Geometric Logic at Sea

Doing a search for the definition of geometric logic, I have discovered that it’s mentioned in the movie The Caine Mutiny, by the notorious character of Captain Queeg:

Ahh, but the strawberries that’s… that’s where I had them. They laughed at me and made jokes but I proved beyond the shadow of a doubt and with… geometric logic… that a duplicate key to the wardroom icebox DID exist, and I’d have produced that key if they hadn’t of pulled the Caine out of action.

The upside of the technique is that it allows you to deduce the existence of real-world objects such as keys. The downside is that it drives you insane.

(More on the context of the quote here.)

Commutativity Theorems for Rings

MathJax doesn’t work with RSS readers, so when I have some more time I will look into using a plug-in instead. Until then, sorry for filling up your RSS feed with dollar signs.

Theo Raedschelders has written a nice sketch of Herstein’s commutativity theorem for rings. It is a generalization of Wedderburn’s theorem that a finite division ring must be a field. The theorem states that if for every pair of elements $a$ and $b$ there exists an $n > 1$ (which can depend on $a$ and $b$) such that
(ab – ba)^n = ab – ba,
then the ring is commutative. What’s surprising about the proof is its indirectness. The proof requires essentially all of Nathan Jacobson’s structure theory for rings.

PlanetMath has a nice summary of known conditions that imply commutativity.


As of a few hours ago, all I know about quandles was that they had something to do with knots. Since in our modern connected age ignorance lasts only as long as you want it to, I decided to find out more. I found this slide deck by Bob McGrail explains both the definition of quandle and the motivation in knot theory in a measly six slides. (The rest of the talk is how to efficiently compute quandles.)

Sam Nelson has a nice introduction to the general area of universal-algebraic invariants from the Notices, called Revolution in Knot Theory. Nelson also discusses the emergence of virtual knots.

Shapley-Folkman-Starr Theorem

While economic theory sometimes uses advanced mathematics, such as Brouwer’s fixed point theorem, it’s less common for economic theory to lead to new mathematical developments. The Shapley-Folkman-Starr Theorem is an example of the latter. Roughly, the theorem states that the (Minkowski) sum of a large number of arbitrary sets in a finite-dimensional vector space will be close to convex. Starr was an economics undergraduate who was working on a term paper on approximating non-convex optimization problems with convex ones. This led to collaboration with Shapley (a game theorist), and Folkman (a mathematician), and the eponymous theorem.

Polytopes with Non-Rational Coordinates

Now that I have external evidence that someone is still hoping for new posts, I thought I better write one.

Here’s a result that not only would have I not have guessed, but I would have assumed the opposite is obviously true. There are convex polytopes that cannot be presented in Rn as a polytope all of with rational coordinates. I would have assumed that this is wrong because you can always take the vertices, and perturb them slightly so that they become rational. This argument doesn’t work in general, but you can prove using other techniques that in 3 dimensions that every convex polytope can be written with rational coordinates. There already exist counterexamples in dimension 4.

The survey paper Non-rational configurations, polytopes, and surfaces, by Günter M. Ziegler, gives an explicit construction in 13 dimensions. The paper provides a decent overview of many other, related, topics.

Model Categories

Sorry for the dearth of posts recently. I’ve been spending time I would ordinarily spend on blogging instead obsessively learning the theory of model categories and related ideas such as (∞,1)-categories. I found the subject completely impenetrable until I read Greg Friedman’s expository article on simplicial sets. (I wrote an earlier post about it here.)

Axiom of Replacement in Category Theory

I had a slightly ironic experience on Math Overflow. A couple of months ago, I started wondering to what extent you could develop category theory “below a cardinal”. When you consider the category of groups (for example), you’re probably not literally interested in groups of arbitrarily large sizes — you just want enough space so that you can perform any operation you need to. I started writing this post here arguing that for concrete categories, sets smaller than a limit cardinal were big enough. Limit cardinals are not usually large cardinals in the sense of set theory, but they’re pretty big — the category of sets smaller than a limit cardinal is closed under the power set operation, for example.

Before I finished the post, I thought I should check the claim and look over some proofs in a category theory book. I realized that, under the usual definition of a diagram in the literature, my proposed restriction would make the category of sets fail to be either complete or cocomplete — even countable diagrams could have to have limits or colimits. You could finesse the issue by changing the definition of diagram, but I thought “No one will stand for that”. Under the standard definition, the construction of limits or colimits requires the Axiom of Replacement, which means that the right condition is inaccessibility, or equivalently you need Grothendieck universes.

So now I thought I understood the big picture. Completeness required replacement, which leads naturally to Groethendieck universes, which explains why the main competitor in textbooks to either Goedel-Bernays or Morse-Kelley set theory is to postulate one or more Grothendieck universes. The only thing that puzzled me was that while people using category theory seemingly made casual use of replacement, people would also argue that replacement is never used in ordinary mathematics. I thought that maybe I was confused on some issue, so I asked on Math Overflow.

It turns out that at least some people really don’t want to use replacement. They would rather change the definition of what it means to be a small diagram so as to be able to avoid the axiom. Avoiding replacement has lots of little consequences. For example, you have to require that the image of a small diagram is a set. Even with the corrected definition, the General Adjoint Functor Theorem becomes false as stated, and you have to strengthen the solution set condition. It means lots of fiddly little details have to be changed. You also no longer have as clean of a distinction between large and small. (You can have categories that are locally small, and have only countably many objects, and yet are not small categories, for example.)

But I could have stuck with my original idea for this post.