It has been argued in the literature that the deflationists’ thesis about the dispensability of truth as an explanatory notion forces them to adopt a conservative theory of truth. I suggest that the deflationists’ claim that the notion of truth is akin to a logical notion should be taken more seriously. This claim casts some doubts on the adequacy of the conservativity requirement, while it also calls for further investigation to assess its philosophical plausibility.

In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows (...) how optimal principles, among those available, may be selected in a justifiable way. (shrink)

This paper is an attempt to convince anti-realists that their correct intuitions against the metaphysical inflationism derived from some versions of mathematical realism do not force them to embrace non-standard, epistemic approaches to truth and existence. It is also an attempt to convince mathematical realists that they do not need to implement their perfectly sound and judicious intuitions with the anti-intuitive developments that render full-blown mathematical realism into a view which even Gödel considered objectionable. I will argue for the following (...) two theses: that realism, in its standard characterization, is our default position, a position in agreement with our pre-theoretical intuitions and with the results of our best semantic theories, and that most of the metaphysical qualms usually related to it depends on a poor understanding of truth and existence as higher-order concepts. (shrink)

There is a basic division in the philosophy of mathematics between realist, ‘platonist’ theories and anti-realist ‘constructivist’ theories. Platonism explains how mathematical truth is strongly objective, but it does this at the cost of invoking mind-independent mathematical objects. In contrast, constructivism avoids mind-independent mathematical objects, but the cost tends to be a weakened conception of mathematical truth. Neither alternative seems ideal. The purpose of this paper is to show that in the philosophical writings of Henri Poincaré there is a coherent (...) argument for an interesting position between the two traditional poles in the philosophy of mathematics. Relying on a semi-Kantian framework, Poincaré combines an epistemological and metaphysical constructivism with a more realist account of the nature of mathematical truth. (shrink)

This paper examines the role of reason in Shepherd's account of acquiring knowledge of the external world via first principles. Reason is important, but does not have a foundational role. Certain principles enable us to draw the required inferences for acquiring knowledge of the external world. These principles are basic, foundational and, more importantly, self‐evident and thus justified in other ways than by demonstration. Justificatory demonstrations of these principles are neither required, nor possible. By drawing on textual and contextual evidence, (...) I will show that Shepherd should have said that we know the first principles of any science, in general, and that “everything which begins to exist must have a cause”, in particular, via intuition, not via reason. Reasoning about such principles can help their self‐evidence shine through in certain cases; their justification, and our being justified in believing them, does not come from this reasoning, however. (shrink)

This paper examines the problem of extending the programme of mathematical constructivism to applied mathematics. I am not concerned with the question of whether conventional mathematical physics makes essential use of the principle of excluded middle, but rather with the more fundamental question of whether the concept of physical infinity is constructively intelligible. I consider two kinds of physical infinity: a countably infinite constellation of stars and the infinitely divisible space-time continuum. I argue (contrary to Hellman) that these do not. (...) pose any insuperable problem for constructivism, and that constructivism may have a useful new perspective to offer on physics. (shrink)

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)

The notion of implicit commitment has played a prominent role in recent works in logic and philosophy of mathematics. Although implicit commitment is often associated with highly technical studies, it remains an elusive notion. In particular, it is often claimed that the acceptance of a mathematical theory implicitly commits one to the acceptance of a Uniform Reflection Principle for it. However, philosophers agree that a satisfactory analysis of the transition from a theory to its reflection principle is still lacking. We (...) provide an axiomatization of the minimal commitments implicit in the acceptance of a mathematical theory. The theory entails that the Uniform Reflection Principle is part of one’s implicit commitments, and sheds light on why this is so. We argue that the theory has significant epistemological consequences in that it explains how justified belief in the axioms of a theory can be preserved to the corresponding reflection principle. The theory also improves on recent analyses of implicit commitment based on truth or epistemic notions. (shrink)

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide (...) variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number of (...) themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...) with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

In ethical discourse, it is common practice to distinguish between normative commitments and descriptive commitments. Normative commitments reflect what a person ought to be committed to, whereas descriptive commitments reflect what a person actually is committed to. While the normative/descriptive distinction is widely accepted as a way of talking about ethical commitments, philosophers have missed this distinction in discussing ontological commitments. In this paper, I distinguish between descriptive ontological commitments and normative ontological commitments and discuss several significant benefits of recognizing (...) this distinction. I argue that just as the normative/descriptive distinction is important for fruitful ethical discourse, so too is it important for fruitful discourse concerning our ontological commitments. And, it constitutes a significant step towards resolving some prominent debates concerning our ontological commitments. (shrink)

The aim of this paper is to challenge Hao Wang's presentation of Gödel's views on the language of mathematics. Hao Wang claimed that the language of mathematics is for Gödel nothing but a sensory tool that helps humans to focus their attention on some abstract objects. According to an alternative interpretation presented here, Gödel believed that the language of mathematics has an important role in acquiring knowledge of the abstract mathematical world. One possible explanation of that role is proposed.

Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize (...) almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments. (shrink)

Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a number of (...) other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

This paper examines the interaction between the philosophy and psychology of concepts and the modal characterization of the deductive concept of logical validity. The concept of logical consequence on which I focus is model-theoretic, where the concept records the property of necessary truth-preservation from the premise of an argument to its conclusion, as well as the condition that, in the class of all possible worlds in which a premise is true, a consequent formula or succedent class of formulas is true, (...) as well. Focusing on the case of logical necessity, I argue that the ability to account for the modal properties of concepts places a desideratum on the explanatory adequacy of theories of mental representation. I outline, then, five approaches to the nature of concepts pursued in philosophy and cognitive science, and argue that the above desideratum does not appear to be satisfiable by the candidate proposals. Given the limits of the foregoing, I appeal to a sixth approach -- namely, cognitivism about epistemic modality, according to which epistemic intensions are semantically imbued functions in the language of thought -- and endeavor to demonstrate how the proposal can sufficiently account for the modal profile of the concept of logical consequence. (shrink)

This paper aims to vindicate the thesis that cognitive computational properties are abstract objects implemented in physical systems. I avail of the equivalence relations countenanced in Homotopy Type Theory, in order to specify an abstraction principle for epistemic intensions. The homotopic abstraction principle for epistemic intensions provides an epistemic conduit into our knowledge of intensions as abstract objects. I examine, then, how intensional functions in Epistemic Modal Algebra are deployed as core models in the philosophy of mind, Bayesian perceptual psychology, (...) and the program of natural language semantics in linguistics, and I argue that this provides abductive support for the truth of homotopic abstraction. Epistemic modality can thus be shown to be both a compelling and a materially adequate candidate for the fundamental structure of mental representational states, comprising a fragment of the Language of Thought. (shrink)

Mark Balaguer ha elaborado una peculiar variante del platonismo matemático –denominada ‘full-blooded platonism’ o ‘FBP’– para solucionar el problema de Benacerraf sobre la inaccesibilidad de las entidades abstractas. Según FBP, todos los objetos matemáticos consistentemente caracterizables existen, aunque de modo contingente. En este trabajo quisiera mostrar que la plenitud ontológica y la contingencia modal no pueden converger en una teoría de objetos matemáticos filosóficamente respetable. Para esto argumento que FBP no cubre algunos factores elementales de confiabilidad epistémica y que envuelve (...) un criterio de plenitud tanto matemática como metafísicamente implausible. (shrink)

"The definitive clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but rather for the honour of the human understanding itself. The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have ; but also the infinite, more than other notion, is in need of clarification." (David Hilbert 1925).

Four requirements are suggested for an axiomatic system S to provide the foundations of category theory: (R1) S should allow us to construct the category of all structures of a given kind (without restriction), such as the category of all groups and the category of all categories; (R2) It should also allow us to construct the category of all functors between any two given categories including the ones constructed under (R1); (R3) In addition, S should allow us to establish the (...) existence of the usual basic mathematical structures and carry out the usual set-theoretical operations; and (R4) S should be shown to be consistent relative to currently accepted systems of set theory. This paper explains how all but parts of (R3) can be met using a system S extending NFU enriched by a stratified pairing operation; to meet more of (R3) a stronger system S∗ is introduced, but there are still some real obstacles to meeting this requirement in full. For (R4) it is sketched how both S and S∗ are shown to be consistent. (shrink)

Quine rejects intensional Platonism and, with it, also rejects attributes (properties) as designations of predicates. He pragmatically accepts extensional Platonism, but conceives of classes as merely auxiliary entities needed to express some laws of set theory. At the elementary logical level, Quine develops an “ontologically innocent” logic of predicates. What in standard quantification theory is the work of variables is in the logic of predicates the work of a few functors that operate on predicates themselves: variables are eliminated. This “predicate (...) functor logic” may be conceived as a peculiar sort of Platonism - ontologically neutral, reduced to schematized linguistic forms. (shrink)

This paper analyzes some examples of diagrammatic proofs in elementary mathematics. It suggests that the cognitive features that allow us to understand such proofs are extensions of the cognitive features that allow us to navigate the physical world.

In the year 1900, the German mathematician David Hilbert gave a dramatic address in Paris, at the meeting of the 2nd International Congress of Mathematicians—an address which was to have lasting fame and importance. Hilbert was at that point a rapidly rising star, if not superstar, in mathematics, and before long he was to be ranked with Henri Poincar´.

The notion of strength has featured prominently in recent debates about abductivism in the epistemology of logic. Following Williamson and Russell, we distinguish between logical and scientific strength and discuss the limits of the characterizations they employ. We then suggest understanding logical strength in terms of interpretability strength and scientific strength as a special case of logical strength. We present applications of the resulting notions to comparisons between logics in the traditional sense and mathematical theories.

Galileo's paradox of infinity involves comparing the set of natural numbers, N, and the set of squares, {n2 : n ∈ N}. Galileo sets up a one-to-one correspondence between these sets; on this basis, the number of the elements of N is considered to be equal to the number of the elements of {n2 : n ∈ N}. It also characterizes the set of squares as smaller than the set of natural numbers, since ``there are many more numbers than squares". (...) As a result, it concludes that infinities cannot be compared in terms of greater--lesser and the law of trichotomy does not apply to them. Cantor's cardinal numbers provide a measure for sets. Cantor gives a definition of the relation greater–lesser between cardinal numbers and establishes the law of trichotomy for these numbers. Yet, when Cantor's theory is applied to subsets of N, it gives that any set can be either finite or of the power ℵ0. Thus, although the set of squares is the subset of N, they are of the same cardinality. Benci, Di Nasso introduces specific numbers to measure sets called numerosities. With numerosities, the following claim is true: numerosity of A < numerosity of B, whenever A ⊈ B. In this paper, we present a simplified version of the theory of numerosities that applies to subsets of N. This theory complies with Galileo's presupposition that when A ⊈ B, then the number of elements in A is smaller than the number of elements in B. Specifically, we show that as the numerosity of N is the number α, the numerosity of the set of squares is the integer part of the number √α, that is ⌊√α⌋, and the inequality ⌊√α⌋ < α holds. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)

One of the important challenges in the philosophy of mathematics is to account for the semantics of sentences that express mathematical propositions while simultaneously explaining our access to their contents. This is Benacerraf’s Dilemma. In this dissertation, I argue that cognitive science furnishes new tools by means of which we can make progress on this problem. The foundation of the solution, I argue, must be an ontologically realist, albeit non-platonist, conception of mathematical reality. The semantic portion of the problem can (...) be addressed by accepting a Chomskyan conception of natural languages and a matching internalist, mentalist and nativist view of semantics. A helpful perspective on the epistemic aspect of the puzzle can be gained by translating Kurt G ̈odel’s neo-Kantian conception of the nature of mathematics and its objects into modern, cognitive terms. (shrink)

In recent discussions concerning the definition of argument, it has been maintained that the word ‘argument’ exhibits the process-product ambiguity, or an act/object ambi-guity. Drawing on literature on lexical ambiguity we argue that ‘argument’ is not ambiguous. The term ‘argument’ refers to an object, not to a speech act. We also examine some of the important implications of our argument by considering the question: what sort of abstract objects are arguments?

The aim of the present paper is to offer a new analysis of the multifarious relations between mathematics and reality. We believe that the relation of mathematics to reality is, just like in the case of the natural sciences, mediated by instruments . Therefore the kind of realism we aim to develop for mathematics can be called instrumental realism. It is a kind of realism, because it is based on the thesis, that mathematics describes certain patterns of reality. And it (...) is instrumental realism, because it pays atten-tion to the role of instruments by means of which mathematics identifies these patterns. The article concludes by offering solutions to some famous semantic paradoxes based on the diagonal construction as corroboration for this claim. (shrink)

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)

In this paper I discuss some issues concerning the semantics of set theory. The set-theoretical nature of the semantics of axiomatic set theory raises a problem of circularity. It is well-known that when we adopt the model-theoretic point of view in the study of mathematical theories we decide to consider primarily structures in their relationship with languages. But for the fundamental structure adopted in a set-theoretic setting, namely the collection of all sets, together with the relation of membership, we would (...) have in that case a structure whose universe would be a set, and at the same time it would have the power of the collection of all sets. This is precisely the origin of the fundamental problem in the semantics of set theory. Apparently, the recourse to an intuitive or pre-theoretic semantics in terms of the iterative conception of sets is unavoidable. However, this strategy doesn´t seem to solve some basic philosophical topics involved and linked to the central semantic problem of set theory, namely, how to understand quantification over the totality of sets. Finally, I put forward some meta-philosophical considerations concerning the semantic problem I deal with. (shrink)

I am a realist of a metaphysical stripe. I believe in an immense realm of "modal" and "abstract" entities, of entities that are neither part of, nor stand in any causal relation to, the actual, concrete world. For starters: I believe in possible worlds and individuals; in propositions, properties, and relations (both abundantly and sparsely conceived); in mathematical objects and structures; and in sets (or classes) of whatever I believe in. Call these sorts of entity, and the reality they comprise, (...) metaphysical. In contrast, call the actual, concrete entities, and the reality they comprise, physical. Physical and metaphysical reality together comprise all that there is. In this paper, it is not my aim to defend realism about any particular metaphysical sort of entity. Rather, I ask quite generally whether and how any brand of realism about metaphysical sorts of entity could be justified? (shrink)

State of the art paper on the topic realism/anti-realism. The first part of the paper elucidates the notions of existence and independence of the metaphysical characterization of the realism/anti-realism dispute. The second part of the paper presents a critical taxonomy of the most important positions and doctrines in the contemporary literature on the domains of science and mathematics: scientific realism, scientific anti-realism, constructive empiricism, structural realism, mathematical Platonism, mathematical indispensability, mathematical empiricism, intuitionism, mathematical fictionalism and second philosophy.

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working (...) mathematician. (shrink)

L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes (...) it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense. (shrink)