Thursday, November 27, 2014

Celebrating abstract nonsense

Berkeley mathematician Edward Frenkel writes in the NY Times:
Alexander Grothendieck, who died on Nov. 13 at the age of 86, was a visionary who captivated the collective psyche of his peers like no one else. ...

Grothendieck’s genius was to recognize that there is a “being” hiding behind a given algebraic equation (or a system of equations) called a scheme. The spaces of solutions are mere projections, or shadows of this scheme. Moreover, he realized that these schemes inhabit a rich world. They “interact” with one another, can be “glued” together and so on. ...

Though one might ask if there are any real-world applications of his work, the more important question is whether having found applications, we also find the wisdom to protect the world from the monsters we create using these applications. Alas, the recent misuse of mathematics does not give us much comfort.

For example, according to published reports, the National Security Agency inserted a back door in a widely used encryption algorithm based on “elliptic curves” — mathematical objects illuminated by Grothendieck’s research. Though that specific algorithm was developed much later, Grothendieck recognized the potential dangers of such misuse of math and sounded the alarm. He was incensed when he learned that IHES, the mathematics institute near Paris where he worked, received funding from the French Ministry of Defense. In protest, he resigned from the institute in 1970 at the height of his power. He had hoped that his colleagues would follow him, but none did.
They did not follow him because he was going crazy. He spent the last 20 years writing unintelligible rants.

Grothendieck’s life story is a wild one. He was the illegitimate son of anarchist parents, and grew up without nationality or parents. He is mainly famous for what mathematicians affectionately call abstract nonsense, and recast the foundations of algebraic geometry in that style. He had a cult-like following of some of the world's smartest men. There are no practical applications, as far as I know, even tho his Wash. Post obituary said:
His contributions to mathematics were often likened to those of Albert Einstein in physics. ... Other scholars came to apply Mr. Grothendieck’s theoretical frameworks to such fields as computer programming, software development, satellite communications, classification systems and the study of biological data.
The analogy to Einstein is strange. Einstein is mainly famous for popularizing relativity theory, and relativity was pivotal for XX century physics. Maybe a better comparison would be to Kurt Goedel, whose work was of pre-eminent foundational importance, but of little consequence to the rest of mathematics or the real world.

Frenkel suggests that Grothendieck might have objected to an NSA-designed random number generator. Maybe so, but that would be more proof of his paranoia. While the news regularly has news of security breeches affecting millions of people, there is no evidence of anyone being adversely affected by that random number generator.

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