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)

The Hyperuniverse Programme, introduced in Arrigoni and Friedman (2013), fosters the search for new set-theoretic axioms. In this paper, we present the procedure envisaged by the programme to find new axioms and the conceptual framework behind it. The procedure comes in several steps. Intrinsically motivated axioms are those statements which are suggested by the standard concept of set, i.e. the `maximal iterative concept', and the programme identi fies higher-order statements motivated by the maximal iterative concept. The satisfaction of these statements (...) (H-axioms) in countable transitive models, the collection of which constitutes the `hyperuniverse' (H), has remarkable 1st-order consequences, some of which we review in section 5. (shrink)

Gödel’s philosophical conceptions bear striking similarities to Cantor’s. Although there is no conclusive evidence that Gödel deliberately used or adhered to Cantor’s views, one can successfully reconstruct and see his “Cantorianism” at work in many parts of his thought. In this paper, I aim to describe the most prominent conceptual intersections between Cantor’s and Gödel’s thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics.

In this paper, I consider an argument for the claim that any satisfactory epistemology of mathematics will violate core tenets of naturalism, i.e. that mathematics cannot be naturalized. I find little reason for optimism that the argument can be effectively answered.

A comprehensive survey of Martin-Löf's constructive type theory, considerable parts of which have only been presented by Martin-Löf in lecture form or as part of conference talks. Sommaruga surveys the prehistory of type theory and its highly complex development through eight different stages from 1970 to 1995. He also provides a systematic presentation of the latest version of the theory, as offered by Martin-Löf at Leiden University in Fall 1993. This presentation gives a fuller and updated account of the system. (...) Earlier, brief presentations took no account of the issues related to the type-theoretical approach to logic and the foundations of mathematics, while here they are accorded an entire part of the book. Readership: Comprehensive accounts of the history and philosophy of constructive type theory and a considerable amount of related material. Readers need a solid background in standard logic and a first, basic acquaintance with type theory. (shrink)

O desenvolvimento da filosofia acadêmica no Brasil é direcionada, entre vários fatores, pelas investigações dos diversos Grupos de Trabalho (GTs) da Associação Nacional de Pós-Graduação em Filosofia (ANPOF). Esses GTs se dividem de acordo com a temática investigada. O GT de Metafísica Analítica é relativamente novo e ainda tem poucos membros, mas os temas nele trabalhados são variados e todos centrais no debate metafísico contemporâneo internacional. A sua investigação se caracteriza pelo rigor lógico e conceitual com o qual aborda esses (...) tradicionais tópicos da metafísica. Embora os assuntos tratados tenham sido, em sua grande maioria, tópicos abordados e discutidos em reuniões remotas do GT, nem todos os membros do nosso grupo estão aqui representados. Esperamos apresentar seus trabalhos em futuros livros. Por enquanto, queremos apenas divulgar ao público acadêmico algumas de nossas produções, trabalhadas e produzidas exclusivamente para este livro. Segundo a caracterização tradicional, com a qual plenamente concordamos, a metafísica se ocupa com a natureza mais íntima e geral da realidade. Essa metafísica é, por um lado, ambiciosa: não pretende apenas expor a realidade segundo o modo como ela se apresenta para nós, mas como ela é ‘em si mesma’. Por outro lado, ela é modesta: ela sabe da sua dívida para com as outras disciplinas, especialmente lógica, linguagem e epistemologia. Além disso, ela se entende como sempre tentativa, provisória, aberta a críticas e revisões. Neste livro, tratamos temas como as categorias ontológicas fundamentais e as suas relações de dependência e de fundamentalidade, a natureza da composição mereológica, da persistência temporal e do próprio tempo, a natureza dos objetos abstratos e dos mundos possíveis, as naturezas do mundo quântico, da causalidade nômica do mundo físico, e, finalmente, de um dos objetos mais característicos da presença humana no nosso mundo físico, que é o objeto de arte. É claro que essa coletânea está longe de apresentar exaustivamente todos os tópicos da metafísica. Tentamos selecionar e organizar os textos da forma mais fluida possível, de modo que eles possam levar tanto a uma reflexão específica quanto a uma reflexão geral sobre a natureza da realidade. Ainda que tenha uma inerente incompletude, o livro serve à finalidade de levar o/a estudante de graduação e pós-graduação a refletir sobre temas contemporâneos da Metafísica Analítica. Justamente por ter esse objetivo de formentar a pesquisa em metafísica analítica nacional, optamos por traduzir muitos termos do inglês que ainda não tem uma tradução canônica, a fim de evitar anglicismos e contribuir para uma leitura mais natural. Muito mais do que leitores complacentes, esse livro pede por leitores críticos, debatedores corajosos, que estejam dispostos a contribuir e a propor novas teorias e argumentos. Agradecemos a todos os membros do GT de Metafísica Analítica da ANPOF e a todos os autores deste livro, que decidiram apresentar publicamente suas pesquisas à academia; aos membros do Grupo de Pesquisa Investigação Filosófica, que decidiram pela publicação deste livro; à Pró-Reitoria de PósGraduação e Pesquisa da Universidade Federal do Amapá, que o financiou; e ao NEPFIL Online e à Editora da UFPel, que o editaram e publicaram. [Guido Imaguire e Rodrigo Reis Lastra Cid, Organizadores] . (shrink)

What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

Contemporary accounts of logic and language cannot give proper treatments of plural constructions of natural languages. They assume that plural constructions are redundant devices used to abbreviate singular constructions. This paper and its sequel, "The logic and meaning of plurals, II", aim to develop an account of logic and language that acknowledges limitations of singular constructions and recognizes plural constructions as their peers. To do so, the papers present natural accounts of the logic and meaning of plural constructions that result (...) from the view that plural constructions are, by and large, devices for talking about many things (as such). The account of logic presented in the papers surpasses contemporary Fregean accounts in its scope. This extension of the scope of logic results from extending the range of languages that logic can directly relate to. Underlying the view of language that makes room for this is a perspective on reality that locates in the world what plural constructions can relate to. The papers suggest that reflections on plural constructions point to a broader framework for understanding logic, language, and reality that can replace the contemporary Fregean framework as this has replaced its Aristotelian ancestor. (shrink)

I propose a new theory of mereology based on a mereological reflection principle. Reflective mereology has natural fusion principles but also refutes certain principles of classical mereology such as Universal Fusion and Fusion Uniqueness. Moreover, reflective mereology avoids Uzquiano’s cardinality problem–the problem that classical mereology tends to clash with set theory when they both quantify over everything. In particular, assuming large cardinals, I construct a model of reflective mereology and second-order ZFCU with Limitation of Size. In the model, classical mereology (...) holds when the quantifiers are restricted to the urelements. (shrink)

Artykuł dotyczy zagadnienia, w jakim sensie można stosować kategorię wyjaśnienia do interpretacji filozofii matematyki Kurta Gödla. Gödel – jako realista matematyczny – twierdzi bowiem, że w wypadku matematyki mamy do czynienia z niezależnymi od nas faktami. Jednym z owych faktów jest właśnie rozwiązywalność wszystkich dobrze postawionych problemów matematycznych – i ten fakt domaga się wyjaśnienia. Kluczem do zrozumienia stanowiska Gödla jest identyfikacja założeń, na których się opiera: metafizyczny realizm: istnieje uniwersum matematyczne, ma ono charakter obiektywny, niezależny od nas; optymizm epistemologiczny: (...) jesteśmy wyposażeni w wystarczająco dobre środki poznawcze, aby uzyskać wgląd w owo uniwersum. Pojęcie rozwiązania problemu matematycznego Gödel rozumie znacznie szerzej niż jako podanie matematycznego dowodu – chodzi raczej o znalezienie wiarogodnych aksjomatów, prowadzących do rozwiązania. Stawiany w artykule problem analizuję na przykładzie hipotezy kontinuum. (shrink)

The article deals with the question of in which sense the notion of explanation can be applied to Kurt Gödel’s philosophy of mathematics. Gödel, as a mathematical realist, claims that in mathematics we are dealing with facts that have an objective character. One of these facts is the solvability of all well-formulated mathematical problems—and this fact requires a clarification. The assumptions on which Gödel’s position is based are: metaphysical realism: there is a mathematical universe, it is objective and independent of (...) us; epistemological optimism: we are equipped with sufficient cognitive power to gain insight into the universe. Gödel’s concept of a solution to a mathematical problem is much broader than of a mathematical proof—it is rather about finding reliable axioms that lead to a solution of the problem. I analyse the problem presented in the article, taking as an example the continuum hypothesis. (shrink)

In this article the problem of unification of mathematical theories is discussed. We argue, that specific problems arise here, which are quite different than the problems in the case of empirical sciences. In particular, the notion of unification depends on the philosophical standpoint. We give an analysis of the notion of unification from the point of view of formalism, Gödel's platonism and Quine's realism. In particular we show, that the concept of “having the same object of study” should be made (...) precise in the case of mathematical theories. In the appendix we give a working proposal of a certain understanding of this notion. (shrink)

Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...) any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity. (shrink)

The notion of a proposition is central to philosophy. But it is subject to paradoxes. A natural response is a hierarchical account and, ever since Russell proposed his theory of types in 1908, this has been the strategy of choice. But in this paper I raise a problem for such accounts. While this does not seem to have been recognized before, it would seem to render existing such accounts inadequate. The main purpose of the paper, however, is to provide a (...) new hierarchical account that solves the problem. (shrink)

This paper investigates the determinacy of mathematics. We begin by clarifying how we are understanding the notion of determinacy before turning to the questions of whether and how famous independence results bear on issues of determinacy in mathematics. From there, we pose a metasemantic challenge for those who believe that mathematical language is determinate, motivate two important constraints on attempts to meet our challenge, and then use these constraints to develop an argument against determinacy and discuss a particularly popular approach (...) to resolving indeterminacy, before offering some brief closing reflections. We believe our discussion poses a serious challenge for most philosophical theories of mathematics, since it puts considerable pressure on all views that accept a non-trivial amount of determinacy for even basic arithmetic. (shrink)

We explore aspects of an experimental approach to mathematical proof, most notably number crunching, or the verification of subsequent particular cases of universal propositions. Since the rise of the computer age, this technique has indeed conquered practice, although it implies the abandonment of the ideal of absolute certainty. It seems that also in mathematical research, the qualitative criterion of effectiveness, i.e. to reach one’s goals, gets increasingly balanced against the quantitative one of efficiency, i.e. to minimize one’s means/ends ratio. Our (...) story will lead to the consideration of some limit cases, opening up the possibility of proofs of infinite length being surveyed in a finite time. By means of example, this should show that mathematical practice in vital aspects depends upon what the actual world is like. (shrink)

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...) theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Gödel's wish to integrate Cantorian set theory within Leibniz' philosophy, it will not be this famous argument of Leibniz'. (shrink)

We document the influence on programming language semantics of the Platonism/formalism divide in the philosophy of mathematics.

In this paper we discuss a particular example of the passage from the informal, but rigorous description of a concept to the axiomatic formulation of principles holding for the concept; in particular, we look at the principles of continuity and lawlike choice in the theory of lawless sequences. Our discussion also leads to a better understanding of the rôle of the so-called density axiom for lawless sequences.

Many philosophers claim to employ intuitions in their philosophical arguments. Others contest that no such intuitions are used frequently or at all in philosophy. This article suggests and defends a conception of intuitions as part of the philosophical method: intuitions are special types of philosophical assumptions to which we are invited to assent, often as premises in argument, that may serve an independent function in philosophical argument and that are not formed through a purely inferential process. A series of philosophical (...) case studies shows that intuitions in these arguments contain the relevant features. The view has implications for philosophical method, offering a compromise between opponents on the divisive debate of the merits of experimental philosophy: experimental philosophy provides an especially useful role in philosophical assumption analysis. (shrink)

In 1928 Edmund Husserl wrote that “The ideal of the future is essentially that of phenomenologically based (“philosophical”) sciences, in unitary relation to an absolute theory of monads” (“Phenomenology”, Encyclopedia Britannica draft) There are references to phenomenological monadology in various writings of Husserl. Kurt Gödel began to study Husserl’s work in 1959. On the basis of his later discussions with Gödel, Hao Wang tells us that “Gödel’s own main aim in philosophy was to develop metaphysics—specifically, something like the monadology of (...) Leibniz transformed into exact theory—with the help of phenomenology.” (A Logical Journey: From Gödel to Philosophy, p. 166) In the Cartesian Meditations and other works Husserl identifies ‘monads’ (in his sense) with ‘transcendental egos in their full concreteness’. In this paper I explore some prospects for a Gödelian monadology that result from this identification, with reference to texts of Gödel and to aspects of Leibniz’s original monadology. (shrink)

Gödel has argued that we can cultivate the intuition or perception of abstractconcepts in mathematics and logic. Gödel's ideas about the intuition of conceptsare not incidental to his later philosophical thinking but are related to many otherthemes in his work, and especially to his reflections on the incompleteness theorems.I describe how some of Gödel's claims about the intuition of abstract concepts are related to other themes in his philosophy of mathematics. In most of this paper, however,I focus on a central (...) question that has been raised in the literature on Gödel: what kind of account could be given of the intuition of abstract concepts? I sketch an answer to this question that uses some ideas of a philosopher to whom Gödel also turned in this connection: Edmund Husserl. The answer depends on how we understand the conscious directedness toward objects and the meaning of the term abstract in the context of a theory of the intentionality of cognition. (shrink)

Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and several works by (...) Gödel, Cantor, Wittgenstein and Weierstrass. We examine several fallacies of intuition and determine how far our intuitive conjectures are limited by the nature of our sense-experience, and by our capacities for conceptualization. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuitions and how the breadth of this new epistemic perspective can be useful in cases where intuition has traditionally been regarded as out of its depth. (shrink)

The paper is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker andGödel will be discussed and analysed. The aim of the paper is to show the influence of phenomenological ideas on the philosophical conceptions concerning mathematics. We shall start by indicating the attachment of Edmund Husserl to mathematics and by presenting the main points of his philosophy of mathematics. Next, works of two philosophers who attempted to apply Husserl’s phenomenological ideas to the (...) philosophy of mathematics, namely Hermann Weyl and Oskar Becker, will be briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel will be studied. (shrink)

This paper is an exploration of the role of intuition in physics. The ways in which intuition is appealed to in physics are not well understood. To the best of my knowledge, there is no analysis of the different contexts in which we might appeal to intuition in physics, nor is there any analysis of the different potential uses to which intuition might be put. In this paper I look to provide data that goes some way to giving a sense (...) of the different contexts in which intuition is appealed to in physics. As I note in the conclusion, there is still much work to be done but I hope that the work here provides us with a first step in the journey to properly understand the use to which intuitions are put in physics and science more generally. (shrink)

The use of tensed language and the metaphor of set ‘formation’ found in informal descriptions of the iterative conception of set are seldom taken at all seriously. Both are eliminated in the nonmodal stage theories that formalise this account. To avoid the paradoxes, such accounts deny the Maximality thesis, the compelling thesis that any sets can form a set. This paper seeks to save the Maximality thesis by taking the tense more seriously than has been customary (although not literally). A (...) modal stage theory, MST, is developed in a bimodal language, governed by a tenselike logic. Such a language permits a very natural axiomatisation of the iterative conception, which upholds the Maximality thesis. It is argued that the modal approach is consonant with mathematical practice and a plausible metaphysics of sets and shown that MST interprets a natural extension of Zermelo set theory less the axiom of Infinity and, when extended with a further axiom concerning the extent of the hierarchy, interprets Zermelo–Fraenkel set theory. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

By taking serious a remark once made by Paul Bernays, namely that an account of the nature of rationality should begin with concept-formation, this article sets out to uncover both the restrictive and the expansive boundaries of rationality. In order to do this some implications of the perennial philosophical problem of the “coherence of irreducibles” will be related to the acknowledgement of primitive terms and of their indefinability. Some critical remarks will be articulated in connection with an over-estimation of rationality (...) – concerning the influence of Kant's view of human understanding as the formal law-giver of nature (the supposedly “rational structure of the world”), and the apparently innocent (subjectivist) habit to refer to experiential entities as 'objects'. The other side of the coin will be highlighted with reference to those kinds of knowledge transcending the limits of concept-formation – culminating in formulating the four most basic idea-statements philosophy can articulate about the universe. What is found “in-between” these (restrictive) and (expansive) boundaries of rationality will then briefly be placed within the contours of a threefold perspective on the self-insufficiency of logicality – as merely one amongst many more dimensions conditioning human life. Although the meaning of the most basic logical principles – such as the logical principles of identity, non-contradiction and sufficient reason – will surface in our analysis, exploring some of the complex issues in this respect, such as the relationship between thought and language, will not be analysed. The important role of solidarity – as the basis of critique – will be explained and related both to the role of immanent criticism in rational conversation and the importance of acknowledging what is designated as the principle of the excluded antinomy (which in an ontic sense underlies the logical principle of non-contradiction). The last section of our discussion will succinctly illuminate the proper place of the inevitable trust we ought to have in rationality – while implicitly warning against the rationalistic over-estimation of it (its degeneration into a rationalist “faith in reason”). Our intention is to enhance an awareness of the reality that rationality is embedded in and borders on givens which are not open to further “rational” exploration – givens that both condition (in a constitutive sense) and transcend the limits of conceptual knowledge. Some of the distinctions and insights operative in our analysis are explained in Strauss 2000 and 2003. Yet, most of the systematic perspectives found in this analysis of rationality are only developed in this article for the first time. Since a different study is required to discuss related problems and results found within cognitive science, it cannot be discussed within one article. S. Afr. J. Philos. Vol.22(3) 2003: 247–266. (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. 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 naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize (...) in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...) These anchors add enough to the content of mathematical expressions to forestall the radical kind of indeterminacy that model-theoretic arguments are purported to show, while still leaving room for a plausible, moderate kind of indeterminacy. (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 ambigu-ity. Drawing on literature on lexical ambiguity we argue that ‘argument’ is not ambiguous. The term ‘argu-ment’ 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 incompleteness theorems constitute the mathematical core of Gödel’s philosophical challenge. They are given in their “most satisfactory form”, as Gödel saw it, when the formality of theories to which they apply is characterized via Turing machines. These machines codify human mechanical procedures that can be carried out without appealing to higher cognitive capacities. The question naturally arises, whether the theorems justify the claim that the human mind has mathematical abilities that are not shared by any machine. Turing admits that (...) non-mechanical steps of intuition are needed to transcend particular formal theories. Thus, there is a substantive point in comparing Turing’s views with Gödel’s that is expressed by the assertion, “The human mind infinitely surpasses any finite machine”. The parallelisms and tensions between their views are taken as an inspiration for beginning to explore, computationally, the capacities of the human mathematical mind. (shrink)

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

In Grundgesetze, Vol. II, §91, Frege argues that ‘it is applicability alone which elevates arithmetic from a game to the rank of a science’. Many view this as an in nuce statement of the indispensability argument later championed by Quine. Garavaso has questioned this attribution. I argue that even though Frege's applicability argument is not a version of ia, it facilitates acceptance of suitable formulations of ia. The prospects for making the empiricist ia compatible with a rationalist Fregean framework appear (...) thus much less dim than expected. Nonetheless, those arguing for such compatibility eventually face an hardly surmountable dilemma. (shrink)

We consider certain predicative classes with respect to their bearing on set theory, namely on its semantics, and on its ontological power. On the one hand, our predicative classes will turn out to be perfectly suited for establishing a nice hierarchy of metalanguages starting from the usual set theoretical language. On the other hand, these classes will be seen to be fairly inappropriate for the formulation of strong principles of infinity. The motivation for considering this very type of classes is (...) a reasonable philosophy of set theory. Familiarity is assumed only with basic concepts of both set theory and its philosophy. (shrink)

The foregoing considerations have shown that on the Fregean model, no descriptive rendition of the meaning of a word, and no feature of the subject's psychological state, will be sufficient to answer the question of how reference takes place. Reference is determined by an independent semantical object, and the mind is limited by its perceptual access to this external semantical realm. The psychological and epistemic states of the language user will be causally influenced by this perceptual contact, and such causal (...) influences will enable the subject to evince the appropriate behavioral signs of comprehension. But the internal effects produced by the perception of meaning are not in principle sufficient to individuate their semantical causes, and hence some of the prominent criticisms of the traditional theory are simply not applicable to Frege's system. (shrink)

In recent work Bruno Whittle has presented a new challenge to the Cantorian idea that there are different infinite cardinalities. Most challenges of this kind have tended to focus on the status of the axioms of standard set theory; Whittle’s is different in that he focuses on the connection between standard set theory and intuitive concepts related to cardinality. Specifically, Whittle argues we are not in a position to know a principle I call the Quantificational Hume Principle, which connects the (...) application of intuitive, quantificational cardinality concepts to claims involving the existence of functions. This paper responds to Whittle’s skeptical arguments by providing an argument justifying the QHP. (shrink)

The Platonic description of the cognitive development of the human being is a crucial part of his philosophy. This account emphasizes not only the existence of phases of rational growth but also the need that the cognitive progress of the individuals is investigated further. I will reconstruct what rational growth is for Plato in light of the deliberate choice of the philosopher to leave incomplete his schematization of human intellectual development. I will argue that this is a means chosen by (...) Plato to stimulate his readers to use his text as a starting point for an enquiry into the capabilities of human intellection. I will illustrate the presence of two phases in the process of human rational growth, highlighting the different techniques utilized by Plato to promote cognitive development during each of them. When the individuals are in the first phase of intellectual growth, they are rationally stimulated via conventional discursive material transmitted in written form. Human rational progress at the level of sophistication of theoretical adulthood is promoted through mathematics. (shrink)

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of (...) this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections. (shrink)

In this paper, I argue against Penelope Maddy's set-theoretic realism by arguing (1) that it is perfectly consistent with mathematical Platonism to deny that there is a fact of the matter concerning statements which are independent of the axioms of set theory, and that (2) denying this accords further that many contemporary Platonists assert that there is a fact of the matter because they are closet foundationalists, and that their brand of foundationalism is in radical conflict with actual mathematical practice.

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.