Philosophy of Mathematics

In this paper, I argue that a naturalist approach in philosophy of mathematics justifies a pluralist conception of set theory. For the pluralist, there is not a Single Universe, but there is rather a Multiverse, composed by a plurality of universes generated by various set theories. In order to justify a pluralistic approach to sets, I apply the two naturalistic principles developed by Penelope Maddy (cfr. Maddy (1997)), UNIFY and MAXIMIZE, and analyze through them the potential of the set theoretic (...) multiverse to be the best framework for mathematical practice. According to UNIFY, an adequate set theory should be foundational, in the sense that it should allow one to represent all the currently accepted mathematical theories. As for MAXIMIZE, this states that any adequate set theory should be as powerful as possible, allowing one to prove as many results and isomorphisms as possible. In a recent paper, Maddy (2017) has argued that this two principle justify ZFC as the best framework for mathematical practice. I argue that, pace Maddy, these two principles justify a multiverse conception of set theory, more precisely, the generic multiverse with a core (GMH). (shrink)

In this paper, I develop a new interpretation of the order of nature, its function, and its implications in Margaret Cavendish’s philosophy. According to the infinite balance account, the order of nature consists in a balance among the infinite varieties of nature. That is, for Cavendish, nature contains an infinity of different types of matter: infinite species, shapes, and motions. The potential tumult implicated by such a variety, however, is tempered by the counterbalancing of the different kinds and motions of (...) matter against one another, or what Cavendish calls the “poising” of nature’s actions by their opposites. The infinite balance account of order offers insight into a central notion of Cavendish’s system and bears important implications for other interpretive issues. To wit, the account resolves the standing issue of whether there is genuine disorder in Cavendish’s universe and elucidates the nature of her opposition to atomism. Nevertheless, the interpretation faces an epistemic challenge, insofar as Cavendish appears to deny knowledge of infinity. I argue that, for Cavendish, our knowledge of the order of nature is conceptual and non-empirical, revealing limits to her apparent empiricism. (shrink)

In a series of works, Jody Azzouni has defended deflationary nominalism, the view that certain sentences quantifying over mathematical objects are literally true, although such objects do not exist. One alleged attraction of this view is that it avoids various philosophical puzzles about mathematical objects. I argue that this thought is misguided. I first develop an ontologically neutral counterpart of Field’s reliability challenge and argue that deflationary nominalism offers no distinctive answer to it. I then show how this reasoning generalizes (...) to other philosophically problematic entities. The moral is that puzzle avoidance fails to motivate deflationary nominalism. (shrink)

Dummett’s notion of indefinite extensibility is influential but obscure. The notion figures centrally in an alternative Dummettian argument for intuitionistic logic and anti-realism, distinct from his more famous, meaning-theoretic arguments to the same effect. Drawing on ideas from Dummett, a precise analysis of indefinite extensibility is proposed. This analysis is used to reconstruct the poorly understood alternative argument. The plausibility of the resulting argument is assessed.

ABSTRACT Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a (...) key neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter. (shrink)

In a forthcoming paper in this journal, entitled “Bad company objection to Joongol Kim’s adverbial theory of numbers”, Namjoong Kim presents an ingenious Russell-style paradox based on an analogue of Kim’s definition of the number 1, and argues that Kim’s theory needs to provide a criterion of demarcation between acceptable and unacceptable definitions of adverbial entities. This paper addresses this ‘bad company’ objection and some other related issues concerning Kim’s adverbial theory by clarifying the purposes and uses of the formal (...) framework of the theory. (shrink)

This is a contribution to a symposium on Amie Thomasson’s Ontology Made Easy (2015). Thomasson defends two deflationary theses: that philosophical questions about the existence of numbers, tables, properties, and other disputed entities can all easily be answered, and that there is something wrong with prolonged debates about whether such objects exist. I argue that the first thesis (properly understood) does not by itself entail the second. Rather, the case for deflationary metaontology rests largely on a controversial doctrine about the (...) possible meanings of ‘object’. I challenge Thomasson's argument for that doctrine, and I make a positive case for the availability of the contested, unrestricted use of ‘object’. (shrink)

Prominent constructive theories of sets as Martin-Löf type theory and Aczel and Myhill constructive set theory, feature a distinctive form of constructivity: predicativity. This may be phrased as a constructibility requirement for sets, which ought to be finitely specifiable in terms of some uncontroversial initial “objects” and simple operations over them. Predicativity emerged at the beginning of the 20th century as a fundamental component of an influential analysis of the paradoxes by Poincaré and Russell. According to this analysis the paradoxes (...) are caused by a vicious circularity in definitions; adherence to predicativity was therefore proposed as a systematic method for preventing such problematic circularity. In the following, I sketch the origins of predicativity, review the fundamental contributions by Russell and Weyl and look at modern incarnations of this notion. (shrink)

Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predica-tivity have since taken a life of their own, (...) but have also produced a deeper understanding of the notion of predicativity, therefore witnessing the " light logic throws on problems in the foundations of mathematics. " (Feferman 1998, p. vii) Predicativity has been at the center of a considerable part of Feferman's work: over the years he has explored alternative ways of explicating and analyzing this notion and has shown that predicative mathematics extends much further than expected within ordinary mathematics. The aim of this note is to outline the principal features of predicativity, from its original motivations at the start of the past century to its logical analysis in the 1950-60's. The hope is to convey why predicativity is a fascinating subject, which has attracted Feferman's attention over the years. (shrink)

This is a tutorial for the Minlog Proof Assistant, version 5.0.

In 1907 Einstein had the insight that bodies in free fall do not “feel” their own weight. This has been formalized in what is called “the principle of equivalence.” The principle motivated a critical analysis of the Newtonian and special-relativistic concepts of inertia, and it was indispensable to Einstein’s development of his theory of gravitation. A great deal has been written about the principle. Nearly all of this work has focused on the content of the principle and whether it has (...) any content in Einsteinian gravitation, but more remains to be said about its methodological role in the development of the theory. I argue that the principle should be understood as a kind of foundational principle known as a criterion of identity. This work extends and substantiates a recent account of the notion of a criterion of identity by William Demopoulos. Demopoulos argues that the notion can be employed more widely than in the foundations of arithmetic and that we see this in the development of physical theories, in particular space-time theories. This new account forms the basis of a general framework for applying a number of mathematical theories and for distinguishing between applied mathematical theories that are and are not empirically constrained. (shrink)

This book analyses the impact computerization has had on contemporary science and explains the origins, technical nature and epistemological consequences of the current decisive interplay between technology and science: an intertwining of formalism, computation, data acquisition, data and visualization and how these factors have led to the spread of simulation models since the 1950s. -/- Using historical, comparative and interpretative case studies from a range of disciplines, with a particular emphasis on the case of plant studies, the author shows how (...) and why computers, data treatment devices and programming languages have occasioned a gradual but irresistible and massive shift from mathematical models to computer simulations. -/- . (shrink)

O tendinţă relativ nouă în filosofia contemporană a matematicii este reprezentată de nemulţumirea manifestată de un număr din ce în ce mai mare de filosofi faţă de viziunea tradiţională asupra matematicii ca având un statut special ce poate fi surprins doar cu ajutorul unei epistemologii speciale. Această nemulţumire i-a determinat pe mulţi să propună o nouă perspectivă asupra matematicii – una care ia în serios aspecte până acum neglijate de filosofia matematicii, precum latura sociologică, istorică şi empirică a cercetării matematice (...) şi care acordă astfel o atenţie deosebită practicii matematice. (shrink)

I present an argument that for any computer-simulated civilization we design, the mathematical knowledge recorded by that civilization has one of two limitations. It is untrustworthy, or it is weaker than our own mathematical knowledge. This is paradoxical because it seems that nothing prevents us from building in all sorts of advantages for the inhabitants of said simulation.

A way to argue that something plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this (...) paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it. (shrink)

A new trend in the philosophical literature on scientific explanation is that of starting from a case that has been somehow identified as an explanation and then proceed to bringing to light its characteristic features and to constructing an account for the type of explanation it exemplifies. A type of this approach to thinking about explanation – the piecemeal approach, as I will call it – is used, among others, by Lange (2013) and Pincock (2015) in the context of their (...) treatment of the problem of mathematical explanations of physical phenomena. This problem is of central importance in two different recent philosophical disputes: the dispute about the existence on non-causal scientific explanations and the dispute between realists and antirealists in the philosophy of mathematics. My aim in this paper is twofold. I will first argue that Lange (2013) and Pincock (2015) fail to make a significant contribution to these disputes. They fail to contribute to the dispute in the philosophy of mathematics because, in this context, their approach can be seen as question begging. They also fail to contribute to the dispute in the general philosophy of science because, as I will argue, there are important problems with the cases discussed by Lange and Pincock. I will then argue that the source of the problems with these two papers has to do with the fact that the piecemeal approach used to account for mathematical explanation is problematic. (shrink)

The simulation hypothesis proposes that all of reality is an artificial simulation. In this article I describe a simulation model that derives Planck level units as geometrical forms from a virtual (dimensionless) electron formula $f_e$ that is constructed from 2 unit-less mathematical constants; the fine structure constant $\alpha$ and $\Omega$ = 2.00713494... ($f_e = 4\pi^2r^3, r = 2^6 3 \pi^2 \alpha \Omega^5$). The mass, space, time, charge units are embedded in $f_e$ according to these ratio; ${M^9T^{11}/L^{15}} = (AL)^3/T$ (units = (...) 1), giving mass M=1, time T=$2\pi$, length L=$2\pi^2\Omega^2$, ampere A = $(4\pi \Omega)^3/\alpha$. We can thus for example create as much mass M as we wish but with the proviso that we create an equivalent space L and time T to balance the above. The 5 SI units $kg, m, s, A, K$ are derived from a single unit u = sqrt(velocity/mass) that also defines the relationships between the SI units; kg = $u^{15}$, m = $u^{-13}$, s = $u^{-30}$, A = $u^{3}$, $k_B = u^{29}$. To convert MLTA from the above $\alpha, \Omega$ geometries to their respective SI Planck unit numerical values (and thus solve the dimensioned physical constants $G, h, e, c, m_e, k_B$) requires an additional 2 unit-dependent scalars. Results are consistent with CODATA 2014. The rationale for the virtual electron was derived using the sqrt of momentum P and a black-hole electron model as a function of magnetic-monopoles AL (ampere-meters) and time T. (shrink)

I prove that invoking the univalence axiom is equivalent to arguing 'without loss of generality' (WLOG) within Propositional Univalent Foundations (PropUF), the fragment of Univalent Foundations (UF) in which all homotopy types are mere propositions. As a consequence, I argue that practicing mathematicians, in accepting WLOG as a valid form of argument, implicitly accept the univalence axiom and that UF rightly serves as a Foundation for Mathematical Practice. By contrast, ZFC is inconsistent with WLOG as it is applied, and therefore (...) cannot serve as a foundation for practice. (shrink)

Strict finitism is the position that only those natural numbers exist that we can represent in practice. Michael Dummett, in a paper called Wang’s Paradox, famously tried to show that strict finitism is an incoherent position. By using the Sorites paradox, he claimed that certain predicates the strict finitist is committed to are incoherent. More recently, Ofra Magidor objected to Dummett’s claims, arguing that Dummett fails to show the incoherence of strict finitism. In this paper, I shall investigate whether Magidor (...) is successful in preventing Dummett from proving the incoherence of strict finitism. Though not all the counterarguments Magidor presents are successful, she does in the end manages to corner Dummett. There remains a small opportunity for Dummett to insist on the incoherence of strict finitism, but this is a very small opening. The final conclusion of this paper is that Dummett cannot logically prove the incoherence of strict finitism, even though not all hope is lost. (shrink)

An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., the counting routine).

Il y a cent ans paraissaient un ouvrage de logique qui a marqué considérablement les études dans ce domaine tout au long du XXe siècle, soit que l’on en poursuivit le projet, soit, au contraire, que l’on en critiqua la démarche. Les Principia Mathematica de Russell et Whitehead sont donc une œuvre majeure sur laquelle il n’était par inutile de revenir en ce début du XXIe siècle. Certes la question à laquelle ils étaient censés répondre, c'est-à-dire celle d’un fondement rigoureux (...) et solide des mathématiques sur la logique formelle paraît aujourd’hui dépassée. Les travaux de Gödel dans les années trente du siècle précédent ont d’une certaine façon mis fin aux ambitions du formalisme et du logicisme tels qu’ils s’exprimaient dans cet ouvrage. Cependant, de nombreuses autres questions ont été soulevées par ce texte que l’ouvrage ici présenté tente d’élucider. Les articles qui le composent reprennent, entre autres, le débat qui opposa à la démarche de Russell aussi bien l’intuitionnisme que les travaux de Lesniewski. Le travail présenté ici montre combien demeure aujourd’hui vivante la philosophie de la logique en langue française, ce dont on peut se réjouir. (shrink)

لا شك أن مفارقات زينون في الحركة قد تم تناولها – تحليلاً ونقدًا – في كثيرٍ من أدبيات العلم والفلسفة قديمًا وحديثًا، حتى لقد ساد الظن بأن ملف المفارقات قد أغُلق تمامًا، لاسيما بعد أن نجح الحساب التحليلي في التعامل منطقيًا مع صعوبات الأعداد اللامتناهية، لكن الفرض الأساسي لهذا البحث يزعم عكس ذلك؛ أعني أن الملف مازال مفتوحًا وبقوة – خصوصًا على المستوى الرياضي الفيزيائي – وأن إغلاقه النهائي قد لا يتم في المستقبل القريب. من جهة أخرى، إذا كانت فكرة (...) وجود العالم ذاته – مصدر مشكلاتنا الفلسفية – لا معنى لها إلا في إطار نسقٍ من التصورات التي تُصاغ بالضرورة في لغة، ومن ثم يُصبح العالم كيانًا كوّنته أساليبنا اللغوية في وصفه، فإن ما يواجهنا دومًا هو السؤال الكانطي المُعضل: إذا كانت اللغة ليست مرآة عاكسة للواقع الفعلي – أو للشيء في ذاته – وإنما للواقع كما تدركه الذات وتؤسلبه، ألا يعني ذلك أن وجودنا ذاته ووجود العالم مجرد فكرة تقطع الصلة بين العوالم الممكنة والعالم الفعلي؟. (shrink)

CraigWilliam Lane.* * God andObjects – The Coherence of Theism : Aseity.Springer, 2017. ISBN: 978-3-319-55383-2 ; 978-3-319-55384-9. Pp. xv + 540.

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...) mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz (...) and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic. (shrink)

The enhanced mathematical indispensability argument, proposed by Alan Baker (2005), argues that we must commit to mathematical entities, because mathematical entities play an indispensable explanatory role in our best scientific theories. This article clarifies the doctrines that support this argument, namely, the doctrines of naturalism and confirmational holism.

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

This essay uses a mental files theory of singular thought—a theory saying that singular thought about and reference to a particular object requires possession of a mental store of information taken to be about that object—to explain how we could have such thoughts about abstract mathematical objects. After showing why we should want an explanation of this I argue that none of three main contemporary mental files theories of singular thought—acquaintance theory, semantic instrumentalism, and semantic cognitivism—can give it. I argue (...) for two claims intended to advance our understanding of singular thought about mathematical abstracta. First, that the conditions for possession of a file for an abstract mathematical object are the same as the conditions for possessing a file for an object perceived in the past—namely, that the agent retains information about the object. Thus insofar as we are able to have memory-based files for objects perceived in the past, we ought to be able to have files for abstract mathematical objects too. Second, at least one recently articulated condition on a file’s being a device for singular thought—that it be capable of surviving a certain kind of change in the information it contains—can be satisfied by files for abstract mathematical objects. (shrink)

In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.

According to the iterative conception of set, each set is a collection of sets formed prior to it. The notion of priority here plays an essential role in explanations of why contradiction-inducing sets, such as the Russell set, do not exist. Consequently, these explanations are successful only to the extent that a satisfactory priority relation is made out. I argue that attempts to do this have fallen short: understanding priority in a straightforwardly constructivist sense threatens the coherence of the empty (...) set and raises serious epistemological concerns; but the leading realist interpretations---ontological and modal interpretations of priority---are deeply problematic as well. I conclude that the purported explanatory virtues of the iterative conception are, at present, unfounded. (shrink)

I argue that absolutism, the view that absolutely unrestricted quantification is possible, is to blame for both the paradoxes that arise in naive set theory and variants of these paradoxes that arise in plural logic and in semantics. The solution is restrictivism, the view that absolutely unrestricted quantification is not possible. -/- It is generally thought that absolutism is true and that restrictivism is not only false, but inexpressible. As a result, the paradoxes are blamed, not on illicit quantification, but (...) on the logical conception of set which motivates naive set theory. The accepted solution is to replace this with the iterative conception of set. -/- I show that this picture is doubly mistaken. After a close examination of the paradoxes in chapters 2--3, I argue in chapters 4 and 5 that it is possible to rescue naive set theory by restricting quantification over sets and that the resulting restrictivist set theory is expressible. In chapters 6 and 7, I argue that it is the iterative conception of set and the thesis of absolutism that should be rejected. (shrink)