How can there be alternatives for the foundations of mathematics?

Question

How can set theory and category theory both be plausible theories for the foundations of mathematics? If these two theories are not mathematically equivalent, does it not mean that the rest of mathematics, when taken exclusively with either of these two theories as foundation, will be distinct; that is, unless these two theories are, in some way, equivalent.

Thanks

Answer 1

There are two ways to answer your question, first through different ideas of what 'foundation' might mean, and second about multiple different foundations of the same kind.

• What does foundation mean? Set theory (and logic) are attempts at foundations of mathematical knowledge about how we can know mathematical truths - how do you know that a theorem is true and what does 'true' mean? Set theory does this by allowing a translation from your favorite branch of mathematics to sets, and then formalizing the inference operations on just the set objects.

  Category theory is a foundation for what we know about mathematics - what similarities or commonalities are among disparate areas of mathematics. Take your favorite brach of mathematics, define some objects and some suitable functions and then you get all sorts of theorems for free.

  Of course you can use one of these for the other. You can -logical- foundation of category theory, e.g. the objects and morphisms are implemented with sets. And you can give a -categorical- foundation for set theory/logic, e.g. one can have a category of 'set' with 'functions' as the morphisms.

• How can there be more than one foundation at all, even of the same kind? For any given theory, let's say Euclidean geometry, there is more than one axiomatization that results in the same collection of theories. There can certainly be more than one truth foundation intuitionistic, constructivist, allowing different set axioms, different rules of inference, ZFC, NBG, etc., each allowing desired features and disallowing unwanted features.

Set theory and Category theory are not the only kinds of foundations. And there have been many foundations that attempt to unify ideas. For example, algebra as a whole is a kind of unification, converting all mathematical thinking into rules of symbolic manipulation. Category theory is the latest is just the unification of concepts.

Answer 2

I'm learning about some of this right now! As far as I know, you already need quite a lot of set theory to say topos. An (elementary) topos is is a category closed under a lot of common constructions.

These topos gadgets also have an internal logic (usually intuisionistic), and given for instance a natural number object you can perform many of the well-known constructions of mathematics, such as those of the real numbers (here the Cauchy and Dedekind reals may be different). This is what we do in the topos of sets!

An elementary topos $E$, by the way, is a cartesian closed category (which means it has a terminal object $1$, products, exponentiation, and an adjunction between the two, i.e. $\hom_E(X\times Y,Z)\cong\hom_E(X,Z^Y)$) with a subobject classifier $\Omega$ together with a morphism $\top:1\to\Omega$ called true, such that any monic $f:Y\to X$ is the pullback of a unique morphism $\chi(f):Y\to\Omega$. We also require pullbacks along $\top$ to exist. It follows that $E$ has finite limits and colimits, and that the subobject functor $Sub:E\to Set$ is represented by $\Omega$ (i.e. $Sub(-)\cong\hom(-,\Omega)$).