What, if any, is the role of unprovable/undecidable statements in physics?

Great Debates
1

As I understand it, mathematics admits unprovable and undecidable statements in addition to the usual provable and disprovable.

What, if any, is the place of unprovable/undecidable statements in physics?

I’m thinking of things like the ‘Many Worlds’ interpretation of quantum mechanics. Currently, it would seem to be in the realm of the unprovable although it is possible that will change. If it is unprovable, though, is its study not merely another branch of philosophy (i.e. sophisticated opinion)?

Along the same lines, again as I understand it (and I may well be wrong), Bell’s Theorem seems to indicate that one must accept either that there is no ‘underlying’ reality or that non-locality (faster-than-light travel) exists. I interpret this as an example of undecidability, and thus that further study of the question would not be an exercise in physics or even mathematics so much as it was a philosophical pursuit.

I wonder if a similar situation has also arisen with the information/black hole conundrum (Firewall Paradox).

If there is undecidability or unprovability in physics, then, at some level, physics begins to resemble philosophy, and even theology, all three fields alike in ultimately offering nothing but opinion on matters of existence.

I suppose this is just another way of stating the old saw that ‘science has its limits’ and is just a rehashing of ancient questions expressed using more contemporary concepts (and of course I don’t dispute that science also has its benefits). Still, it’s not clear to me where physics ends and philosophy begins.

2

I don’t have time to give a good response to this right now, but there are several ways undecidability interacts with physics—certain questions about physical systems have undecidable answers, and may even be a kind of ‘resource’ in quantum computation.

I have argued (in print) that there is, in fact, a deep connection between quantum mechanics and undecidability: one can appeal to a diagonalization argument to show that certain measurement outcomes are undecidable. As far as I know, this suggestion goes back to John Wheeler, who however got thrown out of Gödel’s office for posing it to him.

You might also be interested in the current FQXi essay contest on the topic ‘Undecidability, Uncomputability, and Unpredictability’—which, however, as of yet has a rather low signal-to-noise ratio. But in my experience, that usually improves, and there have been some real gems coming out of earlier contests.

More later, when I come back from work.

3

The OP might be concerned with the amount of speculation in modern physics. Here is a quote from an article decrying the lack of evidence for theories supported by the scientific establishment:

4

If by “provable”, the OP includes “disprovable”, then yes, it’s conceivable that the Many Worlds interpretation of quantum mechanics might be disprovable. If, for instance, we find some way to receive communications from alternate histories where random events had different outcomes, that would conclusively disprove the Many Worlds interpretation (as well as leaving the field of physics ripe for something else completely new and exciting). But there is no way that the Many Worlds interpretation could possibly be proven correct, because it has been mathematically proven to give exactly the same measurement outcomes as all other interpretations of quantum mechanics. It is therefore impossible to distinguish between the various interpretations by any scientific means.

This does not mean that the various interpretations of quantum mechanics are useless. Human brains did not evolve to deal with quantum mechanics, and are therefore ill-suited for the task. But models suggest analogies which, even if imperfect, can still make it easier for humans to think about certain problems. And thus, true or not (if that’s even a meaningful descriptor for them), the various interpretations of quantum mechanics are still useful.

5

Speculation should not be confused with philosophy, IMO.

Whether speculation is a good thing or bad and whether there is too much speculation in today’s physics are sensible topics for discussion, of course. But that’s not the topic here.

Karl also posted this statement in another thread:

This thread is another stab at finding a way to support that statement. I find the statement unsupportable, even nonsensical. YMMV.

6

I don’t plan to debate about speculation, but I think philosophy and physics are at least partly interrelated. And so does the author of this article.

7

And the author of that article is arguing a lonely position against what appears to be a majority of physicists. He also notes that “many” philosophers “discount science.”

As I said, YMMV. I’d say that the *results *of physics may influence philosophical thought, since that is true of everything. But nothing *in *physics per se is philosophy and neither are unknown or speculative aspects of science.

8

Apologies for this delayed response.

I am not so much looking for “support” that “all of physics (and mathematics) is, ultimately, philosophy” so much as I am trying to express my rather belated realization that much of (what I believe to be) the frontiers of physics, i.e. grand unification, string theory, the multiverse, the role of information as the ‘stuff’ of the universe, and more, seems to resemble exercises in mathematical proof as much it describes any ‘reality’. And mathematical proofs are all, ultimately, based on certain assumed axioms.

What concerns me is that in the same way that certain hypotheses in mathematics cannot be proven or disproven, or that there exist multiple models satisfying the same axioms, physical theories based on those hypotheses and axioms must be likewise unprovable. In this sense, mathematics and the physics that springs from it, seem almost arbitrary and in that way resemble philosophy (admittedly a philosophy with some real world spinoffs).

Again, this may be very obvious (“science has its limits”) and perhaps should have been more obvious to me, but it wasn’t.

9

I don’t think this follows. Every sufficiently complex system has unprovable propositions in it, even arithmetic. Are you seriously saying that arithmetic therefore leads to unprovable propositions in physics?

Physics is not mathematics; it uses math as a language but limits that usage to known and proven areas that can be applied to physical bodies. I know of no physics that is based on the Continuum Hypothesis, e.g. Nevertheless, infinities can be part of physics.

Do you have any examples at all of a piece of physics based on an unprovable or open question in math? Remember, a mere unknown in physics is not at all the same thing.

10

Possibly. I don’t think it is out of the question that Godel’s Theorem has a physical analogue.

In fact, although I can’t prove it of course, I think an analogue is the mind-body problem.

I didn’t understand them, but Half Man Half Wits links above seem to be just that.

11

Again, I believe this is not a meaningful sentence.

I find it amusing that after writing that you think speculation in physics to be philosophy you refer to a pageful of links to totally speculative articles and want to use them as physics.

But let’s grant your case. Say that some physical questions turn out to be undecidable.

So what? How in the world does that make them philosophy? At the very worst, it means that for some esoteric situations physicists won’t be able to make perfectly accurate calculations. What does that change? What does it mean?

It certainly won’t in any way mean that the results of measurements or equations will be “arbitrary.” Exactly the opposite. It means that they won’t be calculable and so no results will appear at all.

12

Gödel himself believed that his theorem—more accurately, the impossibility of deciding by a fixed procedure whether a certain diophantine equation has an integer solution—shows that the human mind can’t be computable. The idea of a connection has come up many times in the philosophy of mind, from John Lucas to Roger Penrose.

I think these arguments are, in this form, severely flawed; but in point of fact, I have just received (pretty encouraging) referee reports on a paper in which I argue for a different role of non-computability in the human mind.

And of course, a central point of Douglas Hofstadter’s Gödel, Escher, Bach was how the self-referential constructions enabling the Gödelian phenomena find a role in the human mind.

These articles are serious physics by respected authorities in their fields, having appeared in Nature and Foundations of Physics. This isn’t even a particularly niche topic; books have been written on this sort of thing, including even some popular ones.

Most of these works also contain a great deal of philosophy—which should be expected of such highly intertwined and interfertile research areas, of course. Philosophy has always played a foundational role in the formulation of new scientific theories; it’s really only when considering periods in which the consequences of established science are being worked out that one can get away with disregarding philosophy in physics.

But let’s take a simple example. Consider the humble three-body problem. It’s generally conjectured that no analytic solution—that is, a closed-form equation allowing you to solve the system for every point in time given the initial conditions—exists. It’s also conjectured that the system suffices to embed universal computations—that is, you can encode an arbitrary computational problem into the initial configuration, and the natural evolution of the system will perform the computation.

If that’s the case, then the undecidability of the halting problem immediately implies that there is no closed-form solution to the equations describing the system, and that questions such as whether one of the bodies will eventually be expelled from the system do not have a general answer (although they may be answerable in specific cases).

Consequently, due to the possibility of embedding computations within physical systems, undecidability straightforwardly impacts our ability to predict the behavior even of quite simple systems. (Wolfram calls this ‘computational irreducibility’ and famously argued that it would take ‘A New Kind of Science’ to deal with, although he hasn’t found many takers as of yet.)

Certainly, undecidability and quantum mechanics have often been considered closely akin. I’ve already mentioned the famous episode of John Wheeler being thrown out of Gödel’s (who, perhaps due to his friendship to Einstein, never thought all that much of quantum mechanics) office for suggesting that the ‘point of origin’ of the ‘quantum principle’ is ‘the undecidable propositions of mathematical logic’.

But there are many concrete results apart from the two above I’ve already pointed out. Recently, a highly intriguing and complex result was obtained, relating the halting problem to a problem of deciding how well one can do in a given task using quantum entanglement.

There is much more in this area. To the best of my knowledge, Dalla Chiara was one of the first to argue for the application of Gödelian reasoning to the quantum measurement problem; this thread has been continued in many directions. For simple systems, for instance, one can show that quantum measurement outcomes are random if the measurements encode propositions that are undecidable from the axioms encoded in the system’s state—however, these are not systems subject to Gödelian undecidability—although there’s nothing specifically ‘Gödelian’ about undecidability in systems like Peano Arithmetic, it’s just that Gödel showed that, counterintuitively, there’s always undecidability present in such systems, and that they can never be completed.

Anyway, I could go on and on about this, but I hope the above helps to put the sort of work done in this area into perspective. If this is ‘totally speculative’, then well, pretty much all of theoretical science is.

13

The OP reminds me of Faraday’s rhetorical question, “What good is a baby”?

Understanding a role, or lack thereof, for unprovable or undecidable statements in physics, doesn’t actually gain you anything. We aren’t sure what’s provable or undecidable, when we are making choices about what to work on next or how to work on it. It’s like asking, what’s the role of incorrect hypotheses?

Perhaps it is easier to become certain a statement in mathematics is unprovable, than for a statement in physics. But in physics I think if there is a visible path toward proving or deciding something worthwhile, somebody goes down it, and nobody ever deliberately goes down paths they can’t see.

14

I must not have been clear enough if you think I’m decrying speculation in science. It’s trivially true that most science at least starts with speculation, and then piles speculation on top of the areas that are pretty well settled.

Nor do I think that undecidability is a barrier or a purely mental state. No body knows if the Reimann zeta function is true but thousands of papers have been written to show what the consequences would be if it were true, and much good advanced math has come out of this speculation.

My only objection is to calling these unknowns philosophy instead of science. I find the way Karl uses the term to be so loose and shaggy that it encompasses everything that isn’t nailed down. Not that he’s alone in doing so. Some scientists like to dabble in philosophy themselves and the results are very often as bad as when philosophers dabble in science. I’m sure it’s easy to think that when you formulate equations that touch on the fundamentals of existence that you are also solving the mysteries of being. But I remain unconvinced that that the two are linked in this particular way.

15

Our difference may be semantic, but to me, your amusement at KarlGauss wanting to use ‘totally speculative articles’ as physics implies that they aren’t.

Well, it’s a broad field. As Wilfried Sellars put it, “the aim of philosophy, abstractly formulated, is to understand how things in the broadest possible sense of the term hang together in the broadest possible sense of the term.”

In a very real sense, it’s the search for Einstein’s ‘elements of reality’ that ultimately led to the quantum computer; then again, not everybody can be Einstein. ¯_(ツ)_/¯

16

I’m giving up because I don’t see how I could have been clearer. Your interpretation of my words is the opposite of what I meant. Nobody else seems to be objecting, so I’ll just leave them as originally stated.

17

As theoretical physics has advanced, it has become harder and harder to create actual laboratory testing conditions to test very advanced theories. The result is that we have postulates that are “mathematically possible” but impossible to prove under actual testing conditions. One way of “testing” things like this is to look for observable particles or situations that would exist if the original postulate was true. I believe the connection between observable and non observable events is called “Data Crystallization”.