TY - JOUR
AU1 - Carter,, Jessica
AB - ABSTRACT A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed. 1. INTRODUCTION In his introduction to The Philosophy of Mathematical Practice, Paolo Mancosu presents a new direction in the philosophy of mathematics, writing The contributions presented in this book are thus joined by the shared belief that attention to mathematical practice is a necessary condition for a renewal of the philosophy of mathematics. We are not simply proposing new topics for investigation but are also making the claim that these topics cannot effectively be addressed without extending the range of mathematical practice one needs to look at when engaged in this kind of philosophical work. Certain philosophical problems become salient only when the appropriate area of mathematics is taken into consideration. [Mancosu, 2008, p. 2] This raises two (related) questions. The first is: what is meant by ‘practice’? The second is how to characterise this new way of doing philosophy, taking mathematical practice seriously, captured by the title Philosophy of Mathematical Practice. Both questions will be dealt with in this paper. In addition, a range of questions and topics studied under the heading of ‘philosophy of mathematical practice’ will be presented. When I write about the philosophy of mathematical practice — from now on abbreviated PMP — note the following. I do not intend to claim that there is a necessary tension or conflict between ‘philosophy of mathematical practice’ and ‘philosophy of mathematics’ (although some might claim there is). It is certainly the case that some of the first proponents of a reorientation in the philosophy of mathematics thought that the previous way of doing philosophy was wrong. In the end I will propose that philosophy of mathematical practice could simply be philosophy of mathematics, although of a particular kind, where (i) mathematics is taken to be mathematics in every shade and not idealisations of mathematics, and (ii) an extension of methods is allowed, i.e., the possibility of bringing in results and tools from other disciplines. The structure of the paper is as follows. The second section recaps some of the motivations that have been advanced for changing the direction of the philosophy of mathematics. The trend observed here is that motivations have moved from being mainly negative towards the previous way of doing philosophy to a realisation and acceptance that different perspectives are possible when doing philosophy of mathematics. The third section tries to give a picture of the current PMP. In order to do this I identify three major strands, characterised in terms of which assumptions are chosen to be at the core of the investigations, and in what is taken as the field of investigation, i.e., what counts as ‘practice’. Finally there is a discussion on how to understand ‘practice’ based on these strands and a conclusion. 2. MOTIVATIONS Much has been written about the motivations for turning away from traditional philosophy of mathematics (see [Corfield, 2003; Ferreirós and Gray, 2006; Mancosu, 2008] and others mentioned throughout this text). It is not my intention to restate all these considerations; rather the aim is to give a few illustrative examples and to direct attention to the more positive points: first the PMP proponents’ insistence that the study of practice opens up many interesting questions that can be dealt with besides traditionally asked questions; and second a development of the studies of philosophy of mathematics towards reconciliation between different ways of doing philosophy and integration of different methodologies. In the mid-twentieth century a growing dissatisfaction with the concerns of the philosophy of mathematics led to studies of mathematical practice. Motivations grew out of a perceived limitation both of the topics studied and methods used.1 For one thing philosophers were critical towards the one-sided focus on foundational questions, that is, focus on logic, set theory, and the three foundational schools, logicism, formalism and intuitionism. See, e.g., [Putnam, 1967]2 and [Tymoczko, 1998].3 Among these critics Imre Lakatos is often mentioned as the first to engage with the philosophy of mathematical practice. In a famous passage he expressed the following: ‘paraphrasing Kant: the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty’ [1976a, p. 2; emphasis original].4 Lakatos’s Proofs and Refutations [1976a] illustrates how these faults can be remedied. It is based on an extensive case study on the historical development of Euler’s formula for polyhedra. If |$V$| denotes the number of vertices, |$E$|⁠, the number of edges and |$F$|⁠, the number of faces of a given polyhedron, the formula |$V-E+F=2$| holds. According to the account given by Lakatos, the first step consists of giving a proof of the formula so that it becomes a theorem. In the next step the theorem is challenged by the discovery of a number of counterexamples, referred to as ‘monsters’. The monsters are handled by, for example, modifying the definition of the concept of a polyhedron. In a later step it is proposed that only specific types of polyhedra fulfil the formula.5 Towards the end, Lakatos introduces the notion of a ‘proof generated theorem’. This means essentially that one considers the constructions made in the lemmas of the proof and determines which types of polyhedra support these constructions. The theorem is thus valid for such polyhedra. A lesson Lakatos learns from this case study is that the formulations of theorems and their proofs are interrelated in a forever ongoing process. In an earlier paper ‘A renaissance of empiricism’ [Lakatos, 1976b] further argues that the ideal of ‘Euclidean’ mathematics should be given up. This is the picture of mathematics (or science) where one picks out true axioms and places them at ‘the top’. Propositions are then derived from these so that ‘truth flows downward in the system’ [Lakatos, 1976b, p. 205]. Instead Lakatos claims that methodologically mathematics is more like the natural sciences and so discusses which type of statements could be considered as falsifiers of an axiomatic theory. The monsters found in his Proofs and Refutations can be seen as examples of such falsifiers. Lakatos introduces the label quasi-empiricism for views of this sort. Another route to PMP comes from a perceived lack of connection between mathematics and the debate on the ontology of mathematics.6 Two influential articles by Paul Benacerraf [1965; 1973] have set the stage for work in this debate.7 Arguments often start out from assumptions about mathematics without testing whether these correspond to what mathematicians do or think. One such assumption is that mathematical theorems are necessarily true. From this assumption it has been argued (e.g., [Katz, 1998]) that mathematical objects exist by necessity. Another assumption employed is that some mathematical statements are true (accompanied by the argument that truth implies existence). Indispensability arguments consist of (at least) two components. The first notes that mathematics is indispensable to natural science. The second, referred to as ‘confirmational holism’, claims that mathematical statements are themselves confirmed whenever the scientific theories in which they take part are confirmed.8 All these assumptions face similar reactions as above — as do most counter assumptions in the debates on realism and anti-realism (e.g., the fictionalist claim that mathematical statements are not true). PMP may enter the picture in trying to investigate these assumptions in terms of mathematical practice. As an example of addressing questions from ontology by drawing on mathematical practice one could mention the contributions of Penelope Maddy. Her work is informed by studies of the history and practice of set theory. Maddy considers, for example, set theorists’ discussions about which criteria should be used in order to adopt set-theoretical principles such as the continuum hypothesis (CH), Gödel’s constructibility axiom, |$V=L$|⁠, and the question concerning which definable sets of reals are Lebesgue-measurable. These principles all have their origin in analysis. One of the problems that led Cantor to his transfinite set theory (and so the formulation of CH) concerned representations of functions by series of trigonometric functions. Lebesgue measure was introduced in order to extend the type of functions that can be integrated. Maddy’s ‘Indispensability and practice’ [1992] compares (among other things) discussions on the foundations of set theory with components of the indispensability argument. She first argues that a naturalist philosopher of mathematics should not only take the practice of science, but also the practice of mathematics, into account when assessing claims about mathematics. In this spirit she finds that the mathematicians’ discussion of the principles of set theory contradict, for example, confirmational holism. According to Maddy, the adoption of principles does not depend on physical facts, for example, whether a particular version of the continuum has applications in some scientific theory. Instead set theorists refer to arguments that are based on internal mathematical considerations, such as how restrictive a particular principle is. 2.1. Reconciliations — Towards PMP We saw in the previous section that criticism of the foundational perspectives led to an interest in the development of mathematical ideas, that is, the heuristics of mathematics. In addition a number of scholars have demanded that the philosophy of mathematics should pay more attention to the human dimension of mathematics. See for example [Kitcher, 1984] and [Tymoczko, 1998]. The outcomes are positions placing an emphasis on the human agents doing mathematics and dealing with social aspects of mathematics.9 In particular the mathematician Reuben Hersh [1979] has been a strong advocate for a shift of focus to pay attention to social and cultural aspects of mathematical practice. Today scholars within PMP wish to advance a more balanced position. One does not have to adhere either to a position that mathematics is at core human or that it is certain. As we shall see there are scholars in contemporary PMP taking the assumption that mathematical knowledge is human knowledge while still trying to explain its special certainty. In addition some of the previously mentioned reactions indicate that studies of foundations are incompatible with quasi-empiricism (as it was coined by Lakatos). Scholars have since found that both empiricism and quasi-empiricism may be combined with positions on the foundations of mathematics.10 In support of this view see [Schlimm, 2010] for an account of how Pasch combined his foundations with empiricism and Ferreirós’ exposition [2006] of nineteenth-century mathematicians’, e.g., Riemann’s use of the word ‘foundation’. Inspired by Lakatos’s Proofs and Refutations, scholars have started to use (historical) case studies as a means to investigate philosophical questions. This has created a number of interrelations between history and philosophy of mathematics as will be explained in the next section. Case studies on mathematical practice are not limited to historical cases and the study of written texts. As illustrated by the work of Maddy, philosophers also take into account the contemporary practice of mathematicians, for example, what they say about their motivation for adopting various principles. It is therefore clear that ‘practice’ could mean a number of different things depending on what type of question one wishes to investigate and one’s (philosophical) assumptions. I propose that it is possible to ask questions at different levels — and as we shall see under different interpretations of the notion of ‘practice’. At the top level could be questions about the true nature of mathematical objects and our knowledge of these. At a lower level are the questions that arise in the practice of engaging with mathematics. Since there are different ways of engaging with mathematics (as developed in more detail in [Giaquinto, 2005a], there are different types of questions to be asked. There is the activity of mathematicians producing new material (heuristics), the practices of learning and teaching mathematics, and the practices of using mathematics in science or society. 3. CURRENT SCENE: WHAT IS PMP? In order to get a handle on PMP, and investigate whether it is possible to characterise ‘practice’, we start by painting a picture of the contemporary scene. I shall present examples from PMP and, in order to give the presentation some structure, sort them into different categories. I find it is possible to identify three different, sometimes overlapping, strands:11 an agent-based, a historical, and an epistemological strand. These differ in various respects, e.g., in what aspects of ‘practice’ they consider and so in which assumptions are built in, what the aims are for PMP, and which methods are brought in to study the questions posed. What is offered is a rough classification. It is not always possible to place philosophers or a single work in any one of these categories as many have broad interests.12 It should also be noted that other categorisations could be made, see, e.g., [Van Bendegem, 2014]. 3.1. Agent-Based PMP Central to the agent-based strand is the belief that a philosophy of mathematics is impossible without taking into account the human beings doing mathematics. The focus here is thus on the practitioners of mathematics: their activities in developing, using, or learning mathematics. There are two major developments within this strand. One has strong interconnections with sociology and a built-in assumption that mathematics is a social activity. The other is a pragmatic orientation following philosophers like Peirce, Dewey, and Putnam. Depending on the chosen perspective for such a study, different methodologies can be brought in. Sociology is used to study the interactions between human beings, whereas results from cognitive science are used in order to study the cognitive functioning of the individual’s mind conveying the ability to form mathematical concepts. Furthermore there are interrelations with other fields, such as anthropology and mathematics education. 3.1.1. The Social Strand of Agent-Based PMP The sociological strand has grown out of works such as those of the sociologists Bloor [1976], Restivo (e.g., [1993]), and Heintz [2000]. David Bloor and Sal Restivo (in their different ways) have developed pictures of mathematics based on the idea that mathematics is a social activity. Bettina Heintz bases her conclusions about mathematics on a sociological study of mathematicians. Because of Hersh’s insistence that a philosophy of mathematics should account for the fact that human beings do mathematics and since he claims mathematical objects are ‘social-cultural-historical’, I also place Hersh (e.g. [1979]) in this category of PMP. In his latest book he lists the following five points: Mathematics is human. Mathematical knowledge is fallible. There are different versions of proof or rigour. Empirical evidence, numerical experimentation and probabilistic proof all can help us decide what to believe in mathematics. Mathematical objects are a special variety of a social-cultural-historical object [2013, p. 169]. Agent-based, and in particular social PMP, naturally has ties to mathematics education. Mathematics educators have long been frustrated with the ‘dehumanised’ traditional philosophy of mathematics, which some, e.g. Paul Ernest, claim only offer ‘absolutist’ (referring to platonism and formalism) views of mathematics. They find such views are unfortunate when teaching mathematics.13 The exception is Lakatos, who is mentioned as the lone hero.14 Ernest, an influential figure in the philosophy of mathematics education, has contributed with an alternative, a social constructivist picture of mathematics (e.g., [1998]). Work combining, for example, sociology, philosophy, and education might not be characterised as truly philosophical, but rather dealing with (philosophical) questions within the fields of sociology and mathematics education. A number of studies, though, have tried to bridge the gaps among, e.g., the sociology, history, and philosophy of mathematics and a network has been formed with this as one of its goals, namely ‘Philosophy of Mathematics: Sociological Aspects and Mathematical Practice’.15 In addition, several conferences have been organised (mainly in Brussels) in order to advance such studies (see [Van Kerkhove and Van Bendegem, 2007; Van Kerkhove, 2009; Van Kerkhove et al., 2010]). One reason given for rejecting work combining philosophy with other fields is that they depend on or use empirical facts, e.g., findings from sociological or anthropological studies, when formulating positions on mathematics. A recent program labelled ‘Empirical Philosophy of Mathematics’ has tried to develop a method to rationally integrate empirical statements into the philosophy of mathematics. In this program Löwe et al. [2010] first note that philosophy needs some input.16 Input may consist of philosophers’ (often contradictory) intuitions, but sometimes, they claim, philosophers also refer to empirical statements. As an example of such a statement they mention ‘most mathematicians are platonists’. In the framework of ‘Empirical Philosophy of Mathematics’ Eva Wilhelmus has performed a study [Wilhelmus, 2007] of mathematicians’ knowledge ascriptions.17 Interestingly, her findings show that even though mathematicians claim that mathematical knowledge is objective, they seem to hold that the truth value of a theorem may change depending on available proofs or counterexamples. Her study further indicates that the acceptance of the validity of a theorem depends on, e.g., the importance of the theorem or the reputation of the mathematician proving it. The study consists of a questionnaire sent to a number of practising mathematicians. The first part asks about the mathematicians’ conceptions of proofs. The majority of the mathematicians responding to the study characterise a proof as something close to a formal proof, i.e., that it consists of a list of statements, some of which are axioms, and the rest are logical consequences of these. The second section of the questionnaire presents a variety of scenarios. After each part of the story, the respondents are asked to comment on the state of knowledge of the main character of the story. One such story is about a PhD student, John, who finds a proof of his supervisor’s conjecture, the ‘Jones conjecture’. After the proof of the conjecture has been accepted for publication in a major journal, most participants of the study answer that John knows that the theorem is true. The next part of the scenario explains that John a few years after attends a seminar where he realises that he can construct a counterexample to the theorem. As one would expect, a majority of the participants (61%) then claims that John now knows that the theorem is false. What surprises Wilhelmus, however, is their answer to ‘Did John know that the Jones conjecture was true on the morning before the talk’ (p. 14): 71% answer positively. Wilhelmus concludes that mathematicians in general do not use ‘formalizability of a proof’ as a criterion to assess that a theorem is true. 3.1.2. The Pragmatic Strand of Agent-Based PMP The pragmatic line of PMP sees it as central for philosophy to characterise mathematical knowledge — with the qualification that it should be characterised as knowledge possessed by human beings. This can in part be seen as a rejection of the idea of having an ‘epistemology without subjects’ and opening up considering knowledge as a product of human activities. Even though knowledge is thus agent-based, proponents of a pragmatist version of knowledge hold that mathematics cannot simply be reduced to human conventions. Furthermore they reject that mathematical knowledge is something that can only be studied by sociological methods. They argue that at least parts of mathematics are non-arbitrary and that, although mathematics is sometimes based on conventions or hypotheses, mathematical knowledge is still objective. One source of these ideas is C.S. Peirce, who famously characterised mathematics as the necessary reasoning concerning hypothetical states of affairs, i.e., that it is possible to draw necessary conclusions from formed hypotheses (see [Carter, 2014]). A number of scholars have used Peirce’s ideas to develop pragmatist pictures of mathematics. Among them are Michael Otte (also in relation to mathematics education), Gerhard Heinzmann and Fernando Zalamea. A recent attempt at formulating an agent-based theory of knowledge can be found in José Ferreirós’s book Mathematical Knowledge and the Interplay of Practices [2016]. Ferreirós explains how basic arithmetical truths (based on counting practices) are both objective and certain. Other, more advanced, areas of mathematics are based on hypotheses — but such hypotheses are often integrated with other practices of mathematics, and so cannot be entirely arbitrary. One much discussed example is the axiom of choice (AC). Formally it is independent of the axioms of ZF; so in principle one could adopt AC or choose not to. One reason it is widely accepted among contemporary mathematicians is because it has many consequences that they are not willing to give up, consequences that are integral to the practice of, e.g., analysis. A simple example is that AC implies that two different (standard) ways of defining continuity of functions are equivalent.18 Furthermore, Ferreirós, [2011; 2016] has argued that there are very intimate links between AC and the classical conception of the number continuum. Thus some mathematical concepts may be rooted in our practical dealings with the physical world (and others are based on hypotheses), but this does not entail that we can define concepts at random. When trying to capture such ideas mathematically, i.e., when defining new mathematical concepts or introducing new principles, there will often be certain constraints. A view like this is expressed by David Hilbert in Natur und mathematisches Erkennen, where he describes how one is led by experience and thought to the concept of continuum, writing, furthermore it is seen that the formation of concepts in mathematics is constantly led by thought and experience, so that mathematics as a whole is a non-arbitrary closed structure [Hilbert, 1992, p. 5].19 Both agent-based perspectives may find it relevant to study how human beings form mathematical concepts, theories, and knowledge thereof and so draw on results from cognitive science, see, for example, [Giaquinto, 2005b; 2007]).20 Marcus Giaquinto [2007] draws on cognitive theories in order to explain how visualisation may contribute to the formation of beliefs in mathematics. Although cognitive science could be an important contributor to PMP, scholars have made a case that cognitive scientists have not yet been able to account for how (complex) mathematical concepts are formed, see [Doridot and Panza, 2004] and [Ferreirós, 2016, e.g., pp. 65–67]. In addition anthropological studies are drawn upon in combination with cognitive science as seen, for example, in the work of Helen De Cruz. De Cruz [2009] investigates the claim made by a number of philosophers that geometric knowledge is innate or, as proposed by Kant, that geometric knowledge is based on our intuitions of space. She splits this claim into two parts. The first part, ‘do we have innate intuitions of space?’, is compared to findings from cognitive science, neuroscience ,and anthropology. Based on these studies she finds that it is fair to claim that human beings have intuitions of space. Second she considers parts of the history of non-Euclidean geometry in order to argue for her second claim, i.e., that until the nineteenth century it seems reasonable to say that geometry was based on our intuitions of space. In this story she starts with the fifth postulate of Euclid’s geometry and presents aspects of the long history that led to the formulation and acceptance of the non-Euclidean geometries by Lobachevsky, Bolyai, and Gauss in the nineteenth century. De Cruz notes that, in the many attempts at proving that the Euclidean geometry was the only possible, mathematicians often unconsciously based their arguments on principles that are equivalent to the fifth postulate. Another component of her argument consists of the strong resistance to accepting alternative versions of geometry. As illustrated by the above examples, ‘practice’ in agent-based PMP refers mainly to the fact that mathematics is (also) a human activity focussing on the agents, real or idealised, doing mathematics. Furthermore this perspective draws on a number of different methodologies and fields such as empirical studies, cognitive science, and anthropology (as in the studies by Wilhelmus [2007], De Cruz [2009], and Heintz [2000]). 3.2. Historical PMP According to the historical PMP it is a major concern that mathematics has a history and this gives rise to a number of interesting questions that belong to philosophy. Whereas the agent-based strand stresses the fact that human beings do mathematics, one could claim that historical PMP focuses on the outcome of these activities and, most importantly, questions related to how this outcome, that is, mathematics, has been shaped across time. An underlying assumption of this perspective is thus that mathematics is the product of certain activities and not a static theory. Answers to historical questions could involve adopting a number of different perspectives. According to recent understandings of the history of mathematics, cultural and social factors play a more central role; see [Aspray and Kitcher, 1988, pp. 20–31]. Themes within this strand include how history and philosophy influence each other, i.e., that philosophical views may shape mathematics, and that the development of mathematics at any given time may raise philosophical questions. Indeed, given the special status that mathematical knowledge has had throughout time, the development of mathematics has inspired much philosophical thought. The history of mathematics may also be seen as a natural provider of cases for philosophy as was advised by Lakatos. Concerning the nature of the relation between history and philosophy, however, there is disagreement. Opinions on this range from the view that history is inherently philosophically laden [Oliveri, 2010] to a position that history — or studies of practice — are independent of philosophical concerns [Maddy, 1997, pp. 200–205]. Conversely, Moritz Epple — as will be shown below — claims that history makes good use of philosophical categories. Finally Mancosu [2008, pp. 17–18] writes that it is still too early to say anything conclusive concerning this ‘metaphysical relation’. In the following I present a few illustrative examples of the above themes.21 3.2.1. Historical Circumstances for the Development of Philosophical Views It is generally acknowledged that mathematics underwent profound changes during the nineteenth century. The three foundational schools tried each in their own way to deal with these changes. Whereas the mathematical background and historical motivation of Brouwer and Hilbert are often referred to, one is rarely told much about Frege’s. Exceptions to this are Marco Panza [2015] and Jamie Tappenden [2006]. According to Tappenden, the stories that are told about Frege’s motivation are not correct. Tappenden calls them myths and even goes so far as to write that they are ‘wildly wrong’. Stories give a one-sided and narrow account about mathematical developments in the nineteenth century. They focus on the so-called ‘arithmetisation of analysis’ as done by Weierstrass, that is the project of freeing analysis from its geometrical foundation by introducing the real numbers and basing reasoning on epsilon-delta calculations. Since it is possible to base the real numbers on the natural numbers, the mathematical task becomes to found them. The later counter myth states that Frege’s concerns were philosophical rather than mathematical, questioning his motivations for asking for a higher degree of rigour in mathematics. This myth claims that (via the work of Weierstrass) the problems of analysis had already been solved when Frege began his work. Tappenden points out that many problems had been solved in real analysis. Frege’s mathematical work was within the areas of geometry and complex analysis. When Frege writes ‘analysis’, he refers to complex analysis. Tappenden shows that in the last part of the nineteenth century this field was in wild disarray, with diverging definitions of fundamental concepts, in particular that of a function, giving plenty of reasons why one would want to engage in philosophical queries. Tappenden places Frege as a follower of the Riemann tradition (and a critic of Weierstrass).22 With the work of Riemann, geometrical intuition was reintroduced into analysis and it was by no means clear at the time how to define analysis, geometry, and their interrelation. Given the state of complex analysis one understands the urge to supply a foundation. What is particularly worth emphasizing is Tappenden’s agenda of making readers aware of the many other themes under debate, for example: the issue of the fruitfulness of concepts was of paramount importance, this was bound up in intricate and sometimes surprising ways with the development of geometry and geometric interpretations of analysis. [Tappenden, 2006, p. 99] Like Tappenden, Panza takes up issues beyond the traditional accounts of Frege’s logicism. The motive is to explain the role functions play in Frege’s foundational program. In disagreement with Tappenden, Panza rejects the Riemannian influence when it comes to Frege’s notion of a function. One aim of Panza’s paper is to point out crucial differences from the current set-theoretic foundations. A careful reading of Frege’s work as well as a historical study of his mathematical predecessors reveal that Frege’s characterisation of the notion of a function, and the role functions played, are closer to Lagrange’s conception. To both Lagrange and Frege a function is a primitive in their respective foundations. According to Frege the notion of a function is a prerequisite for his system of logic as presented in his Grundgesetze der Arithmetik. For Lagrange functions serve to give analysis an algebraic foundation. In addition both regard a function as somehow associated with its expression, but whereas Lagrange identified a function with its expression, to Frege a function is not an object and so cannot be an expression — although ‘the idea of a function detached from any appropriate expression is merely inconceivable’ [Panza, 2015, p. 92]. The use of the expression is to present the law specifying from which primitives the function is composed. As a consequence the notion of an arbitrary function is not available to Frege. Historical studies may also be based on questions that have philosophical leanings, such as attempts to understand how ‘simplicity’, ‘generality’ [Chemla et al., 2016], or ‘abstractness’ [Marquis, 2015] can be conceived of.23 Other examples include the studies of ‘structure’ in Noether’s, Mac Lane’s, or Grothendieck’s mathematics [McLarty, 2006; 2007; 2008].24 A question that arises in this context is whether to place such studies under the heading of History or Philosophy. The answer depends on the focus of the study. In general historical studies are concerned with the understanding of a given notion, such as ‘generality’, in a particular setting. That is, historians are after the particular, whereas philosophers are more concerned with whatever general lesson can be drawn from particular examples — and if possible providing general features of the notion in question. A historian could be interested in documenting how a notion such as ‘simplicity’ or ‘generality’ is understood and perhaps used as a guideline for research in a particular period, community, or even by a single mathematician. The philosopher interested in capturing a particular notion could occupy herself with how it is understood in different contexts, with the intent of including possible understandings of it. It could also be remarked that both philosophers and historians talk about ‘mathematical practice’ but that they may have very different understandings of what ‘practice’ means in view of the above mentioned differences between aims. 3.2.2. The Influence of Philosophy on Mathematics Conversely to the above it is possible to consider the philosophical beliefs of mathematicians themselves and even investigate how or whether these beliefs influence their mathematics.25 Mathematicians at any given time have (at least implicit) views on what mathematics is, which methods to use, which are the interesting problems to solve, and so on. These may be merely views and not fully fledged philosophical theories. In some cases, though, mathematicians have developed philosophical positions on mathematics, and sometimes it can even be argued that these have influenced their mathematics. Many studies of the (philosophical) views of past mathematicians have been made. Examples include Bolzano [Rusnock, 2000], Hausdorff [Epple, 2006], Pasch [Schlimm, 2010], and Riemann (e.g., [Scholz, 1982; Bottazzin and Tazziolii, 1995]). It is also possible to name a few contemporary mathematicians who have expressed their views in public, such as Reuben Hersh and Alan Connes ([Changeux and Connes, 1995; Connes et al., 2001]; see also the paper by Kanovei, Katz and Mormann [Kanovei et al., 2013] for an interesting evaluation of Connes’ views). As for the question of whether philosophical views influence the mathematics produced, to my knowledge there have not been any systematic studies. There are the obvious examples of constructivists (of various sorts) developing mathematics along constructivist guidelines, like Brouwer [van Atten, 2004], Weyl [1918], and Bishop and Bridges [1985]. 3.2.3. Philosophical Categories of Historical Studies Along the lines of Lakatos’s second dogma, ‘the history of mathematics, lacking the guidance of philosophy, has become blind’, there are also scholars developing what could be denoted philosophical tools in order to understand the history of mathematics. One such example is the adaption by Moritz Epple of categories developed by Rheinberger [1997]. Epple [2004] employs these tools in order to delve deeper into the history of the introduction of the first invariants of knots, which were introduced in the 1920s independently by Alexander at Princeton and Reidemeister in Vienna. A surface, or anachronistic, reading of the first part of their two papers would make it seem like their work led to the same techniques — a belief that has also previously been accepted. Both Alexander and Reidemeister represent knots by a knot diagram.26 In both cases the presentation of the knot gives rise to a matrix with integer entries and certain invariants in the form of integers. By instead using the tools of epistemic configurations, techniques, and objects, Epple is able to show that their ideas were indeed quite different. Epistemic configurations bear some resemblance to Kuhn’s paradigms and are characterised as: the entirety of the intellectual resources that are involved in a particular episode. It comprises the mathematical language, the skills and techniques at the disposal of the mathematician or the group of mathematicians engaged in this research, the set of research topics and open problems under consideration, the horizon of aims and more general heuristic guidelines followed by the researchers, etc. |$\dots$| An epistemic configuration of mathematical research, together with the mathematician(s) working in and with them, thus constitutes a (usually rather small) working unit for the production of mathematical knowledge. [Epple, 2004, p. 148] By taking us further into Alexander’s and Reidemeister’s articles Epple demonstrates that they work within different epistemic configurations. They both use Poincaré’s topology as a background theory. But whereas Alexander treats the knots as Riemann spreads or Riemann surfaces, Reidemeister considers open unbranched coverings of knots or link complements. That is, their objects of investigation, referred to as the epistemic objects, are different. As they have different epistemic objects they employ different techniques to arrive at the particular matrix: Reidemeister uses combinatorial algebra and Alexander linear systems and homology.27 An epistemic configuration can be seen as a characterisation of ‘a practice’. We shall therefore return to comment on this in the next section when discussing the possibility of capturing a common notion of ‘practice’. Summing up, in Historical PMP, ‘practice’ refers in particular to the various ways that mathematics develops. In this strand we have seen examples of how philosophy and history support each other. For example, philosophy contributes with categories to understand history better [Epple, 2004] or philosophical questions may be answered by referring to historical case studies. Historical cases may, for example, help us understand the mathematical and philosophical background of the “crisis” that led to the formulations of the foundational schools as well as the development of structuralist positions in the philosophy of mathematics. 3.3. Epistemological PMP For lack of a better term, I denote the last strand Epistemological PMP. The qualifier ‘epistemological’ is intended to indicate that this strand is in some respects close to traditional philosophy. It tries to avoid assumptions, e.g., that talking about mathematics requires reference to human agents (although this is not precluded). In addition it still wishes to answer traditional questions within the philosophy of mathematics, such as what is the nature of mathematical objects and what constitutes our knowledge of them. It does not, however, entail a traditional way of viewing epistemology in the sense of “a view from nowhere” and the possibility of obtaining absolute knowledge. Furthermore complementing traditional philosophy of mathematics, philosophers in this strand ask for an extension of topics that can be studied under the heading of philosophy of mathematics (so another name could be ‘Extension-of-topics-studied’ PMP). It is noted (e.g., by David Corfield [2003]) that when considering the actual practice of doing mathematics there is a range of other questions that can be asked and are being asked by contemporary mathematicians. In contrast to the traditional methodology there is a stress on ‘mathematical practice’, and case studies are involved when answering questions. ‘Practice’ in this strand is probably best understood in a more abstract way than in the other two, since in principle it could refer to anything that has to do with (real) mathematics. What is important is the aim when taking this perspective, implying a shift of ‘focus as to what is the subject under study, “mathematics” as it manifests itself, not some idealised version or something based on our prejudices’ [Panza, 2003]. In epistemological PMP it is thus recognised that the aim of investigation is to obtain a better understanding of the mathematics we have access to. Characterising the phenomenological approach to the philosophy of mathematics, Mary Leng writes: The motivation for the so-called ‘phenomenological’ approach to the philosophy of mathematics is thus the idea that philosophers should be seriously concerned with ‘real’ mathematics and not some idealized notion of the discipline. [Leng, 2002, p. 3] — and so yet other headings capturing some contributions in this strand could be ‘phenomenological PMP’ or ‘philosophy of real mathematics’.28 Taking this perspective does not from the outset exclude the possibility of obtaining answers concerning the “true nature of mathematics”, although it can be — and indeed has been — debated whether this is possible. Leng finds that according to the phenomenological approach if you want to understand what mathematics is you must first find out what it is that mathematicians do. Conclusions may later be drawn about ontology and epistemology, but that is a further step. [Leng, 2002, p. 5] Based on a case study on the classification of |$C^*$|-algebras she does draw conclusions about the epistemology and ontology of mathematics. To classify |$C^*$|-algebras means to determine up to isomorphism the different types of algebras that are possible to define. This is done by defining invariants, usually “simpler” objects so that any two algebras are isomorphic if and only if their corresponding invariants are isomorphic. The mathematician Georg Elliott has initiated a program where the hope is that |$K$|-theory will contribute to obtaining such invariants. As part of her case study Leng followed two research seminars conducted by Elliott. One purpose of her study was to test Lakatos’s claim that the formulation of a proposition is developed as a response to found counterexamples. The process observed in the seminars did not support this picture. In these Elliott started by formulating and proving a weak version of the theorem in question, gradually strengthening the formulation whenever it was possible to find a proof of it. That is, whereas it still makes sense to say that the theorem is proof generated, it did not develop by formulating a series of counterexamples. Towards the end of her paper Leng argues that an anti-realist picture emerges from the case study. She finds that the mathematicians do not care whether the objects referred to exist or not when accepting a statement as true. What matters is that it is possible to find a convincing proof. In contrast Brendan Larvor [2001] claims that from the perspective of a ‘dialectical philosopher of mathematics’ it is not possible to reach a conclusion on the ontology of mathematics. The ‘dialectical philosopher of mathematics’ is inspired by Lakatos’s writings. Larvor comments that Lakatos’s most important contribution was to turn our attention to the inner life of mathematics and in this context writes that: Whether we adopt fictionalism; or embrace a kind of emergentism in which the mathematics produces itself out of the activities of the mathematicians; or whether we think of progress as ever-closer approximation to a pre-existing Platonic reality, makes no difference to our study of the inner logic of mathematical development. The dialectical stories turn out the same regardless of any ontological commitment. [Larvor, 2001, p. 218] Others, like Oliveri [2010], find that the history of mathematics should be brought in in order to say something about the philosophy of mathematics, but stress that this is only justified if it contributes to the solution of philosophical problems that remain unsolvable by previous methods. Finally there are philosophers who find that the only way to do philosophy of mathematics is to start from mathematics itself, taking a bottom-up approach. See, for example, [Cellucci, 2012] for a critical attitude towards ‘top-down approaches’. Given this lack of consensus, and especially the lack of convincing arguments for either side, the question concerning the range of conclusions possible from, for example, the use of case studies, is something that should be carefully assessed from case to case. In what follows a few more exemplary cases will be presented illustrating the range of practices dealt with in this strand. For clarity they are divided into epistemology, ontology and foundations. 3.3.1. Epistemology in the Epistemological PMP The prime example of work asking questions related to the epistemology of mathematics is [Mancosu, 2008]. In his introduction, Mancosu notes that a number of different epistemological issues can be addressed besides the traditional question of how knowledge of acausal entities is possible (referring to [Benacerraf, 1973]). Topics he proposes to study, orthogonal to those of traditional philosophy, include ‘fruitfulness, evidence, visualization, diagrammatic reasoning, understanding, explanation’ [2008, p. 1]. Among these, the topics of diagrammatic reasoning and the role of visualisation have received much attention. They serve as a good example for illustrating how a topic has been treated from various perspectives. At first philosophers were mainly concerned with questions related to whether reasoning using diagrams is rigorous. Referring to Pasch and Hilbert, scholars claimed that figures should play no justificatory role in proofs. More recently a number of scholars have defended and explained their use, for example, Ken Manders [2008] and John Mumma [2010] in the context of Euclidean geometry. Jon Barwise and John Etchemendy [1996] have shown how a rigorous axiomatic system including diagrams can be set up. More radically, in a recent article Jody Azzouni writes: ‘Talk of pictorial proofs lacking “rigor” is false’ [2013, p. 325]. Azzouni bases his discussion on a pictorial proof of the identity |$\frac{1}{2} + \frac{1}{4} +\frac{1}{8}+ \ldots +\frac{1}{2^n}+ \ldots=1$| and the (history of the) Intermediate Value Theorem in analysis. Besides a focus on proofs and rigour, attention has been drawn to the many other roles visualisation plays in mathematics. Studies have been made on past mathematicians’ views on the use of diagrams. For example, Ivahn Smadja [2012] notes that Hilbert found that geometric intuition is important in mathematics and that he praised Minkowski’s work in Geometry of Numbers where connections between geometry and number theory are fruitfully exploited to develop both fields. Jeremy Heis tries to locate local reasons for why diagrams were given up, not only in analysis, but even in geometry in the nineteenth century. He finds that in projective geometry mathematicians successfully employed the practice of exploiting ‘special features of the drawn figure’ [Heis, 2012, Section III, p. 38], something that is often used as an argument against diagram use. Furthermore he explains how the practice of reasoning using diagrams breaks down once one considers curves of degree three or higher (the duality principle cannot be upheld), a reason that could be given for the rejection of diagrams in further development of the field. Recent studies by Silvia De Toffoli and Valeria Giardino demonstrate how diagrammatic reasoning is a rigorous and integrated practice in contemporary mathematics, e.g., in low-dimensional topology [De Toffoli and Giardino, 2015]. They illustrate how formal expressions in this area make little sense without accompanying pictures or mental images. Some of their examples illustrate the construction of 3-dimensional manifolds such as the torus. These diagrams show which sides are supposed to be joined, or glued, in order to obtain the desired manifold. Formally gluings can be expressed by certain equivalence relations. But the equivalence relations do not contribute to the topological intuition needed to work in the field. Their role is to provide a formal justification for the manipulations done on diagrams. Further examples include a pictorial proof showing certain constructions to be performed on different presentations on 3-manifolds. Whereas the studies of De Toffoli and Giardino illustrate that diagrams play an epistemic role, other scholars have pointed to the many other roles visualisation plays in mathematics, for example, for understanding and heuristics [Giaquinto, 2005b; Carter, 2010; Starikova, 2010].29 For up-to-date treatments of this topic refer to [Giaquinto, 2015] and [Giardino, 2017]. 3.3.2. Ontology in the Epistemological PMP As regards questions related to the ontology of mathematics, the focus on mathematical practice dictates that answers to the traditional questions be based on studies of actual mathematics rather than on philosophical arguments based on assumptions. There have been some attempts at this, e.g. the above-mentioned study on the development of |$C^*$|-algebra by Leng [2002] and myself (e.g. [Carter, 2004; 2013]). Here I will mention the recent study by Jeremy Avigad and Rebecca Morris [2013; 2016] concerning the introduction of the concept of a character in the proof of Dirichlet’s theorem on the number of prime numbers in an arithmetical progression. More precisely the theorem states that if |$m$| and |$k$| are relatively prime (i.e., have no common factors), then the arithmetical progression |$m,\ m+k,\ m+2k,\ \ldots$| contains infinitely many primes. In addition to directing attention to this nice and thorough work, the intention is to highlight how concrete studies, such as this, of the actual practice of mathematics may lead to new questions and concerns. Avigad and Morris find that although characters play an important role in later proofs, there is no mention of them in Dirichlet’s original proof from 1837. A character — in a modern treatment of the theorem — is a function, or more precisely a group homomorphism, that maps elements of an abelian group to the non-zero complex numbers. Avigad and Morris investigate the development of the proofs leading to the modern formulation and the gradual development of characters. In addition they place the development of characters in a wider context as they regard this study as part of a larger quest to understand the development leading to the modern concept of a function. What makes me characterise this work as belonging also to philosophy, is what I conceive of as their motivation for engaging in this study, namely to understand why the concept of a character — or more broadly a function — was introduced into mathematics. Furthermore the detailed case study enables reflection on what it means for something to be counted as an object in mathematics and the benefits achieved by adopting something as an object rather than treating it via other types of representations.30 Avigad and Morris formulate a list of operations and ways of treating them made possible when considering characters as objects in themselves, such as formulating properties of characters, forming sets of characters and being able to sum over them [2013, p. 21]. As for the benefits gained by considering characters as objects, Avigad and Morris point to an increased conceptual understanding: These innovations play an equally important role in fostering a better understanding of Dirichlet’s proof itself, by highlighting key features of the concepts and objects in question, motivating the steps of the proof, and reducing cognitive burden on the reader by minimizing the amount of information that needs to be kept in mind at each step along the way. [ibid., p. 54] 3.3.3. Logic and Foundations in the Epistemological PMP It has been compellingly argued31 that the motivation of many of the researchers in the early foundational schools came from developments of actual mathematics. Similarly, because of developments within mathematics, other types of foundations have been proposed, e.g., within category theory (followed by a fierce debate between set theorists and category theorists). Recently a new program has been formulated, called ‘Univalent foundations’, which is based on surprising links that have been found between Martin Löf type theory and the mathematical discipline homotopy theory. Furthermore, links found between type theory and computer proof assistants are exploited. The new combination is called ‘Homotopy Type Theory’ (HoTT). In brief, Homotopy Type Theory is an interpretation of type theory in homotopy theory, where types are interpreted as spaces and identity of types is expressible via the notion of homotopies between paths. This bridge gives rise to the univalence axiom. One consequence of this axiom is that it allows the identification of isomorphic structures or objects, something that is often done by mathematicians, but does not follow from the Zermelo-Fraenkel axioms for set theory (ZFC). For further details, see the introduction of the HoTTbook [Univalent Foundations Program, 2013] or the articles by James Ladyman and Stuart Presnell [2015; 2016]. As a second example of enquiries within logic and PMP one could mention the work of Colin McLarty [2010] in determining the strength of the system needed in order to prove Fermat’s last theorem (FLT). Whereas the aforementioned example provides a foundation in its traditional sense, McLarty’s results have practical bearings for mathematicians, in the sense that they show that it might be possible to find proofs of FLT within arithmetic. It is well known that Andrew Wiles’s 1995 proof of FLT is very long and uses techniques that go well beyond what one normally counts as belonging to arithmetic. McLarty has pointed out that Wiles’s proof uses tools that go beyond even ZFC. One assumption is the so-called axiom of Universes (U), formulated by Grothendieck. This axiom is needed in order to provide a foundation for the operations performed, and structures considered, in category theory.32 To found these operations a universe is defined in ZFC33 and a category may then be regarded as an element of some universe. Universes go beyond ZFC: Grothendieck has shown that |$U$| is the set |$V_{\alpha}$| for some strongly inaccessible ordinal |$\alpha$| and since ZFC + U models ZFC, the existence of |$U$| cannot be proved from ZFC. According to McLarty, Grothendieck used universes in cohomology theory to show certain results in number theory needed in order to prove the Weil conjectures. It has been proved that universes are not needed for these proofs, but they make things “simpler”, i.e., provide what McLarty calls ‘conceptual order’ [ibid, p. 361]. In the proof of FLT Wiles refers to results that are obtained by using this machinery. McLarty has shown that it should be possible to rework the proof into one that is only based on finite-order arithmetic. There are even hopes that it might be proved that first-order PA suffices, or even only fragments of PA. If such translations of the proof were actually to be written down, they would be longer than the original. The purpose of using the axiom of universes is to organise; so in addition, such a proof would lack this structure. One implication of these results for the practice of doing mathematics is that one can tell that it is in principle possible to find a proof of FLT using only arithmetic. It is not possible to tell, however, that only long proofs are possible from PA. It has not been excluded that someone might find a short arithmetical proof of FLT. A final interesting study, related to the question of how strong a system is needed in order to prove certain theorems, is the programme of reverse mathematics designed by Stephen Simpson and Harvey Friedman (see e.g. [Simpson, 2009]). John Stillwell [2018] gives an accessible introduction to this programme. Placing it in a broader context, Stillwell traces its roots to the discussions about the status of the parallel postulate in Euclidean plane geometry. The main question of reverse mathematics is: ‘Which set existence axioms are needed to prove the theorems of ordinary, non-set-theoretic mathematics?’ [Simpson, 2009, p. 2]. The programme investigates which subsystems of second-order arithmetic are sufficient to prove a number of specific theorems of mathematics. The results considered are taken from many different branches of mathematics, i.e., logic, analysis, and algebra. Subsystems are constructed by restricting the comprehension axiom schemes, that is, the axioms stating which sets exist in the system. Simpson and Friedman establish that a given subsystem is the weakest possible by reversing the normal proof procedure: in addition to proving that a specific theorem follows from the axioms of a subsystem, they also prove that the theorem implies the axioms of the subsystem (modulo the system in which reasoning takes place). As an example, one could mention that it is possible to prove the existence of an algebraic closure of any countable field from the base system, named ‘recursive comprehension’. But in order to prove that this closure is unique a stronger system is required. Further results from algebra that require even stronger systems include structure theorems about abelian groups such as ‘Any reduced Abelian |$p$|-group has an Ulm resolution’ which is provably weaker than the more general statement ‘Any Abelian |$p$|-group has an Ulm resolution’. This result follows from the theorem that ‘any group is a direct sum of a divisible group and a reduced group’.34 It might already be clear from the above, that a wide range of practice, that is cases, is considered under epistemological PMP: they go from considering examples from contemporary mathematics (⁠|$C^*$|-algebra, proof of Fermat’s theorem) to exploiting and even developing contemporary mathematics as is done in homotopy type theory. Note also the different perspectives chosen in the cases mentioned; whereas McLarty studies the presentation of the proof of FLT, Leng is interested in how proofs are found in analysis. Furthermore historical studies are done for various purposes, e.g., the studies mentioned on Hilbert, projective geometry, and the development of characters. Finally there are also studies of (historical) texts independent of historical concerns as for example, the studies mentioned of Euclid’s Elements. 4. PRACTICE We return to the question of whether it is possible to describe ‘PMP’ through a characterisation of ‘practice’. In the light of the different interpretations of ‘practice’ one is tempted to say that the answer is no. When referring to practice in social and pragmatic PMP, I stressed the practitioners and their activities. In historical PMP, it was noted that the starting point is the product of the activities of mathematicians, but that the emphasis is on the relation between various factors and activities and how they contribute to this product. In addition in historical PMP the temporality of mathematics acts as a central underlying assumption. Finally for epistemological PMP practice is understood in a more abstract way as referring to a methodology in general that takes seriously ‘mathematical practice’ in a rather broad sense. In support of this observation, Van Bendegem [2014] identifies eight different perspectives within the practice turn of mathematics (as held by the mavericks, naturalists, sociologists, cognitive scientists, in education, etc.) that he finds are in opposition to each other. In this section, however, I shall suggest some unifying characterisations. There are a few attempts in the literature to characterise ‘a practice’. Philip Kitcher [1984] and later José Ferreirós [2016] have both described ‘practice’ by drawing on the disciplinary matrices of Thomas Kuhn.35 As noted the characterisation of an epistemic configuration by Moritz Epple could also be taken as a candidate — although his motive for introducing it is different. Kitcher in The Nature of Mathematical Knowledge [1984] breaks with what he refers to as a traditional account of knowledge in a number of ways. First he rejects an ‘apriorist’ account of mathematical knowledge. Second he wishes to take into account that mathematical knowledge depends on a community of knowers and third, he bases his account on examples drawn from the history of mathematics. Kitcher defends the position that to ‘understand the epistemological order of mathematics one must understand the historical order’ [ibid., p. 5]. The main case study that informs the description of knowledge concerns the development of differential and integral calculus from around 1600 until its rigorous treatment by, e.g., Cauchy and Weierstrass during the nineteenth century. One of Kitcher’s points in this story is that the need for rigourisation of mathematics came from mathematics itself and not from ‘apriorist epistemological ideas’ [ibid., p. 246]. One major problem was that most theories depended on an inconsistent handling of the infinitesimals. This problem was solved by the introduction of limits by Cauchy and the real numbers by, for example, Weierstrass during the nineteenth century. Based on these concerns Kitcher defines a practice in terms of a quintuple, |$\langle L, M, S, Q, R\rangle$|⁠. In the tuple |$L$| stands for language, |$M$| for meta-mathematical views, |$S$| for the set of accepted statements, |$Q$| for important questions pursued by the ‘practice’, and |$R$| for accepted reasoning. One aim of his project is to argue that changes in mathematics can be explained as transitions from one practice to another by alterations in one or more of these components. Furthermore he argues that such transitions are ‘rational’ and as such contribute to knowledge. It should be noted that his characterisation of a practice holds a number of assumptions that renders it inappropriate for a general definition.36 The first assumption is that it is a practice without practitioners. Kitcher refers to ‘practices of idealised agents’ and so (real) agents are left out. This in part opposes the view of agent-based PMP. In addition Kitcher wishes to argue for an anti-realist view of mathematics and so he excludes mathematical objects from the tuple. Overall the inclusion of all five components renders the characterisation too fine grained to pass as a general characterisation of ‘a practice’. In addition, such level of detail would make one wonder why some categories are included whereas others, for example ‘objects’, are left out. Ferreirós [2016], discussing various views on ‘practice’, stresses, in opposition to Kitcher, that practice should be taken in the sense of ‘the activities of human agents’ and that knowledge does not make sense without the knowers, i.e., the human agents (p. 21). In order to analyse knowledge he thus considers the pair, framework-agent, where a framework is taken to include something like Kitcher’s four-tuple, |$\langle L,S,Q,R\rangle$|⁠. ‘M’, meta-mathematical views, are excluded from this list. Ferreirós instead places them with the agent. A further important qualification is that Ferreirós holds that it does not make sense to claim that practices are all-encompassing and unique at any given time. He bases his account on the fact that ‘several different levels of practices and knowledge are coexistent’ (p. 25) and that their links and interplay are crucial to mathematical knowledge. In support of these claims Ferreirós draws on a number of historical case studies from the theories of numbers, in particular the real numbers, set theory, and geometry. One major development considered is the transformation of geometry in the hands of Euclid to the theory of the real numbers and the further development leading to contemporary set theory. As in Ferreirós’s framework-agent pair, an epistemic configuration as developed by Epple consists in part of the human agents, in this case the mathematicians developing the results under investigation. In addition it includes the mathematical objects, the epistemic objects, which are not explicitly included in the other two practices. Variations of such characterisations are evidently of value when investigating issues related to the activities of doing mathematics as is manifested by the works of Kitcher, Epple, and Ferreirós. All three include certain assumptions not held by all the strands; so as they stand they are insufficient as a general description of ‘a practice’. But if one makes a generalisation of these, a rough characterisation of ‘a practice’ emerges (one quite similar to Ferreirós’s framework-agent): In general a practice is captured by a tuple consisting of ‘agents’ and ‘mathematics’, which can be written as |$\langle A,M\rangle$|⁠. The component ‘agents’ consists of agents, real or idealised, and their activities and beliefs. ‘Mathematics’ refers to the content of mathematics — theories, theorems, proofs, mathematical models, etc. In general ‘a practice’ consists of relations between agents and what constitutes mathematics. This gives a simple picture of the field characterising some overall structures. In order to capture the richness of each strand this picture should be supplemented with the descriptions given in the previous sections. Agent-based PMP obviously has as a starting point the agent(s) and regards mathematics only in relation to those agents. Historical PMP starts at the level of mathematics and considers the development of a particular part of mathematics through an interrelation between the agents and mathematics. Finally I understand epistemological PMP as starting somewhere in the space set up between the two, with most emphasis on the mathematical case studies (as is seen from the examples given), but with the intent of transcending the mathematical examples and arriving at generalisations, or perhaps even idealisations, based on the considerations of actual practice. In this way philosophical categories, concepts, or even a position may be formed, so that one obtains a relation between philosophy and the practice tuple in the sense that the philosophical considerations are based on mathematical practice.37 I propose also to understand agent-based PMP in the sense that the intended outcome of the investigation is a philosophical position based on the assumption of agents doing mathematics. Similarly historical PMP, as we have seen, formulates philosophical positions held by mathematicians, creates philosophical categories to understand historical cases, tries to understand why certain philosophical questions were asked, and investigates how to understand certain concepts related to mathematics. I therefore propose that PMP intends a relation between philosophy and practice tuple consisting of mathematics and agents. A complement to the abstract characterisation of practice can be found in the preface to a collection of articles assembled under the title From practice to results in logic and mathematics [Giardino et al., 2012]. Léna Soler presents different characterisations of ‘practice’ as follows: (a) Practices as processes, dynamical actions and procedures, contrasted with the propositional products of these practices (theories, experimental facts, results of a mathematical theorem, etc.); (b) Ongoing day-to-day, real time actual science (science as it is really practiced) contrasted with science as it is a posteriori reconstructed by practitioners, notably in their publications; (c) Real science opposed to idealized accounts provided by philosophers — most of the time in a derogatory sense of the term ‘idealized’, that is in the sense implying the allegation to have produced a one-sided and truncated, if not a completely deceitful account of what practitioners indeed do and what the science under scrutiny really is|$\dots$|⁠. [Giardino et al., 2012, Preface, p. 2, emphasis original]. Elaborating on (a) one could note that the development of mathematics can be examined at three levels: macro, meso, and micro [Giardino et al., 2012, Introduction, pp. 7–8]. According to my understanding of these levels, the macro level asks questions regarding what can be said about the development of mathematics generally. At the meso level questions regarding research styles, schools, and in general questions relating particular groups can be asked. Finally at the micro level, one may investigate the development of particular concepts, such as the notion of proof at particular times, validity of diagrammatic reasoning, and such like. Items (a) and (b), focussing on processes or day-to-day science in contrast to the products of these processes, fit well with many of the examples presented in the previous section. For example, Epple’s [2004] study went beyond the presented results of Reidemeister and Alexander, looking instead at their epistemic objects and methods. Leng [2002] participated in research seminars in order to see how theorems are developed. Maddy considered the arguments of set theorists about which principles of set theory to adopt. Kitcher’s [1984] and Ferreirós’s [2016] emphasis on the practices of agents for the development of mathematical knowledge is yet another example. In order to accommodate the mentioning of the human element in the last case one could propose a modification of (a) stressing that the focus is not only on processes but on human beings: (d) Focus on the human agents doing mathematics and the implications that this has for philosophy in opposition to regarding mathematics as independent of human activities and mathematical knowledge without knowers. Item (c) resembles the concerns of the Epistemological PMP. All three strands, however, seem to oppose idealised versions of mathematics as a basis for the philosophy of mathematics. Besides (c) above, and in the light of the common motivation of all strands, I propose the following to capture some of the common threads of PMP: (i) Practice — topics: A demand for an extension of questions asked in the philosophy of mathematics; questions ought to be asked from all ranges of mathematical practice and not simply concern ontology and foundations, and (ii) Practice — methods: The methodologies brought in to study and answer these questions should likewise be extended, as long as they rationally contribute to the questions asked. In addition, when it makes sense, results from other sciences, such as history, cognitive science, and sociology, may be used. The proposal is for philosophy of mathematics to mean philosophy of mathematics, where mathematics is taken in a broad sense. This approach opens up a number of different perspectives as we have seen (dealing with the product, that is, the theories presented in any preferred way, the activities of producing, applying, or learning the product as well as the role of the practitioners in all this). Underlying this proposal is the role of philosophy as our way of asking the most fundamental questions and allowing for these questions to concern all aspects of mathematics. From this perspective the philosophy of mathematical practice could be regarded as a proper subset of the philosophy of mathematics stressing ‘practice’.38 This would mean two things: (1) ‘Mathematics’ in the philosophy of mathematics is to be understood not only in an ideal sense. It should also be possible to deal with questions related to mathematics at different levels and different types of activities. (2) Extension of methodologies brought in to deal with these questions. It is here the real challenge lies, namely to uphold philosophical standards (of reasoning). A careful scrutiny of which conclusions can be made from, e.g., empirical studies and to which “level” of investigation they apply is required. 5. CONCLUSION In conclusion it has been proposed that the PMP starts when philosophical investigations are based on studies or considerations of mathematical practice. Taking a ‘bottom-up’ approach, starting with the many contributions within this field, three different strands have been characterised, the agent-based, the historical, and the epistemological PMP. They differ in various respects, in particular on how ‘practice’ is understood. It has been noted that practice can be characterised in terms of (i) focussing on the practitioners, the human beings engaging in mathematics, (ii) as the mathematics itself conceived of as the product of certain events that can be understood historically, or finally (iii) as simply any relevant aspect of mathematics brought in to investigate whatever question has been proposed to study. In order to state in general what PMP is, a general framework is proposed, that is a relation that holds between philosophy and a general practice consisting of agents and mathematical content. The idea is to capture what is taken to count as ‘practice’ in each strand by varying the stress on the components of this tuple. Some versions of PMP emphasise agents whereas versions of the epistemological PMP focus primarily on the mathematics. Furthermore when referring to the philosophy of mathematical practice, it is proposed that the output of studies of such practices belongs to philosophy. In general it is found that PMP raises a number of new questions that can (fruitfully) be treated within philosophy of mathematics. In the text a number of such questions and studies of these have been presented. It should be noted that PMP means a shift in focus as to what is the subject under study, not idealisations of mathematics, but mathematics in the multiple ways it presents itself to us in our various engagements with it. Furthermore acknowledging that the aim of investigation is to obtain a better understanding of mathematics that we practice — and that there is not one true picture of mathematics. Footnotes †Silvia De Toffoli, José Ferreirós, Valeria Giardino, Marcus Giaquinto, Colin McLarty, Marco Panza, and Jean Paul Van Bendegem read drafts, parts of drafts, or responded to my queries: I thank them all for giving valuable advice. Thanks also to the three anonymous referees for their insightful comments and constructive criticism. 1A number of authors also mention the ‘Kuhnean turn’ in the philosophy of science as inspiration for a similar development in the philosophy of mathematics, at the same time regretting that the turn in mathematics has been very slow to come about (see e.g., [Giardino et al., 2012]). It should be noted that even before the mid-twentieth century there were studies taking the ‘practice’ of mathematics into account. In French philosophy, for example, one could in particular mention the work of Cavaillès and Lautman taking (the history of) mathematics as an important part of philosophy; see [Benis Sinaceur, 2006] and [Lautman, 2011]. 2In ‘Mathematics without foundations’ Hilary Putnam objects that the ‘famous “isms” in the philosophy of mathematics’ (p. 7) do not contribute to a clarification of our conceptions of mathematical truth, objects, or necessity. On the foundations of mathematics he further writes ‘I don’t think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs “foundations”|$\,$|’ [1967, p. 5]. 3The volume New Directions in the Philosophy of Mathematics (1986) contains a number of contributions criticising the dominant ‘foundational perspectives’ in the philosophy of mathematics. The editor, Thomas Tymoczko, writes about the contributors’ motivations that they ‘were frustrated by the inability of traditional philosophical formulations to articulate the actual experience of mathematicians’ [Tymoczko, 1998, p. ix]. 4The reference to Kant is to his Critique of Pure Reason, A51/B51. 5Proposals include, e.g., that simple or convex polyhedra are ‘Eulerian’. 6Aldo Antonelli [2001] points to the fact that mathematics often is used as a test case of metaphysicians’ positions on abstract objects and that mathematical facts rarely enter such investigations. Antonelli instead urges philosophers to engage in ‘Mathematical Philosophy’ (so coined by Russell) about which he says: on the other hand, is respectful of, but not subsidiary to, current mathematical practice. It engages the issues, points out conceptual tensions, and highlights unexpected consequences. Mathematical philosophy positions itself neither above nor below mathematics, but rather on a par with it, taking the role of an equal interlocutor [2001, p. 1]. 7Benacerraf’s ‘What numbers could not be’ [1965] has in part inspired a number of structuralist positions, whereas his ‘Mathematical truth’ [1973], arguing that a causal theory of knowledge is incompatible with platonism, is either used to argue for anti-realism or to offer the challenge for realist positions to account for our knowledge of mathematical statements. 8There are in fact a number of different versions of the indispensability argument, but all are based on the assumption that mathematics is indispensable to the theories of natural science. See [Panza and Sereni, 2013, Chs 6, 7]. 9A few scholars have even taken the failure of the foundational schools to provide mathematics with a secure foundation to support a claim that mathematical knowledge is fallible. See, for example, [Ernest, 1998; Oliveri, 2010; Hersh, 2013]. Mancosu [2008] comments on the lack of success of studies turning their back on foundations and reducing mathematics to a cultural product, that they are unable to account for distinguishing features of mathematics such as its special certainty. He writes that ‘Logically trained philosophers of mathematics and traditional epistemologists and ontologists of mathematics felt that the “mavericks” were throwing away the baby with the bathwater’ [2008, pp. 5–6]. See also [Leng, 2002] for a criticism of fallibilist theories of mathematics. 10See Patrick Peccatte’s FOM posting of September 1998 (https://cs.nyu.edu/pipermail/fom/1998-September/002097.html) for an elaboration of this point. 11See also [Kjeldsen and Carter, 2015]. 12Scholars contributing to at least two of these strands include Andy Arana, José Ferreirós, Valeria Giardino, Colin McLarty, Marco Panza, Dirk Schlimm, and others mentioned in the text. 13Ernest [1996] characterises ‘absolutist views’ as being ‘concerned with the epistemological project of providing rigorous systems to warrant mathematical knowledge absolutely.’ 14It has been argued that philosophy plays a key role when teaching and learning mathematics. For one thing, the philosophical beliefs of both teacher and student influence teaching/learning processes (see [Lerman, 1990; Prediger, 2007]). 15See http://www.lib.uni-bonn.de/PhiMSAMP/. 16Even when it is accepted that input could come from mathematics there is disagreement concerning which areas of mathematics are relevant to study in order to say what mathematics is. It could be debated whether the study of arithmetic suffices or whether one should also include other areas of mathematics. For discussions concerning this issue, see [Corfield, 2003, Introduction] and [Tappenden, 2008]. 17[Inglis and Aberdein, 2015] is another exemplary study employing empirical data in a philosophical context. Inglis and Aberdein have systematically studied how mathematicians employ different terms such as ‘beauty’ and ‘simplicity’ to characterise proofs. 18Note that adopting AC (or any other axiom for that matter) because it has consequences one is not willing to give up is only acceptable if it is consistent with, and independent of, one’s underlying foundation. 19The author’s translation of the sentence: Vielmehr zeigt sich, dass die Begriffsbildungen in der Mathematik beständig durch Anschauung und Erfahrung geleitet werden, so dass in großen und ganzen die Mathematik ein willkürfreies Gebilde darstellt. 20Conversely there are also philosophically oriented scholars in cognitive science, employing philosophical positions when developing theories of how human beings form mathematical concepts. An example is Susan Carey (e.g. [Carey, 2004]) who, inspired by Quine’s position on the formation of conceptual schemes, formulates her bootstrapping account on how natural numbers are acquired. Others refer to Peano’s characterisation of the natural numbers. 21For other themes as well as ways that history and philosophy interact, see the volume edited by Breger and Grosholz [2000]. 22Many facts support this. To mention a few, Frege did his doctorate at Göttingen, where Riemann worked. Many of Frege’s friends and teachers, e.g., Abbe and Schering, were followers of the Riemannian tradition. It is also possible to document that Frege regularly taught courses on Riemannian topics and in several places criticised both Weierstrass and his followers Biermann and Kossak. 23Some of the concepts related to epistemology mentioned in the next section under Epistemological PMP can also be taken in this sense. One could also mention the special issue of Philosophia Mathematica (3) 23 No. 2, (2015) devoted to the notion of ‘mathematical depth’. 24A volume edited by Erich Reck and Georg Schiemer [forthcoming] will uncover the roots of contemporary structuralism. It contains chapters on a number of mathematicians and philosophers from the nineteenth and early twentieth century (e.g., Dedekind, Grassmann, Husserl, and Peirce) holding pre-structuralist or even structuralist views. 25There is a long list of well-known cases where philosophers are mathematicians (or vice versa), as for example, Descartes, Leibniz, Newton, Husserl. 26Their knot diagrams look similar except for different conventions adopted to show which strand is on top when lines cross in the diagram. 27These historiographical tools are also useful in describing how the research focus may change; so, for example, what was previously considered to be an epistemic technique becomes the object under study, i.e., the epistemic object. Tinne Kjeldsen [2009] demonstrates that this was a decisive step in the development of the concept of a convex set by Minkowski. She documents a shift of focus from the development of a geometric method to solve certain problems in number theory to the study of the method itself, finally resulting in the singling out of the concept of a convex body. The shift here is from the designing of a method — the epistemic context — which later itself becomes the object of study, the epistemic object. 28The philosophy of real mathematics is also the title of [Corfield, 2003]. 29The concerns mentioned all belong to epistemology. Scholars have also focused on ontological issues related to diagrams, for example, on the status of diagrams in Euclid [Panza, 2012]. Another topic concerns the cognitive origin of Euclidean geometry; see [Hamami and Mumma, 2013]. 30For example, as expressions. 31 See, e.g., [Gray, 2008] for a historical account of the period from the nineteenth century and into the early twentieth when mathematics developed into what is referred to as ‘modern’ mathematics. 32 In category theory one considers categories, such as the category of Abelian groups with group homomorphisms, a totality that is too big to be regarded as a set. In addition constructions are performed on these and maps, the so-called functors, are defined between categories, and functor categories are considered. 33McLarty describes the universe as follows: an uncountable transitive set |$U$| such that |$\langle U, \in\rangle$| satisfies the ZFC axioms in the nicest way: it contains the powerset of each of its elements, and for any function from an element of |$U$| to |$U$| the range is also an element of |$U$|⁠. [2010, p. 359] 34 The system that is equivalent to the theorem ‘Any countable field has a unique algebraic closure is ‘Weak König’s lemma’ (WKL|$_0$|⁠), which states ‘Any infinite subtree of |$2^{< \mathbb{N}}$| has an infinite path’ [Simpson, 2009, p. 36]. ‘Any reduced Abelian |$p$|-group has an Ulm resolution’ is equivalent to ‘Arithmetical transfinite recursion’ whereas ‘Any group is a direct sum of a divisible group and a reduced group’ requires the strongest existence axiom, |$\Pi_1^1$|CA|$_0$|⁠. The systems (in order of strength) are the ‘recursive comprehension axiom’, RCA|$_0$|⁠, ‘weak König’s lemma’, WKL|$_0$|⁠, ‘Arithmetical comprehension’ ACA|$_0$|⁠, ‘Transfinite Recursion’ ATR|$_0$|⁠, and finally |$\Pi_1^1$|CA|$_0$| denoting ‘|$\Pi_1^1$|-comprehension’. In analysis it is, for example, possible to prove the intermediate value theorem in RCA|$_0$| [Simpson, 2009, pp. 77–91] and the Bolzano-Weierstrass theorem is equivalent to ACA|$_0$| [ibid., p. 107]. 35See also [Van Bendegem and Van Kerkhove, 2004] for an extension of Kitcher’s proposal. They introduce the ‘MathPract-structure’ consisting of a number of components in order to characterise the practice of proving theorems. 36See also the discussion of Kitcher’s contribution in [Ferreirós, 2016, Chs 2, 3]. 37This could be written as |$P:\langle A,M\rangle$|⁠. One of the reviewers proposed an extension of this notation: one could write |$\langle A_M,M\rangle$| stressing that the focus is on mathematical agents doing mathematics; or |$\langle A_P,M\rangle$| for philosophical agents studying mathematics. Philosophers of PMP can then be described as a nested couple: |$\langle A_P, \langle A_M, M\rangle\rangle$|⁠. 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TI - Philosophy of Mathematical Practice — Motivations, Themes and Prospects
JF - Philosophia Mathematica
DO - 10.1093/philmat/nkz002
DA - 2019-02-01
UR - https://www.deepdyve.com/lp/oxford-university-press/philosophy-of-mathematical-practice-motivations-themes-and-prospects-0atqJWkBpt
SP - 1
VL - 27
IS - 1
DP - DeepDyve
ER -