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 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 (see my other unedited comment in response to a geometer)
It was probably not intentional, though, and might trigger noone besides snobs like us :)
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...
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).
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.
> 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.
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.
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 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
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)
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.
(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
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.
what progress has been made on understanding Teichmüller theory in the past decade? not a lot...
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.
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.
Or roughly the cost of producing Star Wars IX: The Rise of Skywalker. Kinda wish that money had gone to string theory.
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 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.
What would you have them work on? Predatory social media platforms that sell your data to advertisers and commoditize you.
Your entire life summed up probably costs $500k-4m (On average, depending). Some bean counter could probably argue that it isn't worth it.
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
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»?
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.
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?
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.
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.
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.
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.
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.
> 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 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.
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 think the subject at question here is mathematical truth, not "mathematics" whatever that means.
> 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.
>Do you just mean that we must assume something to get the ball rolling
They're called "axioms"
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.
>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".
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.
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.
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?