Mathematical form

During a lecture on linear algebra, I was giving a loose description of orthonormal bases in the space of vibrational modes of a metal string. It seemed noteworthy that the underlying vector space was infinite-dimensional, but was approximated by finite-dimensional spaces in practice, corresponding to ignoring high frequencies. That is to say, it is a prototype example where the thing being approximated is not a number or a function, but a *space*. At this point, I received the question `Can you prove that?’ from a student at the very rear of the lecture hall. I was a bit taken aback, and did not manage to deal with the question in a suitable manner. Immediately afterwards, of course, I regretted not having looked more deeply into the perspective of the student. I do hope to discuss the issues surrounding that question at some point in a separate essay. It was somewhat on my mind when I wrote the short note on education. I was thinking of it also during the last day of classes when, in the course of a brief conversation with some students, I asserted that `mathematics is not primarily about proofs.’ Needless to say, this was also an unsatisfactory statement to make on the fly, and ended up inviting the awkward question of what then it might be about. I didn’t answer, and will not attempt any real answer for quite a few more years. I still fear the sterility that might result from excessive rumination on the nature of mathematics, at least for a mediocre practitioner like myself. But I will pass along another Grothendieck quote:

If there is one thing in mathematics that fascinates me more than anything else (and doubtless always has), it is neither “number” nor “size”, but always form. And among the thousand-and-one faces whereby form chooses to reveal itself to us, the one that fascinates me more than any other and continues to fascinate me, is the structure hidden in mathematical things.

Incidentally, when I direct your attention to a paragraph of this sort, the intention (to the extent that it matters) is not necessarily to express full sympathy with the sentiment therein.



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