I could make excuses about why I haven't posted for two weeks, but that would be boring, so I won't. (Besides saying that John Dies at the End is a really fun novel that you can read online as a way to reduce your available free time.) Let's just say that I don't want this blog to die, and with any luck will have fun posts on an engaging schedule.
"The Unreasonable Effectiveness of Mathematics in the Natural Sciences"
One of the things I've recently tried to nail down is what the ideal interface is between mathematics and physics. Of course, plenty of smart people have wondered about this before; this section's title is taken from a very famous paper by the same name. And I don't mean to reiterate those points, but rather see where they take us.
While talking with a friend, we agreed that neither of us could really conceive of a world in which you could not mathematically describe "how it works." I simply cannot believe that it would even be theoretically possible for the world to not be governed by laws. How would things "know what to do"? There is always a guiding principle, a set of equations or constraints, that explain why the system behaved this way.
People might like to invoke free will here, but to me this kind of inconceivability is exactly what makes free will such a problematic concept. Why did you choose to do this, and not that? You can say that it was the result of signals traveling throughout a biological neural network, according to the laws of physics and the initial conditions of the system. Sure, your actions might not be predetermined, but if so only because of some fundamental randomness in nature (stemming from quantum effects, or whatever is behind them). Even Penrose's speculative theories about consciousness are ultimately grounded in physical laws. Whereas, if you invoke some kind of mystical nonphysical ability to "choose," in a way that is not governed by physics, I have to ask "where did that come from? Why did you end up choosing this over that?" And in the end, you'll be reduced to something silly, like "just because" something circular like "because I wanted to." But why? Why isn't that behavior describable? In other words, even if there were some mystical choice-force at work, I should be able to describe it via a set of laws. Essentially we are talking about causality here: and free will (and gods, etc.) is not just some kind of interesting causality violation that we can theorize about, but something that stands entirely outside of the causal structure and the ability to theorize at all. So, that's why I cannot conceive of free will as more than an illusion.
That was a bit of a tangent, but if we're generous we could instead think of it as an example of the fact that our way of thinking is so entirely dependent upon the idea of mathematical laws governing the world. Why we have such a conviction is more of a philosophical question, and why it works so well is metaphysical, but it's important to note that we do. And given this, I ask, what is the right way to describe the universe mathematically?
Some Examples
Ultimately, we seem to derive our mathematical formalisms from how we view the world. For example, it may seem obvious that "of course the world should be invariant under general coordinate transforms!" or "of course things of unequal mass should fall at the same speed!" But ultimately, even though these are beautiful principles with even deeper reasons behind them, it's possible to imagine a world in which this were not the case. And the whole point of my section above was that we could still describe such a world by mathematical laws—we would have to be able to—even though they would be different ones entirely.
To further drive home this point, we note that such "of course!" moments break down at other levels. Most conspicuously, I don't see anyone going "of course nature should evolve complex probability amplitudes over time, instead of actual probabilities!" or "of course a point particle should have intrinsic angular momentum!" Even the ones that people often think of as an "of course!" are really not very good, in the end: "of course everything should ultimately be a point particle!" has, in the last few decades, come under much greater scrutiny, and now we're hoping for something more like loops in spacetime or vibrating strings.
So really, saying that one formalism is "better" or "makes more sense" for describing the universe isn't really possible. I often fall into the trap of thinking that general relativity, with its nice tensor equations based in differential geometry and mapping directly to events and gravity-as-spacetime-geometry, has a "better" mathematical framework than quantum theory. I mean, who really likes having concepts such as "operators on a Hilbert space" or "probability amplitudes" be a fundamental part of your theory? But in the end there really is no difference: each is a way of mapping the real world into certain quantities that obey observed laws. In one case, these laws fit naturally into the framework of differential geometry, and the quantities we work with and think of as "real" end up being tensors and their ilk. In the other case, they fit into the linear algebra formalism, and we work with operators on these spaces and the associated eigenstuffs.
That said, in the end we do want to have a single theory, one that fits into a single mathematical framework. There are lots of people who think they have the right framework, and I'm not really able to judge or even point you in the right direction, since I'm still attempting to master our two main theories separately before worrying about the right way to combine them. But the point is that we hope that what we come up with is grounded in some beautiful mathematics, something that isn't just a hodgepodge of what we already have—"start with our linear algebra, add some differential geometry to the right places, and poke it with a stick until they all fit together."
The Basis of All Mathematics
I seem to have neglected to talk about one of the areas of mathematics that plays a fundamental role in modern physics, namely group theory/group representation theory. The fact that we find this all over the place, from quantum mechanics's operator representations to the standard model's gauge groups to general relativity's diffeomorphism groups, indicates that it's going to end up as a crucial part of our ultimate mathematical framework. This is probably a rather obvious thing to say, but it has some interesting implications.
Because ultimately, groups and their representations and the transformations among them are just a small facet of the much more all-encompassing category theory. Category theory can essentially be used to describe all of mathematics, as it encompasses in the most general sense the idea of "mathematical object," "relationship," and "transformation." Indeed, a subset of category theory, called topos theory, can replace axiomatic set theory as the foundations of mathematics. And, even though the conventional foundations of mathematics are some of my favorite areas to study, in the end this is really a good thing. For when you ultimately trace your physical theory back to its very deepest roots, I personally would rather not have the epiphany be "Oh! The entire time, we were simply talking about sets, building them all up from this list of axioms!" One more along the lines of "oh! we were talking about relations between generalized mathematical objects!" seems much better.
Certainly, categories have come up in the work of lots of mathematical physicists. John Baez has a great page about them. Chris Isham is known for applying them to quantum theory (you have to love the title of one of his relevant papers, "The Representation of Physical Quantities With Arrows"). Indeed, a quick search through the arXiv finds 26 papers on topos theory (in the physics sections). So, wouldn't it be cool if this ended up being physics? If to describe our universe in the most general, fundamental level, wouldn't it be neat if we had to go to the most general, fundamental level of mathematics? It would be an adventure!
No Anthropic Reasoning Necessary?
One of the most frustrating counterpoints to my above "I must have laws!" viewpoint is the emergence of arbitrary constants throughout the standard model. (And elsewhere, of course—the gravitational constant G is an obvious example, but also things like the number of space dimensions, the number of time dimensions, and so forth.) Where do these things come from? A popular solution these days is to invoke anthropic reasoning, which is a pretty feeble attempt at giving us an explanation. The problems with it have been detailed in numerous places, so I won't go into it, but I think that most people will agree with me that it would be nice if we could get all those constants to emerge naturally, without invoking the anthropic principle.
Well, if one looks through category theory, one finds many instances in which something emerges "naturally." For example, I was shown the "natural" definition of a general product in category theory. I'm afraid I'm stepping outside of my knowledge-zone here, but I am assured by both Wikipedia and people who have taken the appropriate courses that many things emerge naturally in category theory, in that there is only one way to reasonably define them.
Wouldn't it be nice if we could do this with the universe? I'm not sure if this even makes sense to say, but wouldn't it be great if there was only one "natural" definition of the universe as a category-theoretic structure? How much of this would be based on what we observe, I wonder—would there be room for speculation as to "if we lived in a universe that were different in these ways, then we'd expect a different natural structure"—or would we simply come to the conclusion that nothing else would make sense? That the math just "doesn't let" anything else have the possibility of existing? I think this is every physicist's dream, in some sense (although I know a lot of them these days seem to be unfortunately smitten by anthropic reasoning).
It's an interesting idea. In the end, it'll be a fun adventure, however it turns out. That is, no matter what the ultimate interface between mathematics and physics ends up being, I will really enjoy spending my life exploring that interface, and trying to pin it down into a simple set of principles and structures.
