The relationship between mathematics and physics is a topic that has always fascinated me, and today I noticed two interesting blog postings related to the topic. The first was Ben Webster’s posting inspired by a recent XKCD comic. The discussion in the comment section is well worth reading, especially the contributions from Terry Tao.

Over at Backreaction, Sabine Hossenfelder discusses an interview with Max Tegmark from the latest issue of Discover magazine entitled Is the Universe Actually Made of Math?. Much of the discussion is about Tegmark’s comments on how he dealt with the potential danger to his career caused by his unconventional publications on the “Mathematical Universe Hypothesis”. This says that

Our external physical reality is a mathematical structure

I’ve always had an extreme case of mixed feelings about this, thinking that Tegmark manages to bring together the extremely deep and the extremely dumb. He embeds this as “Level IV”, the highest level, of the multiverse, and multiverse mania is one reason he has gotten attention for this and not had it dismissed as crackpotism. The idea he is pursuing is that any mathematical structure can be thought of as a “universe”, and we just happen to be in some random one of these. This seems to me to be pretty much content-free, and the attempts to fit it into more conventionally popular multiverse studies don’t help.

At the same time, this does get at an incredibly deep problem, that of the relationship between mathematical structures and physical reality. Some of the central mathematical structures that mathematicians have discovered have turned out to be identical to those found by physicists pursuing models of fundamental physics. This has happened in several very striking ways over the years. Thinking of the universe as a mathematical structure has turned out to be extremely fruitful, both for mathematics and for physics.

What is important though is that not all mathematical structures are equally important, central, or interesting, and this is the crucial point that Tegmark seems to me to be missing. Once you learn enough mathematics, you find certain recurring themes and deep structures throughout the subject. What fascinates me is that these often also turn out to be central in theoretical physics. Tegmark just accepts every mathematical structure as equally important, creating a huge undifferentiated multiverse where we occupy some random anthropically acceptable point. But the evidence is that the mathematical structure we inhabit is a very special one, sharing features of the very special structures that mathematicians have found to be at the core of modern mathematics. Why this is remains a great mystery, one well worth pursuing from both the mathematician’s and physicist’s points of view.