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Case study: Creative math – How AI fakes proofs

tomaszmachnik.pl

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Re: Case study: Creative math – How AI fakes proofs

#61
post #43
post #37

The simpler and I think correct conclusion is that the LLM simply does not reason in our sense of the word. It mimics the reasoning pattern and try to get it right but could not.

What do you make of human failures to reason then?

Humans who fail to reason correctly with similar frequency aren't good at solving that task, same as LLMs. For the N-th time, "LLM is as good at this task as a human who's bad at it" isn't a good selling point.

Re: Case study: Creative math – How AI fakes proofs

#62
I’m not a coder, but I’ve been working extensively on the philosophical aspects of AI. Many technical people are influenced by an algorithmic view of intelligence, primarily because this aligns with programming and the general understanding of reasoning. However, pattern recognition, which is fundamental to LLMs, is not algorithmic. Consider this: a LLM constructs a virtual textual world where landscapes and objects are represented as text, and words are the building blocks of these features. It’s a vast 700+D mathematical space, but visualizing it as a virtual reality environment can help us comprehend its workings. When you provide a prompt, you essentially direct the LLM’s attention to a specific region within this space, where an immense number of sentences exist in various shapes and forms (textual shapes). All potential answers generated by the LLM are contained within this immediate landscape, centered around your prompt’s position. They are all readily available to the LLM at once.

There are certain methods (I would describe them as less algorithmic and more akin to selection criteria or boundaries) that enable the LLM to identify a coherent sequence of sentences as a feature closer to your prompt within this landscape. These methods involve some level of noise (temperature) and other factors. As a result, the LLM generates your text answer. There’s no reasoning involved; it’s simply searching for patterns that align with your prompt. (It’s not at all based on statistics and probabilities; it’s an entirely different process, more akin to instantly recognizing an apple, not by analyzing its features or comparing it to a statistical construct of “apple.”)

When you request a mathematical result, the LLM doesn’t engage in reasoning. It simply navigates to the point in its model’s hyperspace where your prompt takes it and explores the surrounding area. Given the extensive amount of training text, it will immediately match your problem formulation with similar formulations, providing an answer that appears to mimic reasoning solely because the existing landscape around your prompt facilitates this.

A LLM operates more like a virtual reality environment for the entire body of human-created text. It doesn’t navigate the space independently; it merely renders what exists in different locations within it. If we were to label this as reasoning, it’s no more than reasoning by analogy or imitation. People are right to suspect LLMs do not reason, but I think the reason (pun intended) for that is not that they simply do some sort of statistical analysis. This "stochastic parrots" paradigm supported by Chomsky is actually blocking our understanding of LLMs. I also think that seeing them as formidable VR engines for textual knowledge clarifies why they are not the path to AGI. (There is also the embodiment problem which is not solvable by adding sensors and actuators, as people think, but for a different reason)

Re: Case study: Creative math – How AI fakes proofs

#63

This is the classic 'plausible hallucination' problem. In my own testing with coding agents, we see this constantly—LLMs will invent a method that sounds correct but doesn't exist in the library. The only fix is tight verification loops. You can't trust the generative step without a deterministic compilation/execution step immediately following it. The model needs to be punished/corrected by the environment, not just…

Yes, and better still the AI will fix its mistakes if it has access to verification tools directly. You can also have it write and execute tests, and then on failure, decide if the code it wrote or the tests it wrote are wrong, snd while there is a chance of confirmation bias, it often works well enough

> decide if the code it wrote or the tests it wrote are wrong

Personally I think it's too early for this. Either you need to strictly control the code, or you need to strictly control the tests, if you let AI do both, it'll take shortcuts and misunderstandings will much easier propagate and solidify.

Personally I chose to tightly control the tests, as most tests LLMs tend to create are utter shit, and it's very obvious. You can prompt against this, but eventually they find a hole in your reasoning and figure out a way of making the tests pass while not actually exercising the code it should exercise with the tests.

Re: Case study: Creative math – How AI fakes proofs

#64
post #19

Somewhat ironic that the author calls out model mistakes and then presents https://tomaszmachnik.pl/gemini-fix-en.html - a technique they claim reduces hallucinations which looks wildly superstitious to me. It involves spinning a whole yarn to the model about how it was trained to compete against other models but now it's won so it's safe for it to admit when it doesn't know something. I call this a superstition beca…

Is there a term for "LLM psychology" like this? If so, it seems closer to a soft science than anything definitive.

Re: Case study: Creative math – How AI fakes proofs

#65
it seems to me like this is very much an artefact of the left-to-right top-down writing method of the program. Once its committed to a token earlier in its response it kinda just has to go with it. Thats why im so interested in those LLM models that work more like stable diffusion, where they can go back and iterate repeatedly on the output.

Re: Case study: Creative math – How AI fakes proofs

#66
post #19

Somewhat ironic that the author calls out model mistakes and then presents https://tomaszmachnik.pl/gemini-fix-en.html - a technique they claim reduces hallucinations which looks wildly superstitious to me. It involves spinning a whole yarn to the model about how it was trained to compete against other models but now it's won so it's safe for it to admit when it doesn't know something. I call this a superstition beca…

Thanks for the feedback. In my stress tests (especially when the model is under strong contextual pressure, like in the edited history experiments), simple instructions like 'if unsure, say you don't know' often failed. The weights prioritizing sycophancy/compliance seemed to override simple system instructions. You are right that for less extreme cases, a shorter prompt might suffice. However, I published this verbo…

You could try replacing "if unsure..." with "if even slightly unsure..." or so. The verbosity and anthropomorphism is unnecessary.

Re: Case study: Creative math – How AI fakes proofs

#67
post #54

Earlier quoted context omitted.

Definitely not. "Probability" does not mean "maybe yes, maybe not, let me assign some gut feeling value measuring how much I believe something to be the case." The mathematical field of probability theory has very precise notions of what a probability is, based in a measurable probability space. None of that applies to what you are suggesting. The Riemann Hypothesis is a conjecture that's either true or not. More pre…

It's time that mathematics need to choose it's place. Physical world is grainy and probabilistic at quantum scale and smooth amd deterministic at larger scale. Computing world is grainy and deterministic at its "quantum" scale (bits and pixels) and smooth and probabilistic at larger scale (AI). Human perception is smooth and probabilistic. Which world does mathematics model or represent? It has to strongly connect to…

Please elaborate what “quantum scale” means if possible.

Re: Case study: Creative math – How AI fakes proofs

#68

You don’t need a test to know this we already know there’s heavy reinforcement training done on these models so it optimizes for passing the training. Passing the training means convincing the person rating the answers and that the answer is good. The keyword is convince. So it just needs to convince people that’s it’s right. It is optimizing for convincing people. Out of all answers that can convince people some can…

Yet people often forget this. We don't have mathematical models of truth, beauty, or many abstract things. Thus we proxy it with "I know it when I see it." It's a good proxy for lack of anything better but it also creates a known danger: the model optimizes deception. The proxy helps it optimize the answers we want but if we're not incredibly careful they also optimize deception. This makes them frustrating and poten…

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Re: Case study: Creative math – How AI fakes proofs

#69

This is the classic 'plausible hallucination' problem. In my own testing with coding agents, we see this constantly—LLMs will invent a method that sounds correct but doesn't exist in the library. The only fix is tight verification loops. You can't trust the generative step without a deterministic compilation/execution step immediately following it. The model needs to be punished/corrected by the environment, not just…

So you want the program to always halt at some point. How would you write a deterministic test for it?

Re: Case study: Creative math – How AI fakes proofs

#70
post #66

Earlier quoted context omitted.

Thanks for the feedback. In my stress tests (especially when the model is under strong contextual pressure, like in the edited history experiments), simple instructions like 'if unsure, say you don't know' often failed. The weights prioritizing sycophancy/compliance seemed to override simple system instructions. You are right that for less extreme cases, a shorter prompt might suffice. However, I published this verbo…

You could try replacing "if unsure..." with "if even slightly unsure..." or so. The verbosity and anthropomorphism is unnecessary.

That's not obviously true. It might be, but LLMs are complex and different styles can have quite different results. Verbosity can also matter: sheer volume in the context window does tend to bias LLMs to follow along with it, as opposed to following trained-in behaviours. It can of course come with it's own problems, but everything is a tradeoff.
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