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A Prologue
Apr. 20th, 2019 05:10 pmAn odd and, perhaps, rather interesting thing happened this morning. What I'm going to talk about here is not that thing, but only a prologue to it.
There is a branch of mathematics known as category theory. (It is also sometimes referred to as "abstract nonsense", which I am willing to regard as an affectionate nickname. Not all agree.) It arose after WWII; the University of Chicago, where I got my doctorate, was a hotbed of the subject. (One of its creators, Saunders MacLane, taught at Chicago when I was there. I will refrain from talking about him; that would lead to a long digression. But I liked him.)
Geometry is the mathematics of shape. Calculus is the mathematics of change. Category theory is the mathematics... of mathematics. Categories, the objects of study in that field, come in all shapes and sizes, but many branches of mathematics can be identified with categories. (There is a category of finite-dimensional real vector spaces, a category of groups, a category of *sets*,...) (When I say "all sizes", I mean it in the mathematician's sense: many categories are too large to be sets.) (There are also some very small categories; for instance, any graph - in the sense of "graph theory" - can be regarded as a category.) (Do you see why I don't want *additional* digressions?)
Let me give you an example of the power of category theory. A year or two ago, I was teaching courses in group theory and advanced linear algebra in the same semester, back to back. As it happened, I covered the Fundamental Theorem of Finite Abelian Groups in the one class at the same time that I was going over the Jordan Canonical Form in the other. It struck me that the proofs, though different in detail, had the same structure. Now, the category of finite Abelian groups and the category of finite-dimensional complex vector spaces are two examples of "Abelian categories", and it turns out that there is a single theorem, true in any Abelian category (well, there are some restrictions), which manifests, in the one case, as the FTFAG and, in the other, as JCF. It's amazing, because the two theorems are superficially very different, but from the viewpoint of category theory, they're indistinguishable.
To give another example: there is another kind of category, known as a "topos". Roughly speaking, a topos is a category that "looks like" the category of sets. Crucially, analogues of elements ("internal elements") and subsets ("subobjects") are available, and it's possible to ask, "Is this internal element in this subobject?" However, depending on the topos, the answer to that kind of question may not be restricted to "Yes/No". Each topos has an associated logic, which may be multivalued, and the logical constructions of negation ("not"), conjuction ("and"), disjunction ("and/or"), and implication ("if/then") can be carried out. Most interestingly, there are toposes where infinitesimals - the things that Newton and Leibniz really wanted to use in calculus, but couldn't because they are logically impossible, under classical logic - become possible. (An infinitesimal is a number x such that the statement "x = 0" is neither true nor false. This is impossible in Boolean logic, but certain toposes allow it!)
Anywho. Strange and wondrous things happen in category theory. It encourages the wildest of speculations and the highest levels of abstraction. I know no more than the rudiments - and, specifically, the rudiments as of the early '80s; the field has advanced greatly since then. Still, I'm reasonably comfortable with it, even admitting that the epithet "abstract nonsense" bears some truth.
This morning, a bit of category theory leaped at me from hiding and bit me. I'll talk about that in another post.
There is a branch of mathematics known as category theory. (It is also sometimes referred to as "abstract nonsense", which I am willing to regard as an affectionate nickname. Not all agree.) It arose after WWII; the University of Chicago, where I got my doctorate, was a hotbed of the subject. (One of its creators, Saunders MacLane, taught at Chicago when I was there. I will refrain from talking about him; that would lead to a long digression. But I liked him.)
Geometry is the mathematics of shape. Calculus is the mathematics of change. Category theory is the mathematics... of mathematics. Categories, the objects of study in that field, come in all shapes and sizes, but many branches of mathematics can be identified with categories. (There is a category of finite-dimensional real vector spaces, a category of groups, a category of *sets*,...) (When I say "all sizes", I mean it in the mathematician's sense: many categories are too large to be sets.) (There are also some very small categories; for instance, any graph - in the sense of "graph theory" - can be regarded as a category.) (Do you see why I don't want *additional* digressions?)
Let me give you an example of the power of category theory. A year or two ago, I was teaching courses in group theory and advanced linear algebra in the same semester, back to back. As it happened, I covered the Fundamental Theorem of Finite Abelian Groups in the one class at the same time that I was going over the Jordan Canonical Form in the other. It struck me that the proofs, though different in detail, had the same structure. Now, the category of finite Abelian groups and the category of finite-dimensional complex vector spaces are two examples of "Abelian categories", and it turns out that there is a single theorem, true in any Abelian category (well, there are some restrictions), which manifests, in the one case, as the FTFAG and, in the other, as JCF. It's amazing, because the two theorems are superficially very different, but from the viewpoint of category theory, they're indistinguishable.
To give another example: there is another kind of category, known as a "topos". Roughly speaking, a topos is a category that "looks like" the category of sets. Crucially, analogues of elements ("internal elements") and subsets ("subobjects") are available, and it's possible to ask, "Is this internal element in this subobject?" However, depending on the topos, the answer to that kind of question may not be restricted to "Yes/No". Each topos has an associated logic, which may be multivalued, and the logical constructions of negation ("not"), conjuction ("and"), disjunction ("and/or"), and implication ("if/then") can be carried out. Most interestingly, there are toposes where infinitesimals - the things that Newton and Leibniz really wanted to use in calculus, but couldn't because they are logically impossible, under classical logic - become possible. (An infinitesimal is a number x such that the statement "x = 0" is neither true nor false. This is impossible in Boolean logic, but certain toposes allow it!)
Anywho. Strange and wondrous things happen in category theory. It encourages the wildest of speculations and the highest levels of abstraction. I know no more than the rudiments - and, specifically, the rudiments as of the early '80s; the field has advanced greatly since then. Still, I'm reasonably comfortable with it, even admitting that the epithet "abstract nonsense" bears some truth.
This morning, a bit of category theory leaped at me from hiding and bit me. I'll talk about that in another post.
