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Modern logic emerged in the period from 1879 to the Second World War. In the post-war period what we know as classical first-order logic largely replaced traditional syllogistic logic in introductory textbooks, but the main development has been simply enormous growth: The publications of the Association for Symbolic Logic, the main professional organization for logicians, became ever thicker. While 1950 saw volume 15 of the Journal of Symbolic Logic, about 300 pages of articles and reviews and a six‑page member list, 2000 saw volume 65 of that journal, over 1,900 pages of articles, plus volume 6 of the Bulletin of Symbolic Logic, 570 pages of reviews and a sixty‑page member list. Of so large a field, the present survey will have to be ruthlessly selective, with no coverage of the history of informal or inductive logic, or of philosophy or historiography of logic, and slight coverage of applications. Remaining are five branches of pure, formal, deductive logic, four being the branches of mathematical logic recognized in Barwise 1977, first of many handbooks put out by academic publishers: set theory, model theory, recursion theory, proof theory. The fifth is philosophical logic, in one sense of that label, otherwise called non-classical logic, including extensions of and alternatives to textbook logic. For each branch, a brief review of pre‑war background will be followed by a few highlights of subsequent history. The references will be a mix of primary and secondary sources, landmark papers and survey articles.
Nations love to go to war, argue leftist pacifists in democratic Western societies. By this account, governments cannot resist the temptation to assert their selfish interests by violent means. Political leaders use the language of just war to hide their real intentions: imperialistic domination and economic profit. These pacifists charge that with the Iraq war, the disastrous consequences of such politics have become hideously clear again.
§1. I propose to address not so much Gödel's own philosophy of mathematics as the philosophical implications of his work, and especially of his incompleteness theorems. Now the phrase “philosophical implications of Gödel's theorem” suggests different things to different people. To professional logicians it may summon up thoughts of the impact of the incompleteness results on Hilbert's program. To the general public, if it calls up any thoughts at all, these are likely to be of the attempt by Lucas  and Penrose  to prove, if not the immortality of the soul, then at least the non-mechanical nature of mind. One goal of my present remarks will be simply to point out a significant connection between these two topics.
But let me consider each separately a bit first, starting with Hilbert. As is well known, though Brouwer's intuitionism was what provoked Hilbert's program, the real target of Hilbert's program was Kronecker's finitism, which had inspired objections to the Hilbert basis theorem early in Hilbert's career. (See the account in Reid .) But indeed Hilbert himself and his followers (and perhaps his opponents as well) did not initially perceive very clearly just how far Brouwer was willing go beyond anything that Kronecker would have accepted. Finitism being his target, Hilbert made it his aim to convince the finitist, for whom no mathematical statements more complex than universal generalizations whose every instance can be verified by computation are really meaningful, of the value of “meaningless” classical mathematics as an instrument for establishing such statements.
Philosophy is a subject in which there is very little agreement. This is so almost by definition, for if it happens that in some area of philosophy inquirers begin to achieve stable agreement about some substantial range of issues, straightaway one ceases to think of that area as part of “philosophy,” and begins to call it something else. This happened with physics or “natural philosophy” in the seventeenth century, and has happened with any number of other disciplines in the centuries since. Philosophy is left with whatever remains a matter of doubt and dispute.
Philosophy of mathematics, in particular, is an area where there are very profound disagreements. In this respect philosophy of mathematics is radically unlike mathematics itself, where there are today scarcely ever any controversies over the correctness of important results, once published in refereed journals. Some professional mathematicians are also amateur philosophers, and the best way for an observer to guess whether such persons are talking mathematics or philosophy on a given occasion is to look whether they are agreeing or disagreeing.
One major issue dividing philosophers of mathematics is that of the nature and existence of mathematical objects and entities, such as numbers, by which I will always mean positive integers 1, 2, 3, and so on. The problem arises because, though it is common to contrast matter and mind as if the two exhausted the possibilities, numbers do not fit comfortably into either the material or the mental category.
Some philosophers approach mathematics saying, “Here is a great and established branch of knowledge, encompassing even now a wonderfully large domain, and promising an unlimited extension in the future. How is mathematics, pure and applied, possible? From its answer to this question the worth of a philosophy may be judged.”
Other philosophers approach mathematics in a quite different spirit. They say, “Here is a body, already large and still being extended, of what purports to be knowledge. Is it knowledge, or is it delusion? Only philosophy and theology, from their standpoint prior and superior to that of mathematics and science, are worthy to judge.” While this inquisitorial conception of the relation between philosophy and science is less widely held today that it was in Cardinal Bellarmine's time, it continues to have many distinguished advocates.
Prominent among these is Michael Dummett, who has repeatedly advanced arguments for the claim that much of current mathematical theory is delusory and much of current mathematical practice is in need of revision – arguments for the repudiation, within mathematical reasoning, of the canons of classical logic in favor of those of intuitionistic logic. While nearly everything Dummett has written is pertinent in one way or another to his case for intuitionism, there are two texts especially devoted to stating that case: his much anthologized article (Dummett 1973a) on the philosophical basis of intuitionistic logic; and the concluding philosophical chapter of his guidebooks (Dummett 1977) to the elements of intuitionism.
The present volume contains a selection of my published philosophical papers, plus two items that have not previously appeared in print. Excluded are technical articles, co-authored works, juvenilia, items superseded by my published books, purely expository material, and reviews. (An annotated partial bibliography at the end of the volume briefly indicates the contents of such of my omitted technical papers as it seemed to me might interest some readers.) The collection has been divided into two parts, with papers on philosophy of mathematics in the first, and on other topics in the second; references in the individual papers have been combined in a single list at the end of the volume. Bibliographic data for the original publication of each item reproduced here are given source notes on pp. xi—xiii, to which the notes of personal acknowledgment, dedications, and epigraphs that accompanied some items in their original form have been transferred; abstracts that accompanied some items have been omitted.
It has become customary in volumes of this kind for the author to provide an introduction, relating the various items included to each other, as an editor would in an anthology of contributions by different writers. I have fallen in with this custom. The remarks on the individual papers in the introduction are offered primarily in the hope that they may help direct readers with varying interests to the various papers in the collection that should interest them most.
If one is interested in how best to formulate and motivate axioms for set theory, it is worthwhile to take another look at the early history of the subject, right back to the work of its founder, Cantor. Cantor's definition of a set was “any collection into a whole” of “determinate, welldistinguished” sensible or intelligible objects. According to a well-known quip of van Heijenoort, this definition has had as much to do with the subsequent development of set theory as Euclid's definition of point – “that which hath no part” – had to do with the subsequent development of geometry. But in fact the notion of a many made into a one, which is what Cantor's definition makes a set to be, will repay some study.
In order to give concrete meaning to Cantor's abstract definition, our study should begin with a look at the context in which Cantor first felt it desirable or necessary to introduce the notion of set. As is well known, Cantor's general theory of arbitrary sets of arbitrary elements emerged from a previous theory of sets of points on the line or real numbers. This itself emerged from work on Fourier series. The technical details of Cantor's theorems on this topic are irrelevant for present purposes, but the general form of his results should be noted.
A word on terminology may be useful at the outset, since it is pertinent to many of the papers in this collection, beginning with the very first. The label “realism” is used in two very different ways in two very different debates in contemporary philosophy of mathematics. For nominalists, “realism” means acceptance that there exist entities, for instance natural or rational or real numbers, that lack spatiotemporal location and do not causally interact with us. For neo-intuitionists, “realism” means acceptance that statements such as the twin primes conjecture may be true independently of any human ability to verify them. For the former the question of “realism” is ontological, for the latter it is semantico-epistemological. Since the concerns of nominalists and of neo-intuitionists are orthogonal, the double usage of “realism” affords ample opportunity for confusion.
The arch-nominalists Charles Chihara and Hartry Field, for instance, are anti-intuitionists and “realists” in the neo-intuitionists' sense. They do not believe there are any unverifiable truths about numbers, since they do not believe there are any numbers for unverifiable truths to be about. But they do believe that the facts about the possible production of linguistic expressions, or about proportionalities among physical quantities, which in their reconstructions replace facts about numbers, can obtain independently of any ability of ours to verify that they do so.