Is mathematics incapable of describing physics in its entirety?

[dom_quixote]

This Physics Problem is Unsolvable



This is the question presented in Sabine Hossenfelder's video. What would be the consequences of this statement?

 

[sophiecentaur]

This should not be upsetting anyone. Maths is only ever a model. For instance, there are many integrals which do not have an analytic solution but that doesn't mean anything special; the numerical solution will still tell you the levels of FM sidebands.

I found the title "Maths Fail" a bit annoying, as if 'they got it wrong again'. It's more like a gauche social medium posting, along the lines of "Einstein got it wrong".

 

[Motore]

FYI this is the paper she references:
https://arxiv.org/abs/2410.02600

 

[DrClaude]

I found the title "Maths Fail" a bit annoying, as if 'they got it wrong again'.

A video by Sabine Hossenfelder with a ragebate title? Color me surprised!

 

[PeroK]

Is YouTube incapable of describing physics in its entirety?!

 

[sophiecentaur]

FYI this is the paper she references:
https://arxiv.org/abs/2410.02600

I can't see a "fail" in that abstract. I looks to me like a bit of clever maths which I would probably find interesting if my Maths could tackle it. B ut does it actually link to any Physical situation?

video by Sabine Hossenfelder with a ragebate title?

I hadn't bumped into her till now. She appears all over PFA, I see.

 

[gmax137]

What would be the consequences of this statement?

Did you watch the video? Did you see the part where she says what the consequences would be?

 

[DaveC426913]

If mathematics were capable of describing physics completely, I'd think that would put a lot of computational physicists out of work.

 

[dom_quixote]

Did you watch the video? Did you see the part where she says what the consequences would be?

Yes, I watched the video. Since I found it interesting, I posted it here on PF to expand my knowledge.

 

[gmax137]

Sorry for being abrupt, @dom_quixote but your post sounded like "watch this video and tell me what it says." I must have been in a grumpy mood.

 

[dom_quixote]

Sorry for being abrupt, @dom_quixote but your post sounded like "watch this video and tell me what it says." I must have been in a grumpy mood.

No problem, mate. I liked my colleagues' answers, as they were instructive and funny !

 

[Filip Larsen]

Undecidability as part of computation theory has been studied for 40+ years, so the interesting part here is mostly if it is even possible to establish a physical system exhibiting undecidability in some properties, as the referenced paper seems to claim. Not that I in any way understand the details in the physical system the paper models, but if it needs to be infinitely large for the link to undecidability to "work", then I kind of agree with Sabine's conclusion that this particular proposed physical system, while probably interesting to study, is not readily physical realizable.

I recall coming away from the topic at university (as part of study in to chaos and computational science in the late 80-ties) with the same understanding, namely that undecidabilty only emerge in cases where you have an infinite supply of problems. E.g. for the halting problem, if you limit the input set of programs to any finite size, then you no longer (as far as I know) can prove that it is impossible to write a finite (but possibly larger than any input program) analyzer program to compute if one of the input programs halt.

 

[sophiecentaur]

Undecidability as part of computation theory

etc. It strikes me that computational theory can, in no way be certain about the total Science of the Universe. We cannot 'know' the limits of knowledge and any mathematical model can only exist within the limits of our observations.
Mathematics is only self - referential so there is no guarantee that results can be universal. This question is philosophical and not scientific.
There is absolutely no need to be upset about this. The Tree of the knowledge of good and evil is a very early idea and we do not need to stick with it as a valid one.

 

[Filip Larsen]

This question is philosophical and not scientific.

Since science it trying to make finite models that compactly can predict an in principle infinite amount of observations of relevant natural phenomenons, I think it is a relevant question if the natural world is such that you always can represent an infinite set of observations with a finite theory. I assume most people in science at some level just assume that laws of nature in principle apply to infinite systems even if we in practice only observe and apply them to finite systems.

On the other hand if one is just concerned about "practical" unpredictability (i.e. the inability in practical applications to always be able to predict the state of any isolated physical system no matter how precise you measure its initial state) then the concept of undecidability seems rather theoritic way to go. My (engineer) money would still be on the combination of quantum mechanics and chaos providing this unpredicability, if nothing else then because some physical systems exists that can exhibit unpredicability through chaos and all such physical systems also has an underlying uncertainty limit in their initial state due to quantum effects. What I mean here is that for all practical purposes it already seems we can make physical systems that are unpredictable from a single initial state so worrying about undecidability with its seemingly required inifinities in physical systems on top of that seems to add very little to predicability. But I am of course just waving my engineering hands around here.

 

[sophiecentaur]

But I am of course just waving my engineering hands around here

Me too. You have to keep your feet on the ground. Science is a sort of map-making. You have to have a one -to - one mapping of everything in the Universe to the map and there's nothing to say that the system has to be a set of 'subroutines' which might reduce the amount of data capacity for the map.
You have to take this process far enough to a point where the map has to map itself. After that it's Turtles all the way down.

I guess it's not surprising that, when people become all too aware of the complexity that's needed for the model, they have to reach for a god figure to take away the embarrassment and deal with stuff that's outside of our knowledge. This takes things out of the PF domain.

 

[javisot20]

Sabine, second round,

During several videos she tries to imply that the mathematics that we know is not capable of supporting the physics that we are getting to know. We don't know how extensive physical reality is (we understand that it is finite), but we don't know it with complete certainty. In the case that physical reality is a finite object we have enough mathematics to cover it, but if it is an infinite object in some sense we could have problems.

The questions we ask about physical reality are increasingly more extreme, we want to know what happened at T=0, inside a black hole, if a ToE exists, etc.

 

[DaveC426913]

During several videos she tries to imply that the mathematics that we know is not capable of supporting the physics that we are getting to know.

It would not be the first time our mathematics tools had to expand to help us explain nature.

Matrices as a tool are less than 200 years old. Calabi Yau manifolds are barely 75 years old.

 

[fresh_42]

It would not be the first time our mathematics tools had to expand to help us explain nature.

Matrices as a tool are less than 200 years old. Calabi Yau manifolds are barely 75 years old.

And the list is very, very, very long and I think it began with Leibniz and Newton; if it wasn't an unknown Indian some 5,000 years ago who needed the zero to balance his accounts.

 

[javisot20]

It would not be the first time our mathematics tools had to expand to help us explain nature.

I remember an interview by E. Frenkel this year, who at one point says; "String theory has been bad for physicists, but good for mathematicians". We cannot say that we are always explaining nature, but we expand our mathematics anyway.

(a very interesting interview)

 

[fresh_42]

It's been a long time since $F\sim \ddot x$ and I, too, occasionally thought we might need a new mathematical framework to cover the physics ahead of us. On the other hand, the mathematical vocabulary has become so vast in the meantime and decoupled from physics for decades that I very much doubt that Bine has even an idea of what all is available, let alone an overview. Physics only uses parts of mathematics. Maybe it will take someone who sees new connections rather than new mathematics.

It took 227 years from Newton to Noether and QM has only its 124th birthday these days. It took 106 years from Schwarzschild to the picture of M87, and a video game to understand the importance of human behavior in pandemics for mathematical simulations, 89 years between first SIR models and the corrupted blood incident in WOW. ToE is an euphemism and nobody knows what a promising GUT would look like. Physics is facing problems that currently cannot be investigated, the dark areas, what's behind the EH, inflation, or physics before recombination. I doubt that these are mathematical problems.

 

[bob012345]

I think the only thing wrong is using the word ‘maths’.

 

[gmax137]

I think the only thing wrong is using the word ‘maths’.

Sometimes I can tell where people are from by the words they use.