> if you spend too much time interacting with LLMs, you eventually resemble one
Pretty much. I think people who care about reducing their children's exposure to screen time should probably take care to do the same for themselves wrt LLMs.
Yeah this is something I think a lot of people tend to overlook. People are far too quick to rewrite "we don't know of any reason why it would be impossible" to "we know how to do it" in their heads.
I've done a bunch of theoretical PL work and I find this to be a very surprising result... historically the assumption has been that you need deeply "non-computational" classical axioms to work with the sorts of infinites described in the article. There was no fundamental reason that you could give a nice computational description of measure theory just because certain kinds of much better-behaved infinities map naturally to programs. In fact IIRC measure theory was one of the go to examples for a while of something that really needed classical set theory (specifically, the axiom of choice) and couldn't be handled nicely otherwise.
Much of your comment seems to be about your culture — eg, assuming things about axioms and weighting different heuristics. That we prioritize different heuristics and assumptions explains why I don’t find it surprising, but you do.
From my vantage, there’s two strains that make such discoveries unsurprising:
- Curry-Howard generally seems to map “nice” to “nice”, at least in the cases I’ve dealt with;
- modern mathematics is all about finding such congruences between domains (eg, category theory) and we seem to find ways to embed theories all over; to the point where my personal hunch is that we’re vastly underestimating the “elephant problem”, in which having started exploring the elephant in different places, we struggle to see we’re exploring the same object.
Neither of those is a technical argument, but I hope it helps understand why I’d be coming to the question from a different perspective and hence different level of surprise.
The reason people had these assumptions is because people have been trying (unsuccessfully) to find a constructive interpretation of this stuff for a very long time. Even very fundamental results in measure theory like the Heine-Borel theorem typically require some extension to traditional constructive axioms. Like I absolutely get where you are coming from, but there are a large number of "nice" classical results that definitely do not have constructive counterparts. It's cool that descriptive set theory is not one of them but it's not obvious by any stretch of the imagination, and the pattern you're using to say that it's probably true ("Curry Howard maps nice to nice") is not great process IMO since it would fail in a lot of other cases.
Or maybe consumers do not want to (because it doesn't make economic sense for them, or because they don't want to spend every second of their free time optimizing) spend a significant amount of their time doing this kind of research before every purchase decision? I have never understood this argument that as long as information is technically available companies should be absolved of meeting some minimum level of baseline quality. You're also describing an industry that's unusually transparent (in part because there are not that many models of planes); for a lot of online goods there is absolutely no reasonable way you can know what a particular brand will be like before you buy.
If they don't want to look at the relevant column in Google Flights they don't have to. But money and time are the units of caring. If someone isn't willing to take 60 seconds to research or spend $5, I just don't believe it matters that much to them.
Flights in my mind are already a very high quality product offered at a very reasonable price. I can pay $250 and get across the country faster and more safely than any other means of transportation. I don't necessarily think that extra legroom is part of a "baseline level of quality" we should expect everyone to know you feel is worth the money and automatically provide to you
These are just backdoors as a vulnerability class. It does happen occasionally even in high quality widely used libraries, but it's very rare and usually quite sophisticated (e.g. https://en.wikipedia.org/wiki/XZ_Utils_backdoor in 2024). A lot of these attacks, even when highly sophisticated, only really work when the attack payload can be hidden in a binary--it's very very hard to insert a useful backdoor into the source code of an actively maintained project that receives regular code reviews without anyone noticing (which is not to say it's impossible).
It's also worth noting that the xz exploit didn't even live in the xz source, it was added to the source by the attacker before uploading it to the package build farms. With most of the current package repositories, there's not even an expectation that the bundles you get are derived from a particular tag or commit hash, let alone a detailed chain of custody. I remember RPM in the old days made a big deal out of "pristine sources" to which patches would be applied, but it still has no way to prove or enforce that claim.
I have never met a single person who wanted to learn advanced mathematics who was prevented from doing this by not getting into the college of their choice, and I know a lot of mathematicians. Do you genuinely believe that people who get an 800 on the math portion of the SATs (which is actually a pretty large number of people) are struggling to get admission to university at all?
The mathematicians I know wish the were introduced to more material when they were younger so they could spend more time internalizing different concepts before being forced to specialize by the demands of a PhD.
The more scientifically minded people I know wish they were introduced to more mathematical concepts when they were younger so they could feel more able applying more sophisticated models to their problem domain.
Having a pipeline of somewhat mathematically able citizens is crucial to having an advanced economy. I don’t think the preceding statement is remotely controversial.
Nobody is saying "don't teach math to high schoolers." My point is that this whole admissions rant appears to be entirely about admission to "elite" schools no longer being fully determined by SAT scores (or whatever--I've never been completely clear on what people are actually arguing for), when anyone good enough at math to get a really high SAT score can easily gain admission to a ton of universities with great math departments. As for the real "gifted" kids, there will always be some middle schoolers taking calculus etc. with or without a structured gifted & talented program. The majority of people in these programs are not so far beyond their peers as you seem to think, and my experience in math departments has been that there's a pretty even mix of kids with precocious math backgrounds and people who developed their skills at a later point.
The issue, as I see it, the erosion of the level of mathematical competence seen at university entrance. This inhibits the rate of progress one can make with a student over the period of an undergraduate education, and this reduces the exposure to mathematics of the next generation of educators.
Mathematics education is really, really broken unless your measure is Terry Tao’s are still produced. That’s not the issue. The issue is the breadth of people who can recognize what mathematical proficiency can enable within society, not because some wonk says data shows this, but because they personally have experience as to it has empowered them to perform more capably in their chosen field of endeavour.
Okay. A lot of the blame for this has been previously placed directly at the feet of stuff like AP programs that claim to teach, e.g., calculus to a bunch of people who aren't ready for it; it becomes a prestige thing or expected for admittance to college, which results in a bunch of people being taught to the test and not actually gaining the foundational mathematics education they actually need to understand the subject. I don't see how encouraging this sort of thing actually helps with the problem of universities needing to do remedial math education to people who supposedly know the material already, but it appears to be what you're arguing for. If you're arguing for lower education reform in general, great, but that has little to nothing to do with how highly elite colleges weigh the math portion of SAT scores.
I feel compelled to point out that for people who want to learn mathematics, there are more and higher quality resources than there ever have been before. For the most part they are absolutely free, and unlike virtually every other subject on the planet they are generally not the sort of thing where you can be led astray by misleading material. I simply don't see how such people are being suppressed in any way, or why (from the perspective of advancing the state of the art of mathematics) I should care about the "non Terry Taos" in your words, who are merely above average at math but don't actually intend to pursue it as a career. There are plenty of other skills that are actually eroding at a high rate, or have huge startup and lab costs, or are otherwise underappreciated and underpaid relative to their importance to society; I don't think mathematics is one of them.
My initial reply to you was to only articulate some nebulous idea that there is desire for mathematical understanding that is being underserved by mathematics education.
If standardized tests statistically “reliably” predict mathematical “ability”, the act of removing their gatekeeper role to higher education in our society (that is structured around prestige colleges resulting in prestige future income) amounts to disincentivizing whatever performative mathematical education students endure.
Some people, as children, are drawn to mathematical concepts and, yes, now is a better time than in all history to be such a person. The ambient possibility of this for a given individual is I assume unchanged through history.
Some people, like me, are able to do performative mathematics in school well enough but didn’t particularly care for it. Then we encounter some remarkable teacher and we feel some fortunate and enriched.
The existence of the remarkable teacher is the product of the possibility of a society producing mathematically proficient educators. That means students must somehow encounter them in their education.
I believe increasing the production of such teachers is important, irrespective of the field. I believe mathematical thinking can address this problem. Above, I have sketched the most meagre possible outline of how such thinking prepares to address further modelling of the problem.
Yes, this is contingent on my assumption that mathematics is generally useful in problem solving.
If you don’t accept this, that’s fine. We simply don’t agree. That said, I want to add that I appreciate the time you took to engage with me. I’m someone who believes that a lot of structures in society are poorly conceived, but that is a much longer discussion. At the present moment in time, standardized tests are a minor stupid in comparison to removing them without a much broader vision of how to address certain shortcomings of society.
Pretty much. I think people who care about reducing their children's exposure to screen time should probably take care to do the same for themselves wrt LLMs.
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