Lee Smolin has a new book out last month, co-written with philosopher Roberto Unger, entitled The Singular Universe and the Reality of Time. To get some idea of what he’s up to, there’s a review by Bryan Appleyard at The Sunday Times (non paywalled version here), another Bryan Appleyard piece here, and interviews with John Horgan and at Scientia Salon. In other news about Smolin, he’s one of the winners of this year’s first Buchalter Prize in cosmology.

The book is written in a rather unusual style, with the first two thirds or so by Unger, the rest a shorter contribution from Smolin, together with a section discussing where they disagree. It’s neither a popular science book, nor a technical work of philosophy, but something somewhere in between, best perhaps compared to something one rarely now sees, a work of “Natural Philosophy”. I found the long section by Unger rather hard going and not very rewarding, and realized that I have a fundamental problem with this sort of writing. Arguments about physics and mathematics made in natural language leave me often unable to figure out exactly what is being claimed. Sometimes this is because I’m not familiar enough with a philosphical tradition being invoked and its associated use of terms, sometimes I suspect it’s because natural language is just too imprecise and ambiguous.

The Smolin section is shorter and written with more precision, making it easier to get an idea of what he’s trying to claim. To seriously address all his arguments would be a large project I’m not able to undertake, but here is a list of “hypotheses” or “principles” that he arrives at:

- The uniqueness of the universe.
- The reality of time.
- Mathematics as the study of evoked relationships, inspired by observations of nature.

For the first of these I don’t really disagree. Smolin takes this as an hypothesis of no “multiverse”, an hypothesis that science may be able to confirm or disconfirm. Our current best understanding of science shows no evidence for a multiverse, so anyone who wants to posit one needs to come up with some significant evidence for one, experimental or theoretical, and I haven’t seen that happening. It’s entirely possible that a compelling theory may emerge that naturally implies a multiverse, but that’s not currently the case. Unlike Smolin, I wouldn’t take this as an hypothesis, more just would say that the question of multiple universes is well worth ignoring until someone comes up with a good reason to pay attention.

For the second, one problem is that I’m not exactly sure what it means. I guess that when I hear the word “real” I’m always rather suspicious that a meaningless distinction is being invoked (i.e. is the wave-function “real”?), and start trying to remember what it was I once understood about ontological commitments from reading Quine long ago. Part of what Smolin is referring to I think I’m sympathetic with: the nature of time remains mysterious in a way that space isn’t. While relativity treats them on an equal footing, in quantum theory this is not so clear. My suspicions about this mystery though tend to focus on the analytic continuation between Minkowski and Euclidean signature, which I’d guess is quite different than Smolin’s concerns (see hypothesis three…)

What Smolin seems to have in mind here is the hypothesis that physical laws are not “timeless”, but can evolve in time, with an example the ideas about “Cosmological Natural Selection” he has worked on. One problem with this is that the question then becomes “what law describes the evolution of physical laws?”, with an answer re-introducing “timeless” laws. Smolin refer to this as the “meta-law dilemma” and devotes a chapter to it, but I don’t think he has a convincing solution.

On the third hypothesis, about the nature of mathematics and its relationship to physics, I just fundamentally and radically disagree. For a shorter version of Smolin’s argument, see this essay, which he has recently submitted to the FQXI essay contest. I’ve been writing something about how I see the topic, will blog about it here very soon. What I’m writing isn’t a response to Smolin’s arguments, but a positive argument for the unity of math and physics at the deepest level.

My problems with Smolin’s point of view aren’t especially about his arguments concerning Platonism and whether mathematical objects are “real” (see earlier comments about what’s “real”), they’re about arguments like this one, where he argues that the explanation for the “unreasonable effectiveness of mathematics” in physics is not some deep unity, but just

mathematics is a powerful tool for modelling data and discovering approximate and ultimately temporary regularities which emerge from large amalgamations of elementary unique events.

The argument essentially is that mathematics is nothing more than a calculational tool that just happens to be useful sometimes in physics. This is a common opinion among physicists, and a big problem for me is that here Smolin is not taking a provocative minority point of view, but just reinforcing the strong recent intellectual trend amongst the majority of physicists that the “trouble with physics” is too much mathematics. As I’ve often pointed out, the failures of recent theoretical physics are failures of a wrong physical idea, rather than due to too much mathematics, with the multiverse just an endpoint of where you end up if you throw away all non-trivial mathematical structure in pursuit of a bad idea.

In his essay, Smolin gives a discussion of mathematics itself which I think few mathematicians would recognize, defining it as “the study of systems of evoked relationships inspired by observations of nature”, and consisting in bulk just of elaborations of the concepts of number, geometry, algebra and logic. I started my career in physics departments, and I’m well aware of how mathematics looks from that perspective (even if you have a lot of interest in math, like I did). My experience of moving to work in math departments made clear to me that the typical ideas of physicists about what mathematics is and what mathematicians do are highly naive, with Smolin’s a good example.

I’ll end with just one example of what I see is wrong about the conventional physics view that Smolin represents. A big application of mathematics to physics is the use of the rotation group SO(3). In that case it’s true that many of the applications can be thought of as concerning approximate aspects of complicated physical systems, and derived from working out precisely the implications of our experience dealing with the 3d physical world. But, besides the chapter on angular momentum operators (and thus SO(3) representation theory) in every quantum mechanics textbook, there’s an earlier chapter where the Heisenberg commutation relations are given as fundamental postulates of the theory. A concise way of stating this postulate is that quantization is based on a specific unitary representation of a Lie algebra (the Heisenberg Lie algebra). This is not approximate, but the fundamental definition of what we mean by quantum theory. The structure here is very deep mathematics (appearing for instance in number theory, the theory of theta functions and of Abelian varieties), and is far removed from the kinds of mathematics that one runs into as typical approximate calculational tools when studying physical problems. This is just an example, but there are many others. I don’t think that if you look at them you can sustain the argument that deep mathematics and deep physics are not close cousins with a unity we only partially understand.

Anyway, more detail to come about this…