Incompleteness and Physics

Physics makes heavy use of math, and that means that it inherits some of math's fundamental problems.
In 1931, Austrian-born mathematician Kurt Gödel shook the academic world when he announced that some statements are ‘undecidable’, meaning that it is impossible to prove them either true or false. Three researchers have now found that the same principle makes it impossible to calculate an important property of a material — the gaps between the lowest energy levels of its electrons — from an idealized model of its atoms....

Cubitt and his colleagues showed that for an infinite lattice, it is impossible to know whether the computation ends, so that the question of whether the gap exists remains undecidable.

For a finite chunk of 2D lattice, however, the computation always ends in a finite time, leading to a definite answer. At first sight, therefore, the result would seem to have little relation to the real world. Real materials are always finite, and their properties can be measured experimentally or simulated by computer.

But the undecidability ‘at infinity’ means that even if the spectral gap is known for a certain finite-size lattice, it could change abruptly — from gapless to gapped or vice versa — when the size increases, even by just a single extra atom. And because it is “provably impossible” to predict when — or if — it will do so, Cubitt says, it will be difficult to draw general conclusions from experiments or simulations.
In fairness, there's also a huge gap in getting other more-or-less accurate predictive physical models to predict exactly once all the complications are worked in. As James and Eric H were remarking the other day regarding the Russian airplane, calculating for a vacuum is going to yield very different results than when you input calculations for air resistance on an crumpled airframe. The differences in what must be accounted for across the operation may be so immense as to make the calculations practically impossible.

Yet a practically impossible calculation is still different from one that turns out to be impossible in principle. The one we might hope to overcome with better tools. If it's impossible in principle, a better tool alone won't fix the problem. The principles have to change -- and changing the principles of mathematics while preserving its predictive capacity is not easy.


james said...

This blogger points out, at some length, that physics problems can't be undecideable. I don't think he's correct about countable and uncountable numbers (can an algorithm find all the real numbers?--apparently not), but aside from that he shreds the report nicely. In any event, we have a nice little computer called Nature to make the computation for us each time we do the experiment.

Eric Blair said...

As James points out, I was going to question whether the report is correct itself.

Grim said...

I've been thinking about Cantor's argument for a while. I actually do think it points to something real and of deep importance, but it's clearly a problem in need of a solution. I've pondered several, of which the two leading candidates are these:

1) There's only one (infinite) set of divisions of integers; it just happens to be the case that every integer and the next has that potential infinite between them. Then, though you still couldn't do a diagonal proof of the type Cantor wants, you would still have two infinities of plausibly the same size: no matter how many divisions there are, you'd have the whole set of integers to keep assigning to them, infinitely. The problem is just that we were wrong to think that a diagonal proof should be able to account for the way divisive infinities function. (Chief problem with this solution: Pi is clearly different from any other *.14159...)

2) The potential infinities don't actually exist, as Aristotle claimed, which is why (as the author says) you never run into them in physical reality. This is also why physicists can 'cancel' infinities that crop up in calculations and still get predictive results, even though mathematicians say that you shouldn't do that (because of the different cardinality of infinities). (Chief problem with this solution: it creates a fundamental disconnect between math and physics that needs to be squared with the fact that our best physical models suggest that reality is something like the wave equation. If it isn't "just" that, but that plus something else -- what's the something else?)

It's a serious problem. In any case, the argument as I understand it isn't that the physics problem would be undecidable in nature, but that it would be impossible to design an ideal model that could do it -- in other words, as I read the article, he's claiming that a human-designed science of physics would not be able to perfectly predict the actual function of the world. That seems like a problem, a kind of problem of the limits of what the science could accomplish. (Still a very great deal, even within the limits.)

Ymar Sakar said...

So long as physics can accurately predict physical phenomenon, it doesn't matter if it is resolved correctly or not.

When a new drive system, say a reactionless em drive, is generated by a mad scientist, which violates the known laws of physics, then obviously the laws are incorrect and insufficient to describe the physical reality. Just as Newton's Laws of gravitation doesn't apply to light speed, black holes, or gravitational lensing.

Ymar Sakar said...

in other words, as I read the article, he's claiming that a human-designed science of physics would not be able to perfectly predict the actual function of the world.

To be purely precise, it has NEVER predicted the actual function of the universe (world). At best, it is merely an approximation, a summation equation that closely aligns with the differential results.

Grim said...

No, it never has -- but the goal has always been that it could. What he's talking about is abandoning the goal as impossible in principle. That's important, even if the goal is so distant that we'd probably never get there.

Ymar Sakar said...

Whether it can or cannot, is a matter of abstract philosophy. Not something engineers or physics normally worry about. Mathematicians, sure, theoretical mathematicians often come up with equations that describe incidents that cannot be proven or experimented upon.

For mathematicians, it would probably be something akin to the Three Body Problem.

They want an ideal differential equation, born of calculus, that precisely predicts the relative vectors and positions of the 3 bodies in motion around each other, to match the actual results. Whether the universe was truly created or modified in this fashion or not, is not a question most pragmatists need concern themselves with.

When people talk about "abandoning goals in principle", what they are really referring to is the money, power, and leverage nestled within funding. The "science is settled", so there's no point in pursuing or funding that goal to begin with.

Down that road lies something other than the truth of the universe. Frankly, it is not up to physicists, philosophers, or theoretical mathematicians to determine what the goals of individual mad scientists or other outliers are or should be.

Ymar Sakar said...