these parcels must be linked by a path of solid —in a non-technical sense of “solidity”— matter. For example, two spheres which satisfy only the former condition, and perform perfectly coordinated movements, do not constitute a physical body —in the usual sense. They must also be materially connected. Similarly, a flock of birds which have perfected their joint flight to the extreme does not constitute a material body either —even if the flock is given the 13Recall that criticism developed in Chapters 3 and 4 only affects consistency as a criterion for mathematical knowledge. On the contrary, coherentism is a metaphysical thesis claiming that consistency suffices for existence. For this reason, such criticism does not affect coherentism, as it does not affect FBP regarded just as a metaphysical position. 139 status of an “emergent object”, it is not a material body in the usual sense. Now, let ≡bbe the partial equivalence14 relation on parcels of matter defined by these two conditions. Then, for a material body to exist it is necessary for the parcels forming it to be related by the partial equivalence ≡b. However, it is highly plausible that this is also sufficient for its existence. This gives the following abstraction principle for physical bodies. Let b(u)refer to the body to which the parcel of matter ubelongs. b(u1) = b(u2)↔u1≡bu2. Moreover, similarly to the case of cardinal numbers, rationals and directions, a variety of predicates φ ∗ about material bodies can be introduced via inheritance principles. Thus, the equivalence u1≡bu1is metaphysically sufficient for identities of the form b(u1) = b(u1), and thus for the existence of b(u1) —assuming identity entails existence. Similarly, φ (u1...un)is sufficient for φ ∗(b(u1)...b(un)) to hold.15 Generally speaking, abstraction reveals itself as an explanation of what it takes for some objects to exist —metaphorically speaking, of what the existence of these objects “demands on the world.” Consequently, the biconditional ↔featuring in AP is read as an stronger sufficiency operator ⇒: α ∼ β ⇒f( α ) = f( β ). This reads: the fact that α ∼ β —the specifications α and β are related by the unity relation ∼— is metaphysically sufficient for f( α )and f( β )to referrer to the same abstract —thus, for this abstract to exist. Similarly, φ ( α 1... α n)⇒ φ ∗(f( α 1)... f( α n)) says: the fact depicted by φ ( α 1... α n)is metaphysically sufficient for the fact depicted by φ ∗(f( α 1)... f( α n)) to be the case. This provides the following characterization of sufficiency via abstraction (82, p. 37):16 Consider [an abstraction principle] which purports to provide identity conditions for some kind F of object in terms that presuppose only some antecedently accepted ontology but not Fs. Assume that an agent has an appropriate grasp of the unity relation ∼and uses the criterion and property inheritance principles of the form (Inher) to govern her discourse about Fs. Assume that the agent stands in an appropriate relation to specifications α and β , which are in the field of ∼, and uses these specifications as if to make claims about f( α ) and f( β ). Then: (i) the agent is in fact referring to objects f( α )and f( β ), (ii) α ∼ β ⇒f( α ) = f( β ), (iii) φ ( α 1... α n)⇒ φ ∗(f( α 1)... f( α n)). 14The relation is clearly symmetric and transitive. It must be partial because reflexivity can fail: not all parcels of matter give rise to a well-defined object in the previous sense. For instance, think about fluids or gases. 15The example of material bodies is from Linnebo (82, §2.3). He himself acknowledges that it is a toy case to illustrate how abstraction explains the idea of metaphysical sufficiency. 16The original says “suffices for” instead of ⇒. The same criterion for reference and sufficiency can be found in (81). 140 The strength of the operator ⇒is in the middle of the analytic ⇔and the strict conditional □( φ → ψ ). It must be a less demanding notion than the analytic operator precisely because thin objects take attention away from analytic existence claims. Furthermore, it must be more demanding than the strict conditional. First, because otherwise, the well-known effect of the paradoxes affecting the strict conditional prevents a safe epistemological route to abstracts. Any conditional □( φ → ψ )such that ψ is necessarily true, is true. But, knowledge of φ does not ensure knowledge of ψ . For instance, suppose that set-theoretic truths are necessary truths. Moreover, suppose CH is false. Then, □( φ → ¬CH) will be true even if φ ≡ “Wittgenstein is the author of the Tractatus.” While this is known by every philosopher alive, nobody knows that ¬CH. This suggests that ⇒should be developed as a hyperintensional operator.17 If abstraction elucidates some form of metaphysical explanation, it can be used to characterize the idea of thin objects. Instead of asking whether there are objects whose existence is analytic —according to a more “traditional” account of abstraction—, we can ask which objects —to use the previous metaphor— “make little demands on the world.” Material bodies and similar physical entities are thick because their existence imposes contingent, physical requirements on the world. For the body b(u)to exist, it suffices that u≡u: u≡bu⇒b(u) = b(u). However, the partial equivalence relation ≡binvolves a causal chain of events leading uto be spatiotemporally located in the correct position for u≡uto hold —recall the chain of events leading to the existence of books. Material bodies are thick because they have spatiotemporal location and are causally effective. This kind of objects are called concrete. On the contrary, equinumerosity ≈is not an spatio-temporal relation —provided the second order entities which Xrange over are not spatio-temporally located—, and does not involve any story about causal chains of events. Moreover, if HP is read along the lines of minimalism, then the fact that X≈X suffices for #(X) = #(X). Thus, the fact that the concept Xis equinumerous to itself suffices for the existence of the cardinal number to which #(X)refers: X≈X⇒#(X) = #(X). Consequently, the existence of cardinals #(X)does not involve any substantial, causal relation. Abstract objects —such as cardinal or rational numbers— are objects whose existence does not impose any spatiotemporal demand on the world —they are specified by means of non-spatio-temporal entities and a nonspatio-temporal unity relation. Thin objects are so because little is required for their existence. In the middle of the concrete and the abstract realm, directions and types —such as sentence types— are an example of what is known as quasi-concrete objects. Somehow, these are thicker than numbers, but also thinner than mountains and birds. For instance, parallelism —the equivalence relation that suffices for the existence of directions d(li)— is not a spatiotemporal relation: l∥l⇒d(l) = d(l). 17Linnebo (82, p. 18) suggests a connection between his sufficiency operator and metaphysical grounding. Nevertheless, he also notes an important difference. As in the case of the strict conditional, even if φ grounds ψ it is doubtful that knowledge of φ transfers to knowledge of ψ . Also, abstraction principles seem related to the concept of real definition. Indeed, the mathematical examples provided by (120) are close to abstraction principles. The relation between all these metaphysical concepts remains open. 141 However, directions can be specified by concrete objects as well: some physical objects have direction.18 5.4 CHALLENGES TO ABSTRACTION Abstraction principles must be in good condition if we want to use them to explain mathematical language and mathematical knowledge. However, it is unclear if this is the case. These principles have been opposed by two problems: the Caesar problem and the Bad Company Problem. In this section, I will explain what each of them is and how they can be solved. Thus, I will argue that the best solution to Caesar problem is the piecemeal solution advocated by (82, Ch. 9) and (135). In turn, I will argue that the best solution to the problem of bad company is to adopt a predicative reading of quantifiers and, thus, of abstraction principles. 5.4.1 The Caesar Problem The Caesar Problem arises when one tries to answer the following awkward, but otherwise legitimate question: (1) Is Julius Caesar the number 3? Let’s call questions about mixed identities between abstracts —such as cardinal numbers, rationals or directions— and objects of another sort —for example, Julius Gaius Caesar— Caesar questions. The Caesar Problem is the silence of abstract principles when pressed to answer that kind of question. The problem was originally raised by Frege in the Grundlagen. As he recognizes, either HP or Dir ...will not, for instance, decide for us whether England is the same as the direction of the Earth’s axis —if I may be forgiven an example which looks nonsensical. Naturally, no one is going to confuse England with the direction of the Earth’s axis; but that is no thanks to our definition of direction. (46, §66)19 Abstraction principles only provide abstracts with suitable criteria of identity —but no other kinds of objects. Given any two abstracts f( α )and f( β ), AP returns the condition α ∼ β as a criterion for determining if the abstracts are identical or not —f( α ) = f( β )if and only if α ∼ β .20 However, AP is not powerful enough to tell what happens when the identity is a mixed identity involving an abstract f( α ) and Julius Caesar, England, or a glass of water. Indeed, the problem motivated Frege’s extension-based account of cardinals and the introduction of Basic Law V —leading his project to a contradiction. 18Are the concrete objects all and only the thick objects? Are the abstract objects all only the thin objects? I will leave this question open here. It would be interesting to know if the concepts of thin and thickness induce some ordering between metaphysical categories. 19Frege did not explicitly formulate the question. He originally mentions the undecidability of Cesar questions as a challenge to an alternative definition of number: “we can never —to take a crude example— decide by means of our definitions whether any concept has the number Julius Caesar belonging to it, or whether that same familiar conqueror of Gaul is a number or is not.” (46, §56) 20This does not entail that ∼is decidable. See (97) for more on this question. 142 [Document text truncated for crawler view.]