The key phrase is "kind of thing". It certainly does matter what kinds of things we focus our attention on as a species. I think you would have to be quite cynical to think that progress in spaceflight over the past 60+ years hasn't had a positive impact.
I think this is a false dichotomy. If you're passionate about your craft, you will make a higher quality product. The real division is between those who measure the success of a project in:
- revenue/man-hour, features shipped/man-hour, etc.
- ms response time, GB/s throughput, number of bugs actually shipped to customers, etc.
People in the second camp use AI, but it's a lot more limited and targeted. And yes, you can always cut corners and ship software faster, but it's not going to be higher quality by any objective metric.
Most mathematicians don't understand the fields outside of their specialization (at a research level). Your assumption that intuition and applications are limited to hobbyists ignores the possibility of enabling mathematicians to work and collaborate more effectively at the cutting edge of multiple fields.
>The litmus test will be when, if ever, someone wins a Fields using a proof-assistant in an essential way.
You're assuming that the point of interactive theorem provers is to discover new mathematics. While that's an interesting research area, it seems like the more practical application is verifying proofs one has already discovered through other means.
Exactly this. LLMs really aren't built for discovering new mathematics, especially _interesting_ new mathematics. They're built to try the most obvious patterns. When that works, it's pretty much by definition not interesting.
What LLMs are good at is organizing concepts, filling in detail, and remembering to check corner cases. So their use should help mathematicians to get a better handle on what's terra firma and what's still exploration. Which is great. Proof by it-convinced-other-mathematicians doesn't have a flawless track record. Sometimes major theorems turn out to be wrong or wrong-as-stated. Sometimes they're right, but there's never been a complete or completely correct proof in the literature. The latter case is actually quite common, and formal proof is just what's needed.
LLMs and interactive theorem provers are vastly different. There are AI models that come up with workable formal proofs for ITPs but these aren't your usual frontier models, they're specifically trained for this task.
ITPs are far older than LLMs in general, sure, but that's a pedantic distraction. What everyone is talking about here (both the comments, and the article) are ITPs enriched with LLMs to make the "smart" proof assistants. The LLMs used in ITPs are not vastly different from the usual chatbots and coding assistants. Just a different reinforcement learning problem, no fundamental change in their architecture.
Of course, once LLMs are really good at that, they can be set loose on the entire historical math literature, all 3.5M papers worth. And then LLMs can be trained on these formalized results (the ones that turn out upon attempted formalization to have been correct.)
How good do you think AI will be at proving new results given that training set?
Math is going to change, and change massively. There's a lot of whistling past the graveyard going on from those who are frightened by this prospect.
Relevant philosophy paper: "The Vulnerable World Hypothesis" by Nick Bostrom [0].
In that paper, Bostrom floats the idea that it might be in humanity's best interest to have a strong global government with mass surveillance to prevent technological catastrophes. It's more of a thought experiment than a "we should definitely do this" kind of argument, but it's worth taking the idea seriously and thinking hard about what alternatives we have for maintaining global stability.
Cheap hypersonics don't threaten global stability, they threaten global hegemony. Which is really what I suspect irks most people afraid of them.
We've seen a shift towards cheap offensive capacity that gives middle powers or even smaller actors the capacity to hit hegemons where it hurts, very visible in Ukraine and the Middle East now. This leads to instability only temporarily until you end up in a new equilibrium where smaller players will have significantly more say and capacity to retaliate, effectively a MAD strategy on a budget for everyone.
GP's point was broader than that, it was about technological progress and the possibility of terrorist groups or mentally ill individuals getting their hands on weapons that can easily kill millions of people. That's also what the paper I linked is about.
Consider a future where individuals can relatively easily engineer a pathogen or manufacture a nuclear weapon. It's not hard to imagine how that would threaten global stability.
It has been proven that recurrent neural networks are Turing complete [0]. So for every computable function, there is a neural network that computes it. That doesn't say anything about size or efficiency, but in principle this allows neural networks to simulate a wide range of intelligent and creative behavior, including the kind of extrapolation you're talking about.
I think you cannot take the step from any turing machine being representable as a neural network to say anything about the prowess of learned neural networks instead of specifically crafted ones.
I think a good example are calculations or counting letters: it's trivial to write turing machines doing that correctly, so you could create neural networks, that do just that. From LLM we know that they are bad at those tasks.
So for every computable function, there is a neural network that computes it. That doesn't say anything about size or efficiency
It also doesn't say anything about finding the desired function, rather than a different function which approximates it closely on some compact set but diverges from it outside that set. That's the trouble with extrapolation: you don't know how to compute the function you're looking for because you don't know anything about its behaviour outside of your sample.
No, but unless you find evidence to suggest we exceed the Turing computable, Turing completeness is sufficient to show that such systems are not precluded from creativity or intelligence.
I believe that quantum oracles are more powerful than Turing oracles, because quantum oracles can be constructed, from what I understand, and Turing oracles need infinite tape.
Our brains use quantum computation within each neuron [1].
The difference is quantum oracles can be constructed [1] and Turing oracle can't be [2]: "An oracle machine or o-machine is a Turing a-machine that pauses its computation at state "o" while, to complete its calculation, it "awaits the decision" of "the oracle"—an entity unspecified by Turing "apart from saying that it cannot be a machine" (Turing (1939)."
This is meaningless. A Turing machine is defined in terms of state transitions. Between those state transitions, there is a pause in computation at any point where the operations takes time. Those pauses are just not part of the definition because they are irrelevant to the computational outcome.
And given we have no evidence that quantum oracles exceeds the Turing computable, all the evidence we have suggests that they are Turing machines.
Turing machines grew from the constructive mathematics [1], where proofs are constructions of the objects or, in other words, algorithms to compute them.
Saying that there is no difference between things that can be constructed (quantum oracles) and things that are given and cannot be constructed (Turing oracles - they are not even machines of any sort) is a direct refutation of the very base of the Turing machine theoretical base.
That's an irrelevant strawman. It tells us nothing about how create such a system ... how to pluck it out of the infinity of TMs. It's like saying that bridges are necessarily built from atoms and adhere to the laws of physics--that's of no help to engineers trying to build a bridge.
And there's also the other side of the GP's point--Turing completeness not necessary for creativity--not by a long shot. (In fact, humans are not Turing complete.)
No, twisting ot to be about how to create such a system is the strawman.
> Turing completeness not necessary for creativity--not by a long shot.
This is by far a more extreme claim than the others in this thread. A system that is not even Turing complete is extremely limited. It's near impossible to construct a system with the ability to loop and branch that isn't Turing complete, for example.
>(In fact, humans are not Turing complete.)
Humans are at least trivially Turing complete - to be Turing complete, all we need to be able to do is to read and write a tape or simulation of one, and use a lookup table with 6 entries (for the proven minimal (2,3) Turing machine) to choose which steps to follow.
Maybe you mean to suggest we exceed it. There is no evidence we can.
> P.S. everything in the response is wrong ... this person has no idea what it means to be Turing complete.
I know very well what it means to be Turing complete. All the evidence so far, on the other hand suggests you don't.
> An infinite tape. And to be Turing complete we must "simulate" that tape--the tape head is not Turing complete, the whole UTM is.
An IO port is logically equivalent to infinite tape.
> PDAs are not "extremely limited", and we are more limited than PDAs because of our very finite nature.
You can trivially execute every step in a Turing machine, hence you are Turing equivalent. It is clear you do not understand the subject at even a basic level.
> You can trivially execute every step in a Turing machine, hence you are Turing equivalent. It is clear you do not understand the subject at even a basic level.
LOL. Such projection. Humans are provably not Turing Complete because they are guaranteed to halt.
Judging from what I read, their work is subject to regular hardware constraints, such as limited stack size. Because paper describes a mapping from regular hardware circuits to the continuous circuits.
As an example, I would like to ask how to parse balanced brackets grammar (S ::= B <EOS>; B ::= | BB | (B) | [B] | {B};) with that Turing complete recurrent network and how it will deal with precision loss for relatively short inputs.
Paper also does not address training (i.e., automatic search of the processors' equations given inputs and outputs).
Is there actually a use case for graphing calculators anymore? Desmos provides a great graphing program for free in a web browser. In any professional capacity you would be using MATLAB, Mathematica, or the scientific Python ecosystem.
I mostly remember playing games on my TI-84 in high school. We used it in class maybe once or twice. None of my college classes allowed graphing calculators on tests, so ironically I had to buy a "dumb" calculator even though I owned the fancy one.
I don't think there was ever a solid use case for graphing calculators in school, at least not in my experience? The curriculum didn't make good use of them and I'm not convinced it could have. There's little value in having every kid in the classroom replicate the same plot of y = sin(x) or whatever on a tiny screen. And other than such demonstrations... what are you gonna do with it? It was never flexible or powerful enough for serious math. You weren't going to run circuit or physics simulations on a TI-89.
There are other features that can be useful - scientific notation, symbolic solver, unit conversions, etc - but graphing as such always seemed like a gimmick.
I think it's more of a not-entirely-rational appeal to parents: "if my kid has a top-notch calculator for high school / college, maybe they're gonna be better at math". And kids did not object, but in the end, mostly just sideloaded games and horsed around.
> You weren't going to run circuit or physics simulations on a TI-89.
Well, I wrote a couple of programs that were useful for quite a while. They involved electromagnetism and changing frames of reference. I definitely was able to do quite a lot of Physics with my Ti-89.
>DR is a Danish public-service radio and television broadcasting company. Founded in 1925 as a public-service organization, it is Denmark's oldest and largest electronic media enterprise.
Humans are complex. It's possible for someone to want to do good and at the same time want to promote/market their product and make a profit. I don't see a contradiction there.
How do you call a marketing campaign that does not deliver on what it promised? I have no problem with anthropic trying to create good will around their products but this particular campain aiming to find good will around people doing open source was an outright lie that did not deliver what it promised and this was all done on HN.
When a company lies for something that trivial, it does not inspire trust
It's an outright lie because they haven't greenlit your personal project after two weeks? Did it occur to you that maybe they just got a lot of applications and are prioritizing other projects or still working through a backlog?
reply