Anyway, thanks for the interest and the confirmation!

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Finally (I think it was finally), Grothendieck consolidated all of this with the notion of a scheme.

I’m very much in favor of what you say in this post. But I’m not sure that mathematical essences are ever revealed to us in a final “ultimate” form.

In particular, while your history of algebraic geometry is excellent, it may have stopped a little too soon, because after Grothendieck gave his famous definition of a scheme as a ringed space pieced together out of spectra of commutative rings, he gave a much simpler and more general definition in which a scheme is simply a functor from commutative rings to sets.

It seems nobody has ever put much work into studying this definition, because Grothendieck had already shot too far ahead of his time and exhausted everyone who was trying to keep up. Even now, most mathematicians still find his *first* definition of scheme to be painfully abstract and general. But his later definition is beautiful and simple, and may someday be taken up and given serious study.

“Although we often think that theorems are the main product of mathematics, it seems that a lot of the time just identifying the “right” structures to be talking about is really the goal.” Precisely!

Raising the topic of essences as you do, the natural next step is to think about natural kinds, laws, etc. I made a start in Mathematical Kinds, or Being Kind to Mathematics. Admittedly, it’s not the most careful piece, but then I often feel close to quitting philosophy of math and don’t want to leave half-formed ideas out.

]]>For many of these concepts though, we don’t use the alternative characterization as a *definition*, but for some (like ordinals) we do. One of my colleagues told me a story about her first model theory class, when the professor mentioned the phrase “complete theory”, and she asked what it was. Rather than saying “a theory that logically entails every formula or its negation”, he responded “a theory, all of whose models have isomorphic ultraproducts”.

My comment on this post is just a minor observation: You say: *“it’s extremely common not to have a single definition for a mathematical object.”*

Indeed, having more than one definition of an object is often how we **understand** the object. As an example, consider a derivative, which can be understood in terms of epsilon-delta arguments, or infinitesimals, or as a functor between appropriate categories, and so on.