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. -- Michael Atiyah
This captivating remark confuses me -- what is the fundamental difference between algebra and geometry? I thought these labels more represented historical trends in mathematical inquiry rather than real technical distinctions. See Difference between algebra and geometry
Atiyah seems to be suggesting a distinction between the reasoning/axioms associated with geometry and those associated with algebra. Is there a difference? If so, what is it?