I'm not expecting quantum computers to happen tomorrow. Are you at least capable of understanding that your claim about "28" being the largest number factored was wrong for the basic reason that Shor's algorithm only factors odd numbers?

There are some complications. Sometimes SCOTUS will rule against a specific motion or part of a ruling, or refuse to hear something, even as another piece of an appeal can occur. One can also file in some circumstances a petition for the Supreme Court to reconsider its own ruling. But this is basically never successful. You also can in some circumstances ask an appeals court to change a prior ruling based on a recent Supreme Court ruling, and this has some success. The circumstances where you can do this sort of thing are complicated. Jean Carroll's lawsuit against Trump is an example of this https://en.wikipedia.org/wiki/E._Jean_Carroll_v._Donald_J._Trump where Trump tried basically everything to try and prevent having to pay out even some ver low percentage strategies. Frankly, if someone other than Trump had done this, I suspect the court response would have been more negative.

Most of this is wrong or missing important context.

Scaling in terms of what? The other user was talking about energy usage. This is pretty obviously not the case. Most forms of quantum computers have to be kept incredibly cold, often in miliKelvin. If there were exponential energy use, then adding even in a few more quantum gates would make that energy total massive. But we don't see that. So by what metric are you claiming exponential scaling effort?

I don't know what "QC effort" is, but the obvious metric for this is just false. For example, Shor's algorithm, which is the algorithm for factoring using a quantum computer, has a number of qubits which scales slightly worse than the square of the number of digits https://en.wikipedia.org/wiki/Shor's_algorithm.

This is highly garbled. First, of all the record for factoring cannot be 28, and it cannot be 28 for a pretty obvious reason. Shor's algorithm only works for *odd numbers*. This is a very basic part of how the algorithm functions. The number you are thinking of is 21, not 28. https://arxiv.org/abs/1111.4147. Your point about that these have been "no-decision algorithm" is not completely accurate, but is approximately so. They did use a compiled circuit which used prior knowledge of the solution to optimize the arrangement of the qubits.

But there are also good reasons that factorization records have not expanded. First, Shor's algorithm has high overhead if you trace out the number of gates, and requires very high coherence to even start getting used. It isn't a useful metric of where things are going. This is like looking at the number of people going to space in the 1950s when no one has gone to space and ignoring that rockets had been steadily improving since the 1930s. The coherence time for quantum computers continues to improve. For a while it was improving at a rate of a factor of 10 roughly every 3 years https://en.wikipedia.org/wiki/Quantum_computing_scaling_laws#Schoelkopf's_law . That has slowed down in the last decade or so, so it is now improving by about a factor of 3 to 5 every 3 years. Similar remarks apply to other metrics like number of gates. And we know that if you can get error levels down and coherence times long then quantum error correcting codes https://en.wikipedia.org/wiki/Quantum_error_correction works. In particular, the threshold theorem https://en.wikipedia.org/wiki/Threshold_theorem says that once your physical error rate is low enough, the logical error rate can be as low as you want, regardless of the size of the computation.

It isn't unreasonable to have this gut reaction to how quantum computers are described frequently to the general public. But one important thing to realize is that they don't let you do magic parallel information processing. There's this way of describing them as "trying all solutions at once," or things like that. But a quantum computer cannot in general do that, since if one has all potential "solutions" one needs some way of making sure that the non-solutions cancel out. In that context. there are some problems where we can do that, such as factoring numbers via Shor's algorithm https://en.wikipedia.org/wiki/Shor's_algorithm. Even there, what is going on is more subtle than just trying all factorizations, as noted by the number of quantum gates needed scaling at slightly worse than the number of binary digits of the number. If it were just trying everything in parallel this would scale close to linear with the total number of gates.

A different related situation to look at which may also be useful is how we strongly suspect that NP problems cannot be efficiently solved on a quantum computer. In particular, that NP is not contained in BQP https://en.wikipedia.org/wiki/BQP, the set of problems which can be done efficiently on a quantum computer. If a quantum computer could do "magic parallel information processing" then we'd expect NP to be contained in BQP.

I have not had time to look at the Helion claims in detail, but I'm not sure why you think Aaronson should do so. His position a few years ago was that many current claims about practical quantum computing were being exaggerated (including very much the claims by D-Wave with their quantum annealing systems) but that in the near future practical quantum computers were likely. Aaronson has been careful to say that even as he's also been one of the people most vocal in arguing that the burden of proof is on skeptics of quantum computers who would claim it would never be possible. It may help to look at Aaronson's recent post, for example what he says here https://scottaaronson.blog/?p=9979.

I'm reasonably confident I have a pretty good understanding of what doing math is like given that I'm a published mathematician. Heck, I'll be a little egotistical and just to my own recent paper directly https://cs.uwaterloo.ca/journals/JIS/VOL29/Zelinsky/zel14.html. But this also isn't terribly relevant.

LLMs have been very good at doing searches. But they've done a lot of things that are pretty obviously not mere searches. For example, in the case of Erdos 1196, the LLM solution constructed a Markov chain using a weighing via the von Mangoldt function https://en.wikipedia.org/wiki/Von_Mangoldt_function. That wasn't in the literature at all. In the case of the Jacobian Conjecture for n=3, the AI constructed a complicated polynomial which was certainly not in the literature, since the entire reason the polynomial is of interest is as a counterexample to the Jacobian conjecture.

It is true that a Fields Medalist does not imply expertise in other areas. But these are people who have also spent time working with these systems. And understanding *how* an LLM functions in this way isn't as important as them being able to see what the system can do. One can evaluate that a black box is useful for specific purposes without knowing anything about how the black box functions. And this is not just the Fields Medalists, this is almost the entire community of mathematicians, including people like the one I linked to who is explicitly very anti-AI. This doesn't mean that they are necessarily correct by itself. But it should be a pretty serious warning that these are not mere "morons" here and that you possibly are just mistaken about what capabilities these systems have.

No one needs to believe in magic to see that these systems are highly useful to do mathematics, At this point even the mathematicians who are unhappy about AI use and are advocating mathematicians not use AI are more than willing to acknowledge that in practice these systems are able to do a lot of math, but think that other ethical concerns mean they should not be used. See e.g. https://proofsandprompts.com/2026/08/09/the-ai-dissenter-viewpoint/ . At a certain point, it should occur to you that if nearly the entire mathematical community thinks these systems are powerful and useful, and that includes Fields Medalists like Terence Tao, James Maynard, and Tim Gowers, that maybe the people you disagree with are not "morons."

Not what is going on. While we initially thought that having these systems try to generate Lean proofs or proofs in another formal language would be a good approach (Deepseek tried that for example), almost all the success has been with models reasoning in natural language and only after they've constructed a natural language proof has there then been attempts to solve the problem in Lean. As you can see here https://cdn.openai.com/pdf/1625eff6-5ac1-40d8-b1db-5d5cf925de8b/unit-distance-cot.pdf there's no use of Lean. And in that case, there could not have been Lean usage, because while the proof was later verified by mathematicians, it uses a whole bunch of algebraic number theory ideas which were not yet defined in Lean when it ran. You can look at other traces as well. The brute force here is not randomly generated constructions but rather trying a whole bunch of different methods and then seeing what works. The Unit Distance conjecture reasoning chain looks pretty close to what I'd expect to see if a mathematician kept a very careful diary of all their attempts on the problem over a multiyear period.

This is not accurate. While it seems better at constructing counterexamples, many of these results have been straight up proofs. Erdos 1196 is one obvious example here https://www.erdosproblems.com/1196 and the lower bound on the percentage of non-trivial zeros of the Riemann zeta function which are on the critical line is another https://www.anthropic.com/research/riemann-zeta. The other thing to keep in mind is that some things people have been calling "counterexamples" are themselves highly sophisticated constructions which essentially amount to difficult proofs in their own right. The Unit Distance Conjecture and the construction of a non-sofic group are both examples of that sort.

The following is a non-exhaustive list of major successes in math by AI, and then my own speculations/predictions. Note that all of these have happened since

Erdos 1196, completely solved by the AI. https://www.erdosproblems.com/1196. This was the first major unambiguous success by an AI where the solution was clearly not the AI just finding something from a really obscure part of its training data. The method involved a clever construction of a Markov chain on the natural numbers with the chain weighted via the von Mangoldt function.

The Unit Distance Conjecture. This is a really cool and easy to understand problem so I'm going to take a moment to talk about what it is. Suppose you want to put n points in the plane and you want to maximize the number of points which are exactly distance 1 away from each other. Note this is the same as just picking some distance k and making lots of points all distance k from each other and then rescaling. Now, the most naive thing to do is put one point in the center of a circle and then put n-1 points on the edge of that circle. That gets you n-1. But if you instead put your n points in a grid, and make it as close to a square as you can get, then you can get around a constant times n^2 points all the same distance. The conjecture was that this was close the best possible. In particular, that for any epsilon>0, you could not do better than O(n^(2+epsilon)). The AI constructed a clever tower of fields to disprove the conjecture. This was previously discussed also on Slashdot here https://science.slashdot.org/story/26/05/21/0351218/openai-claims-it-solved-an-80-year-old-math-problem

The Cycle Double Cover Conjecture https://arxiv.org/abs/2607.15399 was a long-standing problem in graph theory. Curiously, the problem's solution is something that a skilled undergrad could have plausibly come up with but apparently no one did. The proof uses some standard graph theory tools and then connects it to some ideas from linear algebra, but linear algebra is already pretty often used in graph theory.

The Jacobian conjecture for n=3 https://en.wikipedia.org/wiki/Jacobian_conjecture This is a conjecture that was major enough that I've mentioned it when I've taught multivariable calculus. n=2 is still open. This is one where there may have been some amount of human input but we don't know.

OpenAI released a set of 10 major problems here https://openai.com/index/ten-advances-in-mathematics/. I'm not going to go into all 10 of these in detail, but I will note that both the Ramsey problem and the sphere packing problem are not too complicated to understand. I had also given some thought to both of those problems and got nowhere. The sphere packing problem was close to things I thought about in grad school, and the Ramsey problem is close to some things I've been thinking about it. Of the other 8, 6 we're ones I was familiar with simply because they were well known. For the other 2 of those, I've discussed them now with people in those subfields and confirmed that OpenAI's description of these as major problems is not unreasonable. We don't have direct confirmation that there was minimal human input here.

But by far the biggest breakthrough was recently when Anthropic released a proof which increased the percentage of non-trivial zeros of the Riemann zeta function [on the critical line from around 41% to 67.2% . https://www.anthropic.com/research/riemann-zeta. A clarification: You don't get to prove the Riemann hypothesis when you get 100% on the line. The Riemann hypothesis is that all non-trivial zeros live on the critical line. But that's stricter than 100%. An analogy that may help: There are infinitely many perfect squares. But the perfect squares make up 0% of all positive integers.

Finally, we have just two days ago, Claude was able to find an elliptic curve of rank at least 30 (and assuming some highly plausible standard conjectures, exactly rank 30) https://elliptic-rank.icarm.cloud/curve/273. This is one where we don't know how much human input occurred and how much was just the AI. This was done by Levent Alpoge and Ava Howell with Claude after Alpoge was challenged to do this by Alvaro Lozano-Robledo. There's an open question of whether there are elliptic curves over the rationals with arbitrarily high rank. For a long time, the answer was suspected to be yes, with John Tate back in the 1970s predicting it was unbounded. But more recent heuristics (especially a very careful model introduced by Jennifer Park, Bjorn Poonen, John Voight, and Melanie Wood) have suggested it is likely bounded. But those heuristics would also have made the outliers we seem to be seeing the last few years of curves with rank 28, 29 and now 30 to not be likely. This calls those heuristics into question. Right now, there's no consensus in the field on this question, and this discovery is likely going to push people more into the direction of thinking it is unbounded. Also, this is a good example why people should not pay too much attention to my own guesses about things, because I'm previously on the record for thinking it was likely that none of rank 30 existed.

Now, with all of that, what is going to happen? I'm not sure. Earlier I was overly pessimistic about AI systems, saying even just around a year ago that these systems would be highly limited in what they could do and that this was likely a fundamental structural issue about how LLMs are constructed. Obviously, I was wrong there. And there's a temptation to go far in the other direction and expect that AI will be better than humans at math soon. However, I suspect that if that does happen, it will mean we will have made some fundamental new breakthroughs in how these AI systems function. In particular, at least right now, the AI are very good apparently at connecting ideas from disparate areas of math, and of brute forcing a lot of things, in a way pretty close to frankly what human mathematicians often do. But they aren't apparently constructing really deep new theories. The AIs also as they stand seem pretty weak at making novel definitions or conjectures. But many mediocre mathematicians (like say myself) will go through entire careers without constructing any really deep new theories. So we may end up with a situation where the Terence Taos, Peter Scholze or Maryna Viazovska type will still be doing good work, but the rest of us will be functionally irrelevant.

On the other hand, for other tasks where AI got to be close to as good as human, the amount of time it took for the AI to get superhuman once it got to being slightly below the best humans was pretty short. Take chess for example: in 1988, HiTech was beating some grandmasters but would consistently lose to the strongest ones. A few years later, Deep Blue beat Kasparov. And now you can get software which will beat the best humans that can run on a laptop. Similar remarks apply to computer Go. Chess and Go are much more narrow problems than math, but it seems likely or at least plausible that the same pattern will apply here. On the other hand, at least where making new definitions and conjectures are concerned, the LLM AIs are weaker still even mediocre mathematicians. So we have that at least going for us. But this may be also that the AIs just haven't gotten trained well yet at those things. We've had machine based definitions and conjecturing in a non-LLM fashion since the late 1990s (with some work done by Simon Colton on this subject). And right now, LLMs seem not incredibly impressive at definitions and conjectures compare to some of the experiments from around that era. Whether this is a fundamental limitation of the tech given the way it uses training data is something we don't know yet.

Regarding 1. In fact, post-pandemic traffici levels are as high as they were before the pandemic. In fact, traffic levels were already close to normal by October of 2020 in London, while the pandemic was still going on, in part due to people being less willing to take public transit. See https://www.london.gov.uk/who-we-are/what-london-assembly-does/questions-mayor/find-an-answer/changes-london-road-traffic-march-2020. and https://fleetworld.co.uk/uk-road-traffic-back-at-pre-covid-levels/. World-wide, we've seen pretty similar trends. https://inrix.com/blog/2023-global-traffic-scorecard/.

Re 2 and 3: Aside from that being highly exaggerated at what happened, it also isn't relevant to evaluating this study or making this study become "dishonest propaganda." If your claims are accurate, then they would be reasons to see there as being tradeoffs. But it wouldn't be a problem with this study or this study's claim. I'm not even sure what point you are trying to make with point 4. Are you claiming that the reduction in lung issues is due to less use of boilers due to climate change? While climate change is having an impact, it isn't remotely large enough yes to have had that big an impact on heating that way. It isn't what is going on here.

This isn't about AI but about Ukraine. Aside from this being drastically off topic, it is worth articulating the many differences here. The AI companies are taking individual books and scanning so they have training data. They are making use of them. And they are doing so largely with older used books which would often be thrown out anyways. Used book stores throw out old books all the time, as do libraries. Moreover, the books being destroyed for digital scanning are often things which are not just that important, things like random sword-and-sorcery novels from 1975 and car manuals. This is in drastic contrast to the wide-scale deliberate destruction of books and book infrastructure by Russia; they are trying to destroy Ukrainian culture and the ability for children to get an education. In some cases, nearly entire print runs of a book or entire print runs have been destroyed by what Russia has done here. The situations aren't remotely comparable. So instead of making complaints about AI again, maybe show you actually care about books by making a donation as they've asked?

Yes, people are using AI to generate slop, and do things like generate kitten pictures. But there's a lot of extremely productive use. We have a lot of people using it for coding. We have mathematicians using them to prove long-standing open math problems. There are a lot of positive uses here also.

Noise is largely due to gas turbines or diesel generators running. That's again the sort of thing where reasonable regulation can help out. Some other noise claims are also just not plausible; there have been examples of noise complaints about completed data centers which hadn't been turned on yet for example. The heat claims are generally not plausible. If you take the claims seriously, they'd have to be generating far more energy than they are. Meanwhile, people don't object to large parking lots which are constantly taking in sunlight and converting to heat. I'm also not sure how they engage in "destruction of neighborhoods" any more than any other building project.

Not really. Heavy duty training involves intensive energy use. And one of the genuine problems with AI data centers is the electricity use can be very high or very low with minimal warning to the grid operators. This is part of why it pushes up electric costs. I'm not sure what evidence you have that any new data center now is being built for AI. Many of the data centers being objected to are small "Edge Data Centers" used primarily to reduce latency for video streaming, video games, and other activities which require low latency https://www.maysteel.com/why-maysteel/resources/blog/types-of-data-centers-how-do-you-choose-the-right-data-center.

So, first putting solar panels on is a step. It doesn't do everything, but it is a good step. It also has the nice advantage that if there's a serious bust in data centers, there will be already solar panels ready to be connected the other way for the grid. But you are also underestimating the effectiveness of solar panels, for a few reasons. First, almost no data center is used at 100% capacity. The solar panels keep producing even when server racks are getting little use. Second, data centers require corridors for humans to move, office spaces for humans who work there to stay, along with bathrooms and other needed things. Typically, actual server space takes up around 20-25% of the actual space. Now, that's not quite as good as it seems, since data centers are often 2 or 3 floors. But overall, the ratio is still better than you state.

But if you think that's not enough, then you can make it tighter. You can do things like require that their parking lots have solar panels too. Or you can even just require a specific percentage of their power be handled by solar, and then let them figure out the details, including attached solar arrays if needed. But these are all conversations we can have, and far more useful than just saying "No!" very loudly.

There's obviously a massive large-scale backlash here, and it is pretty bipartisan. Part of it is concern about water use (which is largely overblown) and part is concern about increases in electricity cost (which is much more reasonable). But it is clear that two related things are happening: 1) People are reacting this way to all data centers, whether or not they have anything to do with AI. 2) People are blocking data centers rather than trying to take reasonable steps to make them manageable with their concerns, like requiring that they have solar panels. There's room for compromise here, but we're not getting it.