I dont know what any of that means.

It’s all right. Add to your troubles the idea that topology has questionable mathematical foundations—or none at all, as some of its founders believed—and then what? You can say that it contains its own logic, but isnt that the problem? If you claim that mathematics is not a science then you can claim that it need have no referent save itself. When Wittgenstein convinced Russell that all of mathematics was a tautology Russell gave up mathematics.

Is that true?

I dont know. Russell said it was.

Would that be your view?

I dont think the question can be answered. For the present I guess I’d have to say no. But by then I’d already left the building. And the deeper question, which we touched on, is that if mathematical work is performed mostly in the unconscious we still have no notion as to how it goes about it. You can try and picture the inner mind adding and subtracting and muttering and erasing and beginning again but you wont get very far. And why is it so often right? Who does it check its work with? I’ve had solutions to problems simply handed to me. Out of the blue. The locus ceruleus perhaps. And it has to remember everything. No notes. It’s hard to escape the unsettling conclusion that it’s not using numbers.

I dont understand how that would be possible.

That might not be true. That it is that often right. What’s probably true is that only the right answers get reported. At a conference a while back I ran into the historian of the Manhattan Project. A man named David Hawkins. We got to talking about math and he told me that what first got him interested in the subject was the second chapter of Spengler’s Decline of the West. Which is entitled “The Meaning of Numbers.” I asked him what Spengler’s view was and he said that he wasnt sure. That Spengler seemed eager to make a distinction between mathematics as numeracy and mathematics as chronology. Which I thought was already pretty well established in the cardinal and the ordinal but I assumed that Spengler was after something else. But I got the book and read the opening chapter and a bit more here and there. As with the general run of philosophers—if he is one—the most interesting thing was not his ideas but just the way his mind worked. I read a bit more before I gave it up but I thought it one of the more interesting pieces of nonsense I’d come across. I dont think you can call him a crank. He knows too much. And it really is well written. I think I would put him with Schopenhauer as an exemplar of German prose. He makes some odd statements. The mathematics of night? I suppose Grothendieck is capable of saying something like that. But Grothendieck is a great mathematician. You have to take him seriously. Beginning this lengthy study of what he takes to be the meaning of history with an inquiry into the meaning of mathematics is a strategy that modern philosophers might well take into consideration. An enormous amount of Wittgenstein’s work is in mathematics. Very little of it published.