Two possible walls for advancing math with AI: first, it could be that the kind of problems that AI solves are the ones that don't open new avenues to concepts and ideas, and second, problems complexity is usually not linear, and the new and interesting problems to solve could be at a level of complexity beyond what AI can currently solve. Perhaps mathematicians could work to find ways to help AI cope with those walls.
I think OpenLDK allow you to use the Java world for sbcl when you don't want to use Clojure or ABCL. But using OpenLDK requires you to pay the interop cost and when speed of the java libraries is not a sharp constrain.
Here using images is incidental. In fact, what Fable is actually doing is generic encoding/decoding of a message into a medium. Ask it to hide a message in a poem and it should try to apply a strong (high signal-to-noise ratio) family of schemes, for example quantized, margin-rich encoding. Robustness of features depends on the medium, so Fable is selecting strong encoding and robust features tailored to it. This frames AI steganography not as a trick of computer vision, but as an emergent property of a model reasoning about information density across arbitrary media. Here emergent is just surprising general pattern matching and constraint-satisfaction and cross-representation mapping that allows it to express the encoding/decoding across media.
To meaningfully test a forecasting model, use a data-generating process with a high signal-to-noise ratio where a classical method (empirical conditional distributions, i.e., counting) can capture most of the signal. Then check whether the model captures that signal too. If you only test on high-noise processes, any model looks comparable to SOTA because there's little signal to differentiate them. Also testing on a range of signals with different signal-to-noise ratio would allow us to know how much of the signal the model captures in different scenarios.
Perhaps some people chose math because they were very good at it, they could understand one shot the difficult concepts and have not enough money to study other subjects because you have to move city or pay a lot. Those kind of people can move to others field if economics conditions are favourable.
A neuron is a dirty, contingent, blind-evolved part, like a screw in a particular machine. It is necessary for this particular machine, but not conceptual, not a feature. The strong activation on top-k images and the change when it is removed just mean that the screw is necessary for this particular machine, not that it's something else. The essay shows neurons are legible, but legibility isn't the same as meaning. Still, given math job prospects, I love screws, they are really not dirty.
The only way I see creators can continue earning money is by a new market in which the products they create can not be paid in money but in a new coin. I propose that the coin stays strictly internal and not convertible, that fiat enters only from the outside, and that exit is communal, not automatic. The viable solution is a federated network of small, closed credit loops. Inside each loop, creators use a non convertible token for work and tools. Cash enters through dues or grants, and cash out exit grants are decided by community vote based on need.
Small loops keep social trust high. When loops need to trade, they establish simple bilateral treaties with each other. This preserves value, protects human labor, and keeps decisions in local hands.
In chats with models like deepseek the LLM makes many basic math wrong claims. Maybe more advanced models can in special circumstances, guided by the work of humans and Lean, prove theorems. But just like in RL, the brute force approach of failing a million times may not be enough when the reward is far.
In math the reward is finding the decomposition that reduces the problem to simpler cases, and in many cases that reward is very far. Furthermore, the LLM may process the many cases fast, but the decomposition often comes from a human, and the model takes the credit for the proof.
This foundation could provide the LLMs with the decompositions, generated by human ingenuity, that they need. But once supplied, the model can run the cases and get the credit. Hence more credits for LLMs and fewer jobs and worse prospects for humans, even if it isn't an intended consequence.
When opinions conflict, if the first opinion is right and the second opinion is wrong you could end up receiving an unnecessary treatment that could hurt you. So I think you need to know for sure why the first opinion is incorrect, or at least avoid accepting the second opinion blindly, because the first doctor won't be there to push back.
Of course, I know nothing is ever certain here. Still, having two opinions or more data can be valuable, but only when whoever makes the final decision can actually use that information: detecting any bias and deciding with a good balance between risks and rewards.
The essence of von Neumann's trick is that all permutations of independent events have the same probability. So the trick generalizes to any other set of possible results.
Imagine that instead of two numbers you select a million numbers, and apply a zip algorithm to that sequence to obtain a file then you map the first byte of that file using (first byte) mod 9 + 1, intuitively there must not exist any correlation between consecutive tries of this strategy. Looking for this example I realized that instead of using n1+n2 you should use a non commutative operation to improve the strategy, but when using just two numbers the addition suffices.
Some people are used to thinking deeply and concentrating, but are also curious and they are using LLMs to get more information faster. So for them, the use of LLMs seems fruitful. What I wonder is whether the tendency of letting the LLM do some of the work will affect them too, eroding, in the long term, their concentration powers.
What about the following strategy? You first think of a random integer n1 from 1 to 9, then you think of another one n2, perhaps one that you have not used very often, and finally say (n1+n2) mod 9 + 1. I think this strategy could make your selection more random . Perhaps someone could try it and report the results.
Right now there are two top posts one about GLM-Legal and one about Argon. Both score about the same in the Harvey Lab test, but the prices are $0.15 versus $10.55, the $0.15 is Vals AI's estimate using the base model since GLM-Legal has not published prices.
It seems the model is a little verbose, using more tokens to complete a task that other similar models as stated more precisely in the linked article about Artificial Analysis's work on this model.
The non-disclosure of the training set may raise legitimate skepticism about the training set containing tasks similar to those in the test set. That would make the benchmark result moot. There are benchmarks that by design can't be manipulated by incorporating similar tasks in the training set, for example those that dynamically generate tests and only reveal the global result. Testing on one of those would give a strong measure of the strength of this legal AI framework.
I think using adapters for different legal systems could improve results by 2% or 3% at a similar cost (perhaps under 1% of additional parameters) and allow for independent improvements. If the GLM team hasn't already considered this and adopts my suggestion, tips would be welcome.
Since you've posted in HN about the state of Julia in 2024, I think you have the knowledge, and perhaps the desire, to provide information about what the improvement are in Julia with respect to problems expressed in the 2022 long discussion by Yuri Vishnevsky (1).
Anyway the problem I see is that other languages and tools are also improving, for example the author of this post (2) "I am reluctant to shift to Symbolics, as the calculus specifics (limits and integrals) aren't as competitive".
The stated goal is to introduce math concept without computer language or use being a burden. But then I took a quick glimpse at the derivative chapter, and at the top there is "using Sympy", so you don't only use Julia but also sympy. Mathematica and Matlab are the alternative in a university setting in which free licenses for students are available, with strong numerical and symbolic capabilities . I think Julia is a lovely language, but perhaps Julia + Sympy is a difficult proposal for attaining the stated goal. S.Rational(1) + S.Rational(1,3) type of things from sympy are hard to digest.
Edited: Added: I think if you love teaching calculus using Julia then the passion you transmit might compensate for the additional burden of using Julia plus SymPy.
I think that when there are not explicitly stated attribution of liability for decisions that are at the core of a problem the general rule should be that the accountability should be proportional to the additional benefit that each agent or person receives for that decision. That rule is in the context that each agent is responsible to read information and communicate to others in the chain about the risks that decisions entails. The overall structure of an organization should have the goal of making such information about risk available for all agents in the chain in a way that the amount of information can be digested by the agents without much problems.
It seems one of the errors was a sign error. If sign errors have high probability then there is high probability of having other false proofs, moreover sign error are similar to error in basic operations, so there could be many basic type errors. So I think our prior about false proofs is now a posterior with a higher probability for false proofs.
Edited.Added. To compute the posterior one needs to know the probability of detecting such error in a proposed proof.
I am curious to know whether the proofs given by the LLMs are going to provide new hints about related problems that might be solved using the same machinery as the one used in the proofs. Also, I would like to know what is the average ratio between the length of LLMs proofs and the length of a proof that a mathematician can write to explain that proof to another mathematician. It is like a functor between the category of human mathematical concepts and the category of LLM operational concepts used in those proofs.
Code is not solved seems to reflect the idea that using AI badly is a bad idea, and that to use it well you have to be good at making software.
The real tension is in the human AI interface and there are many unknowns. Can a software engineer with weak design skill use AI to produce good code. Will the future make those requisites less important. Will productivity increase with AI stagnate even for the best. Can a new way to interface humans and AI break that wall. Will software be created in a new way using dynamic libraries that AI agents prepare to cover a large scope of problems. Nobody knows yet. The article is strong on what is closed, accountability, ownership, NFRs, slop, and silent on what is open, which is where the argument actually is.
Reading isn't the right comparison. Human memory is lossy and fades while LLM encoding is durable with no degradation. The valuable content that the author provides: the content, style, selection of topics, and more is encoded, written into the LLM weights, and they obtain profit from them (now directly, via ads). No one can compete with that kind of copying and pasting from copyright-protected material. And the scale is what hurts authors most: flooding the market with millions of copies on demand, without paying for it.
Mathematics can give for example theoretical lower and upper bounds for complexity of algorithms. And that information is very valuable for tradeoffs. For example we know that many algorithms have optimal average complexity or asymptotic behavior so they can be a good tradeoff in certain scenarios. Also knowing that some problems don't have a perfect solution allows you to accept essential tradeoffs
One question: LLM might answer correctly many logical and math problems, but I don't see any guarantee that when the LLM context receives new information the answer could get worst. Security in LLM answer is not a monotone increasing function of context size.
A cynical could say that AI could not interpolate the Dedekind cut but it could extrapolate to create a lot of money but just using an imaginary extra point.
The ironic part is that generating a joke like this requires jumping across three distant fields: real analysis, machine learning limits, and VC market cynicism. For an AI to discover that specific overlap through statistical search, the combinatorial space is absurdly huge. Yet a human brain connects them in a fraction of a second. This tiny joke is a micro-proof of the macro-argument: human conceptual leaps routinely bypass exponential search spaces.
Just to add that the Dedekind cut example seems to stand beyond any RL policy used in chess or go, AlphaZero or AlphaProof. In the classical RL there is an state-action space. If the solution requires jumping to a totally difference action space (that must be created) the local policy stalls. Dedekind cut is an example of a out-of-distribution state-space generation.
I think the author did not mention that the way humans construct the knowledge ladder requires low complexity because each step gives power to the next, but the AI with the exponential exploration can give biggers steps but then it stagnates because the next step can be beyond the exponential exploration complexity. In chess a good strategy can be the best tool, in the game of go our experience suggest the same, but it could happen that mathematical thinking requires a type of policy that could be beyond the current ideas. I can not fathom a LLM could conceive from scratch concepts like the real numbers by Dedekind.
For example he could tell you about how the LLM destroyed the main database (source of data for him) and so you should prompt the LLMs for how to avoid losing all your data. Real experiences help but not painless.
IMHO, I think that it could be better if the question about how to learn programming in the age of LLMs were asked to someone who is learning now by using LLMs. Someone who learned programming thirty years ago can perhaps give you only one side of the coin, whereas someone learning today from scratch using LLMs could give you good advice on what the real difficulties are and where the main drawbacks lie. Combining both views would give a better idea of the landscape.