Thursday, April 27, 2006

Mystic, Martyr, Mathematician

Recently, my officemates and I were looking up famous mathematicians' biographies (and no, we don't do this all the time), and we came across Dmitrii Egorov, a Russian mathematician of the early 20th century, now immortalized (to us, at least) for a highly useful theorem of real analysis. (No, I won't go into Egorov's Theorem here.)

Doesn't he look intense? Anyhow, having heard of him, I was greatly amazed to find out how he died:
However Egorov was a deeply religious man and when the Church was repressed after the revolution, Egorov defended them. In 1922-23 there were mass execution of clergy and in 1928 the attack was renewed. Egorov was in a position of power in the Moscow Mathematical Society and he tried to shelter academics who had been dismissed from their posts. He tried to prevent the attempt to impose Marxist methodology on scientists.

In 1929 Egorov was dismissed as director of the Institute for Mechanics and Mathematics and given a public rebuke.

Some time later he was arrested as a "religious sectarian" and put in prison. The Moscow Mathematical Society continued to support Egorov, refusing to expel him, and those who presented papers at the next meeting, including Kurosh, were to be expelled by an "Initiative group" who took over the Society in November 1930. They expelled Egorov denouncing him as "a reactionary and a churchman".

Egorov went on a hunger strike in prison and eventually, by this time close to death, he was taken to the prison hospital in Kazan. Chebotaryov's wife was working as a doctor in the prison hospital and, although it sounds rather unlikely, it is reported that Egorov died at Chebotaryov's home.
Needless to say, I was greatly impressed by this account, and started looking for more on the man. The search led me to an absolutely fascinating essay, A Comparison of Two Cultural Approaches to Mathematics, on the conflicts of the early 20th century over the reality of non-constructible mathematical objects, and how this related to a divide between milieus of French rationalism on the one hand and Russian mysticism on the other.

If that description appealed to you at all, I urge you to forget the rest of what I write, and go read it for yourself. For those of you bored, baffled, or both, by the above, I promise I'll drop the pedantry and try and explain why this is so awesome, and where exactly Egorov fits in.

Essentially, a number of discoveries in the late 19th century threw mathematics into a flurry of controversy. Apart from the logical paradoxes found at that time (which make for another fascinating story), the most earth-shattering claims were staked by the German mathematician Georg Cantor, who once and for all exploded our original naive intuitions of infinity. (For an explanation of what he did and how, this isn't bad.)

One of the pieces of fallout from this was that the most intuitive definitions of mathematical concepts like "real number" or "function" postulated the existence of real numbers or functions that are inaccessible to our finite minds, because they aren't the results of any formula or the limits of any intelligible process. (I want to stop and prove this assertion, but I'll resist the temptation.) Many of the leading French mathematicians balked once they realized this; Henri Lebesgue wrote of it, "Can we convince ourselves of the existence of a mathematical being without defining it? To define always means naming a characteristic property of what is being defined." Ultimately, the authors of the essay argue that the philosophical presuppositions of the leading French mathematicians led them to abandon their early successes in these areas; a widespread Cartesianism and a secularist positivism made it inconceivable to these men that anything could meaningfully exist which was not intelligible. (Um, about dropping the pedantry: mea culpa?)

Emile Picard sneered at those who engaged with such matters:
All problems of this type are caused by a lack of agreement on what existence means. Some believers in set theory are scholastics who would have loved to discuss the proofs of the existence of God with Saint Anselme and his opponent Gaunilon, the monk of Noirmoutiers.
(His derision toward Catholic philosophers and detailed ontology is not incidental to the topic.)

When the French left the field, it was the Russian mathematicians who rescued the field from the tyranny of epistemology over ontology. Egorov and Nikolai Luzin were foremost among them, and their willingness to interact with non-constructible (non-knowable) mathematical objects was not unrelated to their shared religious faith in the God who truly existed but was beyond all human understanding.

UPDATE: Edited to read: (It must be noted that they were Imiaslavtsy or "Name-Worshippers", who believed that God was wholly present in and even identical to the Name of God. Their like were deemed heretics by the Orthodox Church a few years before the Communist Revolution, which adds an additional note of wonder to Egorov's self-sacrificing defense of that Church against the Marxists. I'd say more about this, but I really hadn't heard of them before.)

In the end, the mathematics produced by these Russian scholars was so beautiful, so useful, so eminently true, that it triumphed over the epistemological snobbery of the French school. Today we learn about non-constructible reals, nowhere differentiable continuous functions, and non-measurable sets in undergraduate classes, and we think little of it besides a passing note of weirdness. Thanks to the Russian mystics like Egorov, we have now realized that it is simply sophomoric to deny the existence of something on the sole grounds that we cannot define or comprehend it ourselves.

Well, most of us have realized it. There's still the V=L wackos.

It is said that Egorov died from the effects of his hunger strike and prison treatment, in the house of the mathematician whom the Party chose to replace him at the Moscow Mathematical Society, chanting the name of Jesus. Requiem aeternam dona eis Domine: et lux perpetua luceat eis.

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