I do mathematics to see the invisible.
To exist (in mathematics), said Henri Poincaré, is to be free from contradiction.
(Gower's in 2000 The Importance of Mathematics)
Atiyah
apparently said: "Algebra is the offer made by the devil to the
mathematician. The devil says: I will give you this powerful machine, it will
answer any question you like. All you need to do is give me your soul: give up
geometry and you will have this marvelous machine."
Weyl said: "In these days the angel of topology and the devil of abstract
algebra fight for the soul of each individual mathematical domain."
(source Don Piponi in G+)
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