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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?

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    $\begingroup$ Atiyah saw mathematicians forming two tribes: geometers and algebraists. He saw himself as a member of the superior tribe: a geometer. $\endgroup$ – Angina Seng Jul 25 '20 at 16:39
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    $\begingroup$ I recommend to read the intreduction of David Hilbert's book "Geometry and the Imagination" or its German original "Anschauliche Geometrie" from 1932. $\endgroup$ – Paul Frost Jul 25 '20 at 16:46
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    $\begingroup$ There’s a story, likely apocryphal, that Chevalley and Zariski were discussing curves, and it became clear that there was some misunderstanding. “What do you mean by a curve, anyhow?” says Z to C, and C writes on the board $f(x,y)=0$. Then Z takes the chalk and makes a looping figure on the board. $\endgroup$ – Lubin Jul 25 '20 at 16:56
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    $\begingroup$ Then point, @C.F.G , is that there are (were) two very different approaches to what even such a simple concept as “curve” is about. For the algebraist, it’s an equation, while to the geometer, it’s a picture. And I think that the debate between C and Z goes to the heart of this question and of the quote from Atiyah. $\endgroup$ – Lubin Jul 25 '20 at 17:06
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    $\begingroup$ I found the quote here. I cannot see how this is related to Soul theorem. @C.F.G $\endgroup$ – Arctic Char Jul 25 '20 at 17:57

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