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)

Mathematical apriorists usually defend their view by contending that axioms are knowable a priori, and that the rules of inference in mathematics preserve this apriority for derived statements—so that by following the proof of a statement, we can trace the apriority being inherited. The empiricist Philip Kitcher attacked this claim by arguing there is no satisfactory theory that explains how mathematical axioms could be known a priori. I propose that in analyzing Ernest Sosa’s model of intuition as an (...) intellectual virtue, we can construct an “intuition–virtue” that could supply the missing explanation for the apriority of axioms. I first argue that this intuition–virtue qualifies as an a priori warrant according to Kitcher’s account, and then show that it could produce beliefs about mathematical axioms independent of experience. If my argument stands, this paper could provide insight on how virtue epistemology could help defend mathematical apriorism on a larger scale. (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” (...) of hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

Mathematical apriorists sometimes hold that our non-derived mathematical beliefs are warranted by mathematical intuition. Against this, Philip Kitcher has argued that if we had the experience of encountering mathematical experts who insisted that an intuition-produced belief was mistaken, this would undermine that belief. Since this would be a case of experience undermining the warrant provided by intuition, such warrant cannot be a priori.I argue that this leaves untouched a conception of intuition as merely an aspect of (...) our ordinary ability to reason. Thus the apriorist may still hold that some mathematical beliefs are warranted by intuition. (shrink)

This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as (...) precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy. (shrink)

To what extent can constructive mathematics based on intuitionistc logic recover the mathematics needed for spacetime physics? Certain aspects of this important question are examined, both technical and philosophical. On the technical side, order, connectivity, and extremization properties of the continuum are reviewed, and attention is called to certain striking results concerning causal structure in General Relativity Theory, in particular the singularity theorems of Hawking and Penrose. As they stand, these results appear to elude constructivization. On the philosophical side, it (...) is argued that any mentalist-based radical constructivism suffers from a kind of neo-Kantian apriorism. It would be at best a lucky accident if objective spacetime structure mirrored mentalist mathematics. the latter would seem implicitly committed to a Leibnizian relationist view of spacetime, but is it doubtful if implementation of such a view would overcome the objection. As a result, an anti-realist view of physics seems forced on the radical constructivist. (shrink)

In this paper we shall concentrate on the issue of those ways of knowing in mathematics that have traditionally been taken to support apriorism. We shall do it by critizing pragmatic naturalism in the philosophy of mathematics, and in particular its historical approach in denying any role to apriority in mathematical epistemology. The version of pragmatic naturalism we shall be analyzing is Kitcher’s. In the paper we shall first set out a brief survey of the relevant features of (...) Kitcher’s pragmatic naturalism in the philosophy of mathematics and then indicate the points that provoke our disagreement.U ovome radu ćemo se koncentrirati na pitanje onih načina spoznaje u matematici za koje se tradicionalno smatralo da podržavaju apriorizam. To ćemo učiniti kritizirajući pragmatički naturalizam u filozofiji matematike, posebice njegov povijesni pristup u negiranju ikakve uloge apriorizma u matematičkoj epistemologiji. Verzija pragmatičkog naturalizma koju ćemo razmatrati je Kitcherova. U radu ćemo prvo iznijeti sažeti pregled relevantnih obilježja Kitcherovog pragmatičkog naturalizma u filozofiji matematike te potom naznačiti točke koje izazivaju naše neslaganje.Dans cet article, nous nous concentrerons sur la question des modes de la connaissance en mathématiques qui ont traditionnellement été considérés comme soutenant l’apriorisme. Nous le ferons en critiquant le naturalisme pragmatique dans la philosophie des mathématiques, notamment son approche historique de la négation de tout rôle de l’apriorité dans l’épistémologie mathématique. La version du naturalisme pragmatique que nous analyserons est celle de Philip Kitcher. Dans l’article, nous présenterons d’abord un court examen des aspects pertinents du naturalisme pragmatique de Philip Kitcher dans la philosophie des mathématiques, puis indiquerons les points qui suscitent notre désaccord.Im vorliegenden Paper konzentrieren wir uns auf die Problematik jener Wege der Erkenntnis innerhalb der Mathematik, denen traditionell Unterstützung des Apriorismus beigemessen wird. Wir verwirklichen dies, indem wir den pragmatischen Naturalismus in der Philosophie der Mathematik kritisieren, namentlich dessen historische Herangehensweise bei der Verneinung jeglicher Rolle der Apriorität in der mathematischen Epistemologie. Die Version des pragmatischen Naturalismus, der wir auf den Grund gehen, ist die von Kitcher. In der Abhandlung machen wir erstens einen Streifzug durch die relevanten Wesenszüge von Kitchers pragmatischem Naturalismus in der Philosophie der Mathematik und geben anschließend Anhaltspunkte an, die unseren Dissens hervorrufen. (shrink)

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view (...) can be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)

Electronic computers form an integral part of modern mathematical practice. Several high-profile results have been proven with techniques where computer calculations form an essential part of the proof. In the traditional philosophical literature, such proofs have been taken to constitute a posteriori knowledge. However, this traditional stance has recently been challenged by Mark McEvoy, who claims that computer calculations can constitute a priori mathematical proofs, even in cases where the calculations made by the computer are too numerous to (...) be surveyed by human agents. In this article we point out the deficits of the traditional literature that has called for McEvoy’s correction. We also explain why McEvoy’s defence of mathematical apriorism fails and we discuss how the debate over the epistemological status of computer-assisted mathematics contains several unfortunate conceptual reductions. (shrink)

With his book The Nature of Mathematical Knowledge (1983), Ph. Kitcher, that had been doing extensive research in the history of the subject and in the contemporary debates on epistemology, saw clearly the need for a change in philosophy of mathematics. His goal was to replace the dominant, apriorist philosophy of mathematics with an empiricist philosophy. The current philosophies of mathematics all appeared, according to his analysis, not to fit well with how mathematicians actually do mathematics. A shift in (...) orientation should invoke the more general reflection that causal, genetic factors are as significant for epistemology as logical structure. As I am to a large extent sympathetic with Kitcher's proposal, my aims here will be simple: first I start presenting Kitcher's argument, and then I try to raise some doubts about his contribution. These doubts come probably from my unskilfulness to follow the torrential flow of Kitcher's ideas. (shrink)

I examine Reichenbach’s theory of relative a priori and Michael Friedman’s interpretation of it. I argue that Reichenbach’s view remains at bottom conventionalist and that one issue which separates Reichenbach’s account from Kant’s apriorism is the problem of mathematical applicability. I then discuss Hermann Weyl’s theory of blank forms which in many ways runs parallel to the theory of relative a priori. I argue that it is capable of dealing with the problem of applicability, but with a cost.

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

a priori. Since I ended up defending an unpopular answer to this question—"No"—it’s hardly surprising that people have scrutinized the account, or that many have concluded that I stacked the deck in the first place. Of course, this was not my view of the matter. My own judgment was that I’d uncovered the tacit commitments of mathematical apriorists and that the widespread acceptance of mathematical apriorism rested on failure to ask what was needed for knowledge to be (...) a priori . Nevertheless, my critics have raised important challenges, and have offered rival conceptions that are less demanding. I want to examine their objections to my explication of a priori knowledge, and to explore whether the weaker alternatives succeed in preserving traditional philosophical claims. What follows is a mixture of penitence and intransigence. (shrink)

Recently Carrie S. Jenkins formulated an epistemology of mathematics, or rather arithmetic, respecting apriorism, empiricism, and realism. Central is an idea of concept grounding. The adequacy of this idea has been questioned e.g. concerning the grounding of the mathematically central concept of set (or class), and of composite concepts. In this paper we present a view of concept formation in mathematics, based on ideas from Carnap, leading to modifications of Jenkins’s epistemology that may solve some problematic issues with her (...) ideas. But we also present some further problems with her view, concerning the role of proof for mathematical knowledge. (shrink)

Artykuł jest próbą scharakteryzowania platońskiego idealizmu według wykładni Hermanna Cohena – filozofa w Polsce niemal zapomnianego; założyciela, a zarazem czołowego, obok Paula Natorpa i Władysława Tatarkiewicza, przedstawiciela marburskiej szkoły neokantyzmu. Tok analiz obejmuje cohenowskie tezy postawione przez filozofa w epistemologicznej pracy Platons Ideenlehre und die Mathematik. Postulaty, do których odwołuje się Cohen wiążą refleksję nad klasycznym idealizmem oraz statusem platońskiej idei z refleksją logiczno-matematyczną i zagadnieniem sichere Hypothesis jako hipotezy pewnej – hipotezy o statusie aksjomatu. W następstwie badań okazuje się, (...) że określenie roli matematyki: arytmetyki i geometrii, staje się nieodzowne w zrozumieniu idealizmu zarówno starożytnego Platona, jak również nowożytnego Kanta. Obaj, Platon i Kant, akcentują znaczenie apriorycznej myśli, logiki czystego myślenia, logicznego rozumowania, poznawczej władzy intelektu, nauki oraz krytycyzmu jako drogi poznawczego, a zarazem kształtującego charakter, postępowania podmiotu poznającego Przeprowadzony w artykule wywód, zakorzeniony w historyczno-filozoficznych analizach problemu, w podsumowaniu jest także próbą określenia znaczenia platońsko-neokantowskiego idealizmu dla współczesnych zagadnień naukowo-filozoficznych. (shrink)

L’article a pour but d’analyser la conception de la géométrie et de la mesure présentée dans Intuition et Raisonnement [Hölder 1900], « Les axiomes de la grandeur et la théorie de la mensuration » [Hölder 1901] et La Méthode mathématique [Hölder 1924]. L’article examine les relations entre a) la démarcation introduite par Hölder entre géométrie et arithmétique à partir de la notion de ‘concept donné’, b) sa position philosophique par rapport à l’apriorisme kantien et à l’empirisme et c) le choix (...) du postulat de la continuité de Dedekind parmi les axiomes de la théorie des grandeurs. L’article montre que les choix faits au niveau axiomatique sont reliés au cadre épistémologique et que l’originalité de Hölder consiste surtout dans son analyse au cas par cas des procédures déductives en mathématiques.The aim of the paper is to analyze Hölder’s understanding of geometry and measurement presented in Intuition and Reasoning [Hölder 1900], “The Axioms of Quantity and the Theory of Measurement” [Hölder 1901], and The Mathematical Method [Hölder 1924]. The paper explores the relations between a) Hölder’s demarcation of geometry from arithmetic based on the notion of given concepts, b) his philosophical stance towards Kantian apriorism and empiricism, and c) the choice of Dedekind’s continuity in the axiomatic formulation of the theory of magnitudes. The paper shows that the choices made at the axiomatic level reflect Hölder’s epistemological framework, and individuates the originality of his approach in the case by case analysis of deductive procedures. (shrink)

C. S. Jenkins has recently proposed an account of arithmetical knowledge designed to be realist, empiricist, and apriorist: realist in that what’s the case in arithmetic doesn’t rely on us being any particular way; empiricist in that arithmetic knowledge crucially depends on the senses; and apriorist in that it accommodates the time-honored judgment that there is something special about arithmetical knowledge, something we have historically labeled with ‘a priori’. I’m here concerned with the prospects for extending Jenkins’s account beyond arithmetic—in (...) particular, to set theory. After setting out the central elements of Jenkins’s account and entertaining challenges to extending it to set theory, I conclude that a satisfactory such extension is unlikely. (shrink)

According to Husserl’s self-description, his phenomenological project was “completely apolitical.” Husserl’s phenomenology did not provide a political philosophy in the classical sense, a normative description of a functioning social order and its respective institutional structures. Nor did Husserl have much to say about the day-to-day politics of his time. Yet his reflections on community and culture were not completely without political implications. This article deals with an often-neglected strand of Husserl’s philosophy, namely his critique of liberalism. In this article, liberalism (...) is understood in the manner of Leo Strauss, as a tradition of individualist philosophy emerging from Hobbes’s political thought. As the article shows, Husserl followed many of his contemporaries in criticizing Hobbes’s abstract individualism, which could provide only a preventative function for political institutions. More importantly, Husserl’s engagement with Hobbes can be understood as a kind of ethical counterpart to his analysis of Galileo in the Crisis, as a critique of formal apriorism in the political domain. What Galileo did for physical nature with his “garb of ideas,” Hobbes did for human nature, reducing human sociality to interactions between atomistic individuals. In doing so, Hobbes ended up presenting a “mathematics of sociality” in an empirical garb. Going against the liberal view of the human person, Husserl presented his theory of social ethics founded on an original co-existence of subjects and a theory of human renewal. (shrink)

The Socratic method has a long history in teaching philosophy and mathematics, marked by such names as Karl Weierstra, Leonard Nelson and Gustav Heckmann. Its basic idea is to encourage the participants of a learning group (of pupils, students, or practitioners) to work on a conceptual, ethical or psychological problem by their own collective intellectual effort, without a textual basis and without substantial help from the teacher whose part it is mainly to enforce the rigid procedural rules designed to ensure (...) a fruitful, diversified, open and consensus-oriented thought process. Several features of the Socratic procedure, especially in the canonical form given to it by Heckmann, are highly attractive for the teaching of medical ethics in small groups: the strategy of starting from relevant singular individual experiences, interpreting and cautiously generalizing them in a process of inter-subjective confrontation and confirmation, the duty of non-directivity on the part of the teacher in regard to the contents of the discussion, the necessity, on the part of the participants, to make explicit both their own thinking and the way they understand the thought of others, the strict separation of content level and meta level discussion and, not least, the wise use made of the emotional and motivational resources developing in the group process. Experience shows, however, that the canonical form of the Socratic group suffers from a number of drawbacks which may be overcome by loosening the rigidity of some of the rules. These concern mainly the injunction against substantial interventions on the part of the teacher and the insistence on consensus formation rooted in Leonard Nelson's Neo-Kantian Apriorism. (shrink)

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert (...) style axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics'' assuggested by Penelope Maddy. The answer is ``no'', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind'' andrelating it to the ``one and only axiom'' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper. (shrink)

Ernst Mach was already an international successful experimental physicist and scientist, when he, after professorships for Mathematics and Physics in Graz and Experimental Physics in Prague, took over the chair for “Philosophy, particularly for the History and Theory of the Inductive Sciences”, at the University of Vienna in 1895. This turn from the natural sciences to philosophy was really an exception in the academic field.Given his strong as well as controversial history of influence in philosophy and in the sciences Mach’s (...) own pessimistic statement about the emergence of aprioristic currents at the beginning of the twentieth century is surprising and in need of an explanation.The article deals with Mach’s appointment in Vienna and Mach’s autobiographical fragments on his relation to academic philosophy from a today’s point of view. It appears that Mach can be regarded as a pioneer and predecessor of a topical “historical epistemology”, “history and philosophy of science”, and above all as a theorist and practitioner of inter- and transdisciplinarity. If he is to be regarded a philosopher, it is mainly in the context of naturalism, pragmatism and common sense philosophy – as general theory of research. (shrink)

Article is devoted to the social legitimation of knowledge. We study the contexts of implantation of knowledge products into the body of culture. The author proceeds from the need to study the process of objectification symbolic of object by applying the category of ‘facies‘, the introduction and justification of which on content and formal level were realized by the author in previous works. Such issues as the following are discussed in the article: the main stages of objectification, cognitions, different worlds (...) in science in the trend of unfolding cognitions, the basic principles of object-symbolic interpretation of the nature of knowledge. The author explores these questions by analyzing the basic epistemological trends that can be seen in the works of the greatest philosophers of both modern and recent times, globalizing the principle of a system approach on the basis of its synthesis with the elements of synergetics and semiotics. The author devotes a great place to criticism of Kant’s transcendental apriorism, revealing, however, its positive potential and determining its promising vector beating literally in the essence of contemporary cognitive science. Author takes on orbit of its original methodology conclusions of not only natural sciences and mathematics, but also poetic insights that allows creating of pulsating and at the same time quite certain interdisciplinary context. The main conclusions regarding the content of prerequisites of cognitive synthesis included in the final paragraph of article and are accompanied by convincing examples of modern physics. (shrink)

This is a new critical edition of Berkeley’s 1734 _Three Dialogues_, a text that is deservedly one of the most challenging and beloved classics of modern philosophy. The heart of the work is the dispute between materialism and idealism, two fundamentally opposed positions that are embodied by Hylas and Philonous, the characters in this philosophical drama. The book is packed with brilliant arguments and counter-arguments of an extraordinarily sophisticated nature. Amid all this philosophical swordplay one would think that there could (...) be scant room for the characters to develop any sort of personality. Yet in Berkeley’s hands, and with his literary gifts, the interlocutors are both vivid and funny. The dialogue deals with some of the most important perennial problems of philosophy, including: the materialism-idealism dispute, skepticism in rationalist and empiricist epistemology, the conflict over apriorism and aposteriorism, rationalism versus empiricism, the existence and nature of God, the philosophy of science, philosophy of mathematics, abstract general ideas, the role of perception in human knowledge, and the metaphysics of causation. This edition combines a usefully annotated version of Berkeley’s complete original text with a substantial critical introduction, a chronology of events in Berkeley’s life and career, and supplementary annotated appendices of original sources from thinkers relevant to Berkeley’s work. (shrink)

This is a review of the book Mozg - kultura - socium. Kantianskaya programma v cognitivnykh issledovaniyakh by Bazhanov V.A. The author`s main aim is a reconsideration of some fundamental principles of epistemology and philosophy of science in the light of state-of-the-art advancements of neurosciences. According to the author of the book, social and cultural revolution in neurosciences has been crucially modifying the initial Kantian program of consciousness research and the modes of linguistic representation of its results. In particular, in (...) epistemology the universal concept of transcendental subject has been altered with regard to neurobiological and social data, which concern some preconditions of cognitive activity. In philosophy of science such a shift of research perspective allows to implement a naturalistic interpretation of natural sciences, clarifies the problem of empirical foundations of mathematics and sheds a new light on the problem of apriori elements of mathematical thought. The analysis of evolutionary and neurobiological preconditions of logical thought leads to the sort of revival of psychologism program in logic, which enables author to put forward an original interpretation of the problem of plurality of logical systems. Finally, some topics of social sciences as well as problems of genesis of art, ethical and religious systems, can acquire new interpretation with respect to modern investigations in cognitive science. (shrink)

This paper explores the main philosophical approaches of David Hilbert’s theory of proof. Specifically, it is focuses on his ideas regarding logic, the concept of proof, the axiomatic, the concept of truth, metamathematics, the a priori knowledge and the general nature of scientific knowledge. The aim is to show and characterize his epistemological approach on the foundation of knowledge, where logic appears as a guarantee of that foundation. Hilbert supposes that the propositional apriorism, proposed by him to support mathematics, (...) sustains — on its turn — a general method for the treatment of the problem in other areas such as natural sciences. This method is axiomatic. Broadly speaking, we intend to recover and update the Hilbert’s philosophical thinking about the role of logic for scientific knowledge. (shrink)

This book is written so as to be ‘accessible to philosophers without a mathematical background’. The reviewer can assure the reader that this aim is achieved, even if only by focusing throughout on just one example of an arithmetical truth, namely ‘7+5=12’. This example’s familiarity will be reassuring; but its loneliness in this regard will not. Quantified propositions — even propositions of Goldbach type — are below the author’s radar.The author offers ‘a new kind of arithmetical epistemology’, one which (...) ‘respects certain important intuitions’ 1 : apriorism, realism, and empiricism. The book contains some clarification of these ‘isms’, and some thoughtful critiques of major positions regarding them, as espoused by such representative figures as Boghossian, Bealer, Peacocke, Field, Bostock, Maddy, Locke, Kant, C.I. Lewis, Ayer, Quine, Fodor, and McDowell. The philosophical reader will find some interest and value in these wider-ranging discussions. Our concern in this review, however, is to examine closely the original positive proposal on offer.Arithmetical truths, the author maintains, are conceptual truths. Knowing truths like 7+5=12 involves no ‘epistemic reliance on any empirical evidence’; but that, she says, is not to claim ‘epistemic independence of the senses altogether’. She wants to show that "experience grounds our concepts … and then mere conceptual examination enables us to learn arithmetical truths ." Concepts that are ‘appropriately sensitive’ to ‘the nature of [an independent] reality’ she calls grounded. Because of the role of grounded concepts, ‘arithmetical truths explain our arithmetical beliefs in the right sort of way for those beliefs to count as knowledge’ .In the context of her concentration on the special nature of arithmetical knowledge, the author offers what could strike some bystanders as an unnecessarily over-ambitious account of knowledge tout court. Knowledge, for the author, is "true belief which … ". (shrink)

This essay is part of an attempt to reconcile two extreme views in economics: the (neglected) subjective, apriorist approach and the (standard) objective, scientific (i.e., falsifiable) approach. The Austrian subjective view of value, building on Carl Menger’s theory of value, was developed into a theory of economics as being entirely an a priori theory of action. This probably finds its most extreme statement in Ludwig von Mises’ Human Action (1949). In contrast, the standard economic view has developed into making falsifiable (...) predictions about economic phenomena whereby the truth of the assumptions, especially about economic agents, is relatively unimportant: predictive fecundity is all. This finds an extreme statement in Milton Friedman’s introductory essay in his Essays in Positive Economics (1953). However, many economists fall somewhere between the two extremes, such as McKenzie and Tullock (1978). (shrink)

Epistemological optimists about thought experiments hold that imagination could be under certain conditions source of epistemic justification. Their claim is usually based on one of three dominant conceptions about epistemic value of thought experiments. Apriorism states that imagination may serve as unique a priori source of new synthetic knowledge about the actual world. I argue against this view and show that apriorism is either too weak, or too strong or too vague. Psychologism is viable, yet not fully clear (...) conception about new meta-knowledge obtained by thought experimenting. I compare some interpretations of this position and present reasons for favorizing one of them. Conceptualism considers thought experiments as instruments for cleaning our conceptual systems. I argue that this position is in fact not about epistemic value of thought experiments, but about one specific usage of experimental result. (shrink)

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...

Current debates between behavioural and orthodox economists indicate that the role and epistemological status of first principles is a particularly pressing problem in economics. As an alleged paragon of extreme apriorism, the methodology of Austrian economics in Mises’ tradition is often dismissed as untenable in the light of modern philosophy. In particular, the defence of the so-called fundamental axiom of praxeology—“Man acts.”—by means of pure intuition is almost unanimously rejected. However, in recently resurfacing debates, the extremeness of Mises’ epistemological (...) position has been called into question. Rather than directly engaging in these exegetical discussions, this paper aims to substantiate the possibility and plausibility of conventionalist defences of praxeology per se. The proposed shift includes settling for an analytic fundamental axiom and acknowledging the prima facie tenability of other research programs than praxeology. Since conventionalist praxeology is only moderately aprioristic, mainstream economists and philosophers might be more likely to engage in fruitful discussions with those Austrian scholars who elaborate pragmatic arguments for praxeology instead of invoking pure intuition. (shrink)

The paper aims to establish that Terence Hutchison’s argument in The Politics and Philosophy of Economics (1981) to the effect that the young F.A. Hayek maintained a methodological position markedly similar to that of Ludwig von Mises fails to establish the relevant conclusion. The first problem with Hutchison’s argument is that it is not clear exactly what conclusion he meant to establish with regard to the methodological views of the two paragons of 20th century Austrian economics. Mises (in)famously maintained a (...) rather extreme methodological apriorism. However, Hutchison’s argument does not support the claim that Hayek was ever an apriorist of the Misesian variety. The concept of a priori knowledge that emerges from Hayek’s epistemology – specifically the epistemology implied by Hayek’s work in theoretical psychology – is the direct opposite of Mises’ treatment of a priori knowledge. Simply stated, Hayek conceived of a priori knowledge as fallible and relative, while Mises considered a priori knowledge to be infallible and absolute. Thus, it cannot be maintained – if, indeed, Hutchison meant to establish – that Hayek was a Misesian apriorist during the years in question. What’s more, the paper shows that Hutchison’s argument does not support a weaker interpretation of the relevant conclusion. There are alternative interpretations of the evidence adduced by Hutchison that are both more charitable and more in line with Hayek’s epistemology that undermine Hutchison’s conclusion. (shrink)

An inclusive vision of mathematics-- its beauty, its humanity, and its power to build virtues that help us all flourish. For mathematician Francis Su, a society without mathematical affection is like a city without museums. To miss out on mathematics is to live without experiencing some of humanity's most beautiful ideas. In this profound book, written for a diverse audience but especially for those disenchanted by their past experiences, an award-winning mathematician and educator weaves personal reflections, puzzles, and stories (...) to show how mathematics meets basic human desires and cultivates virtues essential for human flourishing. Readers will explore mathematical concepts-- and see how mathematical thinking can even fulfill such longings as for love, play, freedom, justice, and community. Some lessons come from those who have struggled, including philosopher Simone Weil, whose own mathematical contributions were overshadowed by her brother's, and Christopher Jackson, who discovered mathematics as an inmate in a federal prison. Christopher Jackson's letters to the author appear throughout the book and show how this intellectual pursuit can-- and must-- be open to all. (shrink)

A common response to those who question the Law of Non-Contradiction is that it is impossible to debate such a fundamental law of logic. The reasons for this response vary, but what seems to underlie them is the thought that there is a minimal set of logical resources without which rational debate is impossible. This chapter argues that this response is misguided. First, it defends non-apriorism in logic: the view that logic is in the same epistemic boat as other (...) scientific theories. It then offers an account of logical theory change in terms of this epistemology. The LNC is discussed in terms of this account of logical theory change, and it is shown that rational debate over this law can, and does, proceed. Finally, arguments for and against the LNC are discussed, and how and where non-a priori considerations arise in these arguments are illustrated. (shrink)

We defend a view according to which Austrian economics rests on what can most properly be called an Aristotelian methodology. This implies a realist perspective, according to which the world exists independently of our thinking and reasoning activities; an essentialist perspective, according to which the world contains certain simple essences or natures which may come together in law-like ways to form more complex static and dynamic wholes, and an apriorist perspective, according to which given essences and essential structures are intelligible, (...) in the sense that they ca be grasped non-inductively in our thinking. We show the consequences of this view for an analysis of the thinking of Mises and Hoppe, both of which – we claim – incorporate what we believe to be foreign mixtures of Kantianism in their account of the foundations of Austrian economics. (shrink)

Metaphysics, Mathematics, and Meaning brings together Nathan Salmon's influential papers on topics in the metaphysics of existence, non-existence, and fiction; modality and its logic; strict identity, including personal identity; numbers and numerical quantifiers; the philosophical significance of Godel's Incompleteness theorems; and semantic content and designation. Including a previously unpublished essay and a helpful new introduction to orient the reader, the volume offers rich and varied sustenance for philosophers and logicians.