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The Habit of Clinging to an Ultimate Ground1

The notions of real interest to mathematicians like myself are not on the printed page. They lurk behind the doors of conception. It is believed2 that they will some day emerge and shed so much light on earlier concepts that the latter will disintegrate into marginalia. By their very nature, they elude precise definition, so that on the conventional account they are scarcely mathematical at all. Coming to grips with them is not to be compared with attempting to solve an intractable problem, an experience that drives most narratives of mathematical discovery. What I have in mind harbors a more fundamental obscurity. One cannot even formulate a problem, much less attempt to solve it; the items (notions, concepts) in terms of which the problem would be formulated have yet to be invented. How can we talk to one another, or to ourselves, about the mathematics we were born too soon to understand? Maybe with the help of mindaltering drugs? That’s what you might conclude from the snippet of dialogue from Proof quoted in the last chapter, but you would be wrong. It seems Proof’s author got that idea by reading Paul Hoffman’s biography of Paul Erdős, the Hungarian mathematician famous for finding unexpectedly rich structures in apparently elementary mathematics; for spending nearly all his life wandering