§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.
REALISM VS NOMINALISM
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.
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