“The rest of the world was uncertain what had been proven and what not.”

Until a minute ago I had never heard of Mizar, a project to record as much mathematics as possible in computer-readable form.  The pieces of this project are published in the Journal of Formalized Mathematics.  Here, for instance, is the paper “Non-negative Real Numbers, part I.”

I learned about Mizar when glancing through the publicly available archive of QED, a mailing list from the early 90s devoted to the formalization of mathematics.  It’s interesting to be reminded just how excited people were about the prospects of computerizing large precincts of mathematical practice, an ambition which as far as I can tell has now receded almost entirely from view.

I found the QED archive, in turn, via Math Overflow, which quoted these pointed remarks of Mumford about algebraic geometry in the Italian style:

The best known case is the Italian school of algebraic geometry, which produced extremely good and deep results for some 50 years, but then went to pieces. There are 3 key names here — Castelnuovo, Enriques and Severi. C was earliest and was totally rigorous, a splendid mathematician. E came next and, as far as I know, never published anything that was false, though he openly acknowledged that some of his proofs didn’t cover every possible case (there were often special highly singular cases which later turned out to be central to understanding a situation). He used to talk about posing “critical doubts”. He had his own standards and was happy to reexamine a “proof” and make it more nearly complete. Unfortunately Severi, the last in the line, a fascist with a dictatorial temperament, really killed the whole school because, although he started off with brilliant and correct discoveries, later published books full of garbage (this was in the 30’s and 40’s). The rest of the world was uncertain what had been proven and what not. He gave a keynote speech at the first Int Congress after the war in 1950, but his mistakes were becoming clearer and clearer. It took the efforts of 2 great men, Zariski and Weil, to clean up the mess in the 40’s and 50’s although dredging this morass for its correct results continues occasionally to this day.

Readers with good memories will recall that this is the second time I’ve quoted uncomplimentary remarks about Severi.

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