The 20th century mathematics revolution

Mathematician Frank Quinn wrote in 2012:

The physical sciences all went through "revolutions": wrenching transitions in which methods changed radically and became much more powerful. It is not widely realized, but there was a similar transition in mathematics between about 1890 and 1930. The first section briefly describes the changes that took place and why they qualify as a "revolution", and the second describes turmoil and resistance to the changes at the time.

The mathematical event was different from those in science, however. In science, most of the older material was wrong and discarded, while old mathematics needed precision upgrades but was mostly correct. The sciences were completely transformed while mathematics split, with the core changing profoundly but many applied areas, and mathematical science outside the core, relatively unchanged. The strangest difference is that the scientific revolutions were highly visible, while the significance of the mathematical event is essentially unrecognized.

More of his opinions are here. This is essentially correct. Relativity and quantum mechanics radically changed physics in those decades, and math was similarly changed.

I would not say that the older physics was discarded; previous ideas about Newtonian mechanics, thermodynamics, and electromagnetism are still correct within their domains of applicability. The new physics was a revolution in the sense of a turning-around with new views that permitted understandings that were unreachable with the old views.

The essence of the math revolution was precise definitions and logically complete proofs. It was led by Hilbert and some logicians. It was perfected by Bourbaki.

Of course mathematicians used the axiomatic method since Euclid, but only in the early XXc was it formalized to where proofs had no gray areas at all. The 19c still used infinitesimals and other constructs without rigorous justification.

Quinn goes on to complain that educators are almost entirely still stuck in the previous 19c math. I would add that about 90% of physicists today are also.

Quinn goes on to relate this lack of understanding to a lack of respecte for mathematicians:

Many scientists and engineers depend on mathematics, but its reliability makes it transparent rather than appreciated, and they often dismiss core mathematics as meaningless formalism and obsessive-compulsive about details. This is a cultural attitude that reflects feelings of power in their domains and world views that include little else, but it is encouraged by the opposition in elementary education and philosophy. In fact, hostility to mathematics is endemic in our culture. Imagine a conversation:

A: What do you do?
B: I am a --- .

A: Oh, I hate that. Ideally this response would be limited to such occupations as "serial killer", "child pornographer", and maybe "politician", but "mathematician" seems to work. It is common enough that many of us are reluctant to identify ourselves as mathematicians. Paul Halmos is said to have told outsiders that he was in "roofing and siding"!

Yes, mathematicians are widely regarded as freaks and weirdos. Hollywood nearly always portrays us as insane losers. There was another big movie last year doing that.

A new paper takes issue with how mathematician Felix Klein fits into this picture:

Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn).

Klein is mostly famous for his Erlangen program, an 1872 essay that modernized our whole concept of geometry. He embedded geometries into projective spaces, and then charactized a geometry by its symmetry group and its invariants.

Courant wrote in 1926:

This so-called Erlangen Program, entitled `Comparative Considerations on recent geometrical research' has become perhaps the most influential and most-read mathematical text of the last 60 years.

These ideas were crucial for the development of relativity. Lorentz had correctly figured out how to resolve the paradox of Maxwell's equations and the Michelson-Morley experiment, but what really tied the theory together beautifully was the symmetry group and the invariants, as that made it a non-Euclidean geometry under the Erlangen Program. Such thinking clearly guided Poincare, Minkowski, Grossmann, and Hilbert.

No history of relativity even mentions this, as far as I know. The historians focus on Einstein, who never seemed to even understand this essential aspect of relativity.

The above paper argues that Klein was a very modern mathematician who has been unfairly maligned by Marxist historians. Quinn was duped by those historians.

Quinn wants to distinguish 19c non-rigorous math from XXc axiomatic modern math, the Marxist historians instead distinguish Aryan math, Jewish math, Nazi math, racist math, and modern math. Klein was said to be half-Jewish.

The paper has quotes like these:

"This is compounded by a defect which can be found in many very intelligent people, especially those from the semitic tribe[;]8 he [i.e., Kronecker] does not have sufficient imagination (I should rather say: intuition) and it is true to say that a mathematician who is not a little bit of a poet, will never be a consummate mathematician. Comparisons are instructive: the all-encompassing view which is directed towards the Highest, the Ideal,9 marks out, in a very striking manner, Abel as better than Jacobi, marks out Riemann as better than all his contemporaries (Eisenstein, Rosenhain), and marks out Helmholtz as better than Kirchhoff (even though the latter did not have a droplet of semitic blood)."

"When the National Socialists came to power in 1933, [Bieberbach] attempted to find political backing for his counter-modernist perspective on mathematics, and declared both, intuition and concreteness, to be the inborn characteristic of the mathematician of the German race, while the tendency towards abstractness and unconcrete logical subtleties would be the style of Jews and of the French (Mehrtens 1987). He thus turned countermodernism into outright racism and anti-modernism."

I don't know what to make of this. No one talks this way anymore.