EDITED to emphasize that the idea Kontsevich "took from string theorists" is not as big a deal as title of TFA implies, thanks sno129
On page6 is the crucial^W main idea that Kontsevich took from string theory, the Hodge Diamond
It's memorable, at least :)
En.wiki has a quick explanation
Mirror symmetry translates the dimension number of the (p, q)-th differential form h^(p,q) for the original manifold into h^(n-p,q) of that for the counter pair manifold.
I wouldn't consider the Hodge diamond the "crucial idea from string theory." It's a pretty basic/fundamental concept in geometry and really doesn't a priori have much to do with string theory. The decomposition they give on page 6 probably predates most of the development of string theory.
I "blame" Quantamagazine for this.. upselling string theory via Kontsevich, because I don't think there's anything in this work related to string theory other than the Hodge diamond + related "elementary" symmetries (see my other unedited comment in response to a geometer)
It was probably not intentional, though, and might trigger noone besides snobs like us :)
There's different ways to define "related", here what does Quanta explicitly claim is "related", plausibly without looking maybe they meant historically related but not conceptually related.
Tbf I was maybe a bit indignant over this sentence from TFA:
>The proof _relies_ on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
They should have said "differential geometry", unless you count Kontsevich himself as a string theorist (maybe he does. I don't know)
From the paper, sec3.1.2:
While historically prevalent in the mirror symmetry and Gromov-Witten literature, the complex
analytic or formal analogues of an F-bundle will not be useful for constructing birational invariants
directly
Later on, however:
One largely unexplored aspect of Gromov-Witten theory is its algebraic flexibility..
I guess we can't really not credit the string theorists if Kontsevich can be so inspired by them :)
That's arguably what String Theory is good for, producing interesting, entertaining, and possibly even useful math. What it seems to fail at is making realistically testable predictions about nature that can't be matched or exceeded by simpler competing theories.
No Theory of Everything is going to make realistically testable predictions. That's a problem of the domain, not the theory. The unification energy between the graviton and quantum field theory is on the order of 10^19 GeV, over a dozen orders of magnitude beyond anything we can generate.
We might get lucky that some ToE would generate low-energy predictions different from GR and QFT, but there's no reason to think that it must.
It's not like there's some great low-energy predictions that we're just ignoring. The difficulty of a beyond-Standard-Model theory is inherent to the domain of the question, and that's going to plague any alternative to String Theory just as much.
The testable predictions would be at the places where QM and GR meet. Some examples:
1. interactions at the event horizon of a black hole -- could the theory describe Hawking radiation?
2. large elements -- these are where special relativity influences the electrons [1]
It's also possible (and worth checking) that a unified theory would provide explanations for phenomena and observed data we are ascribing to Dark Matter and Dark Energy.
I wonder if there are other phenomena such as effects on electronics (i.e. QM electrons) in GR environments (such as geostationary satellites). Or possibly things like testing the double slit experiment in those conditions.
You don't need a full fledged theory of quantum gravity to describe Hawking radiation. Quantization of the gravitational field isn't relevant for that phenomenon. Similarly you don't need quantum gravity to describe large elements. Special relativity is already integrated into quantum field theory.
In some ways saying that we don't have a theory of quantum gravity is overblown. It is perfectly possible to quantize gravity in QFT the same way we quantize the electromagnetic field. This approach is applicable in almost all circumstances. But unlike in the case of QED, the equations blow up at high energies which implies that the theory breaks down in that regime. But the only places we know of where the energies are high enough that the quantization of the gravitational field would be relevant would be near the singularity of a black hole or right at the beginning of the Big Bang.
re 2: special relativity is not general relativity - large elements will not provide testable predictions for a theory of everything that combines general relativity and quantum mechanics.
re: "GR environments (such as geostationary satellites)" - a geostationary orbit (or any orbit) is not an environment to test the interaction of GR and QM - it is a place to test GR on its own, as geostationary satellites have done. In order to test a theory of everything, the gravity needs to be strong enough to not be negligible in comparison to quantum effects, i.e. black holes, neutron stars etc. your example (1) is therefore a much better answer than (2)
Re 2 I was wondering if there may be some GR effect as well, as the element's nucleus would have some effect on spacetime curvature and the electrons would be close to that mass and moving very fast.
For geostationary orbits I was thinking of things like how you need to use both special and general relativity for GPS when accounting for the time dilation between the satellite and the Earth (ground). I was wondering if similar things would apply at a quantum level for something QM related so that you would have both QM and GR at play.
So it may be better to have e.g. entangled particles with them placed/interacting in a way that GR effects come into play and measuring that effect.
But yes, devising tests for this would be hard. However, Einstein thought that we wouldn't be able to detect gravitational waves, so who knows what would be possible.
Can't black holes explain Dark Energy? Supposedly there was an experiment showing Black Holes are growing faster than expected. If this is because they are tied to the expansion of the universe (univ. expands -> mass grows), and that tie goes both ways (mass grows -> universe expands), boom, dark energy. I also think that inside the black holes they have their own universes which are expanding (and that we're probably inside one too). If this expansion exerts a pressure on the event horizon which transfers out, it still lines up.
I think that’s highly debatable. For example, dark matter particles with testable properties could be a prediction of a ToE. Or the ToE could resolve the quantum measurement problem (collapse of the wave function) in a testable way.
What's the "quantum measurement problem"? And why is it a problem? I get the wave function collapses when you measure bit. But which part of this do you want to resolve in a testable way?
It’s the question of how the wave function collapses during a measurement. What exactly constitutes a “measurement”? Does the collapse happen instantaneously? Is it a real physical phenomenon or a mathematical trick?
I thought that what constitutes a measurement is well understood; it's just the entanglement between the experiment and the observer, and the process is called decoherence - and the collapse itself is a probabilistic process as a result.
AFAIK an EoT is not required to design experiments to determine if it's a real physical phenomenon vs. a mathematical trick; people are trying to think up those experiments now (at least for hidden variable models of QM).
I'm far from an expert in this field--indeed, I can but barely grasp the gentle introductions to these topics--but my understanding is that calling string theory a "theory of everything" really flatters it. String theory isn't a theory; it's a framework for building theories. And no one (to my understanding) has been able to put forward a theory using string theory that can actually incorporate the Standard Model and General Relativity running in our universe to make any prediction in the first place, much less one that is testable.
Getting into the weeds about what is and is not "A Theory" is an armchair scientist activity, it's not a useful exercise. Nobody in the business of doing physics cares or grants "theory status" to a set of models or ideas.
Some physicists have been trying to build an updated model of the universe based on mathematical objects that can be described as little vibrating strings. They've not been successful in closing the loop and constructing a model that actually describes reality accurately, but they've done a lot of work that wasn't necessarily all to waste.
It's probably either just the wrong abstraction or missing some fundamental changes that would make it accurate.
It would also be tremendously helpful if we had some new physics where there was a significant difference between an experiment and either GR or the standard model. Unfortunately the standard model keeps being proven right.
There's a more basic problem with string theory, which is that it's not a theory. It's a mathematical framework which is compatible with a very wide range of specific physical theories.
About tests of quantum gravity, there have been proposals for feasible tests using gravitationally-induced entanglement protocols:
I don't think that's quite the problem. In mathematics, the word "theory" is often used when referring to particular mathematical frameworks (e.g. Group Theory, Graph Theory, Morse Theory). In that sense I think String Theory is very much a theory. As you imply, in physics, the word "theory" is typically used in a different sense. I'm not a physicist but I presume a physical theory has to be verifiable, consistent with observations, able to predict the behavior of unexplained phenomena. If I understand correctly, the basic problem is that in some quarters string theory is being passed off as a physical theory. I know of pure mathematicians who are interested in string theory and who couldn't care less whether its a physical theory.
The word "theory" doesn't matter in the way you are portraying it as.
Like a book is a book because it's got pages with words on them glued to a spine with covers. It's not "not a book" because the plot makes no sense.
Scientists don't care about what "a theory" is, it's not philosophically important to them. It's just a vague term for a collection of ideas or a model or whatever.
I guess I'm not being clear. I don't care about the word "theory". The point is string theory makes no predictions. It's not just inaccessible energies which make it untestable, but the fact that, as a framework, string theory is compatible with a huge range of possible universes.
Eh, again you're just saying "it's not real because..." whether or not you attach the word theory. It is a space with lots of possible parameters being explored and is not one set of parameters with predictions because all of the sets that have been explored so far are either broken or don't represent reality. That in itself does not mean it doesn't have merit. It's simply incomplete, but given its history it's fair to doubt that it ever will close the loop and have a final form that models our reality.
I'm not sure what you're disagreeing with tbh. If you look up the thread, my point is that the reason string theory is untestable is not simply that high energies are experimentally inaccessible. Rather, string theory doesn't make any definite predictions for those high energies either. It seems you agree with that?
i mean, a theory of everything should at least make retrodictions, which afaik string theory never got to. if someone wants to point me to where someone solved e.g. the hydrogen spectrum using a string theory, then I will be wrong but very happy
> The unification energy between the graviton and quantum field theory is on the order of 10^19 GeV, over a dozen orders of magnitude beyond anything we can generate.
I am not sure what you are refering to. You absolutely can break Lorentz Invariance in string theory[1]. There is a reason why even some string theory researchers call it the theory of anything.
Wellllll. Despite the title, the paper does not make a claim about string theory. The starting point is the "Witten string field theory" which is a field theory engineered to have properties like string theory. Nothing guarantees that theory is exactly like string theory. In addition, the idea is perturbative in nature, there's no guarantee that perturbative effects are in fact realized in the full quantum theory - exoteric cancellations happen often in field theories with many symmetries. This is two degrees of questionable.
So A) the paper isn't actually about string theory and B) it's not clear that the claim it makes is actually correct for the field theory it supposedly applies to.
Well this is just an early example of the lorentz breaking string theory niche. You can find a lot more. In the short time frame where opera had this supposedly faster than light neutrinos, a lot of papers were published in that regard.
For example you can have string theories that lead to finsler spacetimes, which were used to explain the opera results.
I literally have no idea what our conversation has to do with opera results, or what you think that shows, but just lost all interest in continuing this conversation.
That’s just piggybacking on a prediction of special relativity itself. If string theory predicted something novel that’s testable, that would be a lot more noteworthy.
> That’s just piggybacking on a prediction of special relativity itself.
Let me stop you right now to inform you you don't understand how scientific theories are structured. Special relativity is not a prediction of special relativity. Likewise, 1+1=2 isn't a predict of arithmetic, it's the starting point.
If you are suggesting that string theory is somehow more fundamental or powerful than special relativity, and so SR is a mere consequence of ST, that’s a claim that probably requires more explanation or evidence.
No. Special relativity postulates that special relativity is preserved at all scales. It's an axiom. Comes from nowhere. It's assumed.
This is what a theory is: assume XYZ is true, and see how much of the world you can explain. Why is XYZ? That theory doesn't explain it.
Theoretical physics is: what is the smallest set of XYZ assumptions that can explain other theories. So if you can come up with a theory that's internally self-consistent that _predicts_ something which is postulated by another successful theory, that's a very convincing result.
Pardon, but, huh? I very much thought that Lorentz invariance was built into the assumptions of string theory.
Concluding from “A AND B” that “A”, while it does reach a conclusion that is distinct from the assumption, is not impressive.
If string theory does not bake SR into its assumptions, wouldn’t that make the way it is formulated, not manifestly Lorentz invariant? Don’t physicists typically prefer that their theories be, not just Lorentz invariant, but ideally formulated in a way that is manifestly Lorentz invariant?
Of course, not that it is a critical requirement, but it is very much something I thought string theory satisfied. Why wouldn’t it be?
Like, just don’t combine coordinates in ways that aren’t automatically compatible with Lorentz invariance, right?
If you formulate a theory in a way that is manifestly Lorentz invariant, claiming to have derived Lorentz invariance from it, seems to me a bit like saying you derived “A” from “A AND B”.
If string theory isn’t manifestly Lorentz invariant, then, I have to ask: why not??
Lorentz invariance is built into some descriptions of some stringy theories. For example chapter 1 of the Polchinski, you have the 26-dimensional bosonic string which is constructed to be Lorentz invariance. Obviously in this case it's not a "prediction", but then again, it's just a toy-model. Our Universe doesn't have 26 dimensions and doesn't have only bosons.
Ok, so I looked into it a bit, and here’s my understanding:
The Polyakov action is kinda by default manifestly Lorentz invariant, but in order to quantize it, one generally first picks the light cone gauge, where this gauge choice treats some of the coordinates differently, losing the manifest Lorentz invariance. The reason for making this gauge choice is in order to make unitarity clear (/sorta automatic).
An alternative route keeps manifest Lorentz invariance, but proceeding this way, unitarity is not clear.
And then, in the critical dimensions (26 or 10, as appropriate; We have fermions, so, presumably 10) it can be shown that a certain issue (chiral anomaly, I think it was) gets cancelled out, and therefore the two approaches agree.
But, I guess, if one imposes the light cone gauge, if not in a space of dimensionality the critical dimension, the issue doesn’t cancel out and Lorentz invariance is violated? (Previously I was under the impression that when the dimensionality is wrong, things just diverged, and I’m not particularly confident about the “actually it implies violations of Lorentz invariance” thing I just read.)
You understand that this have nothing to do with actual Lorentz invariance, yes? It sounds like you don't really understand the meaning of those terms you're using.
Do you understand what "manifest Lorentz invariance" means?
Of course. Did part of what I said suggest I thought otherwise?
I guess the part about the “when you quantize it after fixing the gauge in a way that loses the manifestness of the Lorentz invariance, if you aren’t in the critical dimension, supposedly you don’t keep the Lorentz invariance” part could imply otherwise? If that part is wrong, my mistake, I shouldn’t have trusted the source I was reading for that part.
I was viewing that part as being part of how you could be right about Lorentz invariance being something derived nontrivially from the theory.
Because, the Polyakov action (and the Nambu-Goto action) are, AIUI, typically initially(at the start of the definition of the theory) formulated in a way that is not just Lorentz invariant, but manifestly Lorentz invariant,
and if there is no step in the process of defining the theory that isn’t manifestly Lorentz invariant, I would think that Lorentz invariance wouldn’t be a nontrivial implication, but something baked into the definition throughout,
so, for it to be a nontrivial implication of the theory, at some point after the definition of the classical action, something has to be done that, while it doesn’t break Lorentz invariance, it “could” do so, in the sense that showing that it doesn’t is non-trivial.
And, I was thinking this would start with the choice of gauge making it no longer manifestly Lorentz invariant.
I trust you have much more knowledge of string theory than I do, so I would appreciate any correction you might have.
It does, but a number of alternative theories of quantum gravity do not. So, if Lorentz invariance is shown to be violated, this would favor those over string theory.
as i know really nothing about the subject, could someone explain why parent was downvoted ? is it for the tone, or the content ? Because, i , having viewed the youtubers in question, had the same opinion about string theory.
The word 'falsifiable' comes from Popper's criterion, which is central to scientific methodology. What it means: if theory predicts something, and later observations show that prediction doesn't hold, then the theory is incorrect.
String theory doesn't work this way, whatever was measured will be explained as an afterthought by free parameter tuning.
Do you mean that have been falsified? Of course, no standing theory delivers falsified predictions, when that happens you throw the theory in the garbage.
Do you mean that can be falsified in principle? In that case String Theory has falsifiable predictions, I gave you one. In principle, we can make experiment that would falsify special relativity. In fact, we've made such experiments in the past and those experiments have never seen special relativity being violated. The test of special relativity are the most precise tests existing in science.
I suspect what they mean is that there is no outcome of an experiment such that, prior to the experiment, people computed that string theory says that the experiment should have such a result, but our other theories in best standing would say something else would happen, and then upon doing the experiment, it was found that things happened the way string theory said (as far as measurements can tell).
But there are such experiments. String theory says that the result of such experiment is: Lorentz invariance not violated.
> but our other theories
This is not how scientific research is done. The way you do it is you a theory, the theory makes predictions, you make experiments, and the predictions fail, you reject that theory. The fact that you might have other theories saying other things doens't matter for that theory.
So string theories said "Lorentz invariance not violated", we've made the experiments, and the prediction wasn't wrong, so you don't reject the theory. The logic is not unlike that of p-testing. You don't prove a theory correct is the experiments agree with it. Instead you prove it false if the experiments disagree with it.
There are no such experimental results satisfying the criteria I laid out. You may be right in objecting to the criteria I laid out, but, the fact remains that it does not satisfy these (perhaps misguided) criteria.
In particular, predicting something different from our best other theories in good standing, was one of the criteria I listed.
And, I think it’s pretty clear that the criteria I described, whether good or not, were basically what the other person meant, and should have been what you interpreted them as saying, not as them complaining that it hadn’t been falsified.
Now, when we gain more evidence that Lorentz invariance is not violated, should the probability we assign to string theory being correct, increase? Yes, somewhat. But, the ratio that is the probability it is correct divided by the probability of another theory we have which also predicts Lorentz invariance, does not increase. It does not gain relative favor.
Now, you’ve mentioned a few times, youtubers giving bad arguments against string theory, and people copying those arguments. If you’re talking about Sabine, then yeah, I don’t care for her either.
However, while the “a theory is tested on its own, not in comparison to other theories” approach may be principled, I’m not sure it is really a totally accurate description of how people have evaluated theories historically.
> But there are such experiments. String theory says that the result of such experiment is: Lorentz invariance not violated.
This is not a new prediction... String theory makes no new predictions, I hear. I don't understand why you need to be told this.
To your point, there exist various reformulations of physics theories, like Lagrangian mechanics and Hamiltonian mechanics, which are both reformulations of Newtonian mechanics. But these don't make new predictions. They're just better for calculating or understanding certain things. That's quite different from proposing special relativity for the first time, or thermodynamics for the first time, which do make novel predictions compared to Newton.
> there exist various reformulations of physics theories, like Lagrangian mechanics and Hamiltonian mechanics, which are both reformulations of Newtonian mechanics
You have no clue what you're talking about. Did you hear this in some youtube video and have been looking to try it on someone?
It has delivered falsifiable postdictions though. Like, there are some measurable quantities which string theory says must be in a particular (though rather wide) finite range, and indeed the measured value is in that range. The value was measured to much greater precision than that range before it was shown that string theory implies the value being in that range though.
Uh, iirc . I don’t remember what value specifically. Some ratio of masses or something? Idr. And I certainly don’t know the calculation.
Score one more for CAM (computer assisted math) and automated proving tools.
In a world where your academic colleagues will only pay attention to your paper if it comes with a Lean or Coq proof (or at least an MM sketch), we would not be in a "it will take years just to understand the paper" type situation.
The title of the paper would then be: "Birational Invariants from Hodge Structures and Quantum Multiplication: the source code".
Other major offender in that space is Mochizuki's [1] "Inter-universal Teichmüller theory".
The very name of the work uses words that make no sense in common English.
Nah, machine verification just tells you that a theorem is true. It does nothing to help you understand it. An Lean formalization of Wiles' proof of FLT would be as incomprehensible to me as Wiles' non-formalized journal article is.
Hey do you want to hear about this cool new result in maths? Let's just speedrun a graduate course in all the prerequisites!
(I more or less do have the background to read these things, but it's super off-putting to start the article about a crazy new proof from a Fields medallist with an introduction to manifolds.)
Do you have the background to do what QM/wiki does, but for HN?
Eg breakdown the explanation of the Hodge diamond on P6 of the paper LI5
Here's how en.wiki (with diagram!!) does it -- which I thought could have been handwaved better in TFA:
Mirror symmetry translates the dimension number of the (p, q)-th differential form h^(p,q) for the original manifold into h^(n-p,q) of that for the counter pair manifold.
I'm reminded of a graduate course in Elliptical Curves where, late in the semester, we took a lecture to speedrun all the prerequisites and ideas of Perelman's [then new] proof of the Poincaré conjecture. It was wild but a lot of fun.
I mean that's pretty much just what Quanta does, if you don't like it I'd recommend reading a different publication. It's their whole shtick, simplifying complex news about science/mathematics just enough so that people completely unfamilar can get a general sense of what happened.
(Shrug) It's a really nice introduction to manifolds, to the extent that's what it is. Really it's just an introduction to rational parameterization, which I'd never heard of but which at first glance seems extremely nifty.
It all gets pretty incomprehensible after that, but that's not the author's fault.
It's 2025, if you want to publish grand claims, and you're initially the only one who understands your own proof, publish a machine readable proof in say MetaMath's .mm format.
You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
Most groundbreaking proofs these days aren’t just cross-discipline but usually involve one or several totally novel techniques.
All that to say: I think you’re dramatically underestimating the difficulty involved in this, EVEN if the author(s) were a(n) expert(s) in machine readable mathematics, which is highly UNlikely given that they are necessarily (a) deep expert(s) in at LEAST one other field.
> You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
people on hn love making these kinds of declarative statements (the one you responded to, not yours itself) - "for X just do Y" as a kind of dunk on the implied author they're responding to (as if anyone asked them to begin with). they absolutely always grossly exaggerate/underestimate/misrepresent the relevance/value/efficacy of Y for X. usually these declarative statements briskly follow some other post on the frontpage. i work on GPU/AI/compilers and the number of times i'm compelled to say to people on here "do you have any idea how painful/pointless/unnecessary it is to use Y for X?" is embarrassing (for hn).
i really don't get even get it - no one can see your number of "likes". twitter i get - fb i get - etc but what are even the incentives for making shit up on here.
It feels good to be smarter than everyone else. You see your upvotes and that's good enough for an ego boost. Been there, done that.
I wish we were a bit more self-critical about this, but it's a tough problem when what brings the community together in the first place is a sense of superiority: prestigious schools, high salaries, impressive employers, supposedly refined tastes. We're at the top of the world, right?
Do selection dynamics require awareness of incentives? I would think that the incentives merely have to exist, not be known.
On HN, that might be as simple as display sort order -- highly engaging comments bubble up to the top, and being at the top, receive more attention in turn.
The highly fit extremes are -- I think -- always going to be hyper-specialized to exploit the environment. In a way, they tell you more about the environment than whatever their content ostensibly is.
isn't it sufficient of an explanation that one has occasionally wasted a ton of time trying to read an article only to discover after studying and interpreting half of a paper that one of the author's proof steps is wholly unjustified?
is it so hard to understand that after a few such events, you wish for authors to check their own work by formalizing it, saving countless hours for your readers, by selecting your paper WITH machine readable proof versus another author's paper without a machine readable proof?
To demonstrate with another example: "Gee, dying sucks. It's 2025, have you considered just living forever?"
To this, one might attempt to justify: "Isn't it sufficient that dying sucks a lot? Is it so hard to understand that having seen people die, I really don't want to do that? It really really sucks!", to which could be replied: "It doesn't matter that it sucks, because that doesn't make it any easier to avoid."
I don't think it matters, to be quite honest. Absolute tractability isn't relevant to what the analogy illustrates (that reality doesn't bend to whims). Consider:
- Locating water doesn't become more tractable because you are thirsty.
- Popping a balloon doesn't become more tractable because you like the sound.
- Readjusting my seat height doesn't become more tractable because it's uncomfortable.
The specific example I chose was for the purpose of being evocative, but is still precisely correct in providing an example of: presenting a wish for X as evidence of tractability of X is silly.
I object to any argument of the form: "Oh, but this wish is a medium wish and you're talking about a large wish. Totally different."
I hold that my position holds in the presence of small, medium, _or_ large wishes. For any kind of wish you'd like!
Unavoidable: expecting someone else to do the connection isn't a viable strategy in semi-adversarial conditions so it has to be bound into the local context, which costs clarity:
- Escaping death doesn't become more tractable because you don't want to die.
This is trivially 'willfully misunderstood', whereas my original framing is more difficult -- you'd need to ignore the parallel with the root level comment, the parallel with the conversation structure thus far, etc. Less clear, but more defensible. It's harder to plausibly say it is something it is not, and harder to plausibly take it to mean a position I don't hold (as I do basically think that requiring formalized proofs is a _practically_ impossible ask).
By your own reckoning, you understood it regardless. It did the job.
It does demonstrate my original original point though, which is that messages under optimization reflect environmental pressures in addition to their content.
I grossly underestimate the value of the time of highly educated people having to decode the arguments of another expert? Consider all the time saved if for each theorem proof pair, the proof was machine readable, you could let your computer verify the proclaimed proof as a precondition on studying it.
That would save a lot of people a lot of time, and its not random peoples time saved, its highly educated peoples time being saved. That would allow much more novel research to happen with the same amount of expert-years.
If population of planet A would use formal verification, and planet B refuses to, which planet do you predict will evolve faster
Currently, in 2025, it is not possible in most fields for a random expert to produce a machine checked proof. The work of everyone in the field coming together to create a machine checked proof is also more work for than for the whole field to learn an important result in the traditional way.
This is a fixable problem. People are working hard on building up a big library of checked proofs, to serve as building blocks. We're working on having LLMs read a paper, and fill out a template for that machine checked proof, to greatly reduce the work. In fields where the libraries are built up, this is invaluable.
But as a general vision of how people should be expected to work? This is more 2035, or maybe 2045, than 2025. That future is visible, but isn't here.
It's interesting that you place it 10 or 20 years from now, given that MetaMath's initial release was... 20 years ago!
So it's not really about the planets not being in the right positions yet.
The roman empire lasted for centuries. If they wanted to do rigorous science, they could have built cars, helicopters, ... But they didn't (in Rome, do as the Romans do).
This is not about the planets not being in the right position, but about Romans in Rome.
I could believe you, an internet stranger. And believe that this problem was effectively solved 20 years ago.
Or I could read Terry Tao's https://terrytao.wordpress.com/wp-content/uploads/2024/03/ma... and believe his experience that creating a machine checkable version of an informal proof currently takes something like 20x the work. And the machine checkable version can't just reference the existing literature, because most of that isn't in machine checkable form either.
I'm going with the guy who is considered one of the greatest living mathematicians. There is an awful lot that goes into creating a machine checkable proof ecosystem, and the file format isn't the hard bit.
20x the work of what? the work of staying vague? there is no upper limit to the "work" savings, why not be 5 times vaguer, then formal verification can be claimed to be 100x more work.
If ultimate readership (over all future) were less than 20 per theorem, or whatever the vagueness factor would be, the current paradigm would be fine.
If ultimate readership (not citation count) were higher than 20 per theorem, its a net societal loss to have the readers guess what the actual proof is, its collectively less effort for the author to formalize the theorem than it would be to have the readers guess the actual proof. As mathematicians both read and author proofs, they would save themselves time, or would be able to move the frontier of mathematics faster. From a taxpayer perspective, we should precondition mathematics funding (not publication) on machine readable proofs, this doesn't mean every mathematician would have to do it, if some hypothetical person had crazy good intuitions, and the rate of results high enough this person could hire people to formalize it for them, to meet the precondition. As long as the results are successfully formalized, this team could continue producing mathematics.
Plus, mathematics isn't just a giant machine of deductive statements. And the proof checking systems are in their infant stages and require huge amounts of efforts even for simple things.
Theres nothing difficult about formalizing a proof you understand.
Formalizing hot garbage supposedly describing a proof can be arbitrarily difficult.
The problem is not a missing library. The number of definitions and lemmas indirectly used is often not that much. Most of the time wasted when formalizing is discovering time and time again that prior authors are wasting your time, sometimes with verifiably false assumptions, but the community keeps sending you around to another gap-filling approach.
> Theres nothing difficult about formalizing a proof you understand.
What are you basing that on? It's completely false.
If that were true, we would have machine proofs of basically everything we have published proofs for. Every published mathematical paper would be accompanied by with its machine-provable version.
But it's not, because the kind of proof suitable for academic publication can easily take multiple years to formalize to the degree it can be verified by computer.
Yes of course a large part depends on formalizing prior authors' work, but both are hard -- the prior stuff and your new stuff.
Your assertion that there's "nothing difficult" is contradicted by all the mathematicians I know.
Sure the history of mathematics used many alternative conceptions of "proof".
The problem is that such constructions were later found to be full of hidden assumptions. Like working in a plane vs on a spherical surface etc.
The advantage of systems like MetaMath are:
1. prover and verifier are essentially separate code bases, indeed the MM prover is essentially absent, its up to humans or other pieces of software to generate proofs. The database just contains explicit axioms, definitions, theorems claims, with proofs for each theorem. The verifier is a minimalistic routine with a minimum amount of lines of code (basically substitution maps, with strict conditions). The proof is a concrete object, a finite list of steps.
2. None of the axioms are hardcoded or optimized, like they tend to be in proof systems where proof search and verification are intermixed, forcing axioms upon the user.
>You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
Is it really known to be the frontier as long as its not verified? I would call the act of rigorous verification the acknowledgement of a frontier shift.
Consider your favorite dead-end in science, perhaps alchemy, the search for alcahest, the search for the philosophers stone, etc. I think nobody today would pretend these ideas were at the frontier, because today it is collectively identified as pseudoscience, which failed to replicate / verify.
If I were the first to place a flag on some mountain, that claim may or may not be true in the eyes of others, but time will tell and others replicating the feat will be able to confirm observation of my flag.
As long as no one can verify my claims they are rightfully contentious, and as more and more people are able to verify or invalidate my claims it becomes clear if I did or did not move the frontier.
One doesn't need to be an expert in machine readable mathematics, to understand how to formalize it to a machine readable form.
If one takes the time to read the free book accompanying the metamath software, and re implements it in about a weekend time, you learn to understand how it works internally. Then playing around a little with mmj2 or so you quickly learn how to formalize a proof you understand. If you understand your own proof its easy to formalize it. One doesn't need to be "an expert in machine readable mathematics".
If one is given an incomplete proof (i.e. where not each step is justified in terms of theorems completely justified before it) there is an amount of bruteforce entropy involved for guessing which intermediate steps weren't jotted down. Of course it takes 20x or more effort if the prover refused to write down certain steps. It even occurs that when pointed out, it takes the original "prover" a lot of time to find a proof for a gap, hence it wasn't originally proven.
If I find my own proofs, or if the proof of someone else is clearly written, formalization is not hard at all.
Let us assume for the sake of this discussion that Wiles' latest proof for FLT is in fact complete, while his earlier proof wasn't. It took Wiles and his helper more than a weekend to close the gap. Imagine no one had challenged the proof or pointed out this gap. Anyone tasked with formalizing it would face the challenge of trying to figure out which result (incorrectly presumed to be already known) was used in a certain step. The formalizer is in fact finishing an unfinished proof.
After succeeding in closing this gap, who else was willing to point at the next gap? There is always some sense of prestige lost when pointing at a "gap" and then observing the original prover(s) close that gap, in a sense they saw how to prove it while the challenger did not. This dynamic is unhealthy. To claim a proof we should expect a complete proof, the burden of proof should lay on the proving claimant not on the verifier.
I am absolutely no expert but I doubt many of the components of this beast are even expressible in anything currently machine readable (perhaps definable is a better word for now).
The article clearly states that there are multiple reading groups across the world attempting to get to grips with each small aspect of the ideas involved. That they even attempt this implies to me that the ideas are considered worth studying by some serious players in the field: the group (its way more than just one Fields toting bloke) have enough credibility for that.
You're the top-rated comment, didn't read the article*, posted the most flippant response possible based on whatever couple sentences you didn't understand, and I'll get downvoted for pointing it out, even if I leave out the "I'll get downvoted" part. Really a shame.
Some random definition of flippant:
"not showing a serious or respectful attitude."
In my view publishing proofs in 2025 without machine readable proofs is quite flippant yes. We have powerful computation systems, with huge amounts of RAM and storage, yet the bulk does most of mathematics on a computer in the form of... high-tech calligraphy? People are wasting each others time by not using formal verification, to me that is disrespectful.
Is pointing out this disrespectful facet of collective behavior disrespectful?
For example are people who point out the problems associated with GHG emissions, really being flippant when they point out the flippant behavior of excess emission?
I think you have a point. The paper has load bearing reliance on other preprints. I think soon we see a workflow where AI (ChatGPT) can identify precise transitions in the argument that do not require full formalization to falsify. Link - https://chatgpt.com/share/693cc655-ca94-800c-870a-a5c78fb10d...
The mathematics here are far beyond me, but it's interesting that Gemini more or less concurs with chatgpt with respect to "load bearing reliance on other preprints".
Gemini's summary (there's much more in the link above that builds up to this):
The mathematics is largely robust but relies on heavy "black box" theorems (Iritani's blowup formula, Givental's equivariant mirror symmetry) to bypass difficult geometric identifications in the non-archimedean setting. The primary instability is the lack of a non-archimedean Riemann-Hilbert correspondence, which limits the "enhanced" atoms theory. The core results on cubic fourfolds hold because the specific spectral decomposition (eigenvalues $0, 9, 9\zeta, 9\zeta^2$) is robust and distinct22, but the framework's extension to finer invariants (integral structures) is currently obstructed.
Over a couple of decades VCs have invested in vanity startups that cost billions of dollars like it's nothing, countless times.
I think half a billion isn't that expensive for a program that searches for a potential "theory of everything" that can profoundly change our understanding of the universe (even if it brings no results!)
We're long past the point where people can claim string theory has contributed nothing. AdS/CFT correspondence helped us understand what happens to information in black holes, and brought us the holographic principle which now is looking like it might potentially be the next big conceptual revolution in physics. Holography is making meaningful predictions in nuclear and plasma physics right now
Holography desperately needs it's own Brian Greene style ambassador to share the good news. In terms of momentum and taking center stage, it's now in the place where String Theory used to be like 10-15 years ago as the frontier idea with all the excitement ever momentum behind it, and it has been borne from the fruit of string theory. It's quite amazing times we're living in but I think there's been no energy in the post covid world to take a breath and appreciate it.
Last i’d heard, ‘they provided an interesting alternative way of thinking of the problem’ but provided no unique insights or additional testable behaviors. The folks using the alternative theories ended up being able to formulate them more directly using other (‘normal’) physics later. Do you have a cite for anything contrary to that?
It doesn’t help that when something does finally seem experimentally provable (Craig Hogan noise for example), but then gets tested and seems disproven, then it gets ‘removed from Canon’ as it were and ‘is not string theory’.
I'd like to express my doubts about your ability to understand their theories more colourfully, but I'm afraid I'm also under close scrutiny around here.
This exact long running issue with string theory is surely my imagination, and I’m the only one who has commented on it. luckily it’s easy to prove me wrong.
I suspect more people worked on solving quadratic equations in what I estimate to be the 1000 years since the problem was formulated, to when it was solved. The ancient Greeks knew that they could solve some quadratic equations but not others, and Al-Khwarizmi came up with the general solution. And then it was even further generalized with complex numbers.
Neil DeGrasse Tyson made an amazing point some years back that string theory costs practically nothing to develop. It takes some human capital to be sure, but in terms of infrastructure investments, it's pencils and paper and some computers. There's no high stakes mega project requiring massive infrastructure investments for questionable returns; no super colliders or gravitational wave detectors.
For a field repeatedly challenged for not bringing testable predictions to bear, the fact that so much of its rich theoretical framework has been able to be worked out with minimal infrastructure investment is a welcome blessing which, I would hope, critics and supporters alike can celebrate.
I wouldn't downplay the opportunity cost of that much human capital. It really is quite a lot, given the obvious talents of the physicists.
I'm not saying I fully agree with the position, but one way of looking at it is that thousands of incredibly smart people got nerd-sniped into working on a problem that actually has no solution. I sometimes wonder if there will ever be a point where people give up on it, as opposed to pursuing a field that bears some mathematical fruit, always with some future promise, but contributes nothing to physics.
There is almost no opportunity cost: The academic pyramid swaps out the lower parts of the hierarchy at a high pace. You might lose a few smart people who become professors but the average sting theory phd goes to finance or whatever field requires absurd amounts of math at the moment.
You do get people who are happy for a few years since they can live their childhood dream of being a physicist before the turn to actual jobs.
Having people work on things that are at the limit of human understanding is an essential part of a modern educational system.
For every professional string theorist, you get hundreds of people who were brought up in an academic system that values rigor and depth of scientific thinking.
That's literally what a modern technological economy is built on.
Getting useful novel results out of this is almost a lucky side effect.
Don't tell him how much money was invested into CERN over the same timespan to conduct experiments with highly uncertain outcomes. Or into gravitational wave detection. It wasn't certain that those waves even exist until the first measurement decades into the program.
12.5 million a year for a hundred people seems reasonable? 125k per person per year. GP still said "a few hundred" - two hundred would drop that value to 62.5k per person
I am not a fan of String Theory, but as far as fringe science theories go, String Theory is probably one of the more innocent ones. If you are going to pour money into a fringe science theory, I would much rather it goes to scientists trying to discover some properties of the universe which may or may not exist (and probably doesn’t exist; lets be honest here), than many of the awful stuff which exists on the fringes of social sciences (things like longtermism or futurism) or on the fringes of engineering (a future Mars colony, AI singularity, etc.).
Yes, I do. It is engineeringly possible, but societally a horror prescription. I maintain that even the moon landing was an engineering dead end, it resulted in no major breakthrough which we wouldn’t have reached otherwise (for much cheaper) and the humanity benefited nothing but bragging rights. It was then used to further nationalism and exceptionalism by one particular society which went on to conduct many horrible acts of atrocities in the decades that followed.
The prospect of a Mars colony would be that except a million times worse. Humanity will never migrate to Mars, we will never live on Mars, we have nothing to gain by living there, and it may even be impossible for us to live a normal human life over there (e.g. we don‘t know if we can even give birth over there). The only thing it will give us are bragging rights to the powerful individuals which achives it first, who will likely use that as political capital to enact horrible policies on Earth, for their own personal benefits, at the cost of everybody else.
Saying "working toward a martian colony" is akin to saying "working toward a way to colonize the solar system". Mars is not very interesting once you have a methodology. The Moon is a much more practical place to start the process. Then aim at the asteroid belt.
Mars costs the same as the Moon to reach and return from (delta-V) and is a much easier environment to stay in, even over as short a period as a month. Mars makes much more sense than the Moon, which has little of interest and isn’t a stepping stone to anywhere.
> Mars costs the same as the Moon to reach and return from
0/1
If you stay out of gravity wells, traveling anywhere in space is the same cost, minus the non-trivial life support issues, which only come into play on a trip to Mars and back.
> (Mars) is a much easier environment to stay in, even over as short a period as a month
0/2
> (the moon) isn’t a stepping stone to anywhere.
0/3
Humanity has gotten there before Mars for the precise reason that it is a stepping stone.
None of what you posted is factually true and, in good faith, I have to wonder why you might believe these things.
Mining asteroids is a goal that makes sense. I can picture a future where spacecrafts are regularly sent to the asteroid belt and come back to earth with some minerals. Living on the moon does not make sense. There is nothing to be gained from humans living in a future moon base. Not any more than cities built in Antarctica, or in orbit with a constellation of ISS like satellites.
We won’t build a city on the Moon, nor Mars, nor any of Jupiter’s moons, nor anywhere outside of Earth, and we won‘t do this even if engineeringly possible, for the exact same reason we won’t build a bubble city inside the Mariana Trench.
Mining asteroids makes no sense whatsoever with any currently imaginable practical tech, especially not economically. The numbers for even the most basic solutions just don't work, and anything cleverer - like adding thrusters to chunks of metal and firing them at the Earth - has one or two rather obvious issues.
The Moon is interesting because it's there, it's fairly close, it's a test bed for off-world construction, manufacturing, and life support, and there are experiments you can do on the dark side that aren't possible elsewhere.
Especially big telescopes.
It has many of the same life support issues as Mars, and any Moon solutions are likely to work on Mars and the asteroids, more quickly and successfully than trying to do the same R&D far, far away.
Will it pay for itself? Not for a long, long time. But frontier projects rarely do.
The benefit comes from the investment, the R&D, the new science and engineering, and the jobs created.
It's also handy if you need a remote off-site backup.
Mining asteroids wouldn’t be for Earth - it would be for satellites or LEO or possibly even Mars, which is a lot closer to the Asteroids than Earth and may need some extra raw materials we don’t want to spend the horrendous cost of lifting out of Earth’s gravity.
The Moon has nothing to offer Mars explorers as everything will be different and solutions for the unique lunar conditions (two weeks of darkness, temperature extremes, moon dust, vacuum) do not apply to Mars at all. It’like saying living under the ocean is good practice for living in the Artic, but we should start under the ocean because it’s closer.
> Mining asteroids makes no sense whatsoever with any currently imaginable practical tech, especially not economically.
With current tech, it's practical enough to extract rocks from a rock. We've already done this on a comet, which I think is much harder to do. With current economics, not practical to fund such an endeavor, even if it was to haul back an asteroid made of solid gold. Regardless, we're discussing the far future, rather than current state.
If raw materials (again, an unknown) continue to become more scarce, it's hard to say what economics might support extra-planetary resource collection. What's for sure, is mining Mars will be harder than mining asteroids for water or metals, et al.
Mining asteroids makes no sense in the current economy with our current technology. But working towards engineering solutions which makes mining asteroids make sense makes sense (if that makes sense).
However, it is much easier to see us send robots to mine these asteroids, or send robots to the moon to build a giant telescope on the dark side (if that makes sense), then it is to see us build cities on the moon to build said telescope, and to mine those asteroids.
You see the difference here is that the end goal of mining asteroids are resources being sent to earth and exploited, while the goal of space settlements are the settlements them selves, that is some hypothetical space expansion is the goal, and that makes no sense, nobodies lives will improve from space expansion (except for the grifters’ during the grift).
> nobodies lives will improve from space expansion (except for the grifters’ during the grift).
Aspiring to goals and accomplishing them makes life worth living to a lot of people. Furthermore, humanity seems to have an innate drive to explore and learn.
Even to those left at home, it's inspirational to think that there are people who are taking steps to explore the universe.
Maybe it won't help anyone live but it will give a lot of people something to live for.
As others have noted, the moon has significant limitations, in terms of resources and atmosphere. I do think it may have utility, not for anything we might consider settlements or habitats, but perhaps domed science outposts.
Posting the paper: https://arxiv.org/abs/2508.05105
EDITED to emphasize that the idea Kontsevich "took from string theorists" is not as big a deal as title of TFA implies, thanks sno129
On page6 is the crucial^W main idea that Kontsevich took from string theory, the Hodge Diamond
It's memorable, at least :)
En.wiki has a quick explanation
Mirror symmetry translates the dimension number of the (p, q)-th differential form h^(p,q) for the original manifold into h^(n-p,q) of that for the counter pair manifold.
(n=4 for the paper, "cubic 4-fold")
https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Ho...
Don't miss the caption from the end of the previous page, what the sum means :)
I wouldn't consider the Hodge diamond the "crucial idea from string theory." It's a pretty basic/fundamental concept in geometry and really doesn't a priori have much to do with string theory. The decomposition they give on page 6 probably predates most of the development of string theory.
I "blame" Quantamagazine for this.. upselling string theory via Kontsevich, because I don't think there's anything in this work related to string theory other than the Hodge diamond + related "elementary" symmetries (see my other unedited comment in response to a geometer)
It was probably not intentional, though, and might trigger noone besides snobs like us :)
There's different ways to define "related", here what does Quanta explicitly claim is "related", plausibly without looking maybe they meant historically related but not conceptually related.
Tbf I was maybe a bit indignant over this sentence from TFA:
>The proof _relies_ on ideas imported from the world of string theory. Its techniques are wholly unfamiliar to the mathematicians who have dedicated their careers to classifying polynomials.
They should have said "differential geometry", unless you count Kontsevich himself as a string theorist (maybe he does. I don't know)
From the paper, sec3.1.2:
While historically prevalent in the mirror symmetry and Gromov-Witten literature, the complex analytic or formal analogues of an F-bundle will not be useful for constructing birational invariants directly
Later on, however:
One largely unexplored aspect of Gromov-Witten theory is its algebraic flexibility..
I guess we can't really not credit the string theorists if Kontsevich can be so inspired by them :)
That's arguably what String Theory is good for, producing interesting, entertaining, and possibly even useful math. What it seems to fail at is making realistically testable predictions about nature that can't be matched or exceeded by simpler competing theories.
No Theory of Everything is going to make realistically testable predictions. That's a problem of the domain, not the theory. The unification energy between the graviton and quantum field theory is on the order of 10^19 GeV, over a dozen orders of magnitude beyond anything we can generate.
We might get lucky that some ToE would generate low-energy predictions different from GR and QFT, but there's no reason to think that it must.
It's not like there's some great low-energy predictions that we're just ignoring. The difficulty of a beyond-Standard-Model theory is inherent to the domain of the question, and that's going to plague any alternative to String Theory just as much.
The testable predictions would be at the places where QM and GR meet. Some examples:
1. interactions at the event horizon of a black hole -- could the theory describe Hawking radiation?
2. large elements -- these are where special relativity influences the electrons [1]
It's also possible (and worth checking) that a unified theory would provide explanations for phenomena and observed data we are ascribing to Dark Matter and Dark Energy.
I wonder if there are other phenomena such as effects on electronics (i.e. QM electrons) in GR environments (such as geostationary satellites). Or possibly things like testing the double slit experiment in those conditions.
[1] https://physics.stackexchange.com/questions/646114/why-do-re...
You don't need a full fledged theory of quantum gravity to describe Hawking radiation. Quantization of the gravitational field isn't relevant for that phenomenon. Similarly you don't need quantum gravity to describe large elements. Special relativity is already integrated into quantum field theory.
In some ways saying that we don't have a theory of quantum gravity is overblown. It is perfectly possible to quantize gravity in QFT the same way we quantize the electromagnetic field. This approach is applicable in almost all circumstances. But unlike in the case of QED, the equations blow up at high energies which implies that the theory breaks down in that regime. But the only places we know of where the energies are high enough that the quantization of the gravitational field would be relevant would be near the singularity of a black hole or right at the beginning of the Big Bang.
re 2: special relativity is not general relativity - large elements will not provide testable predictions for a theory of everything that combines general relativity and quantum mechanics.
re: "GR environments (such as geostationary satellites)" - a geostationary orbit (or any orbit) is not an environment to test the interaction of GR and QM - it is a place to test GR on its own, as geostationary satellites have done. In order to test a theory of everything, the gravity needs to be strong enough to not be negligible in comparison to quantum effects, i.e. black holes, neutron stars etc. your example (1) is therefore a much better answer than (2)
Re 2 I was wondering if there may be some GR effect as well, as the element's nucleus would have some effect on spacetime curvature and the electrons would be close to that mass and moving very fast.
For geostationary orbits I was thinking of things like how you need to use both special and general relativity for GPS when accounting for the time dilation between the satellite and the Earth (ground). I was wondering if similar things would apply at a quantum level for something QM related so that you would have both QM and GR at play.
So it may be better to have e.g. entangled particles with them placed/interacting in a way that GR effects come into play and measuring that effect.
But yes, devising tests for this would be hard. However, Einstein thought that we wouldn't be able to detect gravitational waves, so who knows what would be possible.
Can't black holes explain Dark Energy? Supposedly there was an experiment showing Black Holes are growing faster than expected. If this is because they are tied to the expansion of the universe (univ. expands -> mass grows), and that tie goes both ways (mass grows -> universe expands), boom, dark energy. I also think that inside the black holes they have their own universes which are expanding (and that we're probably inside one too). If this expansion exerts a pressure on the event horizon which transfers out, it still lines up.
No.
I think that’s highly debatable. For example, dark matter particles with testable properties could be a prediction of a ToE. Or the ToE could resolve the quantum measurement problem (collapse of the wave function) in a testable way.
What's the "quantum measurement problem"? And why is it a problem? I get the wave function collapses when you measure bit. But which part of this do you want to resolve in a testable way?
It’s the question of how the wave function collapses during a measurement. What exactly constitutes a “measurement”? Does the collapse happen instantaneously? Is it a real physical phenomenon or a mathematical trick?
I thought that what constitutes a measurement is well understood; it's just the entanglement between the experiment and the observer, and the process is called decoherence - and the collapse itself is a probabilistic process as a result.
AFAIK an EoT is not required to design experiments to determine if it's a real physical phenomenon vs. a mathematical trick; people are trying to think up those experiments now (at least for hidden variable models of QM).
https://en.wikipedia.org/wiki/Measurement_problem
https://en.wikipedia.org/wiki/Quantum_decoherence
I'm far from an expert in this field--indeed, I can but barely grasp the gentle introductions to these topics--but my understanding is that calling string theory a "theory of everything" really flatters it. String theory isn't a theory; it's a framework for building theories. And no one (to my understanding) has been able to put forward a theory using string theory that can actually incorporate the Standard Model and General Relativity running in our universe to make any prediction in the first place, much less one that is testable.
Getting into the weeds about what is and is not "A Theory" is an armchair scientist activity, it's not a useful exercise. Nobody in the business of doing physics cares or grants "theory status" to a set of models or ideas.
Some physicists have been trying to build an updated model of the universe based on mathematical objects that can be described as little vibrating strings. They've not been successful in closing the loop and constructing a model that actually describes reality accurately, but they've done a lot of work that wasn't necessarily all to waste.
It's probably either just the wrong abstraction or missing some fundamental changes that would make it accurate.
It would also be tremendously helpful if we had some new physics where there was a significant difference between an experiment and either GR or the standard model. Unfortunately the standard model keeps being proven right.
There's a more basic problem with string theory, which is that it's not a theory. It's a mathematical framework which is compatible with a very wide range of specific physical theories.
About tests of quantum gravity, there have been proposals for feasible tests using gravitationally-induced entanglement protocols:
https://arxiv.org/abs/1707.06036
I don't think that's quite the problem. In mathematics, the word "theory" is often used when referring to particular mathematical frameworks (e.g. Group Theory, Graph Theory, Morse Theory). In that sense I think String Theory is very much a theory. As you imply, in physics, the word "theory" is typically used in a different sense. I'm not a physicist but I presume a physical theory has to be verifiable, consistent with observations, able to predict the behavior of unexplained phenomena. If I understand correctly, the basic problem is that in some quarters string theory is being passed off as a physical theory. I know of pure mathematicians who are interested in string theory and who couldn't care less whether its a physical theory.
Yes, that's what I meant: it's not a physical theory in the sense of making a well-defined set of predictions about the actual world.
The word "theory" doesn't matter in the way you are portraying it as.
Like a book is a book because it's got pages with words on them glued to a spine with covers. It's not "not a book" because the plot makes no sense.
Scientists don't care about what "a theory" is, it's not philosophically important to them. It's just a vague term for a collection of ideas or a model or whatever.
I guess I'm not being clear. I don't care about the word "theory". The point is string theory makes no predictions. It's not just inaccessible energies which make it untestable, but the fact that, as a framework, string theory is compatible with a huge range of possible universes.
Eh, again you're just saying "it's not real because..." whether or not you attach the word theory. It is a space with lots of possible parameters being explored and is not one set of parameters with predictions because all of the sets that have been explored so far are either broken or don't represent reality. That in itself does not mean it doesn't have merit. It's simply incomplete, but given its history it's fair to doubt that it ever will close the loop and have a final form that models our reality.
I'm not sure what you're disagreeing with tbh. If you look up the thread, my point is that the reason string theory is untestable is not simply that high energies are experimentally inaccessible. Rather, string theory doesn't make any definite predictions for those high energies either. It seems you agree with that?
i mean, a theory of everything should at least make retrodictions, which afaik string theory never got to. if someone wants to point me to where someone solved e.g. the hydrogen spectrum using a string theory, then I will be wrong but very happy
I remember a video talk by Witten where he said string theory predicts the existence of gravity, and nothing else does.
> The unification energy between the graviton and quantum field theory is on the order of 10^19 GeV, over a dozen orders of magnitude beyond anything we can generate.
lol the confidence.
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I am not sure what you are refering to. You absolutely can break Lorentz Invariance in string theory[1]. There is a reason why even some string theory researchers call it the theory of anything.
[1] https://inspirehep.net/literature/262241
Wellllll. Despite the title, the paper does not make a claim about string theory. The starting point is the "Witten string field theory" which is a field theory engineered to have properties like string theory. Nothing guarantees that theory is exactly like string theory. In addition, the idea is perturbative in nature, there's no guarantee that perturbative effects are in fact realized in the full quantum theory - exoteric cancellations happen often in field theories with many symmetries. This is two degrees of questionable.
So A) the paper isn't actually about string theory and B) it's not clear that the claim it makes is actually correct for the field theory it supposedly applies to.
Well this is just an early example of the lorentz breaking string theory niche. You can find a lot more. In the short time frame where opera had this supposedly faster than light neutrinos, a lot of papers were published in that regard.
For example you can have string theories that lead to finsler spacetimes, which were used to explain the opera results.
I literally have no idea what our conversation has to do with opera results, or what you think that shows, but just lost all interest in continuing this conversation.
People are not likely to look for unphysical models, if they're trying to do physics. The connection is social.
That’s just piggybacking on a prediction of special relativity itself. If string theory predicted something novel that’s testable, that would be a lot more noteworthy.
> That’s just piggybacking on a prediction of special relativity itself.
Let me stop you right now to inform you you don't understand how scientific theories are structured. Special relativity is not a prediction of special relativity. Likewise, 1+1=2 isn't a predict of arithmetic, it's the starting point.
If you are suggesting that string theory is somehow more fundamental or powerful than special relativity, and so SR is a mere consequence of ST, that’s a claim that probably requires more explanation or evidence.
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Forgive my non specialist questions here, but doesn’t special relativity predict that special relativity is preserved at all scales?
No. Special relativity postulates that special relativity is preserved at all scales. It's an axiom. Comes from nowhere. It's assumed.
This is what a theory is: assume XYZ is true, and see how much of the world you can explain. Why is XYZ? That theory doesn't explain it.
Theoretical physics is: what is the smallest set of XYZ assumptions that can explain other theories. So if you can come up with a theory that's internally self-consistent that _predicts_ something which is postulated by another successful theory, that's a very convincing result.
Pardon, but, huh? I very much thought that Lorentz invariance was built into the assumptions of string theory.
Concluding from “A AND B” that “A”, while it does reach a conclusion that is distinct from the assumption, is not impressive.
If string theory does not bake SR into its assumptions, wouldn’t that make the way it is formulated, not manifestly Lorentz invariant? Don’t physicists typically prefer that their theories be, not just Lorentz invariant, but ideally formulated in a way that is manifestly Lorentz invariant?
Of course, not that it is a critical requirement, but it is very much something I thought string theory satisfied. Why wouldn’t it be?
Like, just don’t combine coordinates in ways that aren’t automatically compatible with Lorentz invariance, right?
If you formulate a theory in a way that is manifestly Lorentz invariant, claiming to have derived Lorentz invariance from it, seems to me a bit like saying you derived “A” from “A AND B”.
If string theory isn’t manifestly Lorentz invariant, then, I have to ask: why not??
Lorentz invariance is built into some descriptions of some stringy theories. For example chapter 1 of the Polchinski, you have the 26-dimensional bosonic string which is constructed to be Lorentz invariance. Obviously in this case it's not a "prediction", but then again, it's just a toy-model. Our Universe doesn't have 26 dimensions and doesn't have only bosons.
Ok, so I looked into it a bit, and here’s my understanding:
The Polyakov action is kinda by default manifestly Lorentz invariant, but in order to quantize it, one generally first picks the light cone gauge, where this gauge choice treats some of the coordinates differently, losing the manifest Lorentz invariance. The reason for making this gauge choice is in order to make unitarity clear (/sorta automatic).
An alternative route keeps manifest Lorentz invariance, but proceeding this way, unitarity is not clear.
And then, in the critical dimensions (26 or 10, as appropriate; We have fermions, so, presumably 10) it can be shown that a certain issue (chiral anomaly, I think it was) gets cancelled out, and therefore the two approaches agree.
But, I guess, if one imposes the light cone gauge, if not in a space of dimensionality the critical dimension, the issue doesn’t cancel out and Lorentz invariance is violated? (Previously I was under the impression that when the dimensionality is wrong, things just diverged, and I’m not particularly confident about the “actually it implies violations of Lorentz invariance” thing I just read.)
> losing the manifest Lorentz invariance.
You understand that this have nothing to do with actual Lorentz invariance, yes? It sounds like you don't really understand the meaning of those terms you're using.
Do you understand what "manifest Lorentz invariance" means?
Yes? It means the Lorentz invariance is automatic from the form of the expression, does it not?
Yes. But when "Lorentz invariance isn't automatic from the form of the expression" it does NOT follow that you don't have Lorentz invariance.
Of course. Did part of what I said suggest I thought otherwise?
I guess the part about the “when you quantize it after fixing the gauge in a way that loses the manifestness of the Lorentz invariance, if you aren’t in the critical dimension, supposedly you don’t keep the Lorentz invariance” part could imply otherwise? If that part is wrong, my mistake, I shouldn’t have trusted the source I was reading for that part.
I was viewing that part as being part of how you could be right about Lorentz invariance being something derived nontrivially from the theory.
Because, the Polyakov action (and the Nambu-Goto action) are, AIUI, typically initially(at the start of the definition of the theory) formulated in a way that is not just Lorentz invariant, but manifestly Lorentz invariant,
and if there is no step in the process of defining the theory that isn’t manifestly Lorentz invariant, I would think that Lorentz invariance wouldn’t be a nontrivial implication, but something baked into the definition throughout,
so, for it to be a nontrivial implication of the theory, at some point after the definition of the classical action, something has to be done that, while it doesn’t break Lorentz invariance, it “could” do so, in the sense that showing that it doesn’t is non-trivial.
And, I was thinking this would start with the choice of gauge making it no longer manifestly Lorentz invariant.
I trust you have much more knowledge of string theory than I do, so I would appreciate any correction you might have.
It does, but a number of alternative theories of quantum gravity do not. So, if Lorentz invariance is shown to be violated, this would favor those over string theory.
as i know really nothing about the subject, could someone explain why parent was downvoted ? is it for the tone, or the content ? Because, i , having viewed the youtubers in question, had the same opinion about string theory.
Because String Theory hasn't delivered falsifiable predictions, yet keeps expanding to accommodate failure.
The word 'falsifiable' comes from Popper's criterion, which is central to scientific methodology. What it means: if theory predicts something, and later observations show that prediction doesn't hold, then the theory is incorrect.
String theory doesn't work this way, whatever was measured will be explained as an afterthought by free parameter tuning.
What do you mean by "falsifiable"?
Do you mean that have been falsified? Of course, no standing theory delivers falsified predictions, when that happens you throw the theory in the garbage.
Do you mean that can be falsified in principle? In that case String Theory has falsifiable predictions, I gave you one. In principle, we can make experiment that would falsify special relativity. In fact, we've made such experiments in the past and those experiments have never seen special relativity being violated. The test of special relativity are the most precise tests existing in science.
I suspect what they mean is that there is no outcome of an experiment such that, prior to the experiment, people computed that string theory says that the experiment should have such a result, but our other theories in best standing would say something else would happen, and then upon doing the experiment, it was found that things happened the way string theory said (as far as measurements can tell).
But there are such experiments. String theory says that the result of such experiment is: Lorentz invariance not violated.
> but our other theories
This is not how scientific research is done. The way you do it is you a theory, the theory makes predictions, you make experiments, and the predictions fail, you reject that theory. The fact that you might have other theories saying other things doens't matter for that theory.
So string theories said "Lorentz invariance not violated", we've made the experiments, and the prediction wasn't wrong, so you don't reject the theory. The logic is not unlike that of p-testing. You don't prove a theory correct is the experiments agree with it. Instead you prove it false if the experiments disagree with it.
There are no such experimental results satisfying the criteria I laid out. You may be right in objecting to the criteria I laid out, but, the fact remains that it does not satisfy these (perhaps misguided) criteria.
In particular, predicting something different from our best other theories in good standing, was one of the criteria I listed.
And, I think it’s pretty clear that the criteria I described, whether good or not, were basically what the other person meant, and should have been what you interpreted them as saying, not as them complaining that it hadn’t been falsified.
Now, when we gain more evidence that Lorentz invariance is not violated, should the probability we assign to string theory being correct, increase? Yes, somewhat. But, the ratio that is the probability it is correct divided by the probability of another theory we have which also predicts Lorentz invariance, does not increase. It does not gain relative favor.
Now, you’ve mentioned a few times, youtubers giving bad arguments against string theory, and people copying those arguments. If you’re talking about Sabine, then yeah, I don’t care for her either.
However, while the “a theory is tested on its own, not in comparison to other theories” approach may be principled, I’m not sure it is really a totally accurate description of how people have evaluated theories historically.
And, I think, not entirely for bad reasons?
> But there are such experiments. String theory says that the result of such experiment is: Lorentz invariance not violated.
This is not a new prediction... String theory makes no new predictions, I hear. I don't understand why you need to be told this.
To your point, there exist various reformulations of physics theories, like Lagrangian mechanics and Hamiltonian mechanics, which are both reformulations of Newtonian mechanics. But these don't make new predictions. They're just better for calculating or understanding certain things. That's quite different from proposing special relativity for the first time, or thermodynamics for the first time, which do make novel predictions compared to Newton.
> there exist various reformulations of physics theories, like Lagrangian mechanics and Hamiltonian mechanics, which are both reformulations of Newtonian mechanics
You have no clue what you're talking about. Did you hear this in some youtube video and have been looking to try it on someone?
I suppose it's my bad that I've interacted with a troll that might not even be a real human being.
It has delivered falsifiable postdictions though. Like, there are some measurable quantities which string theory says must be in a particular (though rather wide) finite range, and indeed the measured value is in that range. The value was measured to much greater precision than that range before it was shown that string theory implies the value being in that range though.
Uh, iirc . I don’t remember what value specifically. Some ratio of masses or something? Idr. And I certainly don’t know the calculation.
Because a lot of people felt this applied to them (this was the intention) and were hurt. Good on you for being able to articulate it. Respect.
Score one more for CAM (computer assisted math) and automated proving tools.
In a world where your academic colleagues will only pay attention to your paper if it comes with a Lean or Coq proof (or at least an MM sketch), we would not be in a "it will take years just to understand the paper" type situation.
The title of the paper would then be: "Birational Invariants from Hodge Structures and Quantum Multiplication: the source code".
Other major offender in that space is Mochizuki's [1] "Inter-universal Teichmüller theory".
The very name of the work uses words that make no sense in common English.
[1] https://en.wikipedia.org/wiki/Shinichi_Mochizuki
Nah, machine verification just tells you that a theorem is true. It does nothing to help you understand it. An Lean formalization of Wiles' proof of FLT would be as incomprehensible to me as Wiles' non-formalized journal article is.
what progress has been made on understanding Teichmüller theory in the past decade? not a lot...
Hey do you want to hear about this cool new result in maths? Let's just speedrun a graduate course in all the prerequisites!
(I more or less do have the background to read these things, but it's super off-putting to start the article about a crazy new proof from a Fields medallist with an introduction to manifolds.)
Do you have the background to do what QM/wiki does, but for HN?
Eg breakdown the explanation of the Hodge diamond on P6 of the paper LI5
Here's how en.wiki (with diagram!!) does it -- which I thought could have been handwaved better in TFA:
Mirror symmetry translates the dimension number of the (p, q)-th differential form h^(p,q) for the original manifold into h^(n-p,q) of that for the counter pair manifold.
(n=4 for the paper)
https://en.wikipedia.org/wiki/Homological_mirror_symmetry#Ho...
To call this an idea from string theory, is giving string theory too much credit (imho)
I'm reminded of a graduate course in Elliptical Curves where, late in the semester, we took a lecture to speedrun all the prerequisites and ideas of Perelman's [then new] proof of the Poincaré conjecture. It was wild but a lot of fun.
You can always just not read an article, particularly if it triggers you.
I think it's nice someone wrote about this, even if it's super technical and I cannot understand it completely.
I got it for free!
I can tell which of the two of you likely has a more enjoyable life.
Taste, or whatever you want to call this, is orthogonal to enjoyment.
I think Steve Jobs very much enjoyed life, and you know what kind of an attitude he had about things.
We're all wired up differently.
Excellent advice.
I mean that's pretty much just what Quanta does, if you don't like it I'd recommend reading a different publication. It's their whole shtick, simplifying complex news about science/mathematics just enough so that people completely unfamilar can get a general sense of what happened.
(Shrug) It's a really nice introduction to manifolds, to the extent that's what it is. Really it's just an introduction to rational parameterization, which I'd never heard of but which at first glance seems extremely nifty.
It all gets pretty incomprehensible after that, but that's not the author's fault.
(EDIT: I'm sorry, this silly and dumb.)
"You want many folds!" We gottem!
I'm a differential geometer and I approve this message
[dead]
It's 2025, if you want to publish grand claims, and you're initially the only one who understands your own proof, publish a machine readable proof in say MetaMath's .mm format.
You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
Most groundbreaking proofs these days aren’t just cross-discipline but usually involve one or several totally novel techniques.
All that to say: I think you’re dramatically underestimating the difficulty involved in this, EVEN if the author(s) were a(n) expert(s) in machine readable mathematics, which is highly UNlikely given that they are necessarily (a) deep expert(s) in at LEAST one other field.
> You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
people on hn love making these kinds of declarative statements (the one you responded to, not yours itself) - "for X just do Y" as a kind of dunk on the implied author they're responding to (as if anyone asked them to begin with). they absolutely always grossly exaggerate/underestimate/misrepresent the relevance/value/efficacy of Y for X. usually these declarative statements briskly follow some other post on the frontpage. i work on GPU/AI/compilers and the number of times i'm compelled to say to people on here "do you have any idea how painful/pointless/unnecessary it is to use Y for X?" is embarrassing (for hn).
i really don't get even get it - no one can see your number of "likes". twitter i get - fb i get - etc but what are even the incentives for making shit up on here.
It feels good to be smarter than everyone else. You see your upvotes and that's good enough for an ego boost. Been there, done that.
I wish we were a bit more self-critical about this, but it's a tough problem when what brings the community together in the first place is a sense of superiority: prestigious schools, high salaries, impressive employers, supposedly refined tastes. We're at the top of the world, right?
HN is frequently fodder for satire on other forums. Nobody thinks HN users have "refined tastes", or even that they are "smart" for that matter.
Hey, do you mind sharing any of these other forums? I’m trying to make my way up the satire food chain.
> prestigious schools, high salaries, impressive employers, supposedly refined tastes. We're at the top of the world, right?
Being pompous and self obsessed requires none of those things.
> Being pompous and self obsessed requires none of those things.
Sufficient, but not necessary
Do selection dynamics require awareness of incentives? I would think that the incentives merely have to exist, not be known.
On HN, that might be as simple as display sort order -- highly engaging comments bubble up to the top, and being at the top, receive more attention in turn.
The highly fit extremes are -- I think -- always going to be hyper-specialized to exploit the environment. In a way, they tell you more about the environment than whatever their content ostensibly is.
isn't it sufficient of an explanation that one has occasionally wasted a ton of time trying to read an article only to discover after studying and interpreting half of a paper that one of the author's proof steps is wholly unjustified?
is it so hard to understand that after a few such events, you wish for authors to check their own work by formalizing it, saving countless hours for your readers, by selecting your paper WITH machine readable proof versus another author's paper without a machine readable proof?
If wishes were fishes, as they say.
To demonstrate with another example: "Gee, dying sucks. It's 2025, have you considered just living forever?"
To this, one might attempt to justify: "Isn't it sufficient that dying sucks a lot? Is it so hard to understand that having seen people die, I really don't want to do that? It really really sucks!", to which could be replied: "It doesn't matter that it sucks, because that doesn't make it any easier to avoid."
I understand where you're coming from but it's a bad analogy. Formal proofs are extremely difficult but possible. Immortality is impossible.
I don't think it matters, to be quite honest. Absolute tractability isn't relevant to what the analogy illustrates (that reality doesn't bend to whims). Consider:
- Locating water doesn't become more tractable because you are thirsty.
- Popping a balloon doesn't become more tractable because you like the sound.
- Readjusting my seat height doesn't become more tractable because it's uncomfortable.
The specific example I chose was for the purpose of being evocative, but is still precisely correct in providing an example of: presenting a wish for X as evidence of tractability of X is silly.
I object to any argument of the form: "Oh, but this wish is a medium wish and you're talking about a large wish. Totally different."
I hold that my position holds in the presence of small, medium, _or_ large wishes. For any kind of wish you'd like!
Those are all better analogies than the original one you gave, which didn't illustrate your as clearly as they do.
Unavoidable: expecting someone else to do the connection isn't a viable strategy in semi-adversarial conditions so it has to be bound into the local context, which costs clarity:
- Escaping death doesn't become more tractable because you don't want to die.
This is trivially 'willfully misunderstood', whereas my original framing is more difficult -- you'd need to ignore the parallel with the root level comment, the parallel with the conversation structure thus far, etc. Less clear, but more defensible. It's harder to plausibly say it is something it is not, and harder to plausibly take it to mean a position I don't hold (as I do basically think that requiring formalized proofs is a _practically_ impossible ask).
By your own reckoning, you understood it regardless. It did the job.
It does demonstrate my original original point though, which is that messages under optimization reflect environmental pressures in addition to their content.
wishes can be converted to incentives, what if the incentives change such that formally verified proofs were rewarded more and informal "proofs" less?
I grossly underestimate the value of the time of highly educated people having to decode the arguments of another expert? Consider all the time saved if for each theorem proof pair, the proof was machine readable, you could let your computer verify the proclaimed proof as a precondition on studying it.
That would save a lot of people a lot of time, and its not random peoples time saved, its highly educated peoples time being saved. That would allow much more novel research to happen with the same amount of expert-years.
If population of planet A would use formal verification, and planet B refuses to, which planet do you predict will evolve faster
You appear to be deliberately ignoring the point.
Currently, in 2025, it is not possible in most fields for a random expert to produce a machine checked proof. The work of everyone in the field coming together to create a machine checked proof is also more work for than for the whole field to learn an important result in the traditional way.
This is a fixable problem. People are working hard on building up a big library of checked proofs, to serve as building blocks. We're working on having LLMs read a paper, and fill out a template for that machine checked proof, to greatly reduce the work. In fields where the libraries are built up, this is invaluable.
But as a general vision of how people should be expected to work? This is more 2035, or maybe 2045, than 2025. That future is visible, but isn't here.
It's interesting that you place it 10 or 20 years from now, given that MetaMath's initial release was... 20 years ago!
So it's not really about the planets not being in the right positions yet.
The roman empire lasted for centuries. If they wanted to do rigorous science, they could have built cars, helicopters, ... But they didn't (in Rome, do as the Romans do).
This is not about the planets not being in the right position, but about Romans in Rome.
Let's see.
I could believe you, an internet stranger. And believe that this problem was effectively solved 20 years ago.
Or I could read Terry Tao's https://terrytao.wordpress.com/wp-content/uploads/2024/03/ma... and believe his experience that creating a machine checkable version of an informal proof currently takes something like 20x the work. And the machine checkable version can't just reference the existing literature, because most of that isn't in machine checkable form either.
I'm going with the guy who is considered one of the greatest living mathematicians. There is an awful lot that goes into creating a machine checkable proof ecosystem, and the file format isn't the hard bit.
20x the work of what? the work of staying vague? there is no upper limit to the "work" savings, why not be 5 times vaguer, then formal verification can be claimed to be 100x more work.
If ultimate readership (over all future) were less than 20 per theorem, or whatever the vagueness factor would be, the current paradigm would be fine.
If ultimate readership (not citation count) were higher than 20 per theorem, its a net societal loss to have the readers guess what the actual proof is, its collectively less effort for the author to formalize the theorem than it would be to have the readers guess the actual proof. As mathematicians both read and author proofs, they would save themselves time, or would be able to move the frontier of mathematics faster. From a taxpayer perspective, we should precondition mathematics funding (not publication) on machine readable proofs, this doesn't mean every mathematician would have to do it, if some hypothetical person had crazy good intuitions, and the rate of results high enough this person could hire people to formalize it for them, to meet the precondition. As long as the results are successfully formalized, this team could continue producing mathematics.
Plus, mathematics isn't just a giant machine of deductive statements. And the proof checking systems are in their infant stages and require huge amounts of efforts even for simple things.
> mathematics isn't just a giant machine of deductive statements
I know HN can be volatile sometimes, but I sincerely want to hear more about these parts of math that are not pure deductive reasoning.
Do you just mean that we must assume something to get the ball rolling, or what?
I think the point was that it's not a machine.
Stuff that we can deduce in math with common sense, geometric intuition, etc. can be incredibly difficult to formalize so that a machine can do it.
What do you mean with "do it" in
"...etc. can be incredibly difficult to formalize so that a machine can do it." ?
1. do it = search for a proof
2. do it = verify a purported proof?
Deduce. So your #2.
Of course a machine can verify each step of a proof, but that formal proof must be first presented to the machine.
Right. And I said it's incredibly difficult to formalize so that a machine can do it.
I don't understand what you're confused about.
Theres nothing difficult about formalizing a proof you understand.
Formalizing hot garbage supposedly describing a proof can be arbitrarily difficult.
The problem is not a missing library. The number of definitions and lemmas indirectly used is often not that much. Most of the time wasted when formalizing is discovering time and time again that prior authors are wasting your time, sometimes with verifiably false assumptions, but the community keeps sending you around to another gap-filling approach.
> Theres nothing difficult about formalizing a proof you understand.
What are you basing that on? It's completely false.
If that were true, we would have machine proofs of basically everything we have published proofs for. Every published mathematical paper would be accompanied by with its machine-provable version.
But it's not, because the kind of proof suitable for academic publication can easily take multiple years to formalize to the degree it can be verified by computer.
Yes of course a large part depends on formalizing prior authors' work, but both are hard -- the prior stuff and your new stuff.
Your assertion that there's "nothing difficult" is contradicted by all the mathematicians I know.
For one, some geometric proofs by construction can literally involve pictures rather than statements, right?
Sure the history of mathematics used many alternative conceptions of "proof".
The problem is that such constructions were later found to be full of hidden assumptions. Like working in a plane vs on a spherical surface etc.
The advantage of systems like MetaMath are:
1. prover and verifier are essentially separate code bases, indeed the MM prover is essentially absent, its up to humans or other pieces of software to generate proofs. The database just contains explicit axioms, definitions, theorems claims, with proofs for each theorem. The verifier is a minimalistic routine with a minimum amount of lines of code (basically substitution maps, with strict conditions). The proof is a concrete object, a finite list of steps.
2. None of the axioms are hardcoded or optimized, like they tend to be in proof systems where proof search and verification are intermixed, forcing axioms upon the user.
>Do you just mean that we must assume something to get the ball rolling
They're called "axioms"
> mathematics isn't just a giant machine of deductive statements
I think the subject at question here is mathematical truth, not "mathematics" whatever that means.
>You say that like it’s even remotely feasible at the frontier of mathematics and not a monumental group effort to turn even established proofs into such.
Is it really known to be the frontier as long as its not verified? I would call the act of rigorous verification the acknowledgement of a frontier shift.
Consider your favorite dead-end in science, perhaps alchemy, the search for alcahest, the search for the philosophers stone, etc. I think nobody today would pretend these ideas were at the frontier, because today it is collectively identified as pseudoscience, which failed to replicate / verify.
If I were the first to place a flag on some mountain, that claim may or may not be true in the eyes of others, but time will tell and others replicating the feat will be able to confirm observation of my flag.
As long as no one can verify my claims they are rightfully contentious, and as more and more people are able to verify or invalidate my claims it becomes clear if I did or did not move the frontier.
One doesn't need to be an expert in machine readable mathematics, to understand how to formalize it to a machine readable form.
If one takes the time to read the free book accompanying the metamath software, and re implements it in about a weekend time, you learn to understand how it works internally. Then playing around a little with mmj2 or so you quickly learn how to formalize a proof you understand. If you understand your own proof its easy to formalize it. One doesn't need to be "an expert in machine readable mathematics".
Do you have the weekend free? Perhaps you can take this new proof and show us how it is done.
If one is given an incomplete proof (i.e. where not each step is justified in terms of theorems completely justified before it) there is an amount of bruteforce entropy involved for guessing which intermediate steps weren't jotted down. Of course it takes 20x or more effort if the prover refused to write down certain steps. It even occurs that when pointed out, it takes the original "prover" a lot of time to find a proof for a gap, hence it wasn't originally proven.
If I find my own proofs, or if the proof of someone else is clearly written, formalization is not hard at all.
Let us assume for the sake of this discussion that Wiles' latest proof for FLT is in fact complete, while his earlier proof wasn't. It took Wiles and his helper more than a weekend to close the gap. Imagine no one had challenged the proof or pointed out this gap. Anyone tasked with formalizing it would face the challenge of trying to figure out which result (incorrectly presumed to be already known) was used in a certain step. The formalizer is in fact finishing an unfinished proof.
After succeeding in closing this gap, who else was willing to point at the next gap? There is always some sense of prestige lost when pointing at a "gap" and then observing the original prover(s) close that gap, in a sense they saw how to prove it while the challenger did not. This dynamic is unhealthy. To claim a proof we should expect a complete proof, the burden of proof should lay on the proving claimant not on the verifier.
I am absolutely no expert but I doubt many of the components of this beast are even expressible in anything currently machine readable (perhaps definable is a better word for now).
The article clearly states that there are multiple reading groups across the world attempting to get to grips with each small aspect of the ideas involved. That they even attempt this implies to me that the ideas are considered worth studying by some serious players in the field: the group (its way more than just one Fields toting bloke) have enough credibility for that.
You're the top-rated comment, didn't read the article*, posted the most flippant response possible based on whatever couple sentences you didn't understand, and I'll get downvoted for pointing it out, even if I leave out the "I'll get downvoted" part. Really a shame.
* there's way more than one person involved here
But I did read the article!
Some random definition of flippant: "not showing a serious or respectful attitude."
In my view publishing proofs in 2025 without machine readable proofs is quite flippant yes. We have powerful computation systems, with huge amounts of RAM and storage, yet the bulk does most of mathematics on a computer in the form of... high-tech calligraphy? People are wasting each others time by not using formal verification, to me that is disrespectful.
Is pointing out this disrespectful facet of collective behavior disrespectful?
For example are people who point out the problems associated with GHG emissions, really being flippant when they point out the flippant behavior of excess emission?
I think you have a point. The paper has load bearing reliance on other preprints. I think soon we see a workflow where AI (ChatGPT) can identify precise transitions in the argument that do not require full formalization to falsify. Link - https://chatgpt.com/share/693cc655-ca94-800c-870a-a5c78fb10d...
I did the same thing (roughly) with Gemini 3.0 Pro.
Prompt: "Analyze the mathematics in the paper. Find mistakes, note instabilities (if present) and generally criticize as needed."
https://gemini.google.com/share/6860fdaea334
The mathematics here are far beyond me, but it's interesting that Gemini more or less concurs with chatgpt with respect to "load bearing reliance on other preprints".
Gemini's summary (there's much more in the link above that builds up to this):
The mathematics is largely robust but relies on heavy "black box" theorems (Iritani's blowup formula, Givental's equivariant mirror symmetry) to bypass difficult geometric identifications in the non-archimedean setting. The primary instability is the lack of a non-archimedean Riemann-Hilbert correspondence, which limits the "enhanced" atoms theory. The core results on cubic fourfolds hold because the specific spectral decomposition (eigenvalues $0, 9, 9\zeta, 9\zeta^2$) is robust and distinct22, but the framework's extension to finer invariants (integral structures) is currently obstructed.
A few hundred people working on String Theory for about four decades is about $500 million. Hope this proof was worth it.
Over a couple of decades VCs have invested in vanity startups that cost billions of dollars like it's nothing, countless times.
I think half a billion isn't that expensive for a program that searches for a potential "theory of everything" that can profoundly change our understanding of the universe (even if it brings no results!)
Then just call it maths, not physics?
you can still call it whatever you like!
Last time I called it ‘a haven for folks afraid to have testable theories’ I almost got banned!
Didn't string theory create the concept of supersymmetry, which had testable theories? They were proven wrong, but that's a good thing.
We're long past the point where people can claim string theory has contributed nothing. AdS/CFT correspondence helped us understand what happens to information in black holes, and brought us the holographic principle which now is looking like it might potentially be the next big conceptual revolution in physics. Holography is making meaningful predictions in nuclear and plasma physics right now
Holography desperately needs it's own Brian Greene style ambassador to share the good news. In terms of momentum and taking center stage, it's now in the place where String Theory used to be like 10-15 years ago as the frontier idea with all the excitement ever momentum behind it, and it has been borne from the fruit of string theory. It's quite amazing times we're living in but I think there's been no energy in the post covid world to take a breath and appreciate it.
Last i’d heard, ‘they provided an interesting alternative way of thinking of the problem’ but provided no unique insights or additional testable behaviors. The folks using the alternative theories ended up being able to formulate them more directly using other (‘normal’) physics later. Do you have a cite for anything contrary to that?
It doesn’t help that when something does finally seem experimentally provable (Craig Hogan noise for example), but then gets tested and seems disproven, then it gets ‘removed from Canon’ as it were and ‘is not string theory’.
It got retconned to ‘not string theory’ after that, if i remember correctly.
I'd like to express my doubts about your ability to understand their theories more colourfully, but I'm afraid I'm also under close scrutiny around here.
This exact long running issue with string theory is surely my imagination, and I’m the only one who has commented on it. luckily it’s easy to prove me wrong.
Right?
Is it though? The simplest fool can ask questions that stump the wisest of men.
Still waiting for an actual argument instead of passive aggressive ad hominem.
oh have you not seen the rest of the thread?
Where someone else gave an actual concrete info that I could respond to, and I did?
Have you?
Yes?
Then care to actually add some value to the conversation?
No?
Or roughly the cost of producing Star Wars IX: The Rise of Skywalker. Kinda wish that money had gone to string theory.
What would you have them work on? Predatory social media platforms that sell your data to advertisers and commoditize you.
I suspect more people worked on solving quadratic equations in what I estimate to be the 1000 years since the problem was formulated, to when it was solved. The ancient Greeks knew that they could solve some quadratic equations but not others, and Al-Khwarizmi came up with the general solution. And then it was even further generalized with complex numbers.
Neil DeGrasse Tyson made an amazing point some years back that string theory costs practically nothing to develop. It takes some human capital to be sure, but in terms of infrastructure investments, it's pencils and paper and some computers. There's no high stakes mega project requiring massive infrastructure investments for questionable returns; no super colliders or gravitational wave detectors.
For a field repeatedly challenged for not bringing testable predictions to bear, the fact that so much of its rich theoretical framework has been able to be worked out with minimal infrastructure investment is a welcome blessing which, I would hope, critics and supporters alike can celebrate.
I wouldn't downplay the opportunity cost of that much human capital. It really is quite a lot, given the obvious talents of the physicists.
I'm not saying I fully agree with the position, but one way of looking at it is that thousands of incredibly smart people got nerd-sniped into working on a problem that actually has no solution. I sometimes wonder if there will ever be a point where people give up on it, as opposed to pursuing a field that bears some mathematical fruit, always with some future promise, but contributes nothing to physics.
There is almost no opportunity cost: The academic pyramid swaps out the lower parts of the hierarchy at a high pace. You might lose a few smart people who become professors but the average sting theory phd goes to finance or whatever field requires absurd amounts of math at the moment.
You do get people who are happy for a few years since they can live their childhood dream of being a physicist before the turn to actual jobs.
Having people work on things that are at the limit of human understanding is an essential part of a modern educational system.
For every professional string theorist, you get hundreds of people who were brought up in an academic system that values rigor and depth of scientific thinking.
That's literally what a modern technological economy is built on.
Getting useful novel results out of this is almost a lucky side effect.
If all research bore fruit it wouldn't be research.
so like 12.5 million a year? what an incredible self-own.
aside from that, this number is meaningless without context: how much do other fields of research get?
Don't tell him how much money was invested into CERN over the same timespan to conduct experiments with highly uncertain outcomes. Or into gravitational wave detection. It wasn't certain that those waves even exist until the first measurement decades into the program.
12.5 million a year for a hundred people seems reasonable? 125k per person per year. GP still said "a few hundred" - two hundred would drop that value to 62.5k per person
Your entire life summed up probably costs $500k-4m (On average, depending). Some bean counter could probably argue that it isn't worth it.
I am not a fan of String Theory, but as far as fringe science theories go, String Theory is probably one of the more innocent ones. If you are going to pour money into a fringe science theory, I would much rather it goes to scientists trying to discover some properties of the universe which may or may not exist (and probably doesn’t exist; lets be honest here), than many of the awful stuff which exists on the fringes of social sciences (things like longtermism or futurism) or on the fringes of engineering (a future Mars colony, AI singularity, etc.).
Genuinely curious: Why do you consider a future Mars colony to be «awful stuff»?
Yes, I do. It is engineeringly possible, but societally a horror prescription. I maintain that even the moon landing was an engineering dead end, it resulted in no major breakthrough which we wouldn’t have reached otherwise (for much cheaper) and the humanity benefited nothing but bragging rights. It was then used to further nationalism and exceptionalism by one particular society which went on to conduct many horrible acts of atrocities in the decades that followed.
The prospect of a Mars colony would be that except a million times worse. Humanity will never migrate to Mars, we will never live on Mars, we have nothing to gain by living there, and it may even be impossible for us to live a normal human life over there (e.g. we don‘t know if we can even give birth over there). The only thing it will give us are bragging rights to the powerful individuals which achives it first, who will likely use that as political capital to enact horrible policies on Earth, for their own personal benefits, at the cost of everybody else.
Saying "working toward a martian colony" is akin to saying "working toward a way to colonize the solar system". Mars is not very interesting once you have a methodology. The Moon is a much more practical place to start the process. Then aim at the asteroid belt.
Mars costs the same as the Moon to reach and return from (delta-V) and is a much easier environment to stay in, even over as short a period as a month. Mars makes much more sense than the Moon, which has little of interest and isn’t a stepping stone to anywhere.
> Mars costs the same as the Moon to reach and return from
0/1
If you stay out of gravity wells, traveling anywhere in space is the same cost, minus the non-trivial life support issues, which only come into play on a trip to Mars and back.
> (Mars) is a much easier environment to stay in, even over as short a period as a month
0/2
> (the moon) isn’t a stepping stone to anywhere.
0/3
Humanity has gotten there before Mars for the precise reason that it is a stepping stone.
None of what you posted is factually true and, in good faith, I have to wonder why you might believe these things.
Mining asteroids is a goal that makes sense. I can picture a future where spacecrafts are regularly sent to the asteroid belt and come back to earth with some minerals. Living on the moon does not make sense. There is nothing to be gained from humans living in a future moon base. Not any more than cities built in Antarctica, or in orbit with a constellation of ISS like satellites.
We won’t build a city on the Moon, nor Mars, nor any of Jupiter’s moons, nor anywhere outside of Earth, and we won‘t do this even if engineeringly possible, for the exact same reason we won’t build a bubble city inside the Mariana Trench.
Mining asteroids makes no sense whatsoever with any currently imaginable practical tech, especially not economically. The numbers for even the most basic solutions just don't work, and anything cleverer - like adding thrusters to chunks of metal and firing them at the Earth - has one or two rather obvious issues.
The Moon is interesting because it's there, it's fairly close, it's a test bed for off-world construction, manufacturing, and life support, and there are experiments you can do on the dark side that aren't possible elsewhere.
Especially big telescopes.
It has many of the same life support issues as Mars, and any Moon solutions are likely to work on Mars and the asteroids, more quickly and successfully than trying to do the same R&D far, far away.
Will it pay for itself? Not for a long, long time. But frontier projects rarely do.
The benefit comes from the investment, the R&D, the new science and engineering, and the jobs created.
It's also handy if you need a remote off-site backup.
Mining asteroids wouldn’t be for Earth - it would be for satellites or LEO or possibly even Mars, which is a lot closer to the Asteroids than Earth and may need some extra raw materials we don’t want to spend the horrendous cost of lifting out of Earth’s gravity.
The Moon has nothing to offer Mars explorers as everything will be different and solutions for the unique lunar conditions (two weeks of darkness, temperature extremes, moon dust, vacuum) do not apply to Mars at all. It’like saying living under the ocean is good practice for living in the Artic, but we should start under the ocean because it’s closer.
> Mining asteroids makes no sense whatsoever with any currently imaginable practical tech, especially not economically.
With current tech, it's practical enough to extract rocks from a rock. We've already done this on a comet, which I think is much harder to do. With current economics, not practical to fund such an endeavor, even if it was to haul back an asteroid made of solid gold. Regardless, we're discussing the far future, rather than current state.
If raw materials (again, an unknown) continue to become more scarce, it's hard to say what economics might support extra-planetary resource collection. What's for sure, is mining Mars will be harder than mining asteroids for water or metals, et al.
Mining asteroids makes no sense in the current economy with our current technology. But working towards engineering solutions which makes mining asteroids make sense makes sense (if that makes sense).
However, it is much easier to see us send robots to mine these asteroids, or send robots to the moon to build a giant telescope on the dark side (if that makes sense), then it is to see us build cities on the moon to build said telescope, and to mine those asteroids.
You see the difference here is that the end goal of mining asteroids are resources being sent to earth and exploited, while the goal of space settlements are the settlements them selves, that is some hypothetical space expansion is the goal, and that makes no sense, nobodies lives will improve from space expansion (except for the grifters’ during the grift).
> nobodies lives will improve from space expansion (except for the grifters’ during the grift).
Aspiring to goals and accomplishing them makes life worth living to a lot of people. Furthermore, humanity seems to have an innate drive to explore and learn.
Even to those left at home, it's inspirational to think that there are people who are taking steps to explore the universe.
Maybe it won't help anyone live but it will give a lot of people something to live for.
Why colonize Mars? Why not Moon?
The moon has no atmosphere. It is regularly hit by meteorites. Not sure it’s a very safe place to set up a colony.
Not like Mars is an amazing trip either, but the Moon is simply unsafe long term.
Plus Mars has a far more interesting history so the people living there can do more fun science than stare out at dusty grey rock.
As others have noted, the moon has significant limitations, in terms of resources and atmosphere. I do think it may have utility, not for anything we might consider settlements or habitats, but perhaps domed science outposts.
Mars has larger deposits of water and volatiles, which help with early space expansion.
You can start with a single Moon base but generally it isn't worth the mission control investment once you start to build out Mars.