Michael Detlefsen

(1948 - 2019)

Hide Ishiguro, Leibniz’s philosophy of logic and language. 2nd ed. Cambridge:Cambridge University Press, 1990. x + 246pp. £27.50/$49.50 ; £10.95/$16.95 Massimo Mugnai, Leibniz’ theory of relations. Stuttgart:Franz Steiner Verlag, 1992. 291 pp. 96 DM W. A. Wallace, Galileo’s logic of discovery and proof The background, content, and use of his appropriated treatises on Aristotle’s posterior analytics. Dordrecht, Boston, and London:Kluwer, 1992. xxiii + 323 pp. £84, $139, DF1240 W. A. Wallace, Galileo’s logical treatises. A translation, with notes and commentary, of his appropriated Latin questions on Aristotle’s posterior analytics. Dordrecht, Boston, and London:Kluwer, 1992. xix + 239 pp. £65, $;99, DF1185 W. A. Wallace, Galileo’s early notebooks:the physical questions ; Prelude to Galileo: essays on medieval and sixteenth-century sources of Galileo’s thought ; Galileo and his sources: the heritage of the Collegio Romano in Galileo’s science ; and Galileo Galilei:Tractatio de Precognitionibus et Praecognitis and Tractatio de Démonstration Raymond M. Smullyan, GödeVs incompleteness theorems, New York, Oxford:Oxford University Press, 1992. xiii + 139. pp. $29.95 Raymond M. Smullyan, Recursion theory for metamathematics, New York, Oxford:Oxford University Press, 1993. xiv + 163 pp. $29.95 D. MiéVille, Kurt Gödel:Actes du Colloque, Neuchâtel 13-14 juin 1991. Neuchâtel, Switzerland: Centre de Recherches Sémiologiques, Université de Neuchâtel, 1992. iv +200pp. 15 Sfr Geoffrey Hellman, Mathematics without numbers. Oxford:Clarendon Press, 1989. xi + 154 pp. £22.50 Michael Detlefsen Proof, logic and formalization. London and New York:Routledge, 1992. x + 241 pp. £40.00 J. Cavaillès, Método axiomâtico y formalismo. Foreword by S. Ramirez. Translation from the French by C. Alvarez and S. Ramirez. Mexico:Servicios Editoriales de la Facultad de Ciencias, UNAM, 1992. 199pp. No price stated Roberto Poli, Ontologia formale. Genoa:Marietti, 1992, 542 pp. No price stated Jon Barwise and John Etchemendy, The language of first-order logic, 3rd edition, Stanford, CA:CLSI Lecture Notes No. 34, 1992, xiv + 329pp. US $34.95

This volume of essays addresses the main problem confronting an epistemology for mathematics; namely, the nature and sources of mathematical justification. Attending to both particular and general issues, the essays, by leading philosophers of mathematics, raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And of what epistemological importance is the formalizability of proof? The editor, Michael Detlefsen, has brought together an outstanding collection of essays, only one of which has previously appeared. It will be essential for philosophers and historians of mathematics, as well as philosophically inclined logicians and philosophers interested in the nature of reasoning and justification. A companion volume entitled Proof, Logic and Formalization, edited by Michael Detlefsen, is also available from Routledge.

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

The type of completeness Whitehead and Russell aimed for in their Principia Mathematica was what I call descriptive completeness. This is completeness with respect to the propositions that have been proved in traditional mathematics. The notion of completeness addressed by Gödel in his famous work of 1930 and 1931 was completeness with respect to the truths expressible in a given language. What are the relative significances of these different conceptions of completeness for traditional mathematics? What, if any, effects does incompleteness of the type Gödel had in mind have for the evaluation of the Principia Mathematica project and other projects like it?

© The Author [2017]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected] author makes clear that he does not see this book as a contribution to the philosophy of mathematics as traditionally understood. He takes it instead to be an essay about the philosophy of mathematics, one whose purpose is to explain its existence and to make clear the limited extent to which its current and past forms are properly regarded as philosophies of mathematics per se.A little more exactly, the aim is to explain the seemingly ‘perennial’ interest1 western philosophers have had in mathematics 2 and how this became significant enough to give rise to and sustain an identified sub-discipline of philosophy....Hacking identifies two principal reasons for this, both of which he sees as having originated in what might broadly be termed our experience of mathematics. The first of...

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type.

We argue that Löb's Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löb's Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents.

First published in the most ambitious international philosophy project for a generation; the _Routledge Encyclopedia of Philosophy_. _Logic from A to Z_ is a unique glossary of terms used in formal logic and the philosophy of mathematics. Over 500 entries include key terms found in the study of: * Logic: Argument, Turing Machine, Variable * Set and model theory: Isomorphism, Function * Computability theory: Algorithm, Turing Machine * Plus a table of logical symbols. Extensively cross-referenced to help comprehension and add detail, _Logic from A to Z_ provides an indispensable reference source for students of all branches of logic

In the early years of this century, Poincaré and Russell engaged in a debate concerning the nature of mathematical reasoning. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical in character. Russell urged the contrary view, maintaining that (i) the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that (ii) with the aid of recent developments in logic, it is possible to demonstrate its falsity. This refutation of Kant's views consists in showing that every known theorem of mathematics can be proven by purely logical means from a basic set of axioms. In our view, Russell's alleged refutation of Poincaré's Kantian viewpoint is mistaken. Poincaré's aim (as Kant's before him) was not to deny the possibility of finding a logical ‘proof’ for each theorem. Rather, it was to point out that such purely logical derivations fail to preserve certain of the important and distinctive features of mathematical proof. Against such a view, programs such as Russell's, whose main aim was to demonstrate the existence of a logical counterpart for each mathematical proof, can have but little force. For what is at issue is not whether each mathematical theorem can be fitted with a logical ‘proof’, but rather whether the latter has the epistemic features that a genuine mathematical proof has.