Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...) we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism. (shrink)

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify (...) many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today. (shrink)

Kit Fine develops a Fregean theory of abstraction, and suggests that it may yield a new philosophical foundation for mathematics, one that can account for both our reference to various mathematical objects and our knowledge of various mathematical truths. The Limits ofion breaks new ground both technically and philosophically.

Widespread interest in Frege's general philosophical writings is, relatively speaking, a fairly recent phenomenon. But it is only very recently that his philosophy of mathematics has begun to attract the attention it now enjoys. This interest has been elicited by the discovery of the remarkable mathematical properties of Frege's contextual definition of number and of the unique character of his proposals for a theory of the real numbers. This collection of essays addresses three main developments in recent work on Frege's (...) philosophy of mathematics: the emerging interest in the intellectual background to his logicism; the rediscovery of Frege's theorem; and the reevaluation of the mathematical content of The Basic Laws of Arithmetic. Each essay attempts a sympathetic, if not uncritical, reconstruction, evaluation, or extension of a facet of Frege's theory of arithmetic. Together they form an accessible and authoritative introduction to aspects of Frege's thought that have, until now, been largely missed by the philosophical community. (shrink)

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system--Axiom V--which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of (...) the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be "The Basic Laws of Cardinal Number", as Frege understood them. Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, "Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?". (shrink)

We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...) full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory. (shrink)

In the mathematical documents which have come down to us from these peoples, there are no theorems or demonstrations, and the fundamental concepts of ...

Kit Fine’s book is a study of abstraction in a quite precise sense which derives from Frege. In his Grundlagen, Frege contemplates defining the concept of number by means of what has come to be called Hume’s principle—the principle that the number of Fs is the same as the number of Gs just in case there is a one-to-one correspondence between the Fs and the Gs. Frege’s discussion is largely conducted in terms of another, similar but in some respects simpler, (...) proposal—to define the concept of direction by laying down the principle that the direction of a line a is the same as that of line b if and only if lines a and b are parallel. Such principles have come to be known as abstraction principles. More generally, abstraction principles are ones of the shape: ∀α∀β ↔ α ≈ β), where ≈ is an equivalence relation on entities of the type over which α and β vary and, if the principle is acceptable, § is a function from entities of that type to objects. Since ‘the direction of’ is intended to stand for a function from objects of a certain sort to other objects, the Direction Equivalence is a first-order, or in Fine’s terminology objectual, abstraction; since ‘the number of’ is intended to stand for a function from concepts to objects, Hume’s principle is a higher-order, or as Fine says conceptual, abstraction. As is well-known, Frege himself abandoned the idea of defining number implicitly or contextually by means of Hume’s principle—adopting instead an explicit definition of the number of Fs as the extension of the concept concept equinumerous to the concept F— because he thought that an adequate definition should settle the question whether Julius Caesar, for example, is or is not the number of some concept, but that the proposed implicit definition cannot do so. But interest in abstraction principles has revived in the last couple of decades, largely as a result of Crispin Wright’s attempt to show that in spite of the many difficulties confronting it—including the notorious Julius Caesar problem— Hume’s principle can after all serve as a foundation for arithmetic. (shrink)

Richard G. Heck, On the Philosophical Significance of Frege's Theorem. Language, Thought, and Logic, Essays in Honour of Michael Dummett.George Boolos, Is Hume's Principle Analytic?.Charles Parsons, Wright onion and Set Theory.Richard G. Heck, The Julius Caesar Objection.

Frege's theory of real numbers has undeservedly received almost no attention, in part because what we have is only a fragment. Yet his theory is interesting for the light it throws on logicism, and it is quite different from standard modern approaches. Frege polemicizes vigorously against his contemporaries, sketches the main features of his own radical alternative, and begins the formal development. This paper summarizes and expounds what he has to say, and goes on to reconstruct the most important steps (...) which he is likely to have subsequently taken. The various difficulties facing his theory in this reconstruction are outlined, and some surprising consequences drawn about the nature of his logicism. (shrink)

The book is an attempt at explaining to the nation the ideas of Frege's Grundlagen. It is wordy and trite, a paradigm case of a redundant piece of writing. The reader is advised to steer clear of it.

As is perhaps appropriate for a festschrift by philosophers who worked with Dummett, the eleven essays in this volume touch on a wide range of subjects.