An analysis of the counter-intuitive properties of infinity as understood differently in mathematics, classical physics and quantum physics allows the consideration of various paradoxes under a new light (e.g. Zeno’s dichotomy, Torricelli’s trumpet, and the weirdness of quantum physics). It provides strong support for the reality of abstractness and mathematical Platonism, and a plausible reason why there is something rather than nothing in the concrete universe. The conclusions are far reaching for science and philosophy.

Glaucon in Plato's _Republic_ fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We (...) relate his account briefly to mathematical developments by Plato's associates Theaetetus and Eudoxus, and then to the past 200 years' developments in geometry. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. Dummett (...) draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for...

Mathematical platonism is the view that abstract mathematical objects exist. Ontological pluralism is the view that there are many modes of existence. This paper examines the prospects for plural platonism, the view that results from combining mathematical platonism and ontological pluralism. I will argue that some forms of platonism are in harmony with ontological pluralism, while other forms of platonism are in tension with it. This shows that there are some interesting connections (...) between the platonism–antiplatonism dispute and recent debates over ontological pluralism. (shrink)

In "Plato's Ghost" Jeremy Gray presented many connections between mathematical practices in the nineteenth century and the rise of mathematical platonism in the context of more general developments, which he refers to as modernism. In this paper, I take up this theme and present a condensed discussion of some arguments put forward in favor of and against the view of mathematical platonism. In particular, I highlight some pressures that arose in the work of Frege, Cantor, (...) and Gödel, which support adopting a platonist position. The aim of this discussion is to provide an historically informed introduction to the philosophical position of mathematical platonism and to point at some of its mathematical and philosophical roots in the nineteenth century. (shrink)

Many mathematicians are platonists: they believe that the axioms of mathematics are true because they express the structure of a nonspatiotemporal, mind independent, realm. But platonism is plagued by a philosophical worry: it is unclear how we could have knowledge of an abstract, realm, unclear how nonspatiotemporal objects could causally affect our spatiotemporal cognitive faculties. Here I aim to make room in our metaphysical picture of the world for the causal relevance of abstracta.

Are numbers and other mathematical objects "out there" in some philosophically meaningful sense?

Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a (...) deflationary one, there surprisingly turns out to be logical space for a response to the improved challenge where no such space appeared to exist. (shrink)

The present dissertation includes three chapters: chapter one 'Challenges to platonism'; chapter two 'counterparts of non-mathematical statements'; chapter three 'Nominalizing platonistic accounts of the predictive success of mathematics'. The purpose of the dissertation is to articulate a fundamental problem in the philosophy of mathematics and explore certain solutions to this problem. The central problematic is that platonistic mathematics is involved in the explanation and prediction of physical phenomena and hence its role in such explanations gives us good reason (...) to believe that platonism is true. On the other hand, we have grounds, essentially epistemological, for avoiding the use of platonistic mathematics. ;This philosophical tension can be relieved only by showing that good non-platonistic explanations of physical phenomena are available or by showing that the epistemological objections to platonism may be overcome. While both options are discussed in some detail, a certain emphasis is placed on the task of fashioning non-platonistic accounts of physical phenomena to replace the usual platonistic accounts. As is suggested in chapter one, the reason for this emphasis is that the referential and more broadly epistemological objections to platonism seem quite convincing. Of paramount interest in regards to an inquiry into the utility of mathematics is the status of biconditional statements connecting non-mathematical statements with their mathematical counterparts. ;In chapters two and three of the dissertation I address certain key philosophical issues surrounding the status of these biconditionals. The upshot of these chapters is that these biconditionals can be used to illuminate the utility of mathematics; and there is good reason to think an account of the utility of mathematics can be given without acknowledging the truth of mathematics. (shrink)

What is mathematics about? And if it is about some sort of mathematical reality, how can we have access to it? This is the problem raised by Plato, which still today is the subject of lively philosophical disputes. This book traces the history of the problem, from its origins to its contemporary treatment. It discusses the answers given by Aristotle, Proclus and Kant, through Frege's and Russell's versions of logicism, Hilbert's formalism, Gödel's platonism, up to the the current (...) debate on Benacerraf's dilemma and the indispensability argument. Through the considerations of themes in the philosophy of language, ontology, and the philosophy of science, the book aims at offering an historically-informed introduction to the philosophy of mathematics, approached through the lenses of its most fundamental problem. (shrink)

Dan Baras has recently argued for the claim that Theistic Mathematical Platonism fares no better than Mathematical Platonism with respect to explaining why our mathematical beliefs are correlated with mind-independent mathematical truths. In this paper I argue that, insofar as TMP provides a proximate or local explanation for this truth-tracking correlation whereas MP fails to offer any corresponding explanation, Baras’s claim that TMP fares no better than MP with respect to explaining this correlation is (...) false. (shrink)

In his book Wittgenstein on the Foundations of Mathematics, Crispin Wright notes that remarkably little has been done to provide an unpictorial, substantial account of what mathematical platonism comes to. Wright proposes to investigate whether there is not some more substantial doctrine than the familiar images underpinning the platonist view. He begins with the suggestion that the essential platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several different (...) tests for objectivity. The paper finds problems with each of these tests. (shrink)

In his book Wittgenstein on the Foundations of Mathematics, Crispin Wright notes that remarkably little has been done to provide an unpictorial, substantial account of what mathematical platonism comes to. Wright proposes to investigate whether there is not some more substantial doctrine than the familiar images underpinning the platonist view. He begins with the suggestion that the essential platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several different (...) tests for objectivity. The paper finds problems with each of these tests. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant (...) conception of properties endorsed by Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.

A non-classical logic is proposed that extends classical logic and set theory as conservatively as possible with respect to three domains: the logic of natural language, the logcal foundations of mathematics, and the logical-philosophical paradoxes. A universal mechanics of consciousness connects these domains, and its best witness is the liar paradox. Its solution rests formally on a subject-object partition, mentally arising and disappearing perpetually. All deep paradoxes are paradoxes of consciousness. There are two kinds, solvable ones and unsolvable ones. The (...) solvable ones disappear by proper partitions, the unsolvable ones arise by inevitably improper partitions enforced by the natural worldview. The categorial distinctions of subject and object, certainty and truth, consciousness and being, inner freedom and outer constraint, psyche and physis, mental practice and formal theory indicate an immemorial mystical monism of consciousness without which mathematical Platonism could not survive. (shrink)

This book goes far beyond its title. Landry indeed surveys current definitions of “mathematical platonism” to show nothing like them applies to Socrates in Plat.

The demonstration of a loophole-free violation of Bell's inequality by Hensen et al. (2015) leads to the inescapable conclusion that timelessness and abstractness exist alongside space-time. This finding is in full agreement with the triple-aspect monism of reality, with mathematical Platonism, free will and the eventual emergence of a scientific morality.

In our contribution to this special issue on thought experiments and mathematics, we aim to insert theology into the conversation. There is a very long tradition of substantial inquiries into the relationship between theology and mathematics. Platonism has been provoking a consolidation of that tradition to some extent in recent decades. Accordingly, in this paper we look at James R. Brown’s Platonic account of thought experiments. Ultimately, we offer an analysis of some of the merits and perils inherent in (...) framing the use of thought experiments in mathematics and theology in terms of Platonism. (shrink)

Proponents of the explanatory indispensability argument for mathematical platonism maintain that claims about mathematical entities play an essential explanatory role in some of our best scientific explanations. They infer that the existence of mathematical entities is supported by way of inference to the best explanation from empirical phenomena and therefore that there are the same sort of empirical grounds for believing in mathematical entities as there are for believing in concrete unobservables such as quarks. I (...) object that this inference depends on a false view of how abductive considerations mediate the transfer of empirical support. More specifically, I argue that even if inference to the best explanation is cogent, and claims about mathematical entities play an essential explanatory role in some of our best scientific explanations, it doesn’t follow that the empirical phenomena that license those explanations also provide empirical support for the claim that mathematical entities exist. (shrink)

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that (...) we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

My thesis is devoted to an attempt to offer, on behalf of mathematical Platonism, a reply to what may seem to be a powerful objection to it. The objection is this: If there is, as the Platonist supposes, mathematical knowledge of abstract objects, then there is a correlation between our beliefs and the mathematical facts. However, how is such a correlation to be explained given that mathematical objects are a-causal? The worry is that no explanation (...) is possible and that this is a considerable mark against mathematical Platonism. The source of the difficulty here is obvious: standard models for explaining such correlations proceed via the idea that facts cause beliefs, or that beliefs cause facts, or that there was a common cause. But these possibilities are not available to the Platonist given the a-causality of mathematical objects. ;I claim that such a correlation is explicable in a way which neither, contra expectation, renders abstract objects causal nor assigns to us capacities the exercise of which allows to us to explain the correlation on some implausible extension of a perceptual model. The thought is this: that explaining the correlation requires explaining conjunctions of facts and beliefs. But the conjunctions in question are, at least for the Platonism here considered, conjunctions of a brute fact and an explicable fact . However I argue, and this is the core of the thesis, that such conjunctions can always be explained by reiterating the brute fact and citing whatever explains the explicable fact. If this thesis about explanation is accepted, then such correlations are readily explicable. ;The rest of the thesis is devoted to examining various objections to this reply and to the thesis concerning explanation. I argue that they are not fatal. (shrink)

Al final de su libro “La conciencia inexplicada”, Juan Arana señala que la nomología, explicación según las leyes de la naturaleza, requiere de una nomogonía, una consideración del origen de las leyes. Es decir, que el orden que observamos en el mundo natural requiere una instancia previa que ponga ese orden específico. Sabemos que desde la revolución científica la mejor manera de explicar dicha nomología ha sido mediante las matemáticas. Sin embargo, en las últimas décadas se han presentado algunas propuestas (...) basadas en modelos matemáticos que fundamentarían muchos aspectos de la realidad. Dos claros ejemplos provienen de Roger Penrose y Max Tegmark. Esto lleva a pensar en una posición no solo nomológica sino además nomogónica de la matemática. ¿Puede la Naturaleza estar fundada por las matemáticas como señalan algunos físico-matemáticos? Y en ese caso, ¿sería pertinente buscar una nomo-génesis de esta índole en la constitución de la conciencia? -/- At the end of his book “La conciencia inexplicada”, Juan Arana points out that nomology, explanation according to the laws of nature requires a nomogony, an account of the origin of the laws. This means that the order that we can observe in the natural World demands something prior to posit that specific order. Since the scientific revolution we know that the best way to explain that nomology has been through mathematics. Anyway, in recent decades a number of proposals based on mathematical models that found many aspects of reality has been offered. Two clear examples come from Roger Penrose and Max Tegmark. This drives us to think of a position of mathematics as not only nomological but also nomogonical. Can Nature be founded by mathematics as some physicists and mathematicians point out? And, in this case, would be relevant this kind of approach to search a nomo-genesis in the constitution of consciousness? (shrink)

My thesis discusses the unique challenge that platonistic mathematics poses to philosophical naturalism. It has two main parts. ;The first part discusses the three most important approaches to my problem found in the literature: First, W. V. Quine's holistic empiricist defense of mathematical platonism; then, the nominalists' argument that mathematical platonism is naturalistically unacceptable; and finally, a radical form of naturalism, due to John Burgess and Penelope Maddy, which dismisses any philosophical criticism of a successful science (...) such as mathematics. I find faults with all of these approaches. ;The second part attempts to do better. First, I develop an improved epistemological challenge to mathematical platonism. Roughly, this challenge asks for an account of what mathematicians' justification for believing in platonistic mathematics consists in. I argue this challenge is immune to the criticism I leveled against the traditional challenges. To show that it has bite, I apply this challenge to Quine's philosophy of mathematics. I argue this shows that Quine must invoke, in addition to his confirmational holism, a semantic indeterminacy thesis that is so radical as to be implausible. Finally, I attempt to develop a better response to the improved challenge. I defend an unorthodox view of the concept of an object and argue this makes room for a neo-Dedekindian view of mathematical objects which identifies mathematical existence with the logical coherence of the theory describing the structure in question. This view transforms metaphysical and epistemological questions about mathematical objects into corresponding, but more tractable questions about the logical notion of coherence. (shrink)

In his Realism, Mathematics, and Modality, Hartry Field attempted to revitalize the epistemological case against mathematical platontism by challenging mathematical platonists to explain how we could be epistemically reliable with regard to the abstract objects of mathematics. Field suggested that the seeming impossibility of providing such an explanation tends to undermine belief in the existence of abstract mathematical objects regardless of whatever reason we have for believing in their existence. After more than two decades, Field’s explanatory challenge (...) remains among the best available motivations for mathematical nominalism. This paper argues that Field’s explanatory challenge misidentifies the central epistemological problem facing mathematical platonism. Contrary to Field’s suggestion, inexplicability of epistemic reliability does not act as an epistemic defeater. The failure to explain our epistemic reliability with respect to the existence and properties of abstract mathematical objects is simply one aspect of a broader failure to establish that we are epistemically reliable with respect to abstract mathematical objects in the first place. Ultimately, it is this broader failure that is the source of mathematical platonism’s real epistemological problems. (shrink)

The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Math?matiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincar? had already stressed the 'platonistic' orientation of the mathematicians he called 'Cantorian', as opposed to (...) those who (like himself) were 'pragmatist' ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly 'idealism' (which generally designates what we would call 'mathematical realism') and 'empiricism' (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism. (shrink)

The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Mathématiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincaré had already stressed the 'platonistic' orientation of the mathematicians he called'Cantorian', as opposed to those (...) who (like himself) were 'pragmatist' ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly 'idealism' (which generally designates what we would call 'mathematical realism') and 'empiricism' (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism. (shrink)

Mathematical apriorism holds that mathematical truths must be established using a priori processes. Against this, it has been argued that apparently a priori mathematical processes can, under certain circumstances, fail to warrant the beliefs they produce; this shows that these warrants depend on contingent features of the contexts in which they are used. They thus cannot be a priori. -/- In this paper I develop a position that combines a reliabilist version of mathematical apriorism with a (...) platonistic view of mathematical ontology. I argue that this view both withstands the above objection and explains the reliability of a priori mathematical warrant. (shrink)

Mathematics is about numbers, sets, functions, etc. and, according to one prominent view, these are abstract entities lacking causal powers and spatio-temporal location. If this is so, then it is a puzzle how we come to have knowledge of such remote entities. One suggestion is intuition. But `intuition' covers a range of notions. This paper identifies and examines those varieties of intuition which are most likely to play a role in the acquisition of our mathematical knowledge, and argues that (...) none of them, singly or in combination, can plausibly account for knowledge of abstract entities. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. (...) There are mathematical objects. (shrink)

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether (...) class='Hi'>platonism or anti-platonism is true. (shrink)

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. (...) Not only does this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, (...) where virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

Gödel's relation to the work of Plato, Leibniz, Kant, and Husserl is examined, and a new type of platonic rationalism that requires rational intuition, called ...

PurposeThis paper aims to establish that a certain sort of mathematical pluralism is true. MethodsThe paper proceeds by arguing that that the best versions of mathematical Platonism and anti-Platonism both entail the relevant sort of mathematical pluralism. Result and ConclusionThis argument gives us the result that mathematical pluralism is true, and it also gives us the perhaps surprising result that mathematical Platonism and mathematical pluralism are perfectly compatible with one another.

Mathematics tells us there exist infinitely many prime numbers. Nominalist philosophy, introduced by Goodman and Quine, tells us there exist no numbers at all, and so no prime numbers. Nominalists are aware that the assertion of the existence of prime numbers is warranted by the standards of mathematical science; they simply reject scientific standards of warrant.

Mark Balaguer’s project in this book is extremely ambitious; he sets out to defend both platonism and fictionalism about mathematical entities. Moreover, Balaguer argues that at the end of the day, platonism and fictionalism are on an equal footing. Not content to leave the matter there, however, he advances the anti-metaphysical conclusion that there is no fact of the matter about the existence of mathematical objects.1 Despite the ambitious nature of this project, for the most part (...) Balaguer does not shortchange the reader on rigor; all the main theses advanced are argued for at length and with remarkable clarity and cogency. There are, of course, gaps in the account but these should not be allowed to overshadow the sig-. (shrink)

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic (...) test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic. This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference 'Structuralism in physics and mathematics' held in Bristol on 2–3 December, 2006. (shrink)

Mark Balaguer has written a provocative and original book. The book is as ambitious as a work of philosophy of mathematics could be. It defends both of the dominant views concerning the ontology of mathematics, Platonism and Anti-Platonism, and then closes with an argument that there is no fact of the matter which is right.

A perspective in the philosophy of mathematics is developed from a consideration of the strengths and limitations of both logicism and platonism, with an early focus on Frege’s work. Importantly, although many set-theoretic structures may be developed each of which offers limited isomorphism with the system of natural numbers, no one of them may be identified with it. Furthermore, the timeless, ever present nature of mathematical concepts and results itself offers direct access, in the face of a platonist (...) account which generates a supposed problem of access. Crucially too, pure mathematics has its own distinctive method of confirming or validating results - mathematical proof - which supplies a higher level of confidence and objectivity than that available elsewhere. The dichotomy of invention and discovery is too jejune a framework for analysing creative mathematical activity. The Gödelian platonist perspective is evaluated and queried through scrutiny of the part played by mathematical resources and constraints in relation to human activity. It appears that there can be non-causal mathematical explanations and mathematical constraint on purely natural processes. Valuable implications of Quine’s naturalism are explored, but one must be cautious of his thesis of confirmational holism. The distinction between algebraic and non-algebraic mathematical theories usefully contributes to our understanding of the internally differentiated nature of the subject. (shrink)

Benacerraf’s Problem about mathematical truth displays a tension, indeed a seemingly unbridgeable gap, between Platonist foundations for mathematics on the one hand and Hilbert’s ‘finitary standpoint’ on the other. While that standpoint evinces an admirable philosophical unity, it is ultimately an effete rival to Platonism: It leaves mathematical practice untouched, even the highly non-constructive axiom of choice. Brouwer’s intuitionism is a more potent finitist rival, for it engenders significant deviation from standard (classical) mathematics. The essay illustrates three (...) sorts of intuitionistic deviations (weakening, refinement and outright contradiction), and goes on to sketch the technical tools and arguments that engender those clashes with classical mathematics and the philosophical principles underlying those arguments. Those philosophical principles coalesce into a “standpoint” no less unified and no less finitary that Hilbert’s. This intuitionistic standpoint is sufficiently detailed that it itself provides a benchmark for comparing all the rival positions; and it is sufficiently robust that it dissolves the Benacerrafian dichotomy. However, the intuitionistic refutation of the axiom of choice reveals two distinct sub-streams within that intuitionistic standpoint. Distinguishing these streams suggests perhaps that in spite of that robustness, intuitionism has an internal dichotomy parallel to the Benacerrafian split. The Benacerrafian split is a crack in the foundations of mathematics. But I shall argue that this internal intuitionistic dichotomy is not at all a foundational gap; it is rather a special insight about the nature of mathematical thought. (shrink)