Forcing Truth

This thread at Math Overflow has the feel of advanced alien technology. Forcing is a technique for constructing models of set theory where various hypotheses fail. For example, forcing can be used to construct a model of set theory where the continuum hypothesis is violated.

There are some statements whose value cannot be affected by forcing. These statements are known as absolute. Forcing is useless for establishing such a statement is independent, but this can be a virtue. If you can create a model using forcing such that you can prove that an absolute statement is true in that model, then it must already be true in the universe of ordinary sets. The thread gives several specific examples of theorems you can prove this way.

Reversible Markov Chains

Here’s a pretty idea. A Markov chain is one of the simplest forms of dependence in random variables: an infinite sequence of dependent random variables, where the probability distribution of the next random variable only depends on the value of the current random variable. If you reverse the sequence of variables, you get another Markov chain, the reverse Markov chain. Some Markov chains, reversible Markov chains, have the property that when you reverse them, you get back the same chain. Markov chains represent processes that have no history, in that future is determined solely by the present, not the past. A reversible Markov chain not only has no history, but time has no direction.

Here is a draft of a book by Aldous and Fill on the theory of reversible Markov chains.

The Algebra of Possibilities

There is a notion in symbolic dynamics of a “topological Markov chain” that is analogous to a Markov chain in probability theory. It’s occurred to me that you can extend the analogy to a complete analogy with probability theory. We’re still interested in sets of events, but now we’re no longer interested in the probability of an event, but just whether or not an event is possible.

Start with a σ-algebra of sets, as usual. Instead of associating a probability with each set, associate a member of the set {Not Possible, Possible}. The empty set is assigned the value Not Possible, while the whole space is assigned the value Possible. A countable disjoint union of sets is Possible if and only if at least one of the individual sets is Possible.

A measure takes values in the semigroup of the nonnegative real numbers closed under addition. Here, we’ve replaced that semigroup with the semigroup of {Not Possible, Possible} under the commutative binary operation +, with multiplication table:

+ Not Possible Possible
Not Possible Not Possible Possible
Possible Possible Possible

I’ll explain the relationship with topological Markov chains in a future post.

snarXiv

snarXiv is a site the generates parody abstracts for high-energy physics theory papers, a la arXiv. While the abstracts don’t quite make sense, they eerily resemble the real thing.

snarXiv versus arXiv is another site that gives you a random snarXiv and arXiv paper title, and asks you to tell the fake from the real thing. The fake titles are much harder to recognize than the fake abstracts. Initially, I got the first 5 right, but after about 25 I was down to random chance.

Via Not Even Wrong.

Vladimir Arnold, in memoriam

I missed that Vladimir Arnold has died. Arnold was famous for his own contributions to mathematics, but in my opinion he was also the world’s great expositor of mathematics.

When I first encountered the subject of Lie algebras, I thought it was pointless and unmotivated. I also had the impression from high school physics that classical mechanics was built out of a bunch of random facts that were true for no reason, like the conservation of angular momentum. Also, I thought that potential energy was a sort-of a con — that if you can simply declare that a body has potential energy that you can make the law of conservation of energy tautologically true. Reading Arnold’s Mathematical Methods in Classical Mechanics changed all that. Arnold starts with one-dimensional systems like the inverse-square law and harmonic oscillator, and then to three-dimensional systems where he explains how symmetries in the equations of motion lead to conservation laws. Along the way, he explains how Lie groups lead to Lie algebras, and how in particular how rotational symmetries in 3d lead to the Lie algebra of so(3), which physicists use in the guise of the cross-product of vector calculus. He also introduces the Lagrangian and Hamiltonian formulations of classical mechanics. Most importantly, (since you can learn the equivalent from a physics text like Goldstein’s Classical Mechanics), he puts in the language of mathematicians rather than the language of physicists.

Years after I studied the subject of ODEs, I almost bought Arnold’s (expensive) Ordinary Differential Equations just because it was such a beautiful introduction to the subject. Lots of textbooks allude to the dynamical systems viewpoint for ODEs, but his book really communicates that viewpoint.