In this work I study the main tenets of the logicist philosophy of mathematics. I deal, basically, with two problems: (1) To what extent can one dispense with intuition in mathematics? (2) What is the appropriate logic for the purposes of logicism? By means of my considerations I try to determine the pros and cons of logicism. My standpoint favors the logicist line of thought. -/- .

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer (...) a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). (shrink)

According to Gottlob Frege, his logicism died when it was discovered that the underlying theory of extensions is inconsistent. The neo-logicist attempts to found mathematics on other abstraction principles, such as the so-called Hume’s principle that two concepts have the same number if and only if they are equinumerous. This chapter discusses the state of neo-logicism, responding to various objections that have been raised against it.

We examine the sense in which logic is a priori, and explain how mathematical theories can be dichotomized non-trivially into analytic and synthetic portions. We argue that Core Logic contains exactly the a-priori-because-analytically-valid deductive principles. We introduce the reader to Core Logic by explaining its relationship to other logical systems, and stating its rules of inference. Important metatheorems about Core Logic are reported, and its important features noted. Core Logic can serve as the basis for a foundational program that could (...) be called Natural Logicism, an exposition of which will build on the (meta)logical ideas explained here. (shrink)

In this paper, the connection between logicism and the principle of tolerance in Carnap’s philosophy of logic and mathematics is to be presented in terms of the history of its development. Such development is conditioned by two lines of criticism to Carnap’s attempt to combine Logicism and Conventionalism, the first of which comes from Gödel, the second from Alfred Tarski. The presentation will take place in three steps. First, the Logicism of Carnap before the publication of (...) The Logical Syntax of Language will be analyzed in its main features. Second, the relationship between Logicism and the Principle of Tolerance at the time of the publication of Logical Syntax, and thirdly, in the so-called semantical phase of Carnap’s thought will be examined. On the basis of the two preceding points, we will then critically analyze two interpretations of the possibility to maintain a philosophical position like that of Carnap nowadays. (shrink)

The previous Part I of the paper discusses the option of the Gödel incompleteness statement (1931: whether “Satz VI” or “Satz X”) to be an axiom due to the pair of the axiom of induction in arithmetic and the axiom of infinity in set theory after interpreting them as logical negations to each other. The present Part II considers the previous Gödel’s paper (1930) (and more precisely, the negation of “Satz VII”, or “the completeness theorem”) as a necessary condition for (...) granting the Gödel incompleteness statement to be a theorem just as the statement itself, to be an axiom. Then, the “completeness paper” can be interpreted as relevant to Hilbert mathematics, according to which mathematics and reality as well as arithmetic and set theory are rather entangled or complementary rather than mathematics to obey reality able only to create models of the latter. According to that, both papers (1930; 1931) can be seen as advocating Russell’s logicism or the intensional propositional logic versus both extensional arithmetic and set theory. Reconstructing history of philosophy, Aristotle’s logic and doctrine can be opposed to those of Plato or the pre-Socratic schools as establishing ontology or intensionality versus extensionality. Husserl’s phenomenology can be analogically realized including and particularly as philosophy of mathematics. One can identify propositional logic and set theory by virtue of Gödel’s completeness theorem (1930: “Satz VII”) and even both and arithmetic in the sense of the “compactness theorem” (1930: “Satz X”) therefore opposing the latter to the “incompleteness paper” (1931). An approach identifying homomorphically propositional logic and set theory as the same structure of Boolean algebra, and arithmetic as the “half” of it in a rigorous construction involving information and its unit of a bit. Propositional logic and set theory are correspondingly identified as the shared zero-order logic of the class of all first-order logics and the class at issue correspondingly. Then, quantum mechanics does not need any quantum logics, but only the relation of propositional logic, set theory, arithmetic, and information: rather a change of the attitude into more mathematical, philosophical, and speculative than physical, empirical and experimental. Hilbert’s epsilon calculus can be situated in the same framework of the relation of propositional logic and the class of all mathematical theories. The horizon of Part III investigating Hilbert mathematics (i.e. according to the Pythagorean viewpoint about the world as mathematical) versus Gödel mathematics (i.e. the usual understanding of mathematics as all mathematical models of the world external to it) is outlined. (shrink)

This article aims first at showing that Russell's general doctrine according to which all mathematics is deducible ‘by logical principles from logical principles’ does not require a preliminary reduction of all mathematics to arithmetic. In the Principles, mechanics (part VII), geometry (part VI), analysis (part IV–V) and magnitude theory (part III) are to be all directly derived from the theory of relations, without being first reduced to arithmetic (part II). The epistemological importance of this point cannot be overestimated: (...) Russell's logicism does not only contain the claim that mathematics is no more than logic, it also contains the claim that the differences between the various mathematical sciences can be logically justified—and thus, that, contrary to the arithmetization stance, analysis, geometry and mechanics are not merely outgrowths of arithmetic. The second aim of this article is to set out the neglected Russellian theory of quantity. The topic is obviously linked with the first, since the mere existence of a doctrine of magnitude, in a work dated from 1903, is a sign of a distrust vis-à-vis the arithmetization programme. After having shown that, despite the works of Cantor, Dedekind and Weierstrass, many mathematicians at the end of the 19th Century elaborated various axiomatic theories of the magnitude, I will try to define the peculiarity of the Russellian approach. I will lay stress on the continuity of the logicist's thought on this point: Whitehead, in the Principia, deepens and generalizes the first Russellian 1903 theory. (shrink)

Russell's philosophy is rightly described as a programme of reduction of mathematics to logic. Now the theory of geometry developed in 1903 does not fit this picture well, since it is deeply rooted in the purely synthetic projective approach, which conflicts with all the endeavours to reduce geometry to analytical geometry. The first goal of this paper is to present an overview of this conception. The second aim is more far-reaching. The fact that such a theory of geometry was (...) sustained by Russell compels us to question the meaning of logicism: how is it possible to reconcile Russell's global reductionist standpoint with his local defence of the specificities of geometry? * This paper was first presented at the conference ‘Qu'est ce que la géométrie aux époques modernes et contemporaines?’, organized by the Universität Köln and the Archives Poincaré. I would like to thank Philippe Nabonnand for having enlightened me about the issues relative to projective geometry. I would like also to thank Nicholas Griffin, Brice Halimi, Bernard Linsky, Marco Panza, Ivahn Smadja for their helpful discussions. Many thanks also to the two anonymous referees for their useful suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

In this paper, we describe "metaphysical reductions", in which the well-defined terms and predicates of arbitrary mathematical theories are uniquely interpreted within an axiomatic, metaphysical theory of abstract objects. Once certain (constitutive) facts about a mathematical theory T have been added to the metaphysical theory of objects, theorems of the metaphysical theory yield both an analysis of the reference of the terms and predicates of T and an analysis of the truth of the sentences of T. The well-defined terms and (...) predicates of T are analyzed as denoting abstract objects and abstract relations, respectively, in the background metaphysics, and the sentences of T have a reading on which they are true. After the technical details are sketched, the paper concludes with some observations about the approach. One important observation concerns the fact that the proper axioms of the background theory abstract objects can be reformulated in a way that makes them sound more like logical axioms. Some philosophers have argued that we should accept (something like) them as being logical. (shrink)

A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real (...) content of the foundational schools. Hence, a re-thinking from him would be required about them, otherwise he would risk the same reproaches as post-modernists. (shrink)

This paper argues that Weyl's criticism of Dedekind’s principle that "In science, what is provable ought not to be believed without proof." challenges not only a logicist norm of belief in mathematics, but also a realist view about whether there is a fact of the matter as to what norms of belief are correct.

A crucial part of the contemporary interest in logicism in the philosophy of mathematics resides in its idea that arithmetical knowledge may be based on logical knowledge. Here an implementation of this idea is considered that holds that knowledge of arithmetical principles may be based on two things: (i) knowledge of logical principles and (ii) knowledge that the arithmetical principles are representable in the logical principles. The notions of representation considered here are related to theory-based and structure-based notions (...) of representation from contemporary mathematical logic. It is argued that the theory-based versions of such logicism are either too liberal (the plethora problem) or are committed to intuitively incorrect closure conditions (the consistency problem). Structure-based versions must on the other hand respond to a charge of begging the question (the circularity problem) or explain how one may have a knowledge of structure in advance of a knowledge of axioms (the signature problem). This discussion is significant because it gives us a better idea of what a notion of representation must look like if it is to aid in realizing some of the traditional epistemic aims of logicism in the philosophy of mathematics. (shrink)

The virtues of this book are many. It covers a large amount of often-neglected material introduced by Russell in his Principles of Mathematics and elsewhere, and by Russell and Whitehead in Princip...

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

본 연구는 수학철학의 핵심적인 두 유파는 구성주의와 구조주의라는 것을 논증하면서, 양자는 다시 구조-구성주의라는 하나의 유파로 종합될 수 있으며 종합되어야 한다는 것을 논증하고자 한다. 이를 위해 역사-비판적 검토를 수행한다. 역사-비판적 관점에서 볼 때, 수학철학은 광의의 관점에서 10여개의 유파로 구분되지만, 세부적으로는 20여개 이상으로 세분화된다. 그러나 가장 큰 흐름은 수학적 대상들은 인간의 정신으로부터 독립적이고 물질적 대상들로부터도 독립적으로 존재하는 “추상적 실체”라고 주장하는 플라톤주의이고, 이것에 대립적인 것은 심리학적 구성주의이다. 후자에 따르면, 수학적 대상들은 인간정신의 구성의 산물이다. 그러나 여기서 말하는 구성의 근거가 논리적 장치인가 아니면 기호적 상징체계인가 (...) 아니면 경험적 감각-지각인가에 따라, 논리적 구성주의(논리주의), 형식적 구성주의(형식주의), 경험적 구성주의(직관주의) 등으로 세분화된다. 그런데 이러한 3가지 유파들 자체 다시 다양한 변이들로 세분화되는데, 그 중에서 20세기 중반부터 부각되기 시작한 구조주의가 가장 많은 지지를 받고 있다. 구조주의는 수학적 연구의 대상은 더 이상 개별적 대상이 아니라, 구조라는 것을 강조한다. 그러나 구조 그 자체 선험적 실체인가 아니면 인식주체의 창의적 구성의 산물인가에 따라 다시 비제거적 구조주의와 제거적 구조주의로 양분된다. 이런 맥락에서 필자는 구조 그 자체 선험적 실체가 아니라, 인식주관의 구성의 결과라는 것을 주장하고자 한다. 이것이 바로 필자가 정당화하고자 하는 “구조-구성주의”이다. (shrink)

Bertrand Russell's contributions to last century's philosophy and, in particular, to the philosophy of mathematics cannot be overestimated.Russell, besides being, with Frege and G.E. Moore, one of the founding fathers of analytical philosophy, played a major rôle in the development of logicism, one of the oldest and most resilient1 programmes in the foundations of mathematics.Among his many achievements, we need to mention the discovery of the paradox that bears his name and the identification of its logical nature; (...) the generalization to the whole of mathematics of Frege's idea that it is not possible to draw a demarcation line between logic and arithmetic; the programme, carried out with Whitehead, of derivation of mathematics from the logical system of Principia Mathematica ; and the ramified theory of types, devised by Russell to protect the system of PM from the known paradoxes.Although there is an ample literature on these topics, it is quite important to reconsider Russell's contributions to the foundations of mathematics at a time when, as a consequence of the crisis of the classical programmes in the foundations of mathematics, new trends are beginning to develop within the philosophy of mathematics. These are trends which move in a very different direction from that of logicism, intuitionism, and Hilbert's programme.To see this we need to consider that, in spite of profound disagreements on the nature of mathematical activity, on the relationship existing between logic and mathematics, on the causes of and therapies for the paradoxes, etc., logicism, intuitionism, and Hilbert's programme share an important metaphor: the idea that mathematics is an edifice built on unshakable foundations,2 an edifice which makes possible only a cumulative growth of mathematical knowledge.Such a metaphor—which, together with more specific theses belonging to these schools of thought, remained unsupported …. (shrink)

Adapated from talks at the UCLA Logic Center and the Pitt Philosophy of Science Series. Exposition of material from Fixing Frege, Chapter 2 (on predicative versions of Frege’s system) and from “Protocol Sentences for Lite Logicism” (on a form of mathematical instrumentalism), suggesting a connection. Provisional version: references remain to be added. To appear in Mathematics, Modality, and Models: Selected Philosophical Papers, coming from Cambridge University Press.

The question, “Was Aristotle the first logicist?”, may appear anachronistic and elicit skepticism since the doctrine of logicism as a fully-fledged idea emerged only in the nineteenth century in the context of the debates surrounding the foundation of mathematics. Indeed, Bertrand Russell credits Gottlob Frege with being the first in “logicising” mathematics (Russell 1919, p. 7), where the thesis espouses that mathematical concepts and propositions are ultimately reducible to or derivable from a number of fundamental logical concepts (...) and principles. However, anachronistic appearances aside, in a fresh reexamination of some of the specific Aristotelian texts in Metaphysics and Prior Analytics, and especially focusing on Aristotle’s particular remarks on the status and significance of the principle of non-contradiction, one may textually argue for a nascent and burgeoning form of logicism in Aristotle, albeit within a much larger metaphysical context than mathematics. (shrink)

The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these (...) ideas by the principal figures in the history of the subject - Frege, Russell, Ramsey and Carnap - and in doing so illuminate current concerns about the nature of mathematical and theoretical knowledge. Issues addressed include the nature of arithmetical knowledge in the light of Frege's theorem; the status of realism about the theoretical entities of physics; and the proper interpretation of empirical theories that postulate abstract structural constraints. (shrink)

Logicism and the Philosophy of Language brings together the core works by Gottlob Frege and Bertrand Russell on logic and language. In their separate efforts to clarify mathematics through the use of logic in the late nineteenth and early twentieth century, Frege and Russell both recognized the need for rigorous and systematic semantic analysis of language. It was their turn to this style of analysis that would establish the philosophy of language as an autonomous area of inquiry. This (...) anthology gathers together these foundational writings, and frames them with an extensive historical introduction. This is a collection for anyone interested in questions about truth, meaning, reference, and logic, and in the application of formal analysis to these concepts. (shrink)

The period in the foundations of mathematics that started in 1879 with the publication of Frege's Begriffsschrift and ended in 1931 with Gödel's Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I can reasonably be called the classical period. It saw the development of three major foundational programmes: the logicism of Frege, Russell and Whitehead, the intuitionism of Brouwer, and Hilbert's formalist and proof-theoretic programme. In this period, there were also lively exchanges between the various schools (...) culminating in the famous Hilbert-Brouwer controversy in the 1920s. -/- The purpose of this anthology is to review the programmes in the foundations of mathematics from the classical period and to assess their possible relevance for contemporary philosophy of mathematics. What can we say, in retrospect, about the various foundational programmes of the classical period and the disputes that took place between them? To what extent do the classical programmes of logicism, intuitionism and formalism represent options that are still alive today? These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics. (shrink)

PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is (...) radically alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts. (shrink)

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence (...) of individuals rather than on the existence of concepts. (shrink)

We argue that logicism, the thesis that mathematics is reducible to logic and analytic truths, is true. We do so by (a) developing a formal framework with comprehension and abstraction principles, (b) giving reasons for thinking that this framework is part of logic, (c) showing how the denotations for predicates and individual terms of an arbitrary mathematical theory can be viewed as logical objects that exist in the framework, and (d) showing how each theorem of a mathematical theory (...) can be given an analytically true reading in the logical framework. (shrink)

Emmy Noether’s many articles around the time that Felix Klein and David Hilbert were arranging her invitation to Göttingen include a short but brilliant note on invariants of finite groups highlighting her creativity and perspicacity in algebra. Contrary to the idea that Noether abandoned Paul Gordan’s style of mathematics for Hilbert’s, this note shows her combining them in a way she continued throughout her mature abstract algebra.

A little over one hundred years ago , Frege wrote to Russell in the following terms1: I myself was long reluctant to recognize ranges of values and hence classes; but I saw no other possibility of placing arithmetic on a logical foundation. But the question is how do we apprehend logical objects? And I have found no other answer to it than this, We apprehend them as extensions of concepts, or more generally, as ranges of values of functions. I have (...) always been aware that there are difficulties connected with this, and your discovery of the contradiction has added to them; but what other way is there? (shrink)

Throughout most of his career, Rudolf Carnap attempted to articulate an empiricist view. Central to this project is the understanding of how empiricism can be made compatible with abstract objects that seem to be invoked in mathematics. In this paper, I discuss and critically evaluate three moves made by Carnap to accommodate mathematical talk within his empiricist program: the “weak logicism” in the Aufbau; the combination of formalism and logicism in the Logische Syntax; and the distinction between (...) internal and external question characteristic of Carnap’s involvement with modality. As a result, the clear interplay between Carnap’s philosophy of science and his work in the philosophy of mathematics will emerge, as well as some challenges that need to be overcome along the way. (shrink)

The rather unrestrained use of second-order logic in the neo-logicist program is critically examined. It is argued in some detail that it brings with it genuine set-theoretical existence assumptions and that the mathematical power that Hume’s Principle seems to provide, in the derivation of Frege’s Theorem, comes largely from the ‘logic’ assumed rather than from Hume’s Principle. It is shown that Hume’s Principle is in reality not stronger than the very weak Robinson Arithmetic Q. Consequently, only a few rudimentary facts (...) of arithmetic are logically derivable from Hume’s Principle. And that hardly counts as a vindication of logicism. (shrink)

In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, (...) I argue that a nominalist can use the resulting approach to provide a nominalization strategy for mathematics. In this way, mathematical structures can be introduced without ontological costs. And so, if this proposal is correct, we can say that ultimately all the nominalist needs is logic (and, rather loosely, ali the logicist needs is nominalism). (shrink)

What is at stake philosophically for Russell in espousing logicism? I argue that Russell's aims are chiefly epistemological and mathematical in nature. Russell develops logicism in order to give an account of the nature of mathematics and of mathematical knowledge that is compatible with what he takes to be the uncontroversial status of this science as true, certain and exact. I argue for this view against the view of Peter Hylton, according to which Russell uses logicism (...) to defend the unconditional truth of mathematics against various Idealist positions that treat mathematics as true only partially or only relative to a particular point of view. (shrink)

The paper discusses Husserl’s criticism of Frege in Philosophy of Arithmetic (1891) and then his later attitude towards logicism as expressed in Logical Investigations (1900-01). In Philosophy of Arithmetic Husserl holds that logicists offer needless and artificial definitions of notions such as equivalence and number. Frege criticized Husserl’s approach in Philosophy of Arithmetic as psychological, thus shifting the focus of the debate away from logicism. However, Frege’s criticism could be seen to lead Husserl to his later transcendental phenomenological (...) concept of correlation and, with it, to a more developed criticism of logicism. In this regard the division of labor between mathematics and philosophy established in Prolegomena to the Logical Investigations (1900) is central. It implies that in Husserl’s view formal methods fail to shed philosophical insight into the essence of mathematics. In his picture the central notions in need of philosophical clarification are, instead of number and quantity, the notions that are central to abstract structural mathematics. Contrary to Frege who can only elucidate the primitive concepts, Husserl engages in detailed description of their constitution. (shrink)

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving (...) from Gödel’s incompleteness theorem and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable. (shrink)

The aim of the paper is to present the main trends and tendencies in the philosophy of mathematics in the 20th century. To make the analysis more clear we distinguish three periods in the development of the philosophy of mathematics in this century: (1) the first thirty years when three classical doctrines: logicism, intuitionism and formalism were formulated, (2) the period from 1931 till the end of the fifties - period of stagnation, and (3) from the beginning (...) of the sixties till today when new tendencies putting stress on the knowing subject and the research practice of mathematicians arose. (shrink)

One important achievement Rudolf Carnap claimed for his book, The Logical Syntax of Language, was that it effected a synthesis of two seemingly antithetical philosophies of mathematics, logicism and formalism. Reconciling these widely divergent conceptions had been a goal of Carnap’s for several years. But in the years in which Carnap’s synthesis evolved, important intellectual developments influenced the direction of his efforts and, ultimately, the final outcome. These developments were, first of all, the epoch-making theorems proved by Kurt (...) Gödel, which required the abandonment of several theses central to the aims of logicism and formalism. Of far greater significance, in the present context, are the changes in Carnap’s own philosophical outlook, brought about not only by Gödel’s theorems but concurrent discussions within the Vienna Circle as well as his own researches. Consequently, the exact sense in which Carnap attempted the synthesis of logicism and formalism in the Logical Syntax requires careful examination. In what follows below, the evolution of Carnap’s synthesis will be traced, from the first reconciliation he proposed , through the synthesis that appeared with the publication of The Logical Syntax of Language. The aim is to determine which modifications of Carnap’s synthesis were required by Gödel’s theorems, and which were motivated by changes in his own thinking. Although the characteristic theses of both logicism and formalism required profound modifications because of Gödel’s theorems, the philosophical impulses that originally fueled their programs retained much of their former virulence. But the changes in Carnap’s thought that ocurred in the years he was developing his synthesis especially affected his appreciation of the philosophical motivations underwriting the logicist approach, so that much of the philosophical insight that inspired it is lost, and Carnap’s combination of logicism and formalism is a putative synthesis at best. (shrink)

This paper discusses Carnap’s attempts in the late 1920s to provide a formal reconstruction of modern axiomatics.1 One interpretive theme addressed in recent scholarly literature concerns Carnap’s underlying logicism in his philosophy of mathematics from that time, more specifically, his attempt to “reconcile” the logicist approach of reducing mathematics to logic with the formal axiomatic method. For instance, Awodey & Carus characterize Carnap’s manuscript Untersuchungen zur allgemeinen Axiomatik from 1928 as a “large-scale project to reconcile axiomatic definitions (...) with logicism, and transform implicit into explicit definitions.” It is argued that Carnap’s central idea was to balance a Fregean foundational stance with the modern model-theoretic viewpoint introduced in Hilbert’s Grundlagen der Geometrie ). It was also shown in recent literature that Carnap’s attempt to provide a logicist reconstruction of axiomatics is limited in several ways.2 No closer attention, however, has so far been dedicated to some of the details of his proposed reconciliation. (shrink)

Abstractionism, which is a development of Frege's original Logicism, is a recent and much debated position in the philosophy of mathematics. This volume contains 16 original papers by leading scholars on the philosophical and mathematical aspects of Abstractionism. After an extensive editors' introduction to the topic of abstractionism, the volume is split into 4 sections. The contributions within these sections explore the semantics and meta-ontology of Abstractionism, abstractionist epistemology, the mathematics of Abstractionis, and finally, Frege's application constraint (...) within an abstractionist setting. (shrink)

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change (...) as the subject develops? What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century. (shrink)

A detailed discussion of later Wittgenstein's take on the logicism of Frege and Russell. The focus is on Wittgenstein's views in his 1939 Lectures on the Foundations of Mathematics.

In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations$\Omega ^{n}(\bar {\xi })$defined with the help of numeral exponents. Since$Num$(x) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with “insuperable difficulties.” This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that theN-operator$N(\bar {\xi })$and recursively defined arithmetic (...) operators$\Omega ^{n}(\bar {\xi })$are not different in kind, we can address Ramsey’s problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction. (shrink)

Certain advocates of the so-called “neo-logicist” movement in the philosophy of mathematics identify themselves as “neo-Fregeans” (e.g., Hale and Wright), presenting an updated and revised version of Frege’s form of logicism. Russell’s form of logicism is scarcely discussed in this literature and, when it is, often dismissed as not really logicism at all (in light of its assumption of axioms of infinity, reducibility and so on). In this paper I have three aims: firstly, to identify more (...) clearly the primary meta-ontological and methodological differences between Russell’s logicism and the more recent forms; secondly, to argue that Russell’s form of logicism offers more elegant and satisfactory solutions to a variety of problems that continue to plague the neo-logicist movement (the bad company objection, the embarrassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that neo-Russellian forms of logicism remain viable positions for current philosophers of mathematics. (shrink)

This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. The first section focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of (...) such figures in the philosophical and mathematical milieu in which logicist views were first expounded. Section II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, the third section assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics--neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition--which define the prospects of logicism in the years to come. This book represents a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy. (shrink)

Bertrand Russell was one of the best-known proponents of logicism: the theory that mathematics reduces to, or is an extension of, logic. Russell argued for this thesis in his 1903 The Principles of Mathematics and attempted to demonstrate it formally in Principia Mathematica (PM 1910–1913; with A. N. Whitehead). Russell later described his work as a further “regressive” step in understanding the foundations of mathematics made possible by the late 19th century “arithmetization” of mathematics and (...) Frege’s logical definitions of arithmetical concepts. The logical system of PM sought to improve on earlier attempts by solving the contradictions found in, e.g., Frege’s system, by employing a theory of types. In this article, I also consider and critically evaluate the most common objections to Russell’s logicism, including the claim that it is undermined by Gödel’s incompleteness results, and Putnam's charge of “if-thenism”. I suggest that if we are willing to accept a slightly revisionist account of what counts as a mathematical truth, these criticisms do not obviously refute Russell’s claim to have established that mathematical truths generally are a species of logical truth. (shrink)

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In (...) Sect. 3, I raise what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

The most prominent “schools” or programs for the foundations of mathematics that took shape in the first third of the 20th century emerged directly from, or in response to, developments in mathematics and logic in the latter part of the 19th century. The first of these programs, so-called logicism, had as its aim the reduction of mathematics to purely logical principles. In order to understand properly its achievements and resulting problems, it is necessary to review the (...) background from that previous period. (shrink)

Wittgenstein’s philosophy of mathematics is often devalued due to its peculiar features, especially its radical departure from any of standard positions in foundations of mathematics, such as logicism, intuitionism, and formalism. We first contrast Wittgenstein’s finitism with Hilbert’s finitism, arguing that Wittgenstein’s is perspicuous or surveyable finitism whereas Hilbert’s is transcendental finitism. We then further elucidate Wittgenstein’s philosophy by explicating his natural history view of logic and mathematics, which is tightly linked with the so-called rule-following problem (...) and Kripkenstein’s paradox, yielding vital implications to the nature of mathematical understanding, and to the nature of the certainty and objectivity of mathematical truth. Since the incompleteness theorems, foundations of mathematics have mostly lost their philosophical driving force, and anti-foundationalism has become prevalent and pervasive. Yet new foundations have nevertheless emerged and come into the scene, namely categorical foundations. We articulate the foundational significance of category theory by explicating three forms of foundations, i.e., global foundations, local foundations, and conceptual foundations. And we explore the possibility of categorical unified science qua pluralistic unified science, arguing for the categorical unity of science on both mathematical and philosophical grounds. We then turn to an issue in conceptual foundations of mathematics, namely the nature of the concept of space. We elucidate Wittgenstein’s intensional conception of space in relation to Brouwer’s theory of space continua and to the modern conception of space as point-free structure in category theory and algebraic geometry. We finally give a bird’s-eye view of mathematical philosophy from a Wittgensteinian perspective, and further sheds new light on Wittgenstein’s constructive structuralism and his view of incompleteness and contradictions in mathematics. (shrink)

In Part III of his Remarks on the Foundations of Mathematics Wittgenstein deals with what he calls the surveyability of proofs. By this he means that mathematical proofs can be reproduced with certainty and in the manner in which we reproduce pictures. There are remarkable similarities between Wittgenstein's view of proofs and Hilbert's, but Wittgenstein, unlike Hilbert, uses his view mainly in critical intent. He tries to undermine foundational systems in mathematics, like logicist or set theoretic ones, by (...) stressing the unsurveyability of the proof-patterns occurring in them. Wittgenstein presents two main arguments against foundational endeavours of this sort. First, he shows that there are problems with the criteria of identity for the unsurveyable proof-patterns, and second, he points out that by making these patterns surveyable, we rely on concepts and procedures which go beyond the foundational frameworks. When we take these concepts and procedures seriously, mathematics does not appear as a uniform system, but as a mixture of different techniques. (shrink)