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)

In this symposium contribution I critically review the first two chapters, on Frege, in Volume 1 of The Analytic Tradition in Philosophy by Scott Soames.

A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this stance is (...) that it dovetails with the correct response to Benacerraf's other objection to platonism, i.e., his (1973) epistemological objection. (shrink)

This paper examines how the work of Frege was known and received in Poland in the period 1910–1935 (with one exception concerning the later work of Suszko). The main thesis is that Frege's reception in Poland was perhaps faster and deeper than in other countries, except England, due to works of Russell and Jourdain. The works of Łukasiewicz, Leśniewski and Czeżowski are described.

This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert?s Grundlagen der Geometrie to two: (i) Frege?s reliance on what, following Wilfrid Hodges, I call a Frege?Peano language, and (ii) Frege?s view that the sense of an expression wholly determines its reference.I argue that these two preconceptions prevented Frege from achieving the conceptual structure of model theory, whereas Hilbert, at least in his practice, was quite close to the model?theoretic point of view.Moreover, the issues that divided Frege (...) and Hilbert did not revolve around whether one or the other allowed metalogical notions.Frege, e.g., succeeded in formulating the notion of logical consequence, at least to the extent that Bolzano did; the point is rather that even though Frege had certain semantic concepts, he did not articulate them model?theoretically, whereas, in some limited sense, Hilbert did. (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.

O presente manual tem como intenção constituir um guia para uma disciplina introdutória de filosofia da linguagem. Foi elaborado a partir da leccionação da disciplina de Filosofia da Linguagem I na Faculdade de Letras da Universidade do Porto desde 2001. A disciplina de Filosofia da Linguagem I ocupa um semestre lectivo e proporciona aos estudantes o primeiro contacto sistemático com a área da filosofia da linguagem. Pretende-se que este manual ofereça aos estudantes os instrumentos necessários não apenas para acompanhar uma (...) iniciação ao campo da filosofia da linguagem mas também para o estudo e a investigação autónomos posteriores. Para isso é percorrido um trajecto que conduz das intuições pré-teóricas acerca de linguagem, de que todos dispomos, até um conhecimento disciplinar específico, histórico e temático, da filosofia da linguagem. Em termos práticos, são considerados como precedentes da disciplina de Filosofia da Linguagem I – mesmo se, pelo existem actualmente precedências em sentido estrito – as disciplinas de Lógica I e II e de Filosofia do Conhecimento I e II. Os programas dessas disciplinas nos últimos anos estiveram presentes por trás da elaboração do manual. (shrink)

Peter Forrest and D.M. Armstrong have given an argument against a theory of naturalness proposed by David Lewis based on the fact that ordered pairs can be constructed from sets in any of a number of different ways. 1. I think the argument is good, but requires a more thorough defense. Moreover, the argument has important consequences that have not been noticed. I introduce a version of Lewis’s proposal in section one, and then in section two I present and defend (...) my version of the argument. After addressing a worry about my argument in section three, in section four I argue that a similar “argument from arbitrariness” jeopardizes Lewis’s solution to the Kripke/Wittgenstein puzzle of the content of thought. (shrink)

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

This essay consists of two parts. In the first part, I focus my attention on the remarks that Frege makes on consistency when he sets about criticizing the method of creating new numbers through definition or abstraction. This gives me the opportunity to comment also a little on H. Hankel, J. Thomae—Frege’s main targets when he comes to criticize “formal theories of arithmetic” in Die Grundlagen der Arithmetik (1884) and the second volume of Grundgesetze der Arithmetik (1903)—G. Cantor, L. E. (...) J. Brouwer and D. Hilbert (1899). Part 2 is mainly devoted to Hilbert’s proof theory of the 1920s (1922–1931). I begin with an account of his early attempt to prove directly, and thus not by reduction or by constructing a model, the consistency of (a fragment of) arithmetic. In subsequent sections, I give a kind of overview of Hilbert’s metamathematics of the 1920s and try to shed light on a number of difficulties to which it gives rise. One serious difficulty that I discuss is the fact, widely ignored in the pertinent literature on Hilbert’s programme, that his language of finitist metamathematics fails to supply the conceptual resources for formulating a consistency statement qua unbounded quantification. Along the way, I shall comment on W. W. Tait’s objection to an interpretation of Hilbert’s finitism by Niebergall and Schirn, on G. Gentzen’s allegedly finitist consistency proof for Peano Arithmetic as well as his ideas on the provability and unprovability of initial cases of transfinite induction in pure number theory. Another topic I deal with is what has come to be known as partial realizations of Hilbert’s programme, chiefly advocated by S. G. Simpson. Towards the end of this essay, I take a critical look at Wittgenstein’s views about (in)consistency and consistency proofs in the period 1929–1933. I argue that both his insouciant attitude towards the emergence of a contradiction in a calculus and his outright repudiation of metamathematical consistency proofs are unwarranted. In particular, I argue that Wittgenstein falls short of making a convincing case against Hilbert’s programme. I conclude with some philosophical remarks on consistency proofs and soundness and raise a question concerning the consistency of analysis. (shrink)

This dissertation explores the concepts of naturalness, intrinsicality, and duplication. An intrinsic property is had by an object purely in virtue of the way that object is considered in itself. Duplicate objects are exactly similar, considered as they are in themselves. The perfectly natural properties are the most fundamental properties of the world, upon which the nature of the world depends. In this dissertation I develop a theory of intrinsicality, naturalness, and duplication and explore their philosophical applications. Chapter 1 introduces (...) the notions, gives a preliminary survey of some proposed conceptual connections between the notions, and sketches some of their proposed applications. Chapter 2 gives my background assumptions and introduces notational conventions. In chapter 3 I present a theory of naturalness. Although I take ‘natural’ as a primitive, I clarify this notion by distinguishing and explicating various conceptions of naturalness. In chapter 4 I give a theory of various notions related to naturalness, especially intrinsicality and duplication. I show that ‘intrinsic’ and ‘duplicate’ are interde nable, and then give analyses of these and other notions in terms of naturalness. If, as I think likely, naturalness cannot be analyzed, then what is the proper response? David Lewis suggests: accept naturalness as a primitive. I am sympathetic to this proposal, but not to the form Lewis gives it: chapter 5 contains an argument against Lewis’s theory of naturalness. In chapter 6 I reject the idea that naturalness is analyzable in terms of immanent universals. I focus on the work of D. M. Armstrong. I also criticize Armstrong’s arguments against transcendent universals. (shrink)

In this paper I discuss Frege's distinction between objects and concepts and suggest a solution of Frege's paradox of the concept horse. The expression ''the concept horse'' is not eliminated and the concept is not identified with its extension, but the concept is identified with the sense of the corresponding predicate. This solution fits better into a fregean ontology and philosophy of language than alternative solutions and allows for a general answer to the question why Frege's system is infected with (...) Russell's paradox. Russell's paradox is caused by the reification of a concept. Certain problems of modern set theory seem to have a similar cause.Eine weithin sichtbare Warnungstafel muss aufgerichtet. (shrink)

Hilbert’s program was an ambitious and wide-ranging project in the philosophy and foundations of mathematics. In order to “dispose of the foundational questions in mathematics once and for all,” Hilbert proposed a two-pronged approach in 1921: first, classical mathematics should be formalized in axiomatic systems; second, using only restricted, “finitary” means, one should give proofs of the consistency of these axiomatic systems. Although Gödel’s incompleteness theorems show that the program as originally conceived cannot be carried out, it had many partial (...) successes, and generated important advances in logical theory and metatheory, both at the time and since. The article discusses the historical background and development of Hilbert’s program, its philosophical underpinnings and consequences, and its subsequent development and influences since the 1930s. (shrink)

This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...) Tarski’s and Gödel’s work, explaining why the problems they discussed are still unsolved. Finally, the book reports on some of the most influential positions in contemporary philosophy of mathematics, i.e., Maddy’s mathematical naturalism and Shapiro’s mathematical structuralism. Last but not least, the book introduces Biancani’s Aristotelian philosophy of mathematics as this is considered important to understand current philosophical issue in the applications of mathematics. One of the main purposes of the book is to stimulate readers to reconsider the Aristotelian position, which disappeared almost completely from the scene in logic and mathematics in the early twentieth century. (shrink)

The Frege-Russell view is that existence is a second-order property rather than a property of individuals. One of the most compelling arguments for this view is based on the premise that there is an especially close connection between existence and number. The most promising version of this argument is by C.J.F Williams (1981, 1992). In what follows, I argue that this argument fails. I then defend an account according to which both predications of number and existence attribute properties to individuals.

According to the species of neo-logicism advanced by Hale and Wright, mathematical knowledge is essentially logical knowledge. Their view is found to be best understood as a set of related though independent theses: (1) neo-fregeanism-a general conception of the relation between language and reality; (2) the method of abstraction-a particular method for introducing concepts into language; (3) the scope of logic-second-order logic is logic. The criticisms of Boolos, Dummett, Field and Quine (amongst others) of these theses are explicated and assessed. (...) The issues discussed include reductionism, rejectionism, the Julius Caesar problem, the Bad Company objections, and the charge that second-order logic is set theory in disguise. (shrink)

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form (...) of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories. (shrink)

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention to Hilbert’s instrumental formalism. (...) My primary aim there will be to develop its formalist elements more fully. These are, in the main, its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, its departure from the traditional understanding of the basic nature of proof and its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. (shrink)

The paper discusses the emergence of Frege's puzzle and the introduction of the celebrated distinction between sense and reference in the context of Frege's logicist project. The main aim of the paper is to show that not logicism per se is mainly responsible for this introduction, but Frege's constant struggle against formalism. Thus, the paper enlarges the historical context, and provides a reconstruction of Frege's philosophical development from this broader perspective.

Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our access to mathematical truths requires nothing beyond (...) our ability to make logical deductions. Sections 1 and 2 define and motivate deductivism in more detail. Section 3 covers four authors (Russell, Hilbert, Pasch, Curry) who have endorsed deductivism at some point. Section 4 aims to clarify what semantic claim deductivists make. Sections 5–9 review objections to deductivism and possible replies. (shrink)

The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. I intend with this (...) analogy (looking at others' comparisons) to shed a little light on what is going on in mathematics, how one can understand it a bit other than experientially. This intention is philosophical and the way that I am attempting to accomplish it is also philosophical. I shall conclude my attempt to explain how it is possible and even natural for mathematics and fiction to have the analogy they have, taking it for granted, as argued in part I, that they are not to be identified. To this end I shall discuss philosophers' comparisons, mainly those of Hodes, Resnik, Tharp, and Wagner, who are the writers that seem to me to have written most thoughtfully and sufficiently extensively about fiction in making the comparison and of Körner, whose comparison is different. Whether either of these comparisons or the more general analogy is of permanent philosophical interest will have to be decided by philosophers now that they have had a fuller examination. I shall mention some other writers' reference to fiction, but not all; indeed, I am sure that I have not even found all the comparisons that there are. (shrink)

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

This entry introduces the reader to the main ideas in Frege's philosophy of logic, mathematics, and language.

This book offers an historically-informed critical assessment of Dummett's account of abstract objects, examining in detail some of the Fregean presuppositions whilst also engaging with recent work on the problem of abstract entities.

The essays in this volume concern the points of intersection between analytic philosophy and the philosophy of the exact sciences. More precisely, it concern connections between knowledge in mathematics and the exact sciences, on the one hand, and the conceptual foundations of knowledge in general. Its guiding idea is that, in contemporary philosophy of science, there are profound problems of theoretical interpretation-- problems that transcend both the methodological concerns of general philosophy of science, and the technical concerns of philosophers of (...) particular sciences. A fruitful approach to these problems combines the study of scientific detail with the kind of conceptual analysis that is characteristic of the modern analytic tradition. Such an approach is shared by these contributors: some primarily known as analytic philosophers, some as philosophers of science, but all deeply aware that the problems of analysis and interpretation link these fields together. (shrink)