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I recall reading (on this very site, in fact) that there is a mathematician whom argues that we ought to switch from "set" to "order" in the foundations, so as to "recover duality" or some such.

Does anyone know who this is? And/or where they've argued this?

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John Baez once argued that $0$-categories should be preorders rather than sets, but the argument was based on higher category theory. Specifically, that the hierarchy of categories should start at $-2$, and goes:

  • There is just one $(-2)$ category.
  • $(-1)$ categories are equivalent to truth values -- a $(-1)$ category has objects, and the homsets are $(-2)$-categories. Therefore either a $(-1)$ category is empty, or it is non-empty and all objects are isomorphic.
  • $0$ categories should be equivalent to preorders -- a $0$ category would have objects, and the homsets are $(-1)$-categories: i.e. truth values. Working out what the composition law should mean amounts to making the value of $\hom(X,Y)$ mean the truth of $X \leq Y$ where $\leq$ is a preorder.

Then, we recover sets as being analogous to groupoids: they are preorders whose arrows are all isomorphisms. Actually, this gives setoids -- sets with an equivalence relation -- but setoids are equivalent to sets.

(IMO, the rationale that preorders are a better basic notion that set is not implausible even without trying to imagine $(-2)$- and $(-1)$-categories. And I definitely think setoid is a better basic notion than set)

I don't recall him saying anything about duality, though, so this may have nothing to do with what you're remembering.

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  • $\begingroup$ Yes, this is exactly what I was looking for. $\endgroup$ – goblin Jan 31 '14 at 15:34
  • $\begingroup$ Also, why do you think setoid is a better basic notion than set? The way I see, a set is just a weak setoid; i.e. a setoid up to equivalence. $\endgroup$ – goblin Jan 31 '14 at 23:05
  • $\begingroup$ @user18921: Because people use setoids in practice. e.g. consider quotient rings, like $\mathbb{Z} / p \mathbb{Z}$, the field of $p$ elements. When pressed, most people will say it is a set of equivalence classes, or maybe a set of representatives for equivalence classes. But in practice, people almost never use that: they always use ordinary integers for the elements of this ring (and any integer can be used), and write equivalences rather than equations. $\endgroup$ – user14972 Feb 1 '14 at 1:07
  • $\begingroup$ ... also, IMO it's even more far reaching than that. e.g. in the language of integer arithmetic, 1+2 is an integer, and so is 3. I think it's a mildly more accurate description of mathematics to say that 1+2 and 3 are equivalent elements of the integers, than to say 1+2 is 3. $\endgroup$ – user14972 Feb 1 '14 at 1:10
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    $\begingroup$ There's also an idea floating around that I haven't really grokked: that up to equivalence, equality is equivalence. I've been growing into the idea that equality is not talking about what something is, but instead is talking about the point where we stop acknowledging the distinction between equivalent things. $\endgroup$ – user14972 Feb 1 '14 at 1:16

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