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+)