Is mathematics over, or just graduating?
docs.google.com
docs.google.com
I am pretty sure in the next couple of years we'll see more evolved sledgehammers [0], maybe with LLM-like tech playing a role in encoding theorems, that will allow us to prove deeper than we can do manually, then help with the archeology and extraction of learnings from observing the 'trace' of certified proofs. So instead of explore -> understand -> solve, we'll get solve -> compress -> understand.
[0] e.g. https://isabelle.in.tum.de/website-Isabelle2025/dist/Isabell...
https://www.nytimes.com/2026/08/26/technology/bill-gates-ai-...
He was anti: - Machinery replacing handicraft (skill disappeared and poverty followed) - Labor-saving machines replacing human workers (workers became expendable) - Doctors replacing consequences (people lost self-control) - Railways replacing constraint and slowness (harm spread faster)
But he loved the sewing machine.
Gandhi’s test for whether to accept tech-driven change was whether it satisfied primary needs. Interpretation of those needs was certainly not universal. As he wrote: "...it is not the primary need of man to traverse distances with the rapidity of a motor car. The needle on the contrary happens to be an essential thing in life, a primary need.”
That's the airplane in this post. Excitement or fear of the AI airplane is either judged as helping mathematicians (sewing machine) or doing something not quite right (the above machinery/doctor/railway)...
He also seems backwards on the outcomes of things like what you listed. Those things all raised people out of poverty and serfdom.
It’s interesting, though, to see someone as revered as Gandhi having to add epicycles to his beliefs after having seen the other side of it.
I think ghandi speaks much more from a tone of trying to make you think differently, not necessarily being right or wrong, but making you accept or reject him by making you contemplate for a moment his perspective.
In this case you reaffirmed your view of the world that technology is a vector of freedom, but what is the circumstances of that technology? Because when you think of ghandi as an Indian, and that Indians suffered a humanitarian crisis in the hands of the English empire, I can't help but wonder if the colonizers just created the poison than sold them the medicine.
So from his point of view he's right to reject technology that are colonization instruments, but if that same tech would otherwise take place in the world from the needs of the people, then it would be an acceptable technology to adopt. And this argument is timeless, because how many current tech, the social networks, the language models, are actually being used as colonization tools today, by influencing elections or by spreading misinformation? We don't need to go that fast, not when the price to pay is our freedom.
https://www.mkgandhi.org/ebks/hind_swaraj.pdf
I posted more in another comment on this thread but here’s a PDF of his book. Gandhi says in the “A Word of Explanation” section that he would welcome the natural destruction of hospitals. How is that timeless?
Do you want to see beyond the mere meaning of words or do you just want to be a good thesaurus? Because we have the luxury of access to accumulated knowledge. Here I'm appropriating byong chul Hans lingo, but ghandi didn't have access to this vocabulary at his time. That's maybe the best he would have done, and it's a figure that I think is worth it giving the benefit of the doubt and reading him in good faith, but that requires lifting the lenses of skepticism for a little while.
For example he says it would be a good thing if all railways and hospitals ended up being destroyed.
If all hospitals ended up destroyed what would happen to the people who need treatment for bacterial infections? Or need treatment for heart attacks? Or need dialysis? Or get necrotizing fasciitis and need surgery? Or need cancer treatment?
And if all the railways and factories and such are destroyed how do you manufacture medicine and antibiotics and other devices to save lives?
I understand the things I mentioned just now weren’t around in 1908 when Gandhi wrote that essay but who knows what advancements modern AI could bring that we can’t imagine now?
https://www.mkgandhi.org/ebks/hind_swaraj.pdf
This is a PDF of that Gandhi book. Under the “A Word of Explanation” you find Gandhi saying he would welcome the natural destruction of hospitals and railways.
In this respect Gandhi was essentially on the same page as RFK Jr. He thought people should try to lead healthy lives - diet, hygiene, exercise etc. - but ultimately that nature should take its course.
https://en.wikipedia.org/wiki/Flag_of_India
It previously incorporated the symbol of Indian independence and nationalism, the charkha or spinning-wheel. This is not a sewing machine. A spinning wheel is part of a machine used to spin textile fibers into yarn.
https://en.wikipedia.org/wiki/Spinning_wheel#Charkha
Gandhi loved the spinning wheel, but he was citing the Singer sewing machine specifically as a "useful device... with a romance" that was perhaps secondary to his love for the actual spinning wheel, which is stated clearly elsewhere in the same document.
For thousands of years, women were formerly respected and revered for their spinning talents and skills, in pagan and Christian societies alike: such as the Greek goddess of Wisdom, Athena [Minerva] whose symbols included the distaff; this same distaff was also synedoche for the matriarchal aspect of any family, and mentioned in Proverbs 31, as a tool of the industrious and devoted wife.
It is no accident that, in Romance languages, "machina" is a feminine noun. https://en.wiktionary.org/wiki/machina#Latin
- First, airplanes in the first 30 years were extremely rough and dangerous. When Dole hosted a contest for the first transpacific flight in 1927, 6 out of 8 teams failed. Ten people died as a result.
- Second, people made a lot of silly predictions about this new technology. The New York Times extrapolated that people would soon start commuting to New York using personal airplanes.
- Third, people made capital investments based on these wild guesses. For example, the architects of the Empire State Building installed a huge spire to prepare for the upcoming age of airborne traffic (intended to be used as an airship docking station!)
The bottom line is that predicting the trajectory of a new technology is often impossible. As a society, we are currently in the experimental phase of LLM-based AI, testing all the possibilities one by one. Who knows which path leads where.
1869 Scientific American [Major mainstream American science publication]
“When man can make a structure as strong, as rigid, as elastic, as light in proportion to bulk as a goose quill, the problem of flight will be nearly solved.”
https://www.scientificamerican.com/article/is-a-flying-machi...
1888 Joseph Le Conte [Prominent American scientist and UC Berkeley professor]
“A flying-machine… is impossible, in spite of the testimony of the birds.”
https://en.wikisource.org/wiki/Popular_Science_Monthly/Volum...
1902 Lord Kelvin [President of the Royal Society and one of the most prominent physicists of his era]
“No balloon and no aeroplane will ever be practically successful.”
https://zapatopi.net/kelvin/papers/interview_aeronautics_and...
1903 The New York Times [Major mainstream newspaper, 69 days before the Wright brothers’ first powered flight]
“Nature is more successful in applying the law of compensations to the correction of errors of design or development than man has ever been or is ever likely to be.”
https://en.wikisource.org/wiki/The_New_York_Times/1903/10/9/...
> Actually we bet the other way around. History always under estimated technology.
To throw a few more nuggets, being pessimistic about aviation actually paid off. As Buffett famously pointed out in his 1999 talk, there had been zero money made from the aggregate of all stock investments in the airline industry in history.
It was a monumental technological success, but turned out to be a financial bottomless pit.
There may be a finite point at which we've proven and codified everything but the most inane meta-meta-meta-maths though.
It's more of a theoretical corpus for the sake of this discussion. Just think of it as "all math humans have discovered/invented".
> Are all theorems or axioms useful?
Definitively no! Check out the first incompleteness theorem proof by diagonalization. The whole proof is based around generating new "facts" that are completely useless and uninteresting, outside of their utility in the proof itself.
It's actually a great question: to what extent do the incompleteness theorems actually apply to stuff that isn't silly and useless! The same goes for a couple similar proofs in other domains. An important one in Computer Science is Turing's proof that the halting problem can't be solved. There's at least one other important proof that is similar but I can't remember it at the moment.
To massively oversimplify, they can be boiled down to variations of:
> This statement is false.
Which proves that allowing self-referential statements can create a statement that isn't true or false. But they all raise the question: how often does this actually happen with statements we actually care about?
Maaaybe.
On the other hand, proofs was the main thing that distinguished good mathemeticians. It was the way of sorting out who should be a math professor.
It's probably a bit of a blow for math professors. For almost everyone down the "food chain" though - who cares? I'd argue that mathematical thinking is still very useful, even if you offload the heavy lifting to power tools. Math teachers still need to exist. Math tests probably still make sense - you want to incentivise learning, and test to see who the brightest or most studious students are. Math will still be used for physics, economics, etc - no one actually cared about most of the new proofs anyway, it was just a fun tournament to pick math professors with a side effect of generating a bit new knowledge.
At some point, maybe humans don't need to make decisions. We can just sit in VR racks being fed and entertained by AI like we're in The Matrix or Wall-E, as they fill some age-old instruction to make the humans happy and healthy. But until then, humans still need to make decisions, and mathematical thinking makes us better at that.
Kind of an odd statement to make. The ICCF (international correspondence chess federation) lost a good chunk of players but still maintains a decent base of active players. There's also many orders of magnitude more playing correspondence chess in local federations or on online sites which all offer correspondence like time controls.
Computers definitely changed it, because now the best players are simply those who are best at dealing with chess engines, but it's still very much alive. The games played there also regularly studied by high level over-the-board players because it provides as close as we can currently get to the objective assessment of things like openings.
For instance it's extremely interesting to see things like the fact that the King's Indian Defense does okay in correspondence in spite of all engines insisting white has a practically decisive edge after about 10 moves in most lines.
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Of course there's an interesting nuance that 'back in the day' a good correspondence player was almost certainly a good chess player. Whereas now a days its a different skillset altogether. The currently top rated correspondence player is a 20 year old who started playing correspondence 2 years ago, with no title and no rating in normal chess.
And I think we may see something similar in math. I'd wager that, at least in the longrun, the skill set required to effectively apply LLMs to mathematics may have less overlap than would be expected with the skillset to make progress in mathematics in the 'traditional' way.
I sketch, really badly, and I enjoy the process of doing that. The fact that an AI can make a better picture is irrelevant. Photography can do the same. I still enjoy sketching.
But I don't have to make my living from sketching. If I did, I would be more concerned and looking at how that changes.
I can see how both things can be true: mathematics is not "over", but professional mathematicians are looking at the skillset that they have carefully cultivated over their career and realised that those skills are probably not what will serve them best in the years to come. Tomorrow's mathematicians are not probably not going to be the same sort of people as today's mathematicians.
Same as us software devs. Software development isn't "over", by any means, but Software Development as a profession is changing extremely rapidly and the skills that got us this far are probably not the ones we need for the next decade.
RSI will mean that you'll be in the VR rack sooner than you might think, assuming we unlock the good ending.
Observational astronomy is the most easily automated field, survey telescopes already scan the skies by automatic schedules.
https://rubinobservatory.org/explore/how-rubin-works/lsst/ca...
The change happening now is already significant and isn't slowing.
I am not predicting anything, but it would not be surprising if in 10 years (or less) machines were pushing mathematics forward at a rate that left mathematicians the role of finding ways to translate results to other humans. And it wouldn't be surprising if machines were better at that than mathematicians too.
The faster things go, the more people's horizons seem to shrink. Things are moving fast enough that nobody on Earth knows what 10 years from now will look like.
- if AI is so powerful that it is a superset of human capabilities in that it can not only solve questions but also come up with useful abstractions, then we just simply don't need mathematicians!
- if AI is not that powerful but can automate in-distribution tasks, then that allows mathematicians to explore concepts much faster. Overall positive thing for mathematics
We are obviously in the second camp (at least for now). Your job as a mathematician is to offload as much in-distribution task as possible to the LLM and just focus on the rest.
Let's fight that by making OP sound silly. Without citing personal experience. Using vague arguments (Sorry dang!!)
Modern jets turned commercial pilots into bullshit jobbers! But drones turned jet fightin' into a bullshit job! (And you can argue some Ukrainians should do it in a biplane--- turn subsonic drone piloting into a bullshit job!
https://youtu.be/gBxQixJfOzk )
In aviation, it's the parasailers and the wingsuiters who find most meaning in what they do. That's because planes can't really do what they do. Planes can never do what we really want to do... Which is... Fly like a bird.
LLMs will probably never help us navigate the universe like a god.
(I'm still hoping that some future biotech will help us fly like a bird, etc, ... That does sound much more likely than Language Light and Magic (LLM))
Have you utilized an airplane recently?
- Only a tightly regulated industry permits you to buy a single-use ticket before the brief upheaval your personal privacy. This isn't a one time scan, it's one for each ticket generally!
- You don't ride airplanes every day, it's a tightly controlled portal to far away. Maybe some "very special" people have job subsidized commuter flights daily, but they're a minority.
Are we suggesting, and supporting, that commercial AI unfolds the way commercial air travel has? Yikes!!!
This is not the future of commercial AI. We need to toss this airplane analogy and use something more accessible.
Radio.
Radio changed the world 100 years ago, now a significant portion of the human population is armed with at least one radio device they use daily.
Side ponder: developing the applications of radio arguably led to a slew of intermediate and sibling developments resulting in the machines we used to build machine learning. Prove me wrong!
If you're still with me, then we can attempt to extrapolate together: if 100 years of radio applications cumulated into the smartphone, then, what will 100 years of AI accumulate to? What is the "smartphone" of AI in the future?
Electricity is to the SoC pocket or desktop computer. Radio is to the machine learning and LLM. Smartphone is to, what?
Complete the puzzle and win the prize of enlightenment! The radio analogy is the best way to speculate and extrapolate about our future.
Another automation if you don't like the radio analogy is refrigeration. Remember all the industry of ice workers that were displaced by the refrigerator? What happened to them?
What about all the immigrants that built railroads in America during the 19th century? What happened to them? They settled in the towns that grew the result of the railroad work and hopefully found purpose using the railroads they helped build. You know, if they didn't die first.
All the programmers and engineers who were laid off "because of AI" need to settle in one of the "AI railroad towns" and get to work. The railroad is built, the next one is built automatically, so make your living using the railroad you helped build.
FWIW I have one of these railroad job towns, but, I didn't help build the first railroad.
Sounds revolting. Everything being turned into the industry is soulless and uninspiring. I know a lot of people worship this future but I for one hate it. I for one don't see the point in creating this intellectual power where there will always, always be someone to abuse it simply because of our instinct to be greedy.
Maybe your AI prevents the greedy AI from achieving their goals.
"Good guy with an AI"
I'm not sure there's a choice.
Duh? There is nothing left in the world to be excited about, it has been completely and totally strip mined for profit. We are sitting here in a post discussing the commoditization and industrialization of MATHEMATICS and you are confused that people are 'pessimistic'?
> But it will not last. Mathematics is about to graduate. It will grow into a full-scale industry and become the true infrastructure of all scientific and technological progress
If your thesis is "that will not actually happen" then say that instead.
It's okay to acknowledge things are worse than they used to be. It informs on the direction that we need to go to improve things. Not everything needs to be forwards just because some billionaires have willed it onto the rest of us.
It's impossible due to the moderation fallacy and the "Law of Averages."
Imagine if you bought up every single movie studio in the country, locked everyone else out of the market, and exclusively made terrible movies - and I mean exclusively, like you review them all before release and if you see any glow of inspiration you have it cut, the people involved fired, and you put their friends on a list. Lets say you didn't do this all at once, but gradually - not that gradually, but over let's say 15 years.
Movie reviewers would continue to say that about half of movies were good and about half of them were bad. Reviewers who said that all movies were bad, and explained patiently why they were bad so well that you used their thoughts to come up with even worse movies would be pilloried as hopeless cynics and snobs.
"Oh, it's so easy to look down your nose at everything." "This is all just poisonous snark."
A lot of people just pride themselves on being in the middle. They like to hate everyone who is too happy or who is too angry, and they define those things by who surrounds them at any given time. Reality to them is the mean of all known claims. It's the sum of all personality conflicts, and also somehow necessarily zero. And that's where they're comfortable, claiming to be zero.
If you see a bunch of things in the world are going well and you're happy about it, and they're absorbed in a media environment that at that time is determined to have them believe the opposite (maybe they want to fight a war), those are the same people that will tell you you have your head in the sand and you don't understand that the world is burning. If you reply that you don't believe that the world is burning, they'll say "you're such a cynic, you think everyone is lying to you."
It's just sort of a mindless gradient following to death. Never defend looking at the world accurately, and never get involved in conversations with people who argue "oh, please!" or "come on!" and try to turn a conversation about reality into a conversation about your optimism or cynicism. They're not trying to figure out the world through reason, they're trying to figure out what is considered "reasonable" within any particular social and cultural environment.
So how was it in the 90s? Well some things were better, but other things were worse. Rich people may do bad things but they do good things. You're upset about the bad things they do, but you don't appreciate the good things enough. And you say the 90s were great, and of course they were, but I wouldn't trade what I have now for the 90s because you're forgetting how bad it was. It's the universe as a Buttigieg speech. No references to actual events or conditions necessary. Just the mean, give or take, more or less.
I believe in people, not tech. Tech past a certain point just makes things worse and always goes beyond a point of diminishing returns where its innovation becomes a mere arms race.
Other sites beckon.
Let's fight that, not by rejecting tech, but by making OP sound silly. Without citing personal experience. Using vague arguments. Repeating oneself. (Sorry dang!!)
Modern jets turned commercial pilots into bullshit jobbers! But drones turned jet fightin' into a bullshit job! (And you can argue some Ukrainians should Santos-Dumont it in a biplane, turning subsonic drone piloting into a bullshit job
https://youtu.be/gBxQixJfOzk )
In aviation, it's the parasailers and the wingsuiters who find most meaning in what they do. That's because planes can't really do what they do. Planes can never do what we really want to do... Which is... Fly like a bird.
LLMs will probably never help us navigate the universe like a god. That's what math promised us!!
(I'm still hoping that some future (bio)Tech will help us fly like a bird, etc, ... That does sound much more likely than Language Light and Magic (LLM))