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)
Frege's logicism consists of two theses: (1) the truths of arithmetic are truths of logic; (2) the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G (...) are equinumerous is the central plank in the neo-logicist argument for (1) and (2). I defend this position against two objections (a) Hume's principle canot be both a logical truth as required by (1) and also have the ontological import required by (2); and (b) the use of Hume's principle by the logicist is in effect an ontological proof of a kind which is not valid. (shrink)
This paper has two separate aims, with obvious links between them. First, to present Charles S. Peirce and the pragmatist movement in a historical framework which stresses the close connections of pragmatism with the mainstream of philosophy; second, to deal with a particular controversial issue, that of the supposed logicistic orientation of Peirce's work.
Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)
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 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. The irresistible metaphor is that pure abstract objects [...] are no more than shadows cast by the syntax of our discourse. And the aptness of the metaphor is enhanced by the reflection that shadows are, after their own fashion, real. (Crispin Wright [1992], p. 181-2) But I feel conscious that many a reader will scarcely recognise in the shadowy forms which I bring before him his numbers which all his life long have accompanied him as faithful and familiar friends; (Richard Dedekind [1963], p. 33). (shrink)
Simple-type theory is widely regarded as inadequate to capture the metaphysics of mathematics. The problem, however, is not that some kinds of structure cannot be studied within simple-type theory. Even structures that violate simple-types are isomorphic to structures that can be studied in simple-type theory. In disputes over the logicist foundations of mathematics, the central issue concerns the problem that simple-type theory fails to assure an infinity of natural numbers as objects . This paper argues that the problem of infinity (...) is based on a metaphysical prejudice in favor of numbers as objects — a prejudice that mathematics can get along without. (shrink)
The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)
In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
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?’ (16–20 April 2007), 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)
Frege's logicism consists of two theses: (1) the truths of arithmetic are truths of logic; (2) the natural numbers are objects. In this paper I pose the question: what conception of logic is required to defend these theses? I hold that there exists an appropriate and natural conception of logic in virtue of which Hume's principle is a logical truth. Hume's principle, which states that the number of Fs is the number of Gs iff the concepts F and G (...) are equinumerous is the central plank in the neo-logicist argument for (1) and (2). I defend this position against two objections (a) Hume's principle canot be both a logical truth as required by (1) and also have the ontological import required by (2); and (b) the use of Hume's principle by the logicist is in effect an ontological proof of a kind which is not valid. (shrink)
This paper introduces and evaluates two contemporary approaches of neo-logicism. Our aim is to highlight the differences between these two neo-logicist programmes and clarify what each projects attempts to achieve. To this end, we first introduce the programme of the Scottish school – as defended by Bob Hale and Crispin Wright1 which we believe to be a..
The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? (...) 2 Framework 3 GOOD candidates, indefinite extensibility 4 The framework of (RV) alone, or almost alone 5 The axioms 6 Brief closing. (shrink)
Harold Jeffreys' ideas on the interpretation of probability and epistemology are reviewed. It is argued that with regard to the interpretation of probability, Jeffreys embraces a version of logicism that shares some features of the subjectivism of Ramsey and de Finetti. Jeffreys also developed a probabilistic epistemology, characterized by a pragmatical and constructivist attitude towards notions such as ‘objectivity’, ‘reality’ and ‘causality’. 1 Introductory remarks 2 The interpretation of probability 3 Jeffreys' probabilistic epistemology.
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.
Most advocates of the so-called “neologicist” 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 lights of its assumption of axioms of infinity, reducibiity and so on). In this paper I have three aims: firstly, to identify more clearly (...) the primary metaontological 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 embarassment of richness objection, worries about a bloated ontology, etc.); thirdly, to argue that Neo- Russellian forms of neologicism remain viable positions for current philosophers of mathematics. (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 (...) class='Hi'>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)
You’ll be pleased to know that I don’t intend to use these remarks to comment on all of the papers presented at this conference. I won’t try to show that one paper was right about this topic, that another was wrong was about that topic, or that several of our conference participants were talking past one another. Nor will I try to adjudicate any of the discussions which took place in between our sessions. Instead, I’ll use these remarks to make (...) two simple points: one about logicism in the 20th century and one about neo-logicism here at the start of the 21st. (shrink)
In this paper I examine the prospects for a successful neo–logicist reconstruction of the real numbers, focusing on Bob Hale's use of a cut-abstraction principle. There is a serious problem plaguing Hale's project. Natural generalizations of this principle imply that there are far more objects than one would expect from a position that stresses its epistemological conservativeness. In other words, the sort of abstraction needed to obtain a theory of the reals is rampantly inflationary. I also indicate briefly why this (...) problem is likely to reappear in any neo–logicist reconstruction of real analysis. (shrink)
Logicism Lite counts number‐theoretical laws as logical for the same sort of reason for which physical laws are counted as as empirical: because of the character of the data they are responsible to. In the case of number theory these are the data verifying or falsifying the simplest equations, which Logicism Lite counts as true or false depending on the logical validity or invalidity of first‐order argument forms in which no numbertheoretical notation appears.
An outline is given of the development of logicism from the publication of the first edition of Whitehead and Russell's Principia mathematica (1910-1913) through the contributions of Wittgenstein, Ramsey and Chwistek to Russell's own modifications made for the second edition of the work (1925) and the adoption of many of its logical techniques by the Vienna Circle. A tendency towards extensionalism is emphasised.
The aim here is to describe how to complete the constructive logicist program, in the author’s book Anti-Realism and Logic, of deriving all the Peano-Dedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neo-Fregean done so.
According to the received view, Russell rediscovered about 1900 the logical definition of cardinal number given by Frege in 1884. In the same way, we are told, he stated and developed independently the idea of logicism, using the principle of abstraction as the philosophical ground. Furthermore, the role commonly ascribed in this to Peano was only to invent an appropriate notation to be used as mere instrument. In this paper I hold that the study of Russell's unpublished manuscripts and (...) Peano's (and disciples) writings (as part of a larger investigation only pointed out here) shows, on the contrary, that Russell obtained the method to transform definitions by abstraction into nominal definitions and the general logicist idea from Peano (and his school). The only original insight from Russell, the principle of abstraction, partially derived from Moore's early philosophy, was finally abandoned because it was not of practical use. (shrink)
The present paper is a contribution to the history of logic and its philosophy toward the mid-20th century. It examines the interplay between logic, type theory and set theory during the 1930s and 40s, before the reign of first-order logic, and the closely connected issue of the fate of logicism. After a brief presentation of the emergence of logicism, set theory, and type theory (with particular attention to Carnap and Tarski), Quine’s work is our central concern, since he (...) was seemingly the most outstanding logicist around 1940, though he would shortly abandon that viewpoint and promote first-order logic as all of logic. Quine’s class-theoretic systems NF and ML, and his farewell to logicism, are examined. The last section attempts to summarize the motives why set theory was preferred to other systems, and first orderlogic won its position as the paradigm logic system after the great War. (shrink)
The faults of logical empiricist accounts of metascientific discourse are examined through a study of the modifications Carnap makes to his version of the program over four decades. As empiricists acquiesced on the distinction between theory and observation, Carnap attempted to retain and insulate an equally suspect sharp distinction between the theoretic and the pragmatic. Carnap's later philosophy was understood as a modification of the program in the direction of pragmatism. But neither the key notion of "external questions" nor an (...) instrumentalist understanding of "pragmatic utility" are genuinely compatible with pragmatism. This underlines the need to clarify what is unique to pragmatic views of cognitive evaluation and normative discourse as they affect scientific reasoning. I conclude by suggesting how pragmatic conceptions of metascientific discourse can work to correct misconceptions about norm generation and governance that are shared in logicist and historicist accounts. (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)
These questions are addressed in this volume by leading mathematical logicians and philosophers of mathematics.A special section is concerned with constructive ...
Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out (...) to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’. (shrink)
Inferentialism is explained as an attempt to provide an account of meaning that is more sensitive (than the tradition of truth-conditional theorizing deriving from Tarski and Davidson) to what is learned when one masters meanings.
Although the notion of logical object plays a key role in Frege's foundational project, it has hardly been analyzed in depth so far. I argue that Marco Ruffino's attempt to fill this gap by establishing a close link between Frege's treatment of expressions of the form ‘the concept F’ and the privileged status Frege assigns to extensions of concepts as logical objects is bound to fail. I argue, in particular, that Frege's principal motive for introducing extensions into his logical theory (...) is not to be able to make in-direct statements about concepts, but rather to define all numbers as logical objects of a fundamental kind in order to ensure that we have the right cognitive access to them qua logical objects via Axiom V. Contrary to what Ruffino claims, reducibility to extensions cannot be the ‘ultimate criterion’ for Frege of what is to be regarded as a logical object. (shrink)
Unless you are a Frege scholar, or a philosopher of mathematics, if you are familiar at all with Frege’s work, you are most likely familiar with his groundbreaking work in the philosophy of language. You might know that Frege was a mathematician who sought to establish the covertly logical subject matter of arithmetic, a project whose demands drove Frege to his logical investigations and reflections on language. But most likely the connection between Frege’s mathematical research and his philosophy of language (...) remains elusive for you. (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)
In the opening sentence of his Methods of Logic, W. V. O. Quine writes, “Logic is an old subject, and since 1879 it has been a great one.”1 Quine is referring to the year in which Gottlob Frege presented his Begriffschrift, or “concept-script,” one of the first published accounts of a logical system or calculus with quantification and a function-argument analysis of propositions. There can be no doubt as to the importance of these introductions, and, indeed, Frege’s orientation and advances, (...) if not his particular system, have proven to be highly significant for much of mathematical logic and research pertaining to the foundations of mathematics. Nevertheless, Quine’s assessment, often repeated, obscures other .. (shrink)
Self‐evidently the standard work on the topic its whole title defines, Sir Michael Dummett’s Frege: Philosophy of Mathematics (FPM) is also the most profound and creative discussion in recent decades of the problems confronting the branch of philosophy mentioned after the colon. Chapters 14‐18 and 23‐24 of this book constitute a continuous and challenging diagnosis of these problems.1 They culminate in the proposal that these problems present an impasse that can be escaped only by adopting a constructivist understanding of mathematical (...) generality. Dummett’s case for that conclusion is no less complexly over‐layered than the problems themselves. By contrast my aims in this discussion of his case are limited in various ways, and three of these should be mentioned straightaway. In the first place, I will aim to consider a case that, if sound, would warrant a constructivist understanding of generality in mathematics generally (and so I will not be considering lines of argument specific to set theory, or to those parts of mathematics plausibly dependent on notions intrinsic to set theory). Secondly, I aim to consider a case which, while general in its application within mathematics, is not more general than that (and so would not warrant a broader anti‐realism). Reasons for these first two limitations are discussed in section 1. A third limitation is that I will aim only to understand Dummett’s case, and not to assess it. Perhaps some will think this third limitation calls for explanation or excuse. I think it needs no excuse and that the explanation is obvious. When we are dealing with fundamentally important work by a great philosopher, understanding is often ambition enough. In Michael Dummett’s work, that is what we are dealing with. (shrink)
With the aid of a non-standard (but still first-order) cardinality quantifier and an extra-logical operator representing numerical abstraction, this paper presents a formalization of first-order arithmetic, in which numbers are abstracta of the equinumerosity relation, their properties derived from those of the cardinality quantifier and the abstraction operator.
The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)
In the first two decades of the century Vvedenskij developed and defended what he took to be an original argument in support of the impossibility of metaphysical knowledge. This argument, which he hailed as a "proof," involved an examination of the four laws of thought alone. As it made no appeal to the highly technical analyses found in Kant's first Critique, Vvedenskij considered it to be more efficient and thereby effective than Kant's own arguments. Although Vvedenskij's estimation of his accomplishment (...) actually increased with the passage of time, the âproofâ rested on highly dubious assumptions. (shrink)
In the first two decades of the century Vvedenskij developed and defended what he took to be an original argument in support of the impossibility of metaphysical knowledge. This argument, which he hailed as a proof, involved an examination of the four laws of thought alone. As it made no appeal to the highly technical analyses found in Kant''s first Critique, Vvedenskij considered it to be more efficient and thereby effective than Kant''s own arguments. Although Vvedenskij''s estimation of his accomplishment (...) actually increased with the passage of time, the proof rested on highly dubious assumptions. (shrink)
Frege's analysis of arithmetical knowledge -- Carnap's thesis -- On extending 'empiricism, semantics and ontology' to the realism-instrumentalism controversy -- Carnap's analysis of realism -- Bertrand Russell's The analysis of matter: its historical context and contemporary interest (with Michael Friedman) -- On the rational reconstruction of our theoretical knowledge -- Three views of theoretical knowledge -- Frege and the rigorization of analysis -- The philosophical basis of our knowledge of number -- The 1910 Principia's theory of functions and classes -- (...) Ramsey's extensional propositional functions. (shrink)
Non-empirical principles have always been a subject of interest of philosophers. Authors from different times and traditions agree that principles such as analogy or simplicity are present in the scientific practice. The disagreement comes out when these authors affirm that these principles have an epistemic function, and when they try to present reasons in order to found this statement. The first goal of this paper is to describe these principles and to point out their methodological importance. The second goal is (...) to introduce the discussion between the two most important models of philosophical fundamentation – it means, the logicism and the psychologism. The last goal – the main one – is to defend that the non-empirical principals have, legitimately, an epistemic role in the scientific investigation. (shrink)
For over thirty years I have argued that all branches of science and scholarship would have both their intellectual and humanitarian value enhanced if pursued in accordance with the edicts of wisdom-inquiry rather than knowledge-inquiry. I argue that this is true of mathematics. Viewed from the perspective of knowledge-inquiry, mathematics confronts us with two fundamental problems. (1) How can mathematics be held to be a branch of knowledge, in view of the difficulties that view engenders? What could mathematics be knowledge (...) about? (2) How do we distinguish significant from insignificant mathematics? This is a fundamental philosophical problem concerning the nature of mathematics. But it is also a practical problem concerning mathematics itself. In the absence of the solution to the problem, there is the danger that genuinely significant mathematics will be lost among the unchecked growth of a mass of insignificant mathematics. This second problem cannot, it would seem, be solved granted knowledge-inquiry. For, in order to solve the problem, mathematics needs to be related to values, but this is, it seems, prohibited by knowledge-inquiry because it could only lead to the subversion of mathematical rigour. Both problems are solved, however, when mathematics is viewed from the perspective of wisdom-inquiry. (1) Mathematics is not a branch of knowledge. It is a body of systematized, unified and inter-connected problem-solving methods, a body of problematic possibilities. (2) A piece of mathematics is significant if (a) it links up to the interconnected body of existing mathematics, ideally in such a way that some problems difficult to solve in other branches become much easier to solve when translated into the piece of mathematics in question; (b) it has fruitful applications for (other) worthwhile human endeavours. If ever the revolution from knowledge to wisdom occurs, I would hope wisdom mathematics would flourish, the nature of mathematics would become much more transparent, more pupils and students would come to appreciate the fascination of mathematics, and it would be easier to discern what is genuinely significant in mathematics (something that baffled even Einstein). As a result of clarifying what should count as significant, the pursuit of wisdom mathematics might even lead to the development of significant new mathematics. (shrink)
Due programmi diversi si intersecano nel lavoro di Frege sui fondamenti dell’aritmetica: • Logicismo: l’aritmetica `e riducibile alla logica; • Estensionalismo: l’aritmetica `e riducibile a una teoria delle estensioni. Sia nei Fondamenti che nei Principi, Frege articola l’idea che l’aritmetica sia riducibile a una teoria logica delle estensioni.
Nicolai Hartmann contributed significantly to the revitalization of the discipline of ontology in the early twentieth century. Developing a systematic, post-Kantian critical ontology ‘this side’ of idealism and realism, he subverted the widespread impression that philosophy must either exhaust itself in foundationalist epistemology or engage in system-building metaphysical excess. This essay provides an introduction to Hartmann’s approach in light of the recent translation of his early essay ‘How is Critical Ontology Possible?’ ( 1923 ) In it Hartmann criticizes both the (...) pretensions of epistemology as well as the principal errors of classical ontology, and he proposes a series of correctives that lead to his development of a highly original and elaborate stratified categorial ontology. This introduction explains the most important errors of the ‘old’ ontology, his correctives to them, and further fleshes out these correctives with reference to his mature ontological work. (shrink)
Fitch's basic logic is an untyped illative combinatory logic with unrestricted principles of abstraction effecting a type collapse between properties (or concepts) and individual elements of an abstract syntax. Fitch does not work axiomatically and the abstraction operation is not a primitive feature of the inductive clauses defining the logic. Fitch's proof that basic logic has unlimited abstraction is not clear and his proof contains a number of errors that have so far gone undetected. This paper corrects these errors and (...) presents a reasonably intuitive proof that Fitch's system K supports an implicit abstraction operation. Some general remarks on the philosophical significance of basic logic, especially with respect to neo-logicism, are offered, and the paper concludes that basic logic models a highly intensional form of logicism. (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)
... as 'logicism') that the content expressed by true propositions of arithmetic and analysis is not something of an irreducibly mathematical character, ...
Ontology is the study of what there is, what kinds of things make up reality. Ontology seems to be a very difficult, rather speculative discipline. However, it is trivial to conclude that there are properties, propositions and numbers, starting from only necessarily true or analytic premises. This gives rise to a puzzle about how hard ontological questions are, and relates to a puzzle about how important they are. And it produces the ontologyobjectivity dilemma: either (certain) ontological questions can be trivially (...) answered using only uncontroversial premises, or the uncertainties of ontology are really a threat to the truth of basically everything we say or believe. The main aim of this dissertation is to resolve these puzzles and to shed some light on the discipline of ontology. I defend a view inspired by Carnap’s internal-external distinction about what there is, but one according to which both internal and external questions are fully factual and meaningful. In particular, I argue that the trivial arguments are valid, but they do not answer any ontological questions. Furthermore, I propose an account of the function of our talk about properties, propositions and natural numbers. According to this account our talk about them has no ontological presuppositions for its literal and objective truth. This avoids the ontology-objectivity dilemma, and solves the puzzles about ontology. To do this I look at quantification and noun phrases in general, and at their relation to ontology. I argue that quantifiers are semantically underspecified in a certain respect, and play two different roles in communication. I discuss the relation between syntactic form and information structure, the function of certain non-referential, non-quantificational noun phrases, the uses of bare number determiners, and how arithmetic truths are learned and taught. The more metaphysical issues discussed include: inexpressible properties, logicism about arithmetic, nominalism, Carnap’s view about ontology, the problem of universals, the relationship between ontology and objectivity, different projects within ontology, non-existent.... (shrink)
Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, (...) in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)
This article treats three aspects of Frege's discussions of definitions. First, I survey Frege's main criticisms of definitions in mathematics. Second, I consider Frege's apparent change of mind on the legitimacy of contextual definitions and its significance for recent neo-Fregean logicism. In the remainder of the article I discuss a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend: do the logical structures of the definientia reflect the understanding of arithmetical terms prevailing prior (...) to Frege's analyses? Unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. I conclude by examining some options for conceiving of the status of Frege's logicism in light of this apparent tension, and outline a suggestion for a philosophically fruitful way of resolving this tension. (shrink)
According to Kant, pure intuition is an indispensable ingredient of mathematical proofs. Kant‘s thesis has been considered as obsolete since the advent of modern relational logic at the end of 19th century. Against this logicist orthodoxy Cassirer’s “critical idealism” insisted that formal logic alone could not make sense of the conceptual co-evolution of mathematical and scientific concepts. For Cassirer, idealizations, or, more precisely, idealizing completions, played a fundamental role in the formation of the mathematical and empirical concepts. The aim of (...) this paper is to outline the basics of Cassirer’s idealizational account, and to point at some interesting similarities it has with Kant’s and Peirce’s philosophies of mathematics based on the key notions of pure intuition and theorematic reasoning, respectively. (shrink)
[Ian Rumfitt] Frege's logicism in the philosophy of arithmetic consisted, au fond, in the claim that in justifying basic arithmetical axioms a thinker need appeal only to methods and principles which he already needs to appeal in order to justify paradigmatically logical truths and paradigmatically logical forms of inference. Using ideas of Gentzen to spell out what these methods and principles might include, I sketch a strategy for vindicating this logicist claim for the special case of the arithmetic of (...) the finite cardinals. /// [Timothy Williamson]The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward Löwenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus. (shrink)
I shall make two main claims. My first main claim is that Frege started out with a view of logic that is closer to Kant’s than is generally recognized, but that he gradually came to reject this Kantian view, or at least totally to transform it. My second main claim concerns Frege’s reasons for distancing himself from the Kantian conception of logic. It is natural to speculate that this change in Frege’s view of logic may have been spurred by a (...) desire to establish the logicality of the axiom system he needed for his logicist reduction, including the infamous Basic Law V. I admit this may have been one of Frege’s motives. But I shall argue that Frege also had a deeper and more interesting reason to reject his early Kantian view of logic, having to do with his increasingly vehement anti-psychologism. (shrink)
Frege proposed that his Context Principle—which says that a word has meaning only in the context of a proposition—can be used to explain reference, both in general and to mathematical objects in particular. I develop a version of this proposal and outline answers to some important challenges that the resulting account of reference faces. Then I show how this account can be applied to arithmetic to yield an explanation of our reference to the natural numbers and of their metaphysical status.
Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the (...) question “What is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures. (shrink)
One of the most important philosophical topics in the early twentieth century ? and a topic that was seminal in the emergence of analytic philosophy ? was the relationship between Kantian philosophy and modern geometry. This paper discusses how this question was tackled by the Neo-Kantian trained philosopher Ernst Cassirer. Surprisingly, Cassirer does not affirm the theses that contemporary philosophers often associate with Kantian philosophy of mathematics. He does not defend the necessary truth of Euclidean geometry but instead develops a (...) kind of logicism modeled on Richard Dedekind's foundations of arithmetic. Further, because he shared with other Neo-Kantians an appreciation of the developmental and historical nature of mathematics, Cassirer developed a philosophical account of the unity and methodology of mathematics over time. With its impressive attention to the detail of contemporary mathematics and its exploration of philosophical questions to which other philosophers paid scant attention, Cassirer's philosophy of mathematics surely deserves a place among the classic works of twentieth century philosophy of mathematics. Though focused on Cassirer's philosophy of geometry, this paper also addresses both Cassirer's general philosophical orientation and his reading of Kant. (shrink)
Ordinary English contains different forms of quantification over objects. In addition to the usual singular quantification, as in 'There is an apple on the table', there is plural quantification, as in 'There are some apples on the table'. Ever since Frege, formal logic has favored the two singular quantifiers ∀x and ∃x over their plural counterparts ∀xx and ∃xx (to be read as for any things xx and there are some things xx). But in recent decades it has been argued (...) that we have good reason to admit among our primitive logical notions also the plural quantifiers ∀xx and ∃xx. More controversially, it has been argued that the resulting formal system with plural as well as singular quantification qualifies as ‘pure logic’; in particular, that it is universally applicable, ontologically innocent, and perfectly well understood. In addition to being interesting in its own right, this thesis will, if correct, make plural quantification available as an innocent but extremely powerful tool in metaphysics, philosophy of mathematics, and philosophical logic. For instance, George Boolos has used plural quantification to interpret monadic second-order logic and has argued on this basis that monadic second-order logic qualifies as “pure logic.” Plural quantification has also been used in attempts to defend logicist ideas, to account for set theory, and to eliminate ontological commitments to mathematical objects and complex objects. (shrink)
Three clusters of philosophically significant issues arise from Frege's discussions of definitions. First, Frege criticizes the definitions of mathematicians of his day, especially those of Weierstrass and Hilbert. Second, central to Frege's philosophical discussion and technical execution of logicism is the so-called Hume's Principle, considered in The Foundations of Arithmetic . Some varieties of neo-Fregean logicism are based on taking this principle as a contextual definition of the operator 'the number of …', and criticisms of such neo-Fregean programs (...) sometimes appeal to Frege's objections to contextual definitions in later writings. Finally, a critical question about the definitions on which Frege's proofs of the laws of arithmetic depend is whether the logical structures of the definientia reflect our pre-Fregean understanding of arithmetical terms. It seems that unless they do, it is unclear how Frege's proofs demonstrate the analyticity of the arithmetic in use before logicism. Yet, especially in late writings, Frege characterizes the definitions as arbitrary stipulations of the senses or references of expressions unrelated to pre-definitional understanding. One or more of these topics may be studied in a survey course in the philosophy of mathematics or a course on Frege's philosophy. The latter two topics are obviously central in a seminar in the philosophy of mathematics in general or more specialized seminars on logicism, or on mathematical definitions and concept formation. Author Recommends: 1. Kant, Immanuel. Critique of Pure Reason . Trans. P. Guyer and A. Wood. Cambridge: Cambridge University Press, 1999 [1781, 1787], A7-10/B11-14, A151/B190. In the first Critique , Kant appears to give four distinct accounts of analytic judgments. The initial famous account explains analyticity in terms of the predicate-concept belonging to the subject-concept (A6–7/B11). In this passage, we also find an account of establishing analytic judgments on the basis of conceptual containments and the principle of non-contradiction. (The other accounts are in terms of 'identity' (A7/B1l), in terms of the explicative–ampliative contrast (A7/B11), and by reference to the notion of 'cognizability in accordance with the principle of contradiction' (A151/B190).) 2. Frege, Gottlob. The Foundations of Arithmetic . Trans. J. L. Austin. 2nd ed. Evanston, IL: Northwestern University Press, 1980 [1884], especially sections 1–4, 87–91. Frege here criticizes and reformulates Kant's account of analyticity. Central to Frege's account is the provability of an analytic statement on the basis of (Frege's) logic and definitions that express analyses of (mathematical, especially arithmetical) concepts. 3. Frege, Gottlob. Review of E. G. Husserl. 'Philosophie der Arithmetik I [1894],' in Frege, Collected Papers . Ed. B. McGuinness. Trans. M. Black et al. Oxford: Blackwell, 1984. 195–209. In this review, Frege responds to Husserl's charge that Frege's definitions fail to capture our intuitive pre-analytic arithmetical concepts by claiming that the adequacy of mathematical definitions is measured, not by their expressing the same senses, but merely by their having the same references, as pre-definitional vocabulary. It follows not only that Husserl's criticism is unfounded, but also that there can be alternative, equally legitimate, definitions of mathematical terms. 4. Frege, 'Logic in Mathematics,' in Frege, Posthumous Writings . Trans. P. Long and R. White. Oxford: Blackwell, 1979 [1914]. 203–50. These are a set of lecture notes including, among other things, an account of proper definitions as mere abbreviation of complex signs by simple ones, in contrast to definitions which purport to express the analyses of existing concepts. Frege here claims that if there is any doubt whether a definition purporting to express an analysis succeeds in capturing the senses of the pre-definitional expressions, then the definition fails as an analysis, and should be regarded as the introduction of an entirely new expression abbreviating the definiens . 5. Picardi, Eva. 'Frege on Definition and Logical Proof,' Temi e Prospettive della Logica e della Filosofia della Scienza Contemporanee . i vol. Eds. C. Cellucci and G. Sambin. Bologna: Cooperativa Libraria Universitaria Editrice Bologna, 1988. 227–30. Picardi sets out forcefully the view that unless Frege's definitions capture the meanings of existing arithmetical terms, his logicism cannot have the epistemological significance he takes it to have. 6. Dummett, Michael. 'Frege and the Paradox of Analysis,' in Dummett, Frege and other Philosophers . Oxford: Oxford University Press, 1991. 17–52. Dummett agrees with Picardi's view and analyzes the philosophical pressures that led Frege to the account of definition in 'Logic in Mathematics.' Especially significant is Dummett's claim of the centrality of the transparency of sense – that if one grasps the senses of any two expressions, one must know whether they have the same sense – in Frege's account. 7. Benacerraf, Paul. 'Frege: The Last Logicist,' Midwest Studies in Philosophy . vol. 6. Eds. P. French, T. Uehling, and H. Wettstein. Minneapolis: University of Minnesota Press, 1981. 17–35. Frege's aims, on Benacerraf's reading, are primarily mathematical. Frege was interested in traditional philosophical issues such as the analyticity of arithmetic only to the extent that they can be exploited for the mathematical goal of proving previously unproven arithmetical statements. Hence, Frege never had any serious interest in or need for showing that his definitions of arithmetical terms reflect existing arithmetical conceptions. 8. Weiner, Joan. 'The Philosopher Behind the Last Logicist,' in Frege: Tradition and Influence . Ed. C. Wright. Oxford: Blackwell, 1984. 57–79. Weiner argues that on Frege's view, prior to his definitions of arithmetical terms the references of such expressions are in fact not known by those who use arithmetical vocabulary. Thus, in Foundations , Frege operated with a 'hidden agenda' (263) namely, replacing existing arithmetic with a new science based on stipulative definitions that assign new senses to key arithmetical terms. 9. Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Noûs 29 (1995): 427–67. Tappenden argues that Frege takes his crucial innovation over previous practices and accounts of mathematical concept formation to be the role of quantificational structure made possible by his logical discoveries. 10. Horty, John. Frege on Definitions: A Case Study of Semantic Content . Oxford: Oxford University Press, 2007. A useful interpretation of Frege's views of definition, together with suggestive extensions for resolving the issues framing Frege's views. 11. Shieh, Sanford. 'Frege on Definitions,' Philosophy Compass 3/5 (2008): 992–1012. A more detailed account of Frege's views on definition and the philosophical issues they raise, surveying and discussing critically the main substantive and interpretive issues. Online Materials On Frege http://plato.stanford.edu/entries/frege/ On the Paradox of Analysis http://plato.stanford.edu/entries/analysis/ Sample Syllabus The following is a 3-week module that can be incorporated into fairly focused historically oriented graduate-level seminars on logicism or on the paradox of analysis. It is also possible to compress the material into 2 weeks in an undergraduate or graduate class Frege's thought in general. Week I: Background, Kant on Analyticity; Definition in Foundations , Review of Husserl, and 'Logic in Mathematics' Readings Kant, Immanuel. Critique of Pure Reason , A7–10/B11–14. Frege, Gottlob. The Foundations of Arithmetic , sections 1–4, 87–91. Frege, Gottlob. Review of E. G. Husserl, Philosophie der Arithmetik I. Frege, Gottlob. 'Logic in Mathematics.' Optional Proops, Ian. 'Kant's Conception of Analytic Judgment,' Philosophy and Phenomenological Research LXX, 3 (2005): 588–612. Week II: The Supposed Paradox of Analysis, Picardi and Dummett; Bypassing Traditional Epistemological Issues About Mathematics, Benacerraf Readings Picardi, Eva. 'Frege on Definition and Logical Proof.' Dummett, Michael. 'Frege and the Paradox of Analysis.' Benacerraf, Paul. 'Frege: The Last Logicist.' Optional Tappenden, Jamie. 'Extending Knowledge and 'Fruitful Concepts': Fregean Themes in the Foundations of Mathematics.' Week III: Weiner's Hidden Agenda Interpretation Readings Weiner, Joan. 'The Philosopher Behind the Last Logicist.' Optional Weiner, Joan. Frege in Perspective . Ithaca, NY: Cornell University Press, 1990. Focus Questions 1. To what extent is Frege's account of analyticity in Foundations a rejection, and to what extent an updating, of Kant's view of analyticity? 2. According to Picardi it 'would be incomprehensible' how Frege's proofs tells us anything about the arithmetic we already have unless his 'definitions [are] somehow responsible to the meaning of [arithmetical] sentences as these are understood' (228). Why does she hold this? Why does Dummett agree with her? Do you think Frege's logicism needs to address this worry? 3. What are the major differences and continuities in Frege's discussions of definition in mathematics in Foundations , the review of Husserl and 'Logic in Mathematics'? 4. Frege writes that definitions must prove their worth by being fruitful. He also says that nothing can be proven using a proper definition that cannot be proven without it. Are these claims consistent? Why or why not? 5. Weiner held that in Foundations Frege had 'hidden agenda.' What, according to her, is this agenda? How does this fit with Frege's later views of definition? 6. What are Frege's main complaints about Weierstrass's definitions in 'Logic in Mathematics'? Are these criticisms consistent with Frege's account of 'definition proper' in the same text? Seminar/Project Ideas What, if anything, is the relation between Frege's critique of Hilbert's use of definitions and Frege's later views of definitions? (shrink)
Anything worth regarding as logicism about number theory holds that its fundamental laws – in effect, the Dedekind-Peano axioms – may be known on the basis of logic and definitions alone. For Frege, the logic in question was that of the Begriffschrift – effectively, full impredicative second order logic - together with the resources for dealing with the putatively “logical objects” provided by Basic Law V of Grundgesetze. With this machinery in place, and with the course-of-values operator governed by (...) Basic Law V counting as logical, it is possible for all the definitions involved in the logicist reconstruction of arithmetic and analysis to be fully explicit, abbreviative definitions. Had Frege’s project succeeded, he would therefore have been in position – by his own lights – to regard the axioms of number theory simply as definitional abbreviations of certain theorems of his pure logic. Basic Law V, as every interested party knows, is inconsistent. But twentieth century orthodoxy would have scorned its description as a law of logic in any case, purely on the grounds of its existential fecundity. Contemporary Neo-Fregeanism in the foundations of mathematics does not, in intention at least, pick any quarrel with the idea that pure logic should be ontologically austere. It does however maintain that the existence of the natural numbers and the real numbers as classically conceived, and thereby the truth of the traditional axioms of arithmetic and analysis, may still be known a priori on the basis of logic and definitions. For the purposes of this claim, logic is once again conceived as essentially the system of Begriffschrift. But Basic Law V is superseded by a variety of abstraction principles, of which Hume's Principle is the best known example, which we are regarded as free to lay down as true by way of determination of the meaning of the non-logical vocabulary that they contain. Thus — the idea is — the Dedekind-Peano axioms, for example, may be known, a priori, to be true by virtue of their derivation in pure logic from a principle which may be regarded as stipulatively true, and whose very stipulation may be regarded as conferring content upon the sole item of non-logical vocabulary – the cardinality operator – which it contains and thereby as conferring content upon Hume's Principle itself.. (shrink)
Thirty years after the publication of Thomas Kuhn’s The Structure of Scientific Revolutions, sharp disagreement persists concerning the implications of Kuhn’s "historicist" challenge to empiricism. I discuss the historicist movement over the past thirty years, and the extent to which the discourse between two branches of the historical school has been influenced by tacit assumptions shared with Rudolf Carnap’s empiricism. I begin with an examination of Carnap’s logicism --his logic of science-- and his 1960 correspondence with Kuhn. I focus (...) on problems in the analysis applied to the unit of metascientific study or appraisal, arguing for a reassessment of historicist treatment of the internal/external distinction and historiographic meta-methodology. The critique of objectivism and relativism that eventuates from this re-assessment is a double-edged blade, undercutting both objectivist and relativist treatments of cognitive evaluation and scientific change. I use it to cut across an otherwise diverse group of historicist-influenced writers, including Imre Lakatos, Larry Laudan, H. M. Collins, Stephen Stich. I. Introduction.. (shrink)
Neo-Fregean logicism seeks to base mathematics on abstraction principles. But the acceptable abstraction principles are surrounded by unacceptable (indeed often paradoxical) ones. This is the “bad company problem.” In this introduction I first provide a brief historical overview of the problem. Then I outline the main responses that are currently being debated. In the course of doing so I provide summaries of the contributions to this special issue.
