nicf

Yeah, I'm also not a big fan of ghostwriting in the domains where it's prevalent. For what it's worth, though, academic writing is very much not one of those domains: taking LLM's out of it, if ten years ago it had turned out that the "author" of a math textbook like this one had paid someone else to write the whole book for them, that would be considered major academic misconduct and that person's reputation would have been pretty severely damaged. I do think it makes sense to take the norms of a particular subset of the publishing world into account when thinking about what a reasonable reader expects when they see someone's name on a book.

You're proposing a new way that we could treat authorship that's different from how we currently treat authorship. There's an argument to be had about whether that would be better than our current system. My vote would be no, but I'm not claiming it's obvious one way or the other.

But regardless of whether you think your new system would be better, it's not the current system. Surely the question of whether someone is lying by putting their name on a book they didn't write can't depend on some hypothetical future change to the meaning of authorship that hasn't happened yet.

I'm not the OP, but I can give my answer to this question, or at least one of my answers.

This book is presented as being written by a human. (In this case, a human I happen to know, which would add another dimension to my complaint, so let's imagine that weren't the case.) Maybe someday we'll live in a world where sticking a person's name at the top of a book doesn't carry the implication that they wrote the words in the book. That would make me sad for other reasons, but in any event, we don't live in that world right now, so for now, if you put a person's name on your book, you are claiming that that person wrote the book. If that's not true, you are lying, and like most people, I don't like being lied to.

If it makes you feel better, I've known the author personally for over ten years, and he is both a human being and an excellent teacher. I'm about as confident as I could imagine being that he wrote this book himself. (In order for this to completely reassure you, you of course have to trust me, but I thought I'd add my vote of confidence just in case it helps even a little.)

This book aside, though, the fact that writing is no longer definitive evidence of human thought is one of the things I dislike the most about the LLM era. I very much share your sadness about that.

I read them as making a much weaker claim than this: not that the Lean proof isn't valid, just that it is not actually a formalization of the natural-language proof in the PDF they provided alongside it. I haven't heard any PDE people claim that the Lean proof is invalid, and I have heard things from a lot of them that imply that they think it is valid. (I'm a former research mathematician, but this is very far from my specialty, so I'm not really equipped to evaluate this claim myself.)

By all accounts this result cost far more than a million dollars to produce, and either way OpenAI announced they won't be accepting the prize.

There are definitely some results which have this "a bunch of little hacks" quality you're describing, and while opinions differ I share your intuition that there's something a little disappointing about solving a big problem that way.

But I think FLT is about as far as one can get from that situation! Wiles's work was the culmination of centuries of theory-building work, and the concepts that were developed over that time are far more important than FLT; the thing Wiles actually proved (a special case of something called the "Modularity Theorem", the full version of which was proved a bit later) is itself much more valuable to human understanding of mathematics than FLT. It's certainly very cool that it can be used to answer such a simple question that was open for so long, and it makes for a great headline, but I think if you asked number theorists working in the area they would almost all tell you that they're much more grateful for the theory that came out of this quest than for the mere fact that the quest was completed.

I'm a private math tutor specializing in exactly this sort of material, and I agree with this very strongly. Knowing what "counts" as a proof is one of the most common gaps I see in students who come to me after self-studying, and most students do need some back-and-forth with an expert to really get that skill down. I imagine that LLM's could be very helpful for this if they were used judiciously!

This sounds like a fully general argument against being alive and having experiences; you could equally well call love a "brain hack". I also care a lot about solving practical problems, but the reason it's important to solve them is because it leads to human flourishing, which includes participating in all these activities you're deriding. Otherwise why are we doing anything at all?

I've been working for a few years now as a private tutor teaching proof-based math to adults. Maybe this should give me an incentive to join in on some of the cynical replies this has gotten, but that's not actually how I find myself feeling. I love teaching, especially in the one-on-one format, but it really doesn't scale; not everyone can afford a human tutor and there probably aren't nearly enough of them to meet the possible demand.

If a product like this worked well --- and I agree Andrew Ng's involvement seems like a positive sign --- then that would be great! Something I would wonder about from my past experience using LLM's for technical subjects is how to replicate a particular side of the student-teacher relationship. Sometimes a new student of mine will need to be told that they don't actually understand something as well as they think they do, and that they'd be better served by going more slowly than they might have wanted to. This "talking you down gently" conversation is one I'm happy to have (and sometimes it's actually me who's wrong!) but I'd be interested to see whether and how well an LLM run by a company that really wants to retain customers could pull this off.

It's hard to say without an actual product, of course. I'm looking forward to seeing what they come out with.

This was definitely something I saw on HN, but for what it's worth, in my conversations that night with other mathematicians no one brought up Lean proofs or peer review even as a joke. We just copied and pasted the polynomials into a computer algebra system and checked it ourselves and then said "holy shit, I guess the Jacobian conjecture is false". Couldn't have taken more than five minutes after we first saw the tweet.

I worked as a research mathematician for a while, and I've published peer-reviewed math papers. Reading this tweet put me in the strange position of feeling like defending the way peer review works in my (former) field, which I'm not used to doing. Among other problems it goes too slowly, it pays only the participants who provide the least value, and it was designed for a world that hasn't existed for a long time.

But this tweet paints a somewhat misleading picture of the role peer review plays in practice in modern math research. Reading this tweet could leave you with the impression that when someone proves a new result, no one pays any attention until it's gone through peer review and published in a journal. This just isn't true. ArXiv preprints are much more widely read than the journals' version of articles, and they go up essentially as soon as they're written. Certainly no one's waiting the year-plus it would take to get the article published in a journal! And even before that, mathematicians communicate their results to each other through slightly less formal channels, like blogs, conference talks, and regular human word of mouth.

In short, despite the sclerotic and parasitic nature of the journal system, the way ideas get disseminated in the field in practice is actually a lot closer to the ideal prewar picture he's describing in the tweet. I'm led to understand that this was less true before the Internet, but even by the time I started my PhD in 2009 things had been working the way I just described for quite a while.

No, what I'm saying is that I don't agree that taste in mathematics is more uniform than taste in coding! Mathematicians argue about taste all the time. Just as you might look at a piece of code and agree that it compiles and doesn't have any fatal bugs but still think it's badly written, hard to follow, hard to modify, or whatever else, mathematicians judge mathematical work using very similar criteria.

If your complaint is about the type of work that gets you published in a fancy math journal, then I'll happily join you on the barricades. Sure, getting a paper into Annals of Mathematics or whatever is more game than art in the sense I think you mean here.

But "what mathematicians care about" is much, much broader than what gets you published in a fancy journal. Mathematics as a human activity is millennia old, much older than the concept of journals or even universities, and that activity is, to me, very beautiful, worth preserving, and more of an art than a game. The incentive structure of academia for the past few decades has done a pretty bad job at preserving that art form, but that doesn't mean mathematicians as actual human beings don't care about it --- if they didn't, they probably would have chosen a different career.

I've spent some time working both as a math researcher and as a software engineer, and I think this comment actually underrates the similarity between the two fields as they're actually practiced.

Some math research does involve grabbing a single, fully specified conjecture off the shelf and hunting for a proof of it, and it's true that if you manage to solve a long-standing open problem, other mathematicians will be interested no matter how you did it.

But this isn't all of what they do, probably not even most of what they do. Like in software engineering, it's not always obvious which question would be the most useful one to ask. A lot of mathematical work also goes into what we call "theory-building", where you could say that primary work goes into coming up with definitions rather than theorems. Mathematicians also care a great deal about how something is proved; a lot of them are some of the most aesthetically picky people I've ever met. Words like "ugly", "beautiful", "creative", and "boring" are used to describe both definitions and proofs all the time.

From the outside, it can look like all they're doing is pumping out proofs at any cost. But I promise you that when I talk to mathematicians who don't have any experience building software, they have a similarly narrow view of that field as well! Both fields, from the inside, look a lot more human than you might expect.