Wednesday 15 April 2009

Grothendieck's platonism

Alexandre Grothendieck is one of the last century's greatest mathematicians. There is no controversy in describing him as the mathematician who has pursued the most abstract type of mathematics ever. In his mathematics there are usually no numbers directly involved. Most of the time there are even no letters denoting numbers involved. The level of abstraction is breathtaking and, to some, deeply fascinating. Nevertheless, his work does have more concrete implications for relations between numbers. There is a huge collection of articles by and about him on the internet. A brief recent introduction can be found here:

http://www.sciencenews.org/view/generic/id/31898/titl/Math_Trek__Sensitivity_to_the_harmony_of_things


The following passage from Grothendieck's Recoltes et Semailles expresses, in a somewhat poetical way, his scientific realism, in particular mathematical realism. It seems like he is saying that what we call inventions and imagination are in fact very close to what platonists refer to as intuition or 'the mind's eye':

"What makes the quality of a researcher’s inventiveness and imagination is the quality of his attention to hearing the voices of things."

Perhaps Grothendieck is also saying that not only is the platonic realm a static life-less structure which we can tap for information, but he seems to suggest that physical and metaphysical entities are actively revealing themselves in various ways, ways which we of course may or may not pick up.

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