# Michael Hallett

### Contact information

Email: michael [dot] hallett [at] mcgill [dot] ca (Michael Hallett)

Tel.: 514-398-4400 x094813

Office: Ferrier 464

Office hours: Wed 11 a.m. - 12:20 p.m.

### Degrees

BSc (Econ), London School of Economics, University of London, 1972

PhD, London School of Economics, University of London, 1979

### Teaching and research areas

Philosophy and History of Mathematics

Philosophy of Science

History of Logic (in particular Frege, Russell, Hilbert)

Logic and Set Theory

Development of Set Theory

Development of Analytic Philosophy

### Current research projects

### Hilbert:

I am one of four General Editors (with William Ewald, Ulrich Majer and Wilfried Sieg) of a project to publish a large quantity of lecture notes of the German mathematician David Hilbert on the foundations of mathematics and physics, delivered from 1891 until 1933. The edition will be published by Springer-Verlag, and will consist of six volumes. The lectures themselves are given in the original German, while the introductions and editoial material are in English. For titles and descriptions of all the volumes, see Descriptions of the Volumes.

The first volume of this edition is David Hilbert's Lectures on the Foundations of Geometry: 1891-1902, edited by Michael Hallett and Ulrich Majer. (See Descriptions of the Volumes.) This volume is now in press with Springer; to view the contents page, see Geometry Volume . I have written detailed notes and introductions to (a) the so-called Ferienkurse from 1896 and 1898; (b) the long lectures on Euclidean geometry from 1898/1899; (c) the Festschrift of 1899 (the first edition of Hilbert's Grundlagen der Geometrie, reproduced here as Chapter 5); and (d) the lectures on the foundations of geometry of 1902.

The second volume, which I am editing with William Ewald, Wilfried Sieg and Ulrich Majer, is David Hilbert's Lectures on the Foundations of Logic and Arithmetic: 1894 -1917. (See Descriptions of the Volumes.) It is hoped that this volume will be ready for publication early in 2005.

### Papers:

I am currently writing a paper comparing the very different approaches of David Hilbert and Frege to the foundations of mathematics, a paper which begins as an analysis of the Frege-Hilbert correspondence. This will appear in Thomas Ricketts (ed.): The Cambridge Companion to Frege, to be published by Cambridge University Press.

In addition, I am writing a long paper on the early views of Hilbert on the foundations of mathematics, particularly on geometry, and arithmetic ("Hilbert on number, geometry and continuity").

### Books:

1. Hilbert on the Foundations of Mathematics (provisional title), to be published by the Clarendon Press, Oxford, c. 300,000 words.

This book will be a full length study of the development of Hilbert's treatment of the foundations of mathematics, with three central aims: (1) It will attempt to bring out the unified development of Hilbert's thought, and show how Hilbert's early work on the foundations of geometry greatly shaped his later approaches to foundational issues and even to logic, giving rise to mathematical logic as we know it, and show how his work posed most of the central problems in mathematical logic in the first half of the century. (2) It will pay close attention to various elements in Hilbert's thought, e.g., Hilbert's notion of the axiomatic method, and the view of the nature of science which underlies it (here, comparison with the views of Hertz and Mach is especially important); Hilbert's theory of mathematical existence, and (closely linked to this) his reliance on the notion of consistency, his theory of 'ideal elements', and thus his theory of infinity. It will also focus on Hilbert's view of the fundamental role of signs in any communicable intellectual activity, a view which is the basis of Hilbert's theory of consistency proofs. Of great importance in this is an anlysis of Hilbert's relations to other important foundational thinkers of the time, e.g., Cantor, Dedekind and Frege. (3) The book will attempt to situate Hilbert between Kant and Gdel, perhaps (along with Frege and Russell) the two most important thinkers about mathematics in modern times. In particular, the book will argue that much more of Hilbert's theory of the foundations of mathematics is viable than is usually thought.

2. The Philosophy of Mathematics: An Historical Approach. This book (co-authored with William Ewald of the University of Pennsylvania), will be c. 250 pages, and will appear with Cambridge University Press as part of a series of introductory books, edited by Gary Hatfield and Paul Guyer, on the various fundamental fields of philosophy.

The book will cover the central developments from Berkeley's time till the present day. (The series is aimed at senior undergraduates and beginning graduate students, and the books are intended to give an introduction to the central problems of the fields, but in doing so to give some indication of how these problems came to be central. For more details of the series and a list of volumes, see Cambridge University Press.)

### Current teaching

In the academic year 2003-2004, I am teaching Introduction to Deductive Logic (107-210A), Further Topics in Formal Logic (107-411A), Intermediate Logic (107-310B), and a Seminar in the Philosophy of Mathematics (107-510B).

### Selected publications

Cantorian set theory and limitation of size. Oxford: Clarendon Press. (Oxford Logic Guides, Vol. l0. Editor: Dana S. Scott). Foreword by Michael Dummett. Pp.xxii + 343. 1984, reprinted in paperback, with revisions, 1986, 1988.

'Physicalism, reductionism and Hilbert' in Andrew Irvine (ed.): Physicalism in Mathematics, Dordrecht: D. Reidel Publishing Co., 1990, 182-256.

'Hilbert's axiomatic method and the laws of thought', in A. George (ed.): Mathematics and Mind. New York: Oxford University Press, 1994, 158-200.

'Putnam and the Skolem paradox' (with Putnam's reply), in Peter Clark and Bob Hale (eds.): Reading Putnam, Oxford: Basil Blackwell, 1994., 66-97.

'Logic and mathematical existence', in Lorenz Krüger and Brigitte Falkenburg (eds.): Physik, Philosophie und die Einheit der Wissenschaft. Für Erhard Scheibe, Heidelberg: Spektrum Akademischer Verlag, 1995, 33-82.

'Hilbert and logic', in Mathieu Marion and Robert Cohen (eds.): Québec Studies in the Philosophy of Science, Part 1: Logic, Mathematics, Physics and the History of Science, (Boston Studies in the Philosophy of Science, Volume 177), Dordrecht: Kluwer Publishing Co., 1995, 135-87.

The foundations of mathematics 1879-1914 in Thomas Baldwin (ed.): The Cambridge History of Philosophy: 1879-1945, Cambridge: Cambridge University Press, 2003, 128-156.