Hermann Weyl once said “In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.” Discuss.

Hermann Weyl once said “In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics.” Discuss.

Isn’t it the angel of set theory and the devil of category theory? Or are they the same thing?

You forgot the second sentence. “And the devil is winning.”

How is topology/category theory winning?

Why is Abstract Algebra cast as “the devil”?

Reminds me of Atiyah’s comments regarding the “Faustian Offer” in “Mathematics in the 20th Century” (Bull. of the London Math. Soc. 34 (2002) 1-15) (or see duch.mimuw.edu.pl/~sjack/atiyah.ps).

Ford: I wondered the same thing about algebra being the devil, which probably demonstrates that we’re both irretrievably damned.

Instead of learning to solve the problem algebra allows you instead to manipulate symbols. The Devil’s minion, Descartes, allowed people with no geometrical insight to start proving theorems about geometry. You can develop incredible insight into how those symbols behave, but you’re not doing geometry. Even worse, people can be so far removed that they even fail to realise that they are doing geometry.

I for onw revel in my devilishness

make that “one”

You are correct, Sigfpe, that algebra allows geometry to be done by people without geometric intuition. But what is the nature of this intuition? I doubt anyone had or has any *inherent* intuition for non-Euclidean geometry, since our entire every-day experience as children is Euclidean. Physicists only started to think that the Universe may be better modeled with a non-Euclidean geometry after such geometries had been developed by those clever algebraists, Gauss, Bolyai and Riemann. Some other clever people –eg, Frege — simply could not accept that non-Euclidean geometry was more than abstract nonsense, unconnected to anything in the real world.

But isn’t that what’s precisely nice about mathematics that I don’t have to develop the intuition to do geometry, physics, etc. ? Intuition is certainly good to have and I wish I had it for all kinds of subjects, but there’s no methodical way to develop it.

I have met the devil and his name is Nicolas Bourbaki.