Some algebraists (Cartier, Weil, Atiyah, etc.) sometimes speak of geometry as a long history of essentially asking the same question—"what is space, and how would one describe a space uniquely". History that changes not with the question but only with ever more sophisticated tools.
Can you explain how one (esp. physicists with visual intuition interested in extending into abstract mathematics and algebraic geometry) should think about describing a general space, from a historical perspective?
Something like the below:
You might want to study a certain space by
- using algebra (by studying behavior of functions on some field),
- using algebraic varieties (by studying solution sets to systems of algebraic equations)
- with Schemes
- cohomology between objects
- with sheaves
- Topos (by describing categorically how other objects relate to the one you're trying to study)
- Motive theory (unsure but put it here as per Mumford's writings on Grothendieck's work)
Don't take the above list seriously. I am merely regurgitating undigested material without much in-depth understanding. My goal here is to understand how to find a motivation for using more advanced tools in abstract mathematics for studying spaces—usually they are unforgiving in the technical rigor almost from the first paragraph, hence my asking for "historical" perspective.
Thank you in advance!