Prokhor Zakharov
Graduate Poster
- Joined
- May 11, 2016
- Messages
- 1,110
Edited by Agatha:
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Accordingly, here I would like to discuss the value of the branch of mathematics called number theory, which concerns the properties of and relations between integers, and in particular the final achievement of a fully general proof of Fermat's last theorem, the deceptively simple claim that there are no three positive integers that satisfy the equation an + bn = cn for any integer value of n above two.
A good deal of what we now call number theory goes back to antiquity; the proof that the square root of two is irrational is something a lot of students still go through today and a likely apocryphal legend has it that a member of the Pythagorean sect was drowned out of hatred by other members for having achieving this. It really began to come into its own in the 19th century though, and started to be recognized as a distinct branch of mathematics. In fact, the prolific 19th century mathematician Carl Friedrich Gauss once said: "Mathematics is the queen of the sciences and number theory is the queen of mathematics."
However, conspicuously absent during all of this time was much in the way of any applications of number theory. The number theorist Leonard Dickson, whose time came after that of Gauss, but died only a decade after the advent of the digital computer, said: "Thank God that number theory is unsullied by any application." But due to the increasing entrenchment of the digital computer in life in industrial countries, applications of number theory have blown up rapidly in more recent decades for all sorts of calculations. The use of number theory is especially marked in cryptography, and without modern cryptographic methods there wouldn't be many things considered to be of practical value, among them, the entirety of e-commerce. In this way, even accepting the very, very dubious claim that knowledge is of no value for its own sake, all these centuries of work seemed to have paid off after all.
One might still consider all this effort a waste and contend that individuals with more "practical" orientations would have come up with these things anyway. There is a very serious issue with that contention though: it's ahistorical. If there is any example of an elite go-getter with both their feet firmly on the ground, focused only on knowledge of "real" value, who ever came up with such advances as the past centuries have seen in number theory, I'd definitely like to see it. I certainly can't think of any myself, and you would think that people with such a superior, focused worldview would be better thinkers overall.
Now I'd like to turn my attention to Fermat's last theorem. This was stated in a margin of a copy of an ancient mathematical text by the mathematician Pierre de Fermat. He claimed to have a "marvelous proof" of the same that wouldn't fit in the margin (and is widely considered to have been mistaken), but never wrote it down. Fermat's last theorem was finally proven in a truly general form by Andrew Wiles in 1994, after over 350 years of effort by many others as well. This event is considered a great milestone in mathematical history, but I am not aware of any practical applications of Fermat's last theorem as such, except to other claims in pure mathematics. I am under the impression that, along the way, there were developments in what are called elliptic curves that have applications to cryptography presently, that were vital for the proof. But that doesn't mean that Fermat's last theorem per se has any practical applications. What is the value of such a thing to someone with a "pragmatic" outlook, in scare quotes? Was this 350+ years full of wasted time? If it were some sort of multigenerational business venture to yield practical, profitable outcomes, I can tell you everyone involved would have taken a bath on it, thus far at least. Maybe Fermat's last theorem will have practical applications one day, maybe not. The question of course is whether the value of the theorem is solely contingent on whether these applications ever arise.
To draw things to a close, I want to say that even if such things as Fermat's last theorem are "useless" per se, which I of course do not believe, it seems as though it is often the case that the only way to find the practical applications you might want is by not looking for them at all and that people who fancy themselves "pragmatic" when they deride "useless" knowledge are actually just a different word: myopic.
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