What is Hermann Weyl's point?

[pivoxa15]

Hermann Weyl stated

"In these days the angel of toplogy and the devil of abstract algebra fight for the soul of every individual discipline of mathematics."

1) Why does he refer to topology as angel and abstract algebra as the devil?

2) How do they get into every field of maths? I can see how algebra is in many fields and so abstract algebra is a generalisation of intuitive algebra. But how does topology get into it?

3) What about analysis? That is another major field of pure maths and is in many other disciplines of maths?

 

[matt grime]

Who knows - it was, what, 70 years ago.

 

[Hurkyl]

1) Why does he refer to topology as angel and abstract algebra as the devil?

He must have been a geometer.

 

[mathwonk]

if you are a geometer, and have much experience with learning sheaf theory, and cohomology, you will understand what he is saying.

there are even geometric topoologists who dislike algebraic topology. I have tried to teach toric varieties to geometers and topologists who after seeing the definitions via spectra of various rings, asked, "OK, but where is the geometry? how do you get your HANDS on them?"

there is a feeling that algebraic methods take away intuition and render simple arguments too abstract. e.g. do you believe an irreducible non singular affine algebraic curve is really an integrally closed integral domain of krull dimension one?

or that the tangent bundle to a variety is really the set of k[e] valued points where k[e] = k[t](t^2) is the dual numbers? (actually this is fermat's original definition, almost.)

or that a universal family of geometric objects should be regarded as a representable functor?

or that the right way to view a sheaf on a topological space is as a contravariant functor on category defined by the toopology where inclusions are the only morphisms?

You should, as this gives rise to the observation that one can generalize them to categories with more than one map between two objects, leading to the etale topology, and "stacks" where even single points have automorphisms.

these are needed to deal appropriately with local quotient spaces by groups acting with fixed points.

topologists tend to prefer homotopy to homology for this reaon, it is more geometric. Ed Brown Jr. considered his representation thoerem for cohomology as showing that cohomology was better than homology because being representable via homotopy showed that "it occurs in nature".

 

[matt grime]

or that a universal family of geometric objects should be regarded as a representable functor?

only if it is a fine moduli space, right?

topologists tend to prefer homotopy to homology for this reaon, it is more geometric. Ed Brown Jr. considered his representation thoerem for cohomology as showing that cohomology was better than homology because being representable via homotopy showed that "it occurs in nature".

Well, Brown representability is very powerful, or at least its variant Bousfield localization - but you are preaching to the choir with me, Roy.

 

[mathwonk]

well yes matt, i was being fuzzy, but by "universal family" i meant fine moduli space. i.e. i think of a coarse moduli space as the base space for a universal family, but without the family over it.

i was a student of both brown and bousfield, but do not know about bousfield localization.

 

[mathwonk]

i see now that my language was ambiguous. by a universal family of objects of type A, I meant a mapping X-->M whoser fibres are all objects of type A, and such that given any morphism Y-->N whose fibers are objects of type A, there is a unique morphism f:N-->M such that the induced family f*(X)-->N is isomorphic to the given one Y-->N.

I.e. the functor of N, defined by F(N) = famlies of type A over N, is represented by X, i.e. is isomorphic to the functor Hom(N,X).

 

[bmm9]

He wrote an article "Algebra and Topology as Two Roads to Mathematical Comprehension" that you might want to read. It gives some examples of how you can view a problem algebraically and topologically (... you probably got that from the title). I haven't read the whole thing, so I'm not sure if that's where the quote is from.

 

[M Grandin]

"We are left with our symbols"

Hermann Weyl is known having said in his last years "We are left with our symbols".
What did he mean by this?