# Michael Frank Hallett

## Professor & John Frothingham Professor of Logic and Metaphysics

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

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

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

- 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.

## 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 below.

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 Festschriftof 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.

### Decription of Volumes

### General description of the edition.

This six volume collection makes available for the first time the German text of the most important of David Hilbert's unpublished lectures on the foundations of mathematics and physics, together with a scholarly commentary. Hilbert's lectures and his personal interactions with the 'Hilbert circle' exercised a profound influence on the development of twentieth century mathematics and physics. The lecture notes presented, spanning virtually the whole of Hilbert's teaching career, document his intense engagement with the ideas of some of the central figures of modern science (among them Bernays, Boltzmann, Born, Brouwer, Cantor, Courant, Dedekind, Einstein, Frege, Heisenberg, Klein, Lie, Minkowski, PoincarŽ, Russell, von Neumann, Weyl, and Zermelo), and make possible a detailed understanding of the development of his foundational work in geometry, arithmetic, logic, and proof theory, as well as in the theory of relativity, quantum mechanics and statistical physics. The lectures contain many more philosophical, foundational and methodological remarks than does Hilbert's published work. Some of the individual volumes also reprint key published works of Hilbert when these are centrally relevant to the unpublished work presented. (For example, Volume 1 reprints the first edition of Hilbert's celebrated Grundlagen der Geometrie, Volume 3 reprints Bernays's Habilitationschrift and Hilbert and Ackermann's GrundzŸge der theoretischen Logik, Volume 4 reprints Hilbert's fundamental papers on General Relativity Theory.)

### Volume 1: David Hilbert's Lectures on the Foundations of Geometry, 1891-1902

*Edited by Michael Hallett and Ulrich Majer.*

Volume 1 contains six sets of notes for lectures on the foundations of geometry held by Hilbert in the period 1891-1902. It also reprints the first edition of Hilbert's celebrated Grundlagen der Geometrie of 1899, together with the important additions which appeared first in the French translation of 1900. The lectures document the emergence of a new approach to foundational study (the 'axiomatic method'), which concentrates on assessing the logical weight of central propositions by exploiting to the full the method of independence proofs by modelling. This culminates in the lectures of 1898/1899 (the immediate precursor of the 1899 monograph) and 1902. The lectures contain many reflections and investigations which never found their way into print.

### Volume 2: David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1894-1917

*Edited by William Ewald, Michael Hallett, Wilfried Sieg and Ulrich Majer*

Volume 2 focuses on notes for lectures on the foundations of the mathematical sciences held by Hilbert in the period 1894-1917. They document Hilbert's first engagement with 'impossibility' proofs; his early attempts to formulate and address the problem of consistency, first dealt with in his work on geometry in the 1890s; his engagement with foundational problems raised by the work of Cantor and Dedekind; his early investigations into the relationship between arithmetic, set theory, and logic; his advocation of the use of the axiomatic method generally; his first engagement with the logical and semantical paradoxes; and the first formal attempts to develop a logical calculus. The Volume also contains Hilbert's address from 1895 which formed the preliminary version of his famous 'Zahlbericht' (1897).

### Volume 3: David Hilbert's Lectures on the Foundations of Arithmetic and Logic, 1917-1933

*Edited by William Ewald, Wilfried Sieg and Ulrich Majer*

The bulk of Volume 3 consists of six sets of notes for lectures Hilbert gave (often in collaboration with Bernays) on the foundations of mathematics between 1917 and the early 1930s. The notes detail the increasing dominance of the metamathematical perspective in Hilbert's treatment, i.e., the development of modern mathematical logic, the evolution of proof theory, and the parallel emergence of Hilbert's finitist standpoint. The notes are mostly very polished expositions; e.g., the 1917-18 lectures are in effect a first draft of Hilbert and Ackermann's GrundzŸge der theoretischen Logik (1928), reprinted in this Volume. They are thus essential for understanding the development of modern mathematical logic leading up to Hilbert and Bernays's Grundlagen der Mathematik (1934, 1938). Also included is a complete version of Bernay's Habilitationschrift of 1918, only partially published in 1926.

### Volume 4: David Hilbert's Lectures on the Foundations of Physics, 1898-1914: Classical, Relativistic and Statistical Mechanics

*Edited by Ulrich Majer, Tilman Sauer and Klaus BŠrwinkel*

The first part of Volume 4 documents Hilbert's efforts in the period 1898-1910 to base all known physics (including thermodynamics, hydrodynamics and electrodynamics) on classical mechanics. This period closes with a lecture course 'Mechanik der Kontinua' (1911), in which Hilbert considers the consequences of the new principle of special relativity for our understanding of physics. The second part starts with the lecture course 'Kinetische Gastheorie' (1911/12), which introduces a new approach to problems of statistical physics. The lecture course 'Molekulartheorie der Materie' (1913) deals with a topic that was of great importance to Hilbert, returning to it repeatedly. The last lecture course contained in this volume, 'Statistische Mechanik' (1922) presents a very perceptive comparison of the different approaches of Maxwell, Boltzmann, Gibbs etc. to the foundational problems of statistical physics. It is a paradigm of logical analysis and conceptual clarity.

### Volume 5: David Hilbert's Lectures on the Foundations of Physics, 1915-1927: Relativity, Quantum Theory and Epistemology

*Edited by Ulrich Majer, Tilman Sauer and Heinz-JŸrgen Schmidt*

This Volume has three sections, General Relativity, Epistemological Issues, and Quantum Mechanics. The core of the first section is Hilbert's two semester lecture course on 'The Foundations of Physics' (1916/17). This is framed by Hilbert's published 'First and Second Communications' on the 'Grundlagen der Physik' (1915, 1917). The section closes with a lecture on the new concepts of space and time held in Bucharest in 1918. The epistemological issues concern the principle of causality in physics (1916), the intricate relation between nature and mathematical knowledge (1921), and the subtle question whether what Hilbert calls the 'world equations'' are physically complete (1923). The last section deals with quantum theory in its early, advanced and mature stages. Hilbert held lecture courses on the mathematical foundations of quantum theory twice, before and after the breakthrough in 1926. These documents bear witness to one of the most dramatic changes in the foundations of science.

### Volume 6: David Hilbert's Notebooks and General Foundational Lectures

*Edited by William Ewald, Michael Hallett, Wilfried Sieg and Ulrich Majer*

This Volume contains a selection of material exhibiting many of Hilbert's philosophical and foundational views on particular theories and the exact sciences in general, drawn from his private notebooks and from lectures for more general audiences held in the 1920s.

## 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.)

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).