Inaccessible truths and infinite coincidences

It is one of the salient features of an intuitionistic philosophy of mathematics that it denies the possibility that there may be infinite coincidences in mathematics: if a sentence is true, it can only be because there is a finitely expressible reason for it. The same view was expressed from a different standpoint by Hilbert in 1924 when he asserted that in mathematics there is ignorabimus. It has been common since Gödel’s incompleteness theorems to regard this view as no longer open to the platonist. And yet it has been reaffirmed since then by Gödel himself in regard to unsolved problems in set theory such as that of settling the continuum hypothesis. It is therefore a live question whether there is a coherent position which denies this. We should distinguish three possibilities such a position might be intended to allow: first, sentences true accidentally, by infinite coincidence; second, truths which are in principle inaccessible to us and which we cannot even grasp directly via our intuitions about the concepts involved; third, truths for which there is no finitely expressible reason. The first possibility implies the second, and the second implies the third. It is one of the salient features of an intuitionistic philosophy of mathematics that it denies the possibility that there may be infinite coincidences in mathematics. If a sentence is true, then according to the intuitionist account that can only be because there is a reason why it is true, a reason which we — as creative mathematicians — must be capable in principle of grasping. Until 1931 this view — that every true mathematical sentence is in principle capable of being known to be true by us, or at any rate by our ideal counterparts — would also have found support from mathematicians not otherwise sympathetic to constructivism. Frege certainly believed it. ‘In arithmetic,’ he wrote in Grundlagen ([3], §105), ‘we are not concerned with objects which we come to know as something alien from without through the medium of the senses, but with objects given directly to our reason and, as its nearest kin, utterly transparent to it.’ And Hilbert in his 1924 lecture “Über das Unendliche” ([1], p.200) remarked that ‘one of the principal attractions of tackling a mathematical problem is that we always hear this cry within us: There is the problem, find the answer; you can find it just by thinking, for there is no ignorabimus in mathematics.’ But this, of course, was before Gödel had proved his incompleteness theorems. Since 1931 it seems to have been common for those going under