Physics is tied to mathematics, so we have to assume that, by Goedel, that it must be undecidable. We should be able to come up with an experiment whose outcome cannot be derived, but which Nature "knows" how to answer.
@BartoszMilewski wrote: "Physics is tied to mathematics, so we have to assume that, by Goedel, that it must be undecidable."
That's not a proof. I could equally well say "Addition is tied to mathematics so so we have to assume that, by Goedel, that it must be undecidable." But the first-order theory of natural numbers with just addition is decidable: it's called Presburger arithmetic:
So you actually need a certain level of complexity in your first-order theory before undecidability kicks in, and it's actually far from obvious that physics reaches this level of complexity. The people have done so far is prove undecidability of the spectral gap in some systems:
@johncarlosbaez@tiago@BartoszMilewski The original question is so problematic that I consider it gibberish. Mostly because we are perfectly clueless about the relations between nature, physics, and math.
@johncarlosbaez@BartoszMilewski Maybe undecidability proofs for physics actually prove that the reality does not have infinite Turing machine tape (or any other infinite realizations of abstract concepts). But somehow you'd have to establish that space is finite. Maybe Olbers' paradox is sufficient? Of course it's easy to prove non-existence of Turing machines from non-existence of their infinite tape.
Judging from the discussion, there is a general feeling that it should be easy to prove me wrong. And yet there is no obvious proof that Goedel's result cannot possibly lead to an experimentally testable conflict between theory and practice. Why?
@BartoszMilewski I'm going to bet on relative complexity. ISTR the smallest Goedel sentence under ZFC is rather large, and I'm not sure how to set up a natural system with that many controlled variables.
It should also be that the thread doesn't have enough experimental design or proof experience/inspiration.
Finally, maybe "Goedel's result cannot possibly lead to an experimentally testable conflict between theory and practice" is true, but not provable?
@BartoszMilewski I think to provide an explicit case for this, you have to agree on a specific formal system (such as ZFC combined with some accepted physics axioms)? Otherwise someone could simply respond by providing a different formal system that adds an axiom to prove your counterexample.
The second incompleteness theorem should provide one counterexample in any formal system though: your physics formal system cannot prove its own consistency (assuming it can encode peano arithmetic).
@BartoszMilewski This seems rather unlikely because Gödel constructions (at least all the ones I'm aware of) rely on quantification, both variants of which are famously iffy to test experimentally in an interesting way.
@BartoszMilewski I think Physics may be tied to an ultrafinitist version of mathematics where Goedel's incompleteness doesn't work, because some of the number-statements are "too big".
I don't know how that would work exactly, but physics doesn't need to be true for MOST of the Integers.
@BoydStephenSmithJr I thought of that, but it's hard to believe that, for instance, the fine structure constant has only a finite number of significant digits (although for a short period it was thought to be 1/137, and Eddington argued that somehow 137 was magical).
@BartoszMilewski I think it could be a ratio between two large, but not too large Naturals. I don't think it has to have an infinite amount of unpredictable digits.
@BoydStephenSmithJr I always try to avoid wishful thinking (to which we are all very susceptible). So is there a good reason why (\alpha) should be rational, other than our preference for finitarianism?
@BartoszMilewski I don't think it is less (or more) likely than the existence of truly arbitrarily large numbers. But I'm not exactly sure how one would go about assigning likelihoods to such statements.
@BoydStephenSmithJr The notion of "existence" of numbers is Platonic, and I try to avoid Platonism. It made sense for Ancient Greeks, who believed in the Gods on Mound Olympus.
@BartoszMilewski Physics could use a decidable fragment of whatever mathematics we use. I mean, what you said is so generic it could apply to a universe consisting of a single cell automaton with only one state that never changes. You can make up undecidable statements about such a thing, but none of them would really say anything about the physics.
@dpiponi@BartoszMilewski I don’t even understand what it means that physics be “decidable”. Perhaps that we have some axiomatic “language of physics” and in this language there are statements which cannot be proved by this system. One response to this is to say “so what? We do not observe theorems in the real world” and a different response is to doubt the idea that some finite / recursively enumerable axiomatic system might capture all physical phenomena and so doubt that we are in the realm of gödel’s theorem.
I think I’d be more interested to know if one could relate something like the continuum hypothesis to some prediction of a physical theory in the same way people have suggested for the Riemann Hypothesis (though I don’t think take it this would say anything about the truth or falsity of CH, since I have no idea what that means)
@dpiponi@BartoszMilewski I don't either, if physics is written in the language of ZFC set theory, then it will have an undecidable statement - the consistency of ZFC. That doesn't seem to have anything to say about physics, which is why I think it would be more interesting if you could connect any interesting such question (e.g. Goodstein's theorem), to some empirical observation (I don't know whether anything like this is possible in principle)
@BartoszMilewski@dpiponi First, you explain how Gödel's theorem applies to your "definition" of physics and then we can make progress on what we are talking about
@BartoszMilewski@dpiponi I am reluctant to answer, since I think your purpose of asking is only to be argumentative / not relevant to whether Gödel's theorem applies to physics.
For what it is worth - physics is a field of inquiry or human-practice in which we form mathematical models* postulating fundamental theoretical entities (such as force, time, mass, charge, particles etc.) and causal mathematical laws (such as Newton's law or the Schrödinger equation). Such a theory may come with certain ontological commitments towards the postulated entities and will typically describe the means by which one can make empirical observations regarding the predictions of the theoretical entities and laws i.e. describe the means by which the theory is seen to be empirically adequate. The latter description of empirical adequacy might also describe the intended range of applicability of the theory (e.g. one supposes that electromagnetism or statistical mechanics apply to a particular limited range of phenomena).
Whether my definition is sufficient or not, I don't see how it is relevant to the discussion since it doesn't mean that such models make any guarantee about what phenomena are observable (only that the theory is empirically adequate by the lights of its own postulated entities).
*one might contest this is ahistorical, but I am not writing a book on the question
@boarders@dpiponi It looks like we agree on the role of mathematics in physics. So do you think it is it possible that somehow physics is able to isolate the "good parts" of mathematics to build its models and steer away from incompletness?
@BartoszMilewski@boarders It's mathematics that introduces the "bad parts". A claim of the form "for all x..." may be a testable claim because you just need to find one failure case to falsify it. But claims like "there exists x...." may remain forever untestable because your experiment might simply have failed to explore every single place the x could be. These sentences aren't very scientific but you can use them for all kinds of nefarious purposes in mathematics.
@dpiponi@BartoszMilewski@boarders Using a constructivist foundation (e.g. MLTT) gets rid of some of the problems around "there exists", but leaves you without some other common mathematical tools, since you lose double-negation elimination / excluded middle.
@BoydStephenSmithJr@BartoszMilewski@boarders I was convinced a while back that constructive mathematics is more appropriate for physics. I expect that the things you can no longer prove are of no consequence. But sadly many proofs become much harder.
@dpiponi@boarders I wonder if QFT can be used to probe existential statements. For instance, we were able to practically exclude the fourth generation of elementary particles. But, of course, there is always the error bar around any set of experiments.
@dpiponi@boarders . In practice, physicists play loose with mathematics. But, strictly speaking, a good physical model should be built on solid mathematical foundations. So if you can formulate a mathematical statement that is undecidable, it should be possible to design an experiment that would test it in Nature. (I'm not saying it would prove anything besides the inadequacy of our models.)
@BartoszMilewski@boarders But there's no reason to believe every mathematical statement says something about physics. Like in my single state example, you could ask if the number of states (cast to Bool in the usual way) equals the consistency of ZF. This is an undecidable statement but it doesn't correspond to any experiment you could perform on this imaginary universe.
@dpiponi@boarders True, there is no reason to believe that. But there's also no reason to believe that it's possible for physics to steer clear of Goedel's incompletness. Is there a way to decide which parts of mathematics are safe (other than not using natural numbers at all)? Is it possible that all experiments only use the "good parts" of mathematics?
@BartoszMilewski@boarders Is this what you mean by something that couldn't be decided: there is some computer that could be (really, physically) built that if run for long enough could find a contradiction in ZF.
(Just noticed another reply similar to this. Still, I don't know if this is the kind of thing you're talking about.)
@dpiponi@boarders If I were to be more specific, I would consider a process whose outcome would require encoding an undecidable statement in terms of Feynman diagrams.
@dpiponi@boarders Somebody on twitter mentioned the halting problem. We don't know of an experiment that would naively test it, because it would require infinite amount of time. But all physical models assume that at short scales we have infinite detail (except maybe loop quantum gravity).
@dpiponi I think what you call a "universe" is a model. So, yes, you could have models based on decidable mathematics, but they would not describe (the capital) Nature.
@tamas@BartoszMilewski I think you might be able to argue that nature "contains" (in some sense of the word) enough of an arithmetic system that it would satisfy the assumptions of Gödel's first incompleteness theorem, and that doesn't "tie" nature to mathematics in the same way as saying that nature is exactly the mathematical model.
@mebassett@tamas I an not a Platonist, so I don't think nature "contains" mathematics. There is nature, and there a mathematical models that approximate but never capture the entirety of nature.
@BartoszMilewski@tamas I don't think I was describing platonism. I wasn't saying that one can argue that abstract objects exist, but that certain systems in nature might be good models for mathematical models, and if those models "have" "enough" "arithmetic" then some form of the first incompeteness system might hold for it. But I'm really trying to make sense of your verb "tie", as in "physics is tied to mathematics". so maybe you can tell me what I'm telling you that you might mean. :)
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