Saturday, July 5, 2014

Give a Guy a Hammer ...

Mathematician Max Tegmark thinks the fundamental reality is math. A review of Our Mathematical Universe.

The unknown seems to drive us into conniptions, whether one's habit of thought is theology, science, or formal philosophy. The idea that the fundamental reality of our cosmos might be inexlicable is as foreign to the most advanced scientist as it was to the earliest shaman.

So there we are. Physicists are knocking their heads against several walls such as dark energy, the proper interpretation of quantum mechanics, the union of quantum mechanics and relativity / gravity, and of course, the origin of the universe. They have virtually run out of experimental options, the colliders having become as super as they are realistically going to get. What now?

One can sense this fix from recent years of the magazine Scientific American, which runs ever more fanciful articles about the nature of the universe under the heading of physics. Speculation is running rampant, and the field seems to be gradually leaving the orbit of reason. What is time? What is space? Quantum foam, strings, etc.. All worthy questions, but far too speculative and sketchy to be fed to lay readers.

A recent entrant in the cosmic speculation derby is Max Tegmark with a book about how the universe is all a big mathematical structure. It is an excellent book in most respects, very readable and fair on the known science. Even sensible in a pontifical denouement of social policy. He has the most sterling credentials as an MIT physics professor, cosmologist, and protoge of John Wheeler. I should add that I am no expert in the least respect here, so I am just offering an educated lay perspective on the book and its ideas, as presented.

There are excellent aspects also to his cosmological speculations. For instance, he develops a helpful hierarchy of multiverse categories, this being a book largely about multiverses:

Level 1 multiverse: This is the notion that inflation during the big bang gave rise not only to the region of space we can see, but to much more. How much more? Hard to say, but it could be rather enormous, all within the product of the big bang we date to ~13.8 billion years ago.

Level 2 multiverse: Here the additional notion is added that inflation, the key process that we know of from the big bang, could have been a continuous process, not just producing our universe, but many, indeed an infinite number, of others in a process that is still going on. It adds the idea that these others might have different basic physics- different constants, symmetries, etc. Why this would be is due to the unboundedness of our current theories of what might have gone on. So why not everything possible?

Level 3 multiverse: Hugh Everett came up with an interpretation of quantum mechanics that contradicts the Copenhagen interpretation, and posits that the Schrödinger equation never "collapses". It just spawns other realities where events we think occur randomly actually occur in all possibilities, each in its own reality. This does not imply the multiplication of mass and energy into these other universes, but the superposition of an infinity of different possibilities in the mathematical space of quantum mechanics- the Hilbert space- of which we see only one sample at any moment. So it all looks the same as the Copenhagen collapse interpretation.

Level 4 multiverse: This is Tegmark's special theory, where not only does the level 2 multiverse generate an infinity of universes with different laws from some originating ur-structure, but even the most basic mathematical structure- his ultimate reality- can differ to generate alternate inflation (or non-inflation) regimes, of evey possible type. Indeed, he speculates that every computable mathematical structure exists and generates its own multiverse..es.

To be brief, I can easily understand the level 1 multiverse, and don't have a big problem with the level 3 multiverse of quantum mechanics. The others are a different story. Level 2 seems a cop-out, interpreting a lack of knowledge and specification about the universe as a permissive free-for-all where everything possible occurs. The premise is, as Tegmark notes, that our universe has about 32 numbers from which physicists can, in principle, calculate all physical aspects of our universe (not counting the pending conundrums of dark energy and dark matter, among others). And the values of these numbers are, of course, quite important. Any little change here or there would blow us to smithereens. So how did they get set up?

There are two basic approaches. The traditional way was to say god did it, end of story. A slightly more updated version is to look into the matter scientifically and keep hunting for simplifying and unifying theories, especially using mathematics. This has been the job of physics for several centuries, and seems to have arrived at a sizeable set of irreducible particles and forces, but can't seem to break through to a universal theory. The most modern way is to say that all the possibilities occur in all possible universes, of which there are an infinity, and we find ourselves, naturally, only in the one that lets glorious us happen. Ergo, the level 2 multiverse.

What is the prospect of yet more simplifying and unifying insights into the universe(es)? I have no idea. But the multiverse hypotheses seem to give up prematurely, and to what end? Even with a virtual infinity of universes, the chance that we get one that has 32 numbers, some possibly irrational, and thus almost impossible to get precisely right, ranging over countless orders of magnitude, still seems slim. So I am still rooting for a unifying explanation rather than a ramifying one whose sense is saved only by the anthropic principle. And that is really what we are talking about at this point- a rooting interest in where scientific speculation heads, since no evidence to date decides among these possibilities, and evidence may never do so.

Now we get to the weirdest part of the book- the level 4 multiverse, or Tegmark's theory that reality, at its base, is math, not just that it is described by math. And that all possible mathematical structures give rise to their own multi-multiversi, etc., ad infinitum. This is all more than a little fanciful. And his arguments, forming the core of the book and the armature around which so much else is built, are surprisingly weak.

The beginning premise is that external reality exists, separate from us, and even separate from us as observers. This is not at all hard to accept. After all, the universe had to roil and moil for quite some time before we were here to observe it, so the people who posit reality as a figment of our imaginations, or quantum-wise demand observation as the requirement of reality, do not have much to stand on. Then Tegmark goes on with the rest of his argument, which I abridge:

"If we assume that reality exists independently of humans, then for a description to be complete, it must also we well-defined according to nonhuman entities- aliens or supercomputers, say- that lack any understanding of human concepts."
... 
"This means that it [a master theory of everything] must contain no concepts at all! In other words, it must be a purely mathematical theory, with no explanations or 'postulates' as in quantum textbooks ..." 
"Taken together, this implies the Mathematical Universe Hypothesis, i.e. that the external physical reality described by the ToE [theory of everything] is a mathematical structure." 
"This means that our physical world no only is described by mathematics, but that it is mathematical (a mathematical structure), making us self-aware parts of a giant mathematical object. A mathematical structure is an abstract set of entities with relations between them. The entities have no "baggage": they have no properties whatsoever except those relations."

There, in a nutshell, is his argument. Note the slight of hand of getting from a description of reality to the reality itself. He explains himself later on:
"I'm writing is rather than corresponds to here, because if two structures are equivalent, then there's no meaningful sense in which they are not one and the same, as emphasized by Israeli philosopher Marius Cohen."

I can't say that this is convincing, at least to one untutored in the arts. One can also ask whether the starting premise makes any sense. Why must a universe be describable by any entities at all, human or non-human? It could just exist in some way and for some reason we can not understand or describe. The assumption is that there is a theory of everything, which I would certainly like to see. But I don't think it is a given that such a thing exists, let alone that it needs to have the describability property Tegmark claims for it. It could just as well be undescribable, and filled with the relatively arbitrary properties we actually see.

The one thing such a theory must be is consistent enough internally to produce a reality that has the symmetries and durable properties ("laws", constants, etc.) that we see in our versions of physics. And that, of course, is why mathematics is such a useful tool in physics, not because rocks are equations, but because our reality has, necessarily, the kinds of strucures and consistencies that we can use mathematics to describe. The ultimate theory may end up being a beautifully simple equation one can write on a T-shirt (as Tegmark dreams), but we don't know that yet, and it is very hard to see how that could be, with so many simple mathematical structures already known and tested in this respect. Are strings simple? Probably not.

And why one would want to theorize our reality as being a math structure ... that is admittedly beyond me. Tegmark claims that, among other benefits, this gets rid of an infinite regress issue, as we look for ever more fundamental particles and principles. (Though we have reached an end in particle terms, not being able to divide the electrons and quarks any further.) Having the most fundamental "one" be a total abstraction, and indeed every possible total abstraction in his level 4 multiverse, buys finality at the cost of nonsensicality, little better than the turtles or deities of yore. Specifically, it is Platonism revived, thinking that what is in our minds (where math is, exclusively) is the fabric of the universe, not its map. Indeed, one suspects in the end that this book is another edition in the old-as-humanity tradition of seeing the origins of the cosmos in the mirror.


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  • This week in the WSJ: "The more you help unemployed people, the more unemployed people you'll have."
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  • Economic quote of the week, from Joe Stiglitz:
"In fact, Geithner’s attempts to justify what the administration did only reinforce my belief that the system is rigged. If those who are in charge of making the critical decisions are so “cognitively captured” by the 1 percent, by the bankers, that they see that the only alternative is to give those who caused the crisis hundreds of billions of dollars while leaving workers and homeowners in the lurch, the system is unfair."

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