Part I: A review of Christian Szegedy’s “Promising Path”

“Grothendieck’s way of writing is based on an atypical conception of mathematics, described and theorized in texts [that] vigorously bear witness to the unavoidable poetic aspect that motivates scientific work and the surplus of meaning that formalization believes should be eliminated, although this surplus is precisely where the essence of mathematical thought lies.”                 (F. Patras, La pensée mathématique contemporaine, Introduction)1

This week and the next I compare two perspectives on the “the surplus of meaning” and “the essence of mathematical thought,” both implicit in texts about mathematics:  the first by a prominent exponent and prophet of “intelligent computer mathematics,” the second by a prominent number theorist.  It would be too easy to say that the first text takes the position that there is no surplus and that the “essence of mathematical thought” resides in formalization and nothing more, while the second text exemplifies the surplus of meaning as well as what one reviewer of Karen Olsson’s book The Weil Conjectures (to which we return in Part II) called “the poetry and precision of a theorem.”  What, after all, is an “essence”?  Philosophers have worried about this for millenia, with some interruption, asking for example in what sense the essence of Socrates resembles that of being an even prime number, and have failed to arrive at consensus on the essence of “essence.”  The whole confusing history is recorded in the Stanford Encyclopedia of Philosophy’s entry on “Essential vs Accidental Properties” (which, troublingly, includes no references in French).…

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