This is a pretty staggering mistake. The only way for the universe not to be reducible to one set of formulae would be to involve a meddling supernatural and ineffable force. (If it's not ineffable, it's just more formulae.) The goal of a unified theory is to have a single formula.
He definitely didn't, since we know of systems that are capable of proving any statement. It's sufficient for the system to be internally inconsistent.
Outside of that, we'd need to be a little more clear about what we mean by "there are statements that can't be proven in any system". I would bet that it is not the case that there is a single special statement with a fixed meaning which all consistent systems are incapable of proving. In the general case, I would expect that a statement, taken by itself, would constitute a consistent (if uninteresting) system. That system would be easily able to prove its only axiom.
In my mind, the incompleteness theorem says that a consistent system of "sufficient power" cannot prove its own consistency. We could reify that into a statement that looks like "This system is consistent"; that's what I'm trying to rule out above by referring to "a statement with a fixed meaning". "This system" would refer to different things within different systems.
Wikipedia says this:
> The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of natural numbers.
> For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
> The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
This allows for a single statement (say, a result about the natural numbers) that no sufficiently powerful consistent system can prove, but it doesn't require it. The sets of true-but-unprovable statements might be disjoint between two particular systems.
A typical example -- now that I think about it, the only example I have seen -- is the statement "this statement cannot be proved".
Technically, it is not a single statement, because for different axiomatic systems A1, A2... you get different statements "this statement cannot be proved in A1", "this statement cannot be proved in A2" etc.
For any given axiomatic system A, if the system can prove "this statement cannot be proved in A", it contains a contradiction: it just proved something that is not true. But if the system cannot prove "this statement cannot be proved in A", we have an example of a statement that is true but unprovable (in A).
Gödel's trick was inventing a way how a statement can talk about itself. Like, if you say "it is forbidden for a statement to talk about itself", Gödel asks "but surely it is okay for a statement to talk about another statement that it can define unambiguously?" and then creates a statement that sounds like "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A", but when you actually unpack "7461287618276312876873", you get "the statement you get by unpacking the binary string 7461287618276312876873 is not provable in A" again. So it talks about itself without actually using the words "this statement".
What exactly this all means... I wish I would grok, but I don't. But I am pretty sure it means none of the usual "human brain has mystical magical quantum powers beyond mere computation".
axiomatizes its own provability is the following:
Theorems of T can be computationally enumerated.
cannot computationally enumerate its own theorems:
In order to obtain a contradiction, hypothesize theorems
of a foundational theory T are computationally enumerable.
Then procedures that are provably total in T are
computationally by a procedure that is provably total in
T. Consider the Boolean procedure Diagonal defined on
natural number input n to be the result of the complement
of the result of the ith provably total procedure on input
i. The procedure Diagonal differs from every procedure in
the enumeration of provably total procedures in T.
However, by construction, diagonal is a provably total
procedure in T, which is a contradiction.
decidable that formalizes its own provability, the following
proposition is true but unprovable:
Theorems of T are computationally enumerable.
unprovable here:
https://papers.ssrn.com/abstract=3603021
Note that the above result is more powerful than the one
quoted in the post above because theorems of T might not
computationally enumerable, which is the case for the
Dedekind axiomatization of Natural Numbers.
sense that they characterize Natural Numbers up to a unique
isomorphism. However, there is still a proposition about
Natural Numbers that true but unprovable using the
Dedekind/Peano axioms. Further information can be found here:
"Physical Indeterminacy in Digital Computation"
it is not the case that every true proposition is provable in
Dededkind's theory.
no standard models :-)
isomorphism, which is indeed the standard model. However,
Gödel's incompleteness proofs do not work for Dedekind's theory
because there are uncountable axiom instances. (Dedekind's
theory is nevertheless effective because proof checking is
algorithmically decidable.)
Consequently, the first incompleteness result (inferentially
undecidable) has been proved by other means. See the
following for a proof:
https://papers.ssrn.com/abstract=3603021
The above article explains that Gödel's second
incompleteness result is false because the theory can prove
its own consistency.
There's no guarantee the universe is reducible to any of these (although in the likely case the universe is finite in space and time there will be some degenerate case of formulas for everything which "happened" - imo less satisfying than even an approximation via some small set of axiom schemata).
No, this is ridiculous. Not even electromagnetism is a single formula
Expecting we end up with "a single formula" is a ridiculous strawman.
What we need is explain GR with a foundation in QM (or with a common base, which might - but probably won't - be String Theory)
"dF = 0" represents all of Maxwell's (homogeneous) equations in terms of "differential forms".
With the right pattern recognition, the right definitions, and the right paradigm shift, you can compress multiple equations into a single equation.
Maxwell's equations are a nice continuous approximation to whatever it is that really happens.
But it's not a good idea to confuse the map with the territory.
Ah yes that "theory" that ignores the complexity and quirks of reality
1. All possible things won't exist
2. Entirely incompressible areas wont exist becuse that would mean they are completely noise
3. The univese willl have the simplest possbile "generating function", that is, fundamental theory.
Your answer feels dismissive and yet at the same time naive.
2. It's the definition of "incompressible"
3. It's the fundamental principle of algorithmic information theory
But you have also misunderstood what I was saying wrt 'universe'
Consider that this universe could be an instance of the structure that is formed by the algorithm that you are talking about. Consider that all possible alternate structures may also exist. Now consider that there exist 'Garden states' that are unreachable because they are incompressible. So might it not be possible to be conscious in a universe that has no generating function? The mathematical structure representing this universe exist in the abstract, timeless and static. Your life is a slice through this giant structure.
Infinitely many of those structures exist, is there one out there that contains conscious beings that believe they inhabit an orderly universe but they actually inhabit one where the starting state of their universe was a garden state, containing a vast un-generatable structure that has informed the evolution of the universe from afar.
But then it would still have a minimal function. Bugger. What if that universe there existed a capricious demon that used the information in the initial state to... Nope that doesn't work either. Ok you win :(
We have been lucky thus far, but the fundamental nature of luck is that it runs out.
Physics departments are already beginning to turn toward theology. In another two generations will there still be experiments in fundamental physics? What we can be certain of is that they won't close up shop and go home. They will still publish as many papers as required, and graduate post-docs. Ultimately, academic forms do not require factual correction.
1. whether we will find a unified theory.
2. whether it is in theory possible to find a unified theory.
The comment you replied to addresses #2. They're saying basically that if the universe is constrained to some set of knowable rules, then it can be reduced to a set of formulae, which is pretty much the same as saying if the rules are knowable, they're expressible as a set of formulae.
Whether this set of rules is simple enough that we are able to is a separate issue.
There is no guarantee that if a unified theory is possible, it is expressible in mathematics; or even necessarily in any form smaller than the universe itself. We have been lucky for a long time.
That is, if the universe is not ineffable - that is, unless there are unknowable components to the universe - then a "worst case" unified theory can be defined as itself. That is, the universe itself is an expression of itself.
That we can express it in mathematics is a given in that case because mathematics is malleable; all you need to do - though it wouldn't be very useful - is to invent a symbol that represents the universe-as-is that by definition acts how the universe would act in a given context.
Obviously that is not very interesting, but the point it makes is that if the universe is not ineffable, then the question is not if a unified theory is possible, because in that case the universe itself is proof that it is, but how much we can compress such a theory, and whether we can reduce it to something simple.
As such whether or not a theory is possible may seem like it doesn't matter, but it's an important point to make, because it means that anyone making the argument that a unified theory is not possible rather than arguing that a simple unified theory is not possible is in effect arguing the universe is ineffable.
But anyone (e.g., I) can fail to discover one, even if given infinite time trying. It is obviously impossible to distinguish (1) the condition of a conveniently-finite unified theory not being possible from (2) simply failing, collectively, to discover one. It is also impossible, in practice, to distinguish (3) a possible universe-sized theory from, e.g., (4) a necessarily infinite-sized one, having neither. We know that the universe itself has always succeeded at discovering a next state, but do not know whether it might at some time, e.g., find the next state incomputable, and stop evolving.
Insisting that a unified theory conveniently smaller than the universe must be possible, having failed to discover one, is religion. Insisting you have actually got that theory, despite being utterly unable to check its correctness, is religion. That is the religion physics departments are flirting with already.
The very article being discussed claims the very idea that there "may be a deeper, more unified theory" has become completely discredited. It does not call it impossible, but calling the very idea completely discredited skirts very close to it.
And the message you originally replied to criticised a comment above that called the idea a "false idol" and questioned the very possibility of it.
It seems quite clear that while these people may not entirely and completely dismiss the possibility, they see it as unlikely or not worthwhile. My first reply to you was explicitly to point out that the comment you replied to discussed specifically the possibility, not whether or not we'll find one.
> We know that the universe itself has always succeeded at discovering a next state, but do not know whether it might at some time, e.g., find the next state incomputable, and stop evolving.
If so, then this end-state is part of the computation itself that would itself be part of the model. If the universe is not ineffable, such an end-state must be discoverable from the universe itself, and so does not impact the possibility of a unified theory in any sense.
> Insisting that a unified theory conveniently smaller than the universe must be possible, having failed to discover one, is religion.
With the caveat of the assumption that the universe is not ineffable, such an argument certainly can be put forward in a "religious" way, but there are also a whole lot of approaches that could be taken to try to get an idea of an upper bound without knowing the precise details, and while we certainly can not make that claim in the strongest possible interpretation of "must", we can make that claim in the sense that we have strong enough indicators of compressibility to assign it a high probability.
Consider that "all" we're seeking to do is to unify the forces, and depending on who you ask also explain physical constants, within a single framework, not creating a precise simulation of the universe. As such, we can "subtract" any actual randomness - if it exists - and any constant values. We're looking for a model, not an instantiation.
For us to be unable to compress a unified theory pretty much all matter and energy in the universe would need to be integral to the functioning of the universe itself in a way that can't be abstracted away with less complexity.
E.g. "freezing" the universe and moving a few particles around and "resuming" would need to change not just what happens next but how the universe functions, and would need to do so in an irreducible manner - that is, the relationship could not be constant and universal, because if the positioning matters but is following a consistent set of rules that is in any way reducible, then the positions themselves do not need to be included in the universal theory. In that case we can reduce that part of the theory to "here's how to predict what moving mass around will do to e.g. universal constants or relationships between forces".
Now I will accept that it may be possible that a universe that is not ineffable might be possible where a unified theory would be so complex that the shitty fallback of declaring the universe itself the definition would be the smallest option, in about the same sense I'll concede the possibility of Russel's teapot, but while a general compression algorithm capable of compressing every input is impossible, once you take genuinely random output out of a huge data set, a data set that is not compressible becomes increasingly improbably as the size of the data set increases.
In effect, a universal theory smaller than the universe is possible if we can sufficiently for whatever value of precision you want your model to have, compress any information about the universe including the data needed to specify how to integrate it back into the universe (I'm assuming we agree that any compression here needs to include the complexity of the decompressor, as otherwise a special purpose decompressor that includes the universe as a constant can compress the universe to nothing, which would make the discussion pointless).
Now, of course, the question is how much do we need to be able to compress before we'd agree it was "conveniently smaller". But to me at least the extent to which we can e.g. compress information about gravity alone suggests sufficient compressibility that it would seem to be a truly extraordinary claim that we will not be able to achieve similar compressibility in other areas.
But like Russell's teapot, it's possible.
> Insisting you have actually got that theory, despite being utterly unable to check its correctness, is religion. That is the religion physics departments are flirting with already.
We can maybe agree that many people do have strong feelings of which theories will get there and are invested in them more than they might have justification for, that is fair enough. But I've at least never seen anyone suggest they can know for sure, and a lot of large advances have come from people getting extremely invested in untested theories. As long as people don't all get invested in the same untested theories, we'll keep seeing advances.
My interpretation of their "discredited" is very different from yours. I did not take it as saying anything at all about the nature of the universe, as such, but only about the purely human activity of hustling Theories of Everything.
Ineffability is an interesting notion. If you have a theory that would work given complete information, but you can never get complete information, even in principle, is that ineffable? If you would need complete information to complete the theory (say, to choose among 10^500 possible variations) is that ineffable? Or just too hard? Can you really call it a difference?