> The two big discoveries both came from the negligible handful of mathematicians working at OpenAI/Anthropic in spite of many orders of magnitude more mathematicians using them outside of the companies
Well, mathematicians not working for Anthropic/OpenAI are heavily disincentivised from reporting that their discoveries were made using AI. If e.g. the idea that resolved the Mahler conjecture came from AI, it's not like we'd ever know.
> This does not sit well with personal experience and I wonder if it is just one of these questions of AI people being unaware of the level of skill that exists in domains they think have been automated.
I suspect the difficulty here lies more with your reading of the quoted sentence. British grammar school education, for all the years it devotes to the enterprise, does not always succeed in teaching reading comprehension.
You seem to be treating two rather different propositions as though they were one and the same. If text in general is not sufficiently information dense to support decoding some _arbitrary_ signal of provenance, that hardly establishes that no _specific_ passage can carry distinctive markers of provenance.
For example, you can recognize the unmistakable cadence of the California undergraduate. Impressive. Alas, even in your own example, when your British friends are "giving themselves away", you resort to an external signal, beyond the text, to determine provenance! That is, unless the text itself is claiming that its author is British (like the bots who claim they're John Horsetrader from Arkansas oblast).
When you have to decide whether a 2010s era SAT essay was from a SAT prep book author or an LLM prompted to write such an essay, you will struggle to distinguish one from the other. Not all texts have provenance signals. This is what it means for text to simply not be information dense enough to be able to decode some arbitrary signal of provenance from it.
Going from "current mainstream culture is not perfect" to "there should be more experiments in alternative ways of life" requires the assumption namely, that the average experiment is more likely to improve matters than to make them worse. When these experiments go awry, they hurt not only the participants of the experiment (who are themselves often children or others who have no other choice), but also everyone standing nearby.
I don't think the current nuclear doctrines are anywhere close to perfect or best possible. There is surely room for improvement. But I vehemently oppose more countries innovating on nuclear doctrine, because the average outcome of innovation is likely to be worse than the current equilibrium, for bystanders and innovators alike.
Medieval Europeans knew that the fallow-field system was imperfect, but many simultaneous experiments on alternatives would have led to famine, not viable alternatives. Careful experimentation in some monastery gardens is a good thing, but wagering everyone's supper on untested ideas isn't.
The same applies to our own civilization. Western capitalist culture has flaws aplenty. But this does not mean we should throw open the gates to every, or even any, alternative group that comes along.
> What happens to the whole AI value proposition when instead of it being a way to pump out lines of code for crazy cheap it becomes a way for each line of code to become vastly more expensive than it was before?
I would consider that a good thing. Businesses should not able to make money by writing terrible, insecure code.
Anybody who prevents them from doing so surely deserves to capture the additional value created.
Astrology didn't really freeze. The apparent position of celestial objects was important for mundane reasons such as navigation until the 1970s, and the people who compiled nautical almanacs kept doing astrological fortune telling on the side for extra cash, using the very same math for both. Kepler himself cast horoscopes as a side hustle, including for the Holy Roman Emperor, using the same (back then cutting edge) techniques and data that he used for serious work.
Modern ascendant calculations are "correct", in the sense that they'll match what you see in the sky. Apps like AstroMatrix work from NASA data.
As you rightly observed, it doesn't make a difference when it comes to foretelling the future. But it does make a difference if your aim is to make money selling it. The big spenders on these applications often care a great deal about getting the particulars just so.
A horoscope is a fine mixture of fortune-telling bullshit and verifiable astronomical facts. The latter have the form of "where the celestial bodies could be seen at the hour of the client's birth", or "does Jupiter currently appear to be moving forward or backward in the sky".
The average Visual Basic programmer and the world's best mathematician are going to be about equally good at writing the fortune-telling part, but the mathematician will have a much easier time getting the factual part right.
The starting point of casting a horoscope is calculating the apparent locations (this means "where you would have seen them had you looked up there and then") of a whole bunch of celestial objects at the time and place where a particular person was born.
You won't (write software to) do that without knowing a whole bunch of linear algebra and ODEs.
The fortune-telling part is not what needs the math degree.
Astrology is a mixture of factual verifiable information (such as apparent positions of celestial bodies at the time and location a certain person was born) and random baseless divinations.
The "whale" users who account for a disproportionately large percentage of an astrologer's revenue tend to know the factual information surrounding their birth fairly well. An app/astrologer who doesn't get these facts right, even for a handful of clients, will get a bad reputation fairly quickly.
I reckon the same principle would hold in cultural bubbles where reading tea leaves is a customary means of divination. If the client recognizes recognize black tea, but the fortuneteller insists it is rooibos, there won't be much trust in the rest of the prophecy.
Advertising that the horoscope shop uses Haskell is actually a solid business idea. It pre-filters for the sort of dev who will be able to do the math.
> If you consider the "discipline" to be mathematics broadly, even this level of knowledge is actually quite uncommon.
Fwiw, I am in full agreement, and it's commendable when people do have at least a basic understanding of intuitionistic logic.
My remark was simply that the majority in the field don't make this (barely 16 years old) distinction between "proof of negation" and "proof by contradiction", and it has come to be associated with a more introductory or superficial understanding. This is not to suggest that everyone who uses Bauer's terms has a superficial understanding, e.g. Bauer and Escardo are top tier and certainly use it a lot.
I also don't say that this distinction is unique to Bauer. I'm saying he invented and popularized it (I was in fact there in the 2010 thread where it was invented).
With that out of the way:
> It's simply a way to square two widely-held beliefs, even amongst professional mathematicians [...]
>
> I assume you would prefer to correct the first "misconception", by clarifying that only proofs of positive statements that assume the negative are non-constructive.
Well, yes, one should correct the first widely held belief, because it is a baseless misconception (or rather, was a baseless misconception under the reading of everyone before 2010).
There is no necessity to divide it into “two kinds,” or to speak of positive and negative statements though. If one assumes not-X and thereby arrives at a contradiction, then one has indeed established that not-X is not the case. This works the same way in both classical and intuitionistic logic, in both classical and constructive mathematics.
If you have some way of going from not-not-X to X, then you also proved X. The difference between classical logic and intuitionistic logic is that the latter does not admit any general way of going from not-not-X to X.
This is what's actually going on, and it's entirely orthogonal to proofs by contradictions. Redefining "proof by contradiction" to make a common misconception come out right does not help communicate this in any way, since the end of a proof by contradiction is not the only place where double negations are eliminated in classical mathematics.
If anything, it obscures what is going on: mathematicians usually come away with more misconceptions, like "in constructive mathematics you are not allowed to assume a negative". And it makes it a fair bit harder for constructive mathematicians to converse clearly with the rest of the mathematical world.
For the working logician or the constructive mathematician, the distinction that matters is whether an argument uses a constructively invalid instance of the law of excluded middle (or double-negation elimination) or not.
Indeed, long before 2010, they already had perfectly serviceable language for this sort of thing: they said "this proof uses DNE". They do not need a separate, additional term for "this proof of this particular form uses DNE at a very specific place".
> One may scoff at these questions (and many of my colleagues do) but I have personally found them helpful to think about, and also relevant now that logic-based computer proof systems are becoming more important to mathematicians.
Bridges, cited above, coauthored with Bishop the main monograph on constructive analysis. Birkedal, for his part, might fairly be said to have done as much as anyone to shape what we now call modern realizability.
They don't scoff at these questions, they take them rather seriously. Yet like almost all mathematicians AND most other logicians, they chose not to use Bauer's terminology.
> Would you care to enlighten us about any of the subtleties of intuitionistic logic that make this a shibboleth?
It's something of a shibboleth because it reveals the speaker first encountered the field through pop literature like blog posts (there is nothing wrong with that), and has not then spent sufficient time with the primary literature to realize that this is not, in fact, customary terminology used by most of those who work in the discipline proper. So it marks the speaker as somebody likely to have somewhat superficial knowledge of the field.
"It's only proof by contradiction if you prove P by assuming ¬P and deriving a contradiction" was a neologism introduced by Andrej Bauer in a 2010 discussion with Timothy Gowers.
Most mathematicians have never heard of it. Those who have tend to scoff, even in CS and constructive mathematics, and call any proof that "supposes for a contradiction that X" a proof by contradiction.
Take a look at Douglas Bridges calling the sqrt 2 proof a standard proof by contradiction [1], or Lars Birkedal in the proof of Lemma 6.6 here [2].
Bauer is a very productive mathematician who maintains a well-read blog, and it was through that blog that the phrase began to circulate, eventually becoming something of a shibboleth, signaling, perhaps a rather superficial acquaintance with the subtleties of intuitionistic logic.
I use Pangram quite extensively (burning through my 600 token allowance every month). They managed to get their false positive rate impressively low: if Pangram says something is 100% AI-written, you can trust that.
But they need to improve their humanizer dataset. Right now, most models can be given system prompts which cause them to emit text classified as 100% human. It looks like their automated humanizers do worse than these system prompts. Or (alarming if so) they chose not to include ones that would make their product look unreliable.
It seems like everybody (including you) knew precisely what I meant: the models available for ChatGPT Plus or Pro subscribers, i.e. GPT-5.5 Thinking Extended and the latest Pro. I've edited the offending sentence for clarity just in case.
If I got you to be skeptical of AI takes, though, mission accomplished. Exercise your skepticism especially when the takes come from somebody who is trying to sell something.
It's an AI-written slop article, which is hugged to death by HN in any case.
It claims to be an evidence-based investigation, but basically invents the contents of the documents they supposedly investigated, such as the Anthropic Frontier Red Team writeup, from whole cloth.
I don't think deeper engagement with it would promote good discussion.
It's pretty clear at this point that Mythos' capability to discover and exploit zero-day vulnerabilities at scale is but an incremental improvement over existing models like the ones available to OpenAI's Plus/Pro subscribers.
Anthropic tries to create marketing hype around Mythos using two psychological tricks.
1. Put large numbers in the headlines.
"Mythos discovered 271 vulnerabilities in Firefox" makes the model seem extremely capable to the uninitiated.
But it's actually meaningless as a measure of capability _improvement_.
Anthropic gave away $100mil specifically as Mythos credits to these projects and companies (that's $2.5mil per project). Spending the same exorbitant amount of compute analyzing the same codebases in an older model like GPT 5.x Pro would have turned up 260 of these vulnerabilities, or could even have turned up more than 271 ones.
No need to speculate, since this is exactly what we saw in the few code bases where we have such comparisons (like in the curl codebase). Supposedly weaker models, working with a much lower budget, turned up dozens of vulnerabilities. Mythos turned up only one, which ended up as a low severity CVE.
2. Do the whole "too dangerous to release" shtick. This is one of Dario Amodei's favorite moves. When he was vice president of research at OpenAI, he declared GPT-3 (which wasn't able to produce coherent text beyond 3-4 sentences at the time) too dangerous [1] as well.
Long story short, it's the ChatGPT 4.5 situation again: a company trained a model that's too slow and expensive, but not much more capable than what came before. It therefore requires these marketing stunts.
Questions which have never been asked or answered before, but to which practitioners have immediately obvious answers, are dime a dozen in mathematics.
You can find thousands of such questions on Math StackExchange. Take e.g. [1]: never been asked anywhere else, interesting enough, yet answered pretty much immediately by two separate mathematicians.
"Is there a single constant and function with connected domain that can express all of $\log, \exp, \sin, \dots$?" would have made a fine question there too, the type that gets a thorough answer very quickly if anyone bothers to ask it.
> the burden of proof should not lay with the reader
You were the one who made the claim that "this is one of the most significant discoveries in years". Feel free to substantiate that claim first, according to the same standards. Are there any authors who ask this question, and/or suggest that they don't know an answer?
Arnold's proof can be used to show that certain classes of functions are insufficient to express a quintic formula.
These classes can always safely include all single-valued continuous functions (you cannot even write the _quadratic_ formula in terms of arithmetic and single-valued continuous functions!), but also plenty of non-single-valued functions (e.g. the +-sqrt function which appears in the well-known quadratic formula).
Applying Arnold's proof to the class given by arithmetic and all complex nth root functions (also multivalued) gives the usual Abel-Ruffini theorem. But Arnold's proof applies to the class "all elm-expressible functions" without modification.
Well, mathematicians not working for Anthropic/OpenAI are heavily disincentivised from reporting that their discoveries were made using AI. If e.g. the idea that resolved the Mahler conjecture came from AI, it's not like we'd ever know.