MATHEMATICS AND LOGIC

A few years ago I had occasion to set forth certain ideas about the logic of infinity, on the role of infinity in mathematics, and the use made of it since Cantor's time. I explained why I did not consider as legitimate certain methods of reasoning which various eminent mathematicians had believed they could employ.

Naturally I drew some sharp replies. These mathematicians did not believe they had erred; they believed they had the right to do what they had done. The discussion dragged on, not because new arguments arose ceaselessly, but because we kept going around in the same circle, each one repeating what he had just said, seemingly not having heard what the opponent had said. On each occasion, I was sent a new proof of the principle under contention in order, it was said, to be safe from all objection; but this proof was always the same, hardly revised. No conclusion consequently was reached.

If I should say that I was surprised, I would convey a false impression of my psychological acumen. Under these conditions, it hardly seems advisable to repeat once more the same arguments to which I could probably give a new form but which I could not change fundamentally, since it seems to me that my opponents have not even tried to refute them. It seems preferable to seek what can be the origin of this difference of mentality which engenders such divergent views. I have just said that these irreducible divergences had not astonished me, that I had foreseen them from the very beginning. But this does not exempt us from seeking the explanation; it is possible to foresee a fact after repeated experiences, and yet be very hard pressed to explain it.