Reconsidering Mach in the light of the interplay of practices

1 Reconsidering Mach in the light of the Interplay of Practices María de Paz ([email protected]) Department of Philosophy, Logic and Philosophy of Science Universidad de Sevilla Abstract: The aim of this paper is to reconsider Mach’s position in the light of the recent approach on the interplay of mathematical practices provided by José Ferreirós’ book Mathematical Knowledge and the Interplay of Practices. In the first section the main approach of the book is presented, with a characterization of the notion of ‘scientific practice’ and the triple approach used to understand mathematical knowledge: cognitive, pragmatist and historical. In the sections 2, 3 and 4 Mach’s philosophical views are presented according to this epistemological approach. The fifth section is the conclusion. Keywords: Cognitive Approach, History, Philosophy of mathematics, Philosophy of science, Pragmatism In the past few years there has been an interest on highlighting a recent ‘turn’ in epistemology characterized as ‘the practice turn’. This shift considers the previous one carried out by science studies in the 1970s as a reaction to traditional philosophy of science. This was regarded as the attitude according to which “[r]esearchers in science studies [...] were progressively more encouraged to pay attention to scientific practices in meticulous detail and along multiple dimensions, including the material, tacit, and psycho-social ones” 1 . That turn was partly a reaction to a philosophy of science that was very focused on the analysis of scientific theories as completed processes, which disregarded the actual practice of science, the development of experimental practice, the activities that prompted the scientist to the formulation of new explanatory hypotheses, or the practical paths that lead scientists to the development of those theories which were the explicit object of analysis. 1 Léna Soler et al., Science After the Practice Turn in Philosophy, History, and the Social Studies of Science (New York and London: Routledge, 2014), 1. 2 The practice turn in epistemology tries to overcome some problems posed by the science studies turn, such as the excessive focus on experiment which contributed to reinforcing the classical opposition between theory and practice. This was the case, to some extent, of the trend known as ‘New Experimentalism’ 2 , but especially of the Empirical Program of Relativism (EPOR) in their sociological studies of experimental work 3 . One possible way of overcoming this classical opposition is to pay attention to the practice of mathematics, since this is a discipline traditionally linked to theoretical activity. The philosophy of mathematical practice constitutes the focus of attention of much recent research, particularly since the creation of the APMP (Association for the Philosophy of Mathematical Practice) in 2009. In this paper I would like to bring the attention to a specific contribution to this field in order to understand, update and reconsider Mach’s philosophical position. I am talking here about the recent book published by José Ferreirós Mathematical Knowledge and the Interplay of Practices. Although the core of Mach’s philosophical position is not concerned with the history and philosophy of mathematics, I will show that Ferreirós’ approach can enlighten several aspects of Mach’s views. In order to attain this goal, the first section of the paper is devoted to present and summarize the general lines of Ferreirós’ work. In the second and in the third, a connection with Mach’s philosophical perspective will be provided and the forth is intended to offer a conclusion based on the main lines of the paper. 1. A triple approach to the interplay of practices. Despite being specialized on the history and philosophy of mathematics, more specifically on mathematical practice, Mathematical Knowledge and the Interplay of Practices discusses several epistemological and philosophical issues that are important and interesting for philosophers working on how we acquire knowledge, how it is produced, how it becomes a coherent body of systematized and constituted topics and 2 See Ian Hacking, Representing and Intervening, Introductory Topics in the Philosophy of Natural Science (Cambridge: Cambridge University Press, 1983) and Jed Z. Buchwald (ed.) Scientific Practice: Theories and Stories of Physics (Chicago: The University of Chicago Press, 1995). 3 See in particular Harry M. Collins, “‘Son of seven sexes’, The Social Destruction of a Physical Phenomenon”, Social Studies of Science, 11, no. 1 (1981): 33–62, doi: 10.1177/030631278101100103. A good reference is also Andrew Pickering, “Against Correspondence: A Constructivist View of Experiment and the Real”, in Proceedings of PSA 1986, eds. A. Fine and P. Machamer (Pittsburgh: Philosophy of Science Association), vol. 2 (1987): 196–206. 3 how a specific part of knowledge is related to other human activities 4 . In this sense, it could be asserted that the book does not concern only our way of knowing and understanding mathematics, but our knowledge in general, and might be relevant to the knowledge of any other scientific discipline. By asserting this I would not like to deny the specificity of mathematical knowledge –which Ferreirós claims and shows in the book: “mathematics is an essential part and parcel of the theoretical workings of science […], but also distinctive and peculiar, because here the theoretical work becomes autonomous.” 5 I would only like to stress that the book develops a theory of knowledge for a specific field that could be extended to other domains of knowledge or, at least, whose extension should be considered. In fact, this is partly the purpose of this paper, in which I extend Ferreirós’ views in order to understand those of Mach, which were mainly focused on the history and philosophy of physics, rather than on mathematics. The theory of knowledge presented by Ferreirós is made explicit through a triple approach stated at the very beginning of the book: 6 cognitive, pragmatist and historical. Before describing this triple approach and its implications, I would like to focus, first, on an analysis of the notion of practice proposed by the author. Given that my characterization of Mach’s philosophy would be precisely in terms of a practical approach, it would be useful to understand the notion placed at the basis of this conception. Regarding the notion of ‘mathematical practice’ proposed by Ferreirós, first of all I must say that he does not want to introduce a rigid definition, 7 but one that is general enough to encompass the plurality of practices that he wants to deal with. That said let’s see what the author says: “By a ‘practice’ in general I mean a recognizable type of activity that is done –and can be taught and learned– by human agents.” 8 This appears to be a very general notion, but it shows certain constraints given the specificity of Ferreirós’ focus on mathematics. 4 In this part of the paper I use my previous book review of Mathematical Knowledge and the Interplay of Practices published on The Philosophical Quarterly. See María de Paz, “Book review of Mathematical Knowledge and the Interplay of Practices, by José Ferreirós.” The Philosophical Quarterly, 68, no. 271 (2018): 406-408 https://doi.org/10.1093/pq/pqx036 (online first, 2017). 5 José Ferreirós, Mathematical Knowledge and the Interplay of Practices (Princeton: Princeton University Press, 2016), 14. 6 Ibid., 3 7 Ibid., 17. 8 Ibid., 28 4 So, he says: “mathematical practice is what the community of mathematicians do when they employ resources such as frameworks (and other instruments) on the basis of their cognitive abilities to solve problems, prove theorems, shape theories and (sometimes) to elaborate new frameworks.” 9 When we apply the broader notion of practice to mathematical practice, we can see that it is necessarily an activity carried out by human beings –no other community of beings practices mathematics in a recognizable way–; this implies the analysis of a concrete type of knowledge, not an abstraction in which the features of the practitioner could be abstracted. Besides, mathematical practice is something done within a community, so the approach has to consider the work of the single mathematician and her relation to other mathematicians. It is also restricted concerning the object of application. Although he offers a very general notion of practice, when coming to mathematical practice, this is defined to attain concrete goals (solve problems, prove theorems, shape theories), but he adds: “it will always be possible to link such mathematical practices with particular mathematical theories.” 10 This goal restriction affects the approach itself, since he provides a notion of practice useful to develop an epistemology of mathematics, 11 thus, able to say something about the kind of knowledge mathematics provides. Therefore, to define a practice the presence of the practitioner, the community and the goal is clearly needed. In order to fully understand the notion of mathematical practice we have to clarify the notion of framework. There are two precise meanings for this notion in the approach we are trying to characterize. The first is the symbolic framework which refers to the particular kind of characters, i.e. symbols used by mathematicians in order to represent problems and to establish solutions. The second is the theoretical framework identified by propositions, proofs, principles or laws shared by a community, methods or open questions, such as conjectures. As a result, the frameworks are the instruments employed by a mathematician and her community to attain certain results when they are practising mathematics. Given this definition, we have to take into account the link between the framework and the mathematician, that is, what the author calls the ‘agent’, and this pair ‘agent-framework’ is placed at the centre of the practice. 12 This is because the practice is developed within a framework and thanks to a framework, since it constitutes the 9 Ibid., 33 10 Ibid., 33 11 Ibid., 30 12 Ibid., 44 5 resources that the mathematician has to solve certain concrete problems. And also because the agent –the mathematician– is the one who practices, the one who does mathematics. From my point of view, all these ideas become clearer when we take as a whole the approach that Ferreirós provides in the book, that is, the triple perspective that we referred to above: cognitive, pragmatist and historical. First, let us consider the cognitive approach. This refers, as stated by Ferreirós, to an epistemological conception in which the subject is seriously conceived (‘no epistemology without a subject’). 13 The human subject has definite cognitive skills biologically determined, but this biological determination depends not only on the brain’s development but also on the environment, both natural and cultural, in which the subject grows and lives. This environment determines certain features that the agent has and it is because of these particularities that Ferreirós decides to call this subject an ‘agent’. He wants to make explicit the concrete character that the notion of agent has, opposed to the abstractness that traditional philosophical conceptions have attributed to the notion of ‘subject’. The agent is endowed with a body and a social life and can have different restrictions depending on the cultural milieu she inhabits. 14 It is not the same to be a nineteenth century German mathematician than a Babylonian magician trying to establish astronomical measurements and they do not share the same frameworks. Also, the emphasis on the agent marks another important feature: the fact that the subject acts and interacts, this is why Ferreirós characterizes this cognitive approach as agent-based. He wants to show that mathematics is a human product, and thus promote an epistemology against the oblivion of action. 15 So, knowledge, specifically mathematical knowledge, is not something attained by contemplation, but by developing and performing certain actions. Another feature of this cognitive and agentbased approach is the role that the community plays in the development of (mathematical) knowledge. The mathematician is not an isolated individual; she might create proofs or theorems, but it is within a community that mathematics is transmitted, and this is where new developments are accepted, modified, and receive the form in which they will be taught. So, agents by acting and interacting produce and develop knowledge, by performing certain practices, which connects with the second part of the approach mentioned. 13 Ibid., 10. 14 Ibid., 60-61. 15 Ibid., 11. 6 This pragmatist perspective is indeed better characterized, as the author does, as practice-oriented, meaning that the roots of knowledge should be found in practical actions of human beings, hence the important notion of practice defined and discussed above, and also the emphasis on the subject performing this practice who is called ‘the agent’. In this sense, there are proto-mathematical practices which can be considered as technical activities in the physical world such as counting and measuring. These technical activities or practices –which imply the manipulation of concrete objects or human-made instruments– could be situated at the basis of more abstract mathematical practices that are reached in further stages of knowledge. It is the characterization into different knowledge strata (from more technical and basic to more abstract and complex) and the way the different strata relate to each other, which responds to the interplay of practices that names the book. This recourse to technical practices is important for the theory of knowledge proposed, since through a detailed study of some historical cases, the author aims at supporting his philosophical statement that the “roots of knowledge and thought lie deeper than language”, 16 that is, these roots may well lie on the use of instruments and on the development of technical practices (counting and measuring being the best examples for the case of mathematical practice). This permits a link with the third feature, given that there is a clear recourse to a historical survey. The historical approach stresses that in order to understand mathematical knowledge we have to study its history. This implies a vision of mathematics as a historical science, i.e., a science that has a historical development, a diachronic production of concepts that can be explained in a time line; but more importantly, mathematics is a historically developed science in the sense of historically laden. This means accepting the potentially contingent aspects of a science that has too frequently been understood as a priori and necessary. That is why Ferreirós characterizes his approach also as hypothetical. At this point, he traces an important frontier dividing basic from advanced mathematics, as it is only in the latter where the hypothetical elements can be found. The contingency and hypotheticality of mathematics do not imply a potentially relativistic character. Advanced mathematics is dependent on certain choices, but these choices are not arbitrary since they are constrained by the rooting of more abstract practices in more elementary ones – and these last ones in technical practices. This conception justifies the relevance of studying technico-practical 16 Ibid., 13. 7 developments in what we have called above proto-mathematics for the understanding of abstract mathematical concepts – just like in Mach’s historical-critical method, as we will try to show later. To sum up, the interconnection of this triple epistemological approach might be understood when thinking about the meaning of terms: the meaning is determined and acquired by agents through the practical use of concepts and tools in the course of the historical development of science (math in particular). It is now time to see this approach from a Machian perspective. 2. The notion of practice in Mach. The purpose of this and the next two sections is to frame Mach’s philosophical conception within the triple approach presented by Ferreirós. In this sense, part of the idea is to bring Mach’s epistemology up to date, for which in some respects it will imply to introduce foreign elements to reinterpret his claims and not to take strictly concepts and elements provided by Mach himself or even to consider Mach’s contributions with certain flexibility. We would start by rethinking Ferreirós’ notion of practice in a Machian light and then the triple approach will follow. First of all, in order to justify our application of a practical approach to Mach, it is important to consider that, according to Ferreirós “attention to processes, and not merely to more or less perfect results, is characteristic of studies of mathematical practice.” 17 It is precisely this approach to science as a process that can be characterized as proper of Mach’s position: Ordinary thought does not serve pure knowledge, and therefore suffers from various defects that at first survive in scientific thought, which is derived from it. Science only very gradually shakes itself free from these flaws. Any glance at the past will show that progress in scientific thought consists in constant correction of ordinary thought. 18 Indeed, from this quotation we can see that science is a gradual process and not a complete endeavour, it is seen as a progression in which errors are depurated, and it has its origin in the improvement of common ideas (ordinary thoughts). Mach conceives 17 Ibid., 29. 18 Ernst Mach, Knowledge and Error. Trans. T. J. McCormack and P. Foulkes (Dordrecth/Boston: Dordrecht-Reidel publishing company, 1905 [1976]), 2. 8 science as an unfinished task, because each particular science has to take into account the progresses made in other sciences “for the sake of orientation in its own field.” 19 Taking that into account, it is clear that, to Mach, science as a whole is part of a process and thus, the results of science could never be considered as definitive or perfect, but simply as part of the normal development. It is on this basis that we can start interpreting Mach’s position as a practical approach. But in order to bring it closer to the conception that we are dealing with here, we have to be more specific and try to see if it is possible, in some way, to conceive Mach’s notion of practice “as a recognizable type of activity that is done –and can be taught and learned– by human agents.” 20 To attain this goal, we should consider Mach’s studies on mechanics in his famous work Die Mechanik in ihrer Entwickelung historisch-kritisch Dargestellt. As is well known, this book starts with the study of some machines, the lever and the pulley in particular, in order to explain the first proto-mechanical developments. What we want to emphasize here is how he explains the development of the first mechanical machines when referring the work of Archytas of Tarento: “he built some mechanical devices to draw certain curves.” 21 And a few lines later he says: “It is precisely a distinctive feature of the prehistory of mechanics that, it has directed its attention towards the practical uses [...].” 22 These passages reinforce the practical perspective we want to emphasize. But they also point in the direction of understanding Mach’s notion of practice, which has not been philosophically elaborated as Ferreirós’. The construction of the device to draw curves implies a recognizable type of activity done by human agents, since it is peculiar to human beings to construct instruments to develop geometry. But this activity can also be taught and learned, since there must be a technique behind the construction of the instrument that can be transmitted to reproduce the construction of similar instruments. As a result, since the very beginning in the science of mechanics we can find the production and reproduction of an activity as the foundation of mechanical knowledge. And, of course, there is no possibility of identifying an activity, i.e. a practice, without the practitioner –what Ferreirós calls ‘the agent’– Archytas in the case we analyze here. 19 Ibid., 3. 20 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 28. 21 Ernst Mach, Die Mechanik in ihrer Entwickelung historisch-kritisch Dargestellt (Leipzig: Brockhaus, 1883 [1901]), 9 22 Ibid., 9. 9 In the practice referred, we can also see that it aims at a concrete goal –to draw specific types of curves– which requires the existence of an unambiguous object of application (to solve determined problems). The study of this practice is also useful to develop an epistemology of mechanics in the same sense as Ferreirós develops an epistemology of mathematics. This is because Mach’s study is directed to understanding the practices at the basis of the science of statics which is considered as the most ancient of the mechanical researches, whose origin is precisely the development of these machines – the pulley, the lever– and also of primitive automatons. As we see, just like in Ferreirós’ epistemology, certain practices are able to be linked with certain theories. There are still two elements that we need to discuss to bring Mach’s views under Ferreirós’ perspective. The first one is the community and the second the framework. Regarding the community, it is present all over Mach’s work. In his Mechanics, from the very beginning he considers the existence of more than one man developing practices. Thus, he speaks about “Archimedes and his successors.” 23 But it becomes even clearer when he refers to the process of knowledge acquisition characterized in Darwinian terms: “The acquisition of the most elementary knowledge does not devolve upon the individual alone, it is pre-effected in the development of the species.” 24 This statement points at a global community for the development of the science: the community of human beings. And here it can be objected that Ferreiros’ notion of community is much more restricted. And in fact it is, but the restriction comes with the notion of framework. The general feature –and the most relevant– of the notion of community is the fact that science is not developed in isolation and, given the references that Mach made to the human species and several peoples having mechanical practices –Egyptians, Assyrians, etc.– the necessity of the community to develop knowledge and practices is beyond any doubt. We have asserted above that we required a second element to understand Mach’s perspective from Ferreirós’ and this was precisely the framework. The introduction of this notion will allow us to fully explain Mach’s notion of practice according to Ferreirós’ restrictions and to restrict also the notion of community that we have defined in the previous paragraph. Ferreirós’ considers two specific kinds of frameworks: the symbolic and the theoretical. Regarding this, Mach considers the understanding of phenomena as linked to the constitution of specialized jobs and working associations. 23 Ibid., 16 and ff. 24 Ibid., 1. 16 perceive; an idealization of the more relevant elements or sensations selected to satisfy our practical ends or to overcome certain intellectual difficulties. The processes of grouping and establishing connections among elements are varied, but they do not constitute a ‘representation’ of our perception, as we said above. This is because elements are particular, that is, unrepeatable and any idealization or abstraction made by us with the purpose of organizing them, of expressing regularities, or, of constituting laws, is a distancing from or an alienation of perception. 49 As a result, laws, theories and all physical science are not the result of sensations themselves, but of our way of grouping them regarding our cognitive structure which is subordinated to the experience of the species. This constitutes some kind of biological a priori, that we have already mentioned, in the sense that what is a priori for an individual is a posteriori for her ancestors, because they have acquired it during the course of their contact with phenomena. 50 In this way, the economy of thought is related to Machian Darwinism in two senses. First, because it helps science to get the adaptation of thought to facts and of thoughts between each other with the least possible number of symbols. 51 Second, because economic media for adaptation have been acquired by us not as isolated individuals but as members of the species. The concept of economy is so crucial because it permits the connection between the pragmatist and the cognitive approaches. But the arrangement of elements depends also on the interest of the researcher (the scientist or the agent), since depending on how sensations are associated among each other, the research belongs to one science or to another (to physics, to mathematics, to psychology, etc.). This is why Mach defended that physics is a science created with limited and specific purposes –in the same sense as math has concrete aims to Ferreirós. In physics, the scientist has the power of determining her own point of view in order to emphasize the more interesting questions to her. As we referred above, every science, and physics in particular, is part of a larger body of knowledge. Therefore Mach recognizes the plurality and specificity of the sciences, being thus impossible to 49 Cf. María de Paz, Henri Poincaré: Del Convencionalismo a la Gravitación (London: College Publications, 2016), 151. 50 Cf. Rudolph Haller, “Conventionalism and its Impact on Logical Empiricism.” Philosophia Scientiae 3, no. 2 (1998-1999): 100. 51 Given this restriction of media (symbols, propositions, etc.), Mach’s economy has also been interpreted as a variant of Ockham’s razor (cf. Erwin N. Hiebert, “Mach’s Philosophical Use of History.” In Historical & Philosophical Perspectives on Science, ed. Roger Stuewer (Minneapolis: University of Minnesota Press, 1970), 189 and Murat Baç “Structure versus process: Mach, Hertz, and the normative aspect of science.” Journal for General Philosophy of Science 31, no. 1 (2000): 40). 17 attribute to him any reductionist interpretation. This complementarity among the sciences recalls Ferreirós words, when he asserts that mathematical practices “do not live in a vacuum of pure thought” and he goes on: Mathematical ideas always depend on a technical and practical background; they are constantly linked (cognitively) with non-mathematical practices. Among these I count not only technical practices such as counting and measuring, but, especially, scientific practices of modelling that are at the core of the sciences, in which mathematics plays a central constitutive role. In a nutshell, a full explanation of mathematical knowledge and the web of mathematical practices makes it necessary to embed this network within the larger web of scientific practices. 52 In fact, the historical study developed on counting practices such as the Babilonian or the ones developed by tribal societies, whether it is for making business or distributing food, 53 is reminiscent of Mach’s Mechanics and his study of modern abstract mechanical concepts by rooting them in the study of machines constructed to satisfy technical needs (recall that Mach’s book starts with the study of the lever). In both studies we can see the link between specific scientific theories and technical practices –basic arithmetic and primitive statics, respectively. This interaction –or interplay between practices is explained by Ferreirós in terms of thematizing, 54 meaning that the progress in science –in mathematics, concretely– is done by making explicit by means of theory some aspects that have previously been implicit in practice. This perspective is astonishingly similar to Mach’s emphasis on instinctive knowledge, 55 which he also calls ‘involuntary’ and which is previous to conscious –i.e. explicit– scientific knowledge. In Ferreirós book, this is also present when asserting the need to pay attention to non-linguistic forms of knowledge, including tacit forms, 56 as is the case of the use of diagrams in geometry –or diagrammatic thinking. In short: to both authors, the roots of knowledge lie deeper than thought and language. The role played by this instinctive or tacit knowledge implies the interconnection between different levels of practice. To Ferreirós 57 we have operative knowledge of 52 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 163. 53 Ibid., 187. 54 Ibid. 84. 55 Mach, Die Mechanik in ihrer Entwickelung historisch-kritisch Dargestellt, 1. 56 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 46. 57 Ibid., 39. 18 different practices and knowledge strata and they have systematic connections. These connections between levels are not only a guide to developing further knowledge –this is also the reason why we can link specific theories to specific knowledge when making epistemology–, but they also play a role as constraints. There are several constraints with a role in the development of knowledge. Some of them are cognitive 58 and others are intra-mathematical. 59 They restrict the formation of new concepts. In some respects this is close to Mach’s conception of knowledge as a restriction of our expectations: An unprejudiced consideration of the development of natural science makes us see its origin in the fact that we begin by noticing in processes those aspects that are of immediate biological importance to us [hence the cognitive restrictions], and only later progressively extend our interest to aspects, that are moderately important. In the light of this reflection, the following obvious formulation may perhaps be acceptable: in origin, the ‘laws of nature’ are restrictions that under the guidance of our experience we prescribe to our expectations. 60 Despite the similarities, it seems to be some kind of subjective element in Mach’s formulation that is not present in Ferreirós, since the latter introduces the constraints to guarantee the objectivity of mathematical concepts. And we cannot attribute a conception of objectivity to the Machian formation of concepts, since physics is an economic description with the purpose of adaptation, not to explain natural processes. Consequently, science is a description of the elements grouped by the scientists to satisfy our needs, and as such, any statement of existence based on physical laws is as superfluous as naïf. However, as we will try to show later, this does not imply a completely relativistic conception of knowledge and we will require the historical approach to prove this. 4. How to frame Mach in the historical approach. In the practical-pragmatist approach that we have just shown, we can see a good opportunity to emphasize the interplay of practices. But it is even more patent when recourse is made to the historical approach. To apply this to Mach’s perspective, it would be better characterized as historical-critical, which is precisely the method he 58 Ibid., 62. 59 Ibid., 257. 60 Mach, Knowledge and Error, 351. 19 used to study the development of mechanics. This means that history provides the privileged milieu for the understanding and clarification of concepts, guided by a critical –philosophical– perspective. Mach considers that the proper analysis of science to solve problems is not the logical, but the historical one, because the development of scientific concepts depends on historical contingencies. As Cohen asserts, to Mach, “a theory of history must supplement epistemology, and perhaps even complete it.” 61 This relevance played by history requires a diachronic presentation of science in order to show the links between Ehrfahrungen or technical practices and theories or Wissenschaften, creating a net of links. Or in Ferreirós words: The whole history of mathematics can be presented as the gradual development of a network of links connecting different core notions that initially lie separate, i.e., as the creation of a delicate tapestry or spider web establishing bridges between the discrete and the continuum, number and geometry, algebra and analysis –and, later, topology– etcetera. 62 This implies a link between the plurality of the practices that can go beyond the specific domain of a discipline –such as mathematics– transcending thus its limits and connecting with other disciplines. A clear example is the use of mathematical methods in physics. As a result, the interplay of practices is not only from technical or protomathematical practices to more abstract ones (bottom-up), but also between different sciences, in such a way that “conceptual understanding depends on the web of practices and knowledge strata that an agent commands.” 63 This entails that the acquisition of meaning is not dependent only on use or on history, because new connections between practices not previously regarded are able to modify the conceptual understanding of the mathematician. Otherwise concepts will always have the same meaning or will only vary depending on different uses. In the process of concept formation different cognitive resources might apply, such as imagination and abstraction. A good example is the role of idealizations introduced in Euclid’s Elements. 64 This might at first appear as a transit from technical to mathematical practices, but it is a different process than the one we referred above, this is a top-down process in which the constitution of ideal 61 Cohen, “Ernst Mach: Physics, Perception and the Philosophy of Science.”, 142. 62 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 40. 63 Ibid., 101. 64 Ibid., 113 and ff. 20 figures is previous to the constitution of their scientific relations as expressed in the scientific framework that forms the thirteen books of the Elements. It is a fact that Mach considers the variability and contingency of scientific concepts as well. Concepts are functional labels which facilitate the transmission of knowledge. In his study of mechanics he shows how the basic principles of this discipline are fixed by means of simple and convenient conjectures based on the laws of motion, which are groups of elements economically structured as a restriction of our expectations. According to this, laws and concepts are postulated for our convenience, to provide an explanation of a concrete phenomenon, i.e., with the purpose of restricting our field of possibilities for action and to mark the proper course that our thinking should follow to get a better adaptation. The convenience of the principles is based on an economic image to explain phenomena. It is based on pragmatic criteria in order to obtain a solution to orientate ourselves in the world, but only one among the multiplicity of possible enlightening solutions. As a result, scientific theories are conceived as ordered arrangements of elements with functional value directed towards action and adaptation. However, as we mentioned before, scientific concepts are not mere names or relative concepts, “they are stable and rich in content because they are charged of history and experience.” 65 The functional role of concepts is not an arbitrary one, they are not only names, they have empirical but contingent origins and history makes them stable, as happens with some groupings of elements such as the ‘I’. 66 It is true that according to Ferreirós, certain statements of mathematics are simply true or false, such as those of elementary number theory. What makes them true, according to his argument, is the meaning and properties of numbers as they are conceived in the context of basic counting practices. This makes his position different to Mach, since truth is not a value that he wants to include in his theory of knowledge, 67 and we cannot see if he traces a frontier dividing basic from advanced science as Ferreirós does with arithmetic –meaning that truth is applied for basic arithmetic and, as we will show, hypothetical is used for advanced math. The historical contingency is expressed by both authors in the variability of concepts. Ferreirós considers the variation of concepts as dependent on the web of 65 Michel Paty, “Mach et Duhem, l’épistémologie des savants-philosophes.” In Épistémologie et matérialisme ed. Olivier Bloch (Paris: Klinsiek, 1986), 200. 66 Mach, The Analysis of Sensations, 23. 67 Pietro Gori, “Nietzsche as Phenomenalist?” In Nietzsches Wissenschaftsphilosophie: Hintergründe, Wirkungen und Aktualität ed, Helmut Heit, Günter Abel and Marco Brusotti (Berlin: De Gruyter, 2012), 347. 21 practices dominated by an agent and, as a result, a different web of practices implies a different conceptualization. 68 To Mach, concepts have an intrinsic historical and evolutionary variability, explaining thus the different conceptual possibilities depending on the historical background and the environment. Ferreirós names his historical approach also as hypothetical, because of the dependence that advanced mathematics has on certain choices. These hypotheses are constitutive of some mathematical domains, 69 founding hypothetical systems that “arise out of a richly embedded historical development”. 70 Similarly, Mach states that “a proposition in natural science always has a merely hypothetical sense”, 71 meaning that propositions in physics are far from being normative or definitively established. In this sense a new parallelism might be drawn between both authors, particularly if we take into account that Ferreirós states that “the presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory” 72 and he develops a hypothetical conception of mathematics for advanced –not for elementary– math. To Mach, propositions are hypothetical because they cannot be true or false, but they are also constitutive because they provide the most beneficial conception to mankind –human agents– both in biological and cultural respects. 73 They are a condition of possibility –hence their constituiveness– for our orientation in the world. However, the emphasis that Mach places on avoiding any truth conception has too frequently provoked the association of his thought with mere instrumentalism or antirealism. This is denied by Cohen by emphasizing that to an instrumentalist evidence and history are usually not relevant. 74 This is nevertheless problematic since one can think about evidence as empirical adequacy –a concept very dear to instrumentalists such as van Fraassen; and one can consider the role history plays in another well-known instrumentalist like Duhem –Duhem’s case is more controversial given his notion of ‘natural classification’, but to enter into such a dispute would require a different paper. 75 A more interesting perspective has been recently provided by Gori in which he 68 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 103. 69 Ibid., 156. 70 Ibid., 157. 71 Mach, Knowledge and Error, 356. 72 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 154. 73 Cf. Mach, Knowledge and Error, 361. 74 Cohen, “Ernst Mach: Physics, Perception and the Philosophy of Science.”, 141. 75 There are several discussions of Duhem’s natural classification such as Sonia M. Dion, “Pierre Duhem and the inconsistency between instrumentalism and natural classification.” Studies in History and Philosophy of Science 44, no. 1 (2013): 12 and Alfredo Marcos, Pierre Duhem. La filosofía de la ciencia en sus orígenes (Barcelona: PPU, 1988), 153 and ff). 22 characterizes Mach’s position as a hypothetical realism, interpreted as a third way between realism and anti-realism. 76 Gori uses Mach’s Darwinism to mitigate his explicit anti-realist perspectives –i.e. the idea that scientific laws do not correspond to states of facts. Adaptation of thoughts to facts requires some kind of environment to be adapted to. I do not want to introduce here the notion of ‘external world’ as Gori does, because this would mean the return of the distinction between mind and world that Mach wants to dissolve. But in some sense, Gori is right because adaptation implies the existence of certain constraints that might be important in the development of the individual or of the species, and as such, also of science. This idea might bring even closer Mach’s and Ferreirós’ views, and thus we will be very glad in the attainment of our goals for this paper. However, there is no way of avoiding Mach’s assertions regarding the impossibility of distinguishing reality from dream: The question which is often asked, whether the world is real or whether we merely dream it, is devoid of all scientific meaning. Even the wildest dream is a fact as much as any other. If our dreams were more regular, more connected, more stable, they would also have more practical importance for us. 77 Consequently, despite Darwinism, it seems difficult to combine such ideas with a realistic conception, even with a mitigated one. With the historical-critical and hypothetical conception, we come to the end of our analysis. The relevance of history to Mach is emphasized by assertions such as: “let us not let go the guiding hand of history. History has made all; history can alter all. Let us expect from history all”. 78 This places also the emphasis on historical contingency, and we think that the following words could be attributed to Mach, since we find there not only the contingency and the history, but also the interconnection of practices, the recurrence to tacit or instinctive knowledge and the necessary interaction between the plurality of the sciences. But more importantly, it justifies the extension that we have made of Ferreirós’ epistemology to adapt it to Mach’s views: 76 In Gori, “What Does It Mean to Orient Oneself in Science? On Ernst Mach’s Pragmatic Epistemology.” 77 Mach, The Analysis of Sensations, 11. 78 Mach, History and Root of the Principle of the Conservation of Energy, 18. 23 Contingency and the hypothetical are inevitably installed at the heart of modern mathematics, although the highly interconnected web of mathematical and scientific activity (past and present) acts in such a way that it becomes disguised and can even seem invisible. Needless to say, such a viewpoint, if correct, must affect also our epistemological understanding of other sciences, beginning with physics. 79 5. Conclusion. We started this paper by recognizing the recent turn in epistemology in order to present a contemporary perspective that takes seriously the material, tacit, and psychosocial aspects of knowledge. It is a fact that Ferreirós’ approach is one which corresponds to this perspective. But more importantly, we have used this contemporary approach as a fruitful guide to enlighten Mach’s views. This has enabled us to recognize also these features in the thought of an almost forgotten thinker in contemporary philosophy. Mach is usually more remembered as an inspiring figure of the Vienna Circle, rather than by his own work. His books, indeed, are rarely studied and are not present in many philosophy syllabi. Fortunately in 2016, with the occasion of commemorating a century of his passing, new studies have come out –such as Gori’s. In this work, I have tried to show that Mach’s views might have a role in contemporary epistemology by briefly updating and reconsidering them in the light of recent developments in the field. In some respects, the idea of making so much emphasis on ‘recent turns’ –as Soler et al. do– shows more a pretention of being original and thus remembered in the future, than a real innovation or a ground-breaking contribution. I am not denying here the originality of some of the studies present in Soler’s work or in any others, but only claiming for the necessity of remembering the pioneers of the study of scientific practice. This is something very present in Ferreirós’ work, when referring to the works of Dedekind, Riemann or Hilbert. And in this sense, this book could mark a new turn in philosophy, one that considers history seriously and is not ashamed of recognizing the relevance of predecessors or of combining old and new studies. Mach is not quoted all along Ferreirós’ book, and there is no need for it, because, as we said, it is a book about the epistemology of mathematics and Mach was not a relevant figure in this area. In fact, an important contribution of Ferreirós’ approach is his ability to show how a science that is very commonly linked to the idea 79 Ferreirós, Mathematical Knowledge and the Interplay of Practices, 162. 24 of ‘theory’ is also practical. In his philosophy of mathematics, theories and theorising themselves become widely considered and analysed as practice, recognizing that scientific theory itself is better understood in terms of theoretical practices rather than in the classical sense of theory. 80 This is a very important aspect of scientific activity that Mach was not able to encompass within his views. But despite all his problems and his old-fashioned perspectives, Mach’s –as Ferreirós’– is a philosophy based in scientific practice, i.e. a philosophy that takes into account the actual practice of science, which is probably the most serious way of making philosophy of science. We have shown that it is illuminating to analyze Mach’s philosophy with a basis on the triple approach brought up by Ferreirós, although adapted and modified. As we said, Mach’s conception would be better understood as cognitive and evolutionary, pragmatist and practice-oriented (here we choose the same categories), and historicalcritical. There are of course important differences between both authors, but there are also one hundred years of difference between their works. And these two agents have developed their works within different frameworks: Mach was an experimental physicist and Ferreirós has always been a professional philosopher; Mach was a historian of physics and Ferreirós has been –and is– a historian of mathematics; etc. The principles, conjectures and problems that they are confronted with are different beyond any doubt, and so is the context. Their philosophical practices are, consequently, diverse. But they have important elements in common, as we have tried to show in this paper. I would like to finish with a final rapprochement between both: the dissolution of the dichotomy between nature and culture. Ferreirós says: In fact, the nature/culture dichotomy is only valid on a first approximation; when talking of cognitive factors underlying mathematics, cultural elements must be taken into account. Thus, my standpoint is better described as a form of pragmatism than of naturalism. 81 All his work is pervaded by the attempt to avoid simple and classical dichotomies that establish rigid frontiers instead of being helpful for developing a coherent epistemology about a concrete domain of knowledge. The same happens with Mach’s perspective 80 See also the work of Joseph Rouse, How Scientific Practices Matter (Chicago: The University of Chicago Press, 2002) for a practical consideration of theories. 81 Ferreirós, Mathematical Knowldege and the Interplay of Practices, 185-186. 25 when he studies machines, or considers the establishment of a frame of reference to develop the inertia principle: he does not consider these ideas as merely cultural, as opposed to natural. The development of science as a cultural activity is a natural attitude of human beings, it is a way of adapting their thought to facts. A philosophical position which avoids simplistic dichotomies turns out to be fruitful because it is able to approach the real practice of science in all its plurality. 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