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…

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” is the wave-function “real”? ” of course it is but probabilities are not.

A (verbal) description might look as if it is complete without being so, e.g. Russell’s barber; in the real world you go there and check if the barber shaves himself.

Einstein’s positivistic ghost is still much too “real”.

Given that any set of formal rules could be chosen, and the consequences within that set of formal rules worked through, and called mathematics, then the question would seem to be how we decide which sets of formal rules are “interesting”. Additionally, physicists seem often happy to work with a mixture of formal rules and slightly less formal rules, what perhaps might be called “engineering” rules and related to what Lakatos calls Bridge Principles.

Perhaps mathematics should be thought of as something other than or more than just formal rules, but if any set of formal rules is a putative mathematics (which I take to be a large set, and that people over millenia have tried many possibilities), from which we choose whichever sets of formal rules prove to be interesting, or, if we are physicists, “useful” with a judicious addition of engineering rules, then it would seem that evolution would ensure that mathematics will be effective.

I have repeatedly thought that I might put this into an FQXi contest entry, but I have no wish to read the literature I would have to read to put this in academic context, and anyway I don’t much care whether this is what mathematics is or not. Tautology is wonderful: insofar as it continues to be useful, we will continue using it.

Hi Peter, the first Appleyard piece you cited is protected by a paywall. The second will probably sound very impressive to ST readers but is actually quite poor – the argument is terribly muddled by the fact that the author repeatedly conflates theories that are backed by strong empirical evidence with ones that are not. Indeed, the author seems something of cosmology skeptic, which is a different sort of argument

Cormac,

The Sunday Times link is paywalled, but the next link is to the same text, available for free at Appleyard’s site.

“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.”

I see this exactly the opposite way. Time comes from Von Neumann algebras, something that Alain Connes has been preaching for quite a while, but which has been little recognized, I guess.

But where does space come from? Everyone who pretends to know and claims that space is not so mysterious has to answer the question why it comes with three dimensions.

Apologies Peter, I misread the sentence. However, I remain unconvinced by Appleyard’s arguments, I think he’s making a very different point to (say) Ellis and Silk, mainly because he hasn’t really understood the material, what do you think?

Cormac,

I haven’t paid much attention to Appleyard’s views, but I seem to generally disagree with his point of view. For instance, in the review he writes

“In the late 1980s, when physicists were insisting they were on the verge of a “theory of everything”, I found myself using some of the arguments that now appear in this book — I was, primarily, suspicious of the claim that the universe was made of maths.”

as you can see from my essay, my point of view is diametrically opposite.

“Difficult though it is, you should buy it and, whether you read it or not, look after it. This might be one of the most important books of our time. Or not. Right or wrong, like Thomas Piketty’s Capital in the Twenty-First Century, it is an event.” So says Appleyard.

I don’t see why I should spend my time on reading wild speculations which don’t lead anywhere. Wouldn’t it be much better to follow instead the advice “shut up and calculate” and thereby perhaps have a chance to contribute to the advancement of real knowledge?

I disagree with Appleyard and Woit both, especially Appleyard. The discussion of whether Mathematics has (or should have) an unfair and undue influence on physical theories is irrelevant: We don’t choose what’s to be relevant, the Nature does! If we can learn more about Nature by learning more about Math, I say the more power to Math then.

If there was another (equally predictive) alternative to Math for forming theories of Nature, I’d welcome it. But I don’t know what it is, if indeed such an alternative exists. The philosophy certainly does not cut it. Of course religion explains everything, but it predicts absolutely nothing, and it is useless for a Scientific theory. What else is left but Math that Physics can rely on?

Mathematics being “a […] tool for modelling data and discovering approximate and ultimately temporary regularities” automatically follows if one denies the platonic status of any laws of nature. That mathematics is a “powerful tool” is an acknowledgment.

The quote just says (1) what we see is temporary, including the laws, and that (2) we have only an approximate description of what we see, and that (3) mathematics is powerful for describing what we see. All these sound quite fair to me, and also ultimately empirical.

If nature is somehow simple, as it may be because a simple structure is more probable (on some bayesian or complexity theory sense), then a branch of research studying formal structures and aiming for generality easily has structures similar to nature. That the coincidences are sometimes surprising may be because our intuition does not quite grasp the entire thing we have built (mathematics), or its relationship with structures of nature. That the structures are beautiful has more to do with the aspirations of the mathematician than with anything outside.

SO(3) in quantum mechanics puts mathematics in a position more like a language. But of course the whole theory and all the concepts, including three dimensions, invariances implied by the Lie group, etc., may all be approximations. That we have an expressive and elegant language to define a theory still does not give the theory any metaphysical privileges.

(I haven’t read Smolin’s book, which is one reason I might be wrong. Also I’m not a mathematician or a physicist.)

If I ask “why are coordinate systems unreasonably effective in enabling computations about the physical world?” virtually everybody would reply that coordinate systems are not real, they are constructs we impose to enable measurement and calculation.

The relationship between mathematics and physics may be like that of coordinates to measurement and calculation.

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Roberto Unger also submitted an essay in the FQXI contest. Some of his observations are pertinent to this discussion.

“According to the first family of ideas, mathematics is discovery. It is the progressive (or recollected) discovery of truths that exist in a domain of mathematical facts uncomplicated by the vicissitudes and variations of the manifest world.

According to the second family of ideas, mathematics is invention: the free development of a series of conventions of quantitative and spatial reasoning. This conventional practice of analysis maybe rule-guided or even rule bound, but the rules themselves are inventions. There is no closed list of motives for this inventive practice. Some have little or nothing to do with the deployment of mathematical analysis in natural science. Others take this deployment as their goal.” – Unger

Unger’s central point:

“Mathematics is an understanding of nature emptying it out of particularity and temporality: a view of nature without either individual phenomena or time. It empties nature out of them to better focus on one aspect of reality: the recurrence of certain ways in which pieces of the world connect with other pieces. Its subject matter are the structured wholes and bundles of relations that outside mathematics we see embodied only in the time-bound particulars of the manifest world”

As a committed Platonist his essay has given me much to think about.

Smolin’s writing on the subject of the “reality” of time rubs me the wrong way. There’s an unnervingly moralizing tone to a lot of it that recalls creationists railing about the theory evolution damaging public virtue (other even less savory parallels come to mind). That this tone starts creeping into his work around the time he joins FQXi hardly calms my nerves.

Nebulous concerns about the effect that concepts in fundamental physics might have on society at large should be regarded as out of bounds in an intellectually honest debate, as it’s ludicrously parochial to expect nature to shape itself to conform to human preferences.