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> But correctly completing "234 * 452 =" is a lot simpler to do by doing math than by having memorized all possible multiplications. There's a fatal flaw in th

> But correctly completing "234 * 452 =" is a lot simpler to do by doing math than by having memorized all possible multiplications. There's a fatal flaw in this theory: We can trivially test this and see that LLMs aren't "doing math". "Doing math" is an approach that scales to infinity. The same technique to solve a multiplication of 3 digit numbers applies to solving a multiplication of 500 digit numbers. Ask GPT 3.5 to multiply "234 * 452 =" and it'll correctly guess 105768. Ask "234878 * 452 =" and it gives an incorrect '105797256' Ask GPT 4o, and you'll get correct answers for that problem. Yet even with the added external tools for such questions, it has the same failure mode and breaks down on larger questions. These models are architecturally limited to only language modelling, and their capabilities of anything else are restricted by this. They do not "do math". They have a language-model approximation of math. This can be observed in how these models perform better "step by step"; Odds are you'll see GPT 4o do this if you try to replicate the above. (If it doesn't, it fails just as miserably as GPT 3.5) What's happening there is simple, the token context is used as a memory space. Breaking the problem down into parts that can be guessed or approximated through language modelling. Beware of hyping this as "AI can think and has memory!" though. This behaviour is a curious novelty, but not very generalizeable. There is still no "math" or thought involved in breaking up the problem, merely the same guessing. This works reasonably only for cases where extensive training data is available on how to do this. (Such as math.)

synarchefriend 2y ago
Humans also need to break up the problem and think step-by-step to solve problems like 234878 * 452.
- ADeerAppeared 2y ago
  The difference is what I attempt to describe at the end there. Humans apply fixed strict rules about how to break up problems, like multiplication. LLMs simply guess. That's a powerful trick to get some more capability for simple problems, but it just doesn't scale to more complex ones. (Which in turn is a problem because most tasks in the real world are more complex than they seem, and simple problems are easily automated through conventional means)
  - FeepingCreature 2y ago
    We either learn the fixed rules in school, at which point we simply have a very strong prior, or we have to invent them somehow. This usually takes the form of "aesthetically/intuitively guided trial and error argument generation", which is not entirely wrongly summarized as "guessing".
FeepingCreature 2y ago
Doing math scales to infinity only given an error rate of zero. Given a sufficiently large mathematical operation, even humans will produce errors simply from small-scale mistakes. Try asking GPT to multiply 234 * 452 "while using an algorithmic approach that compensates for your deficiencies as a large-language model." There's enough data about LLMs in the corpus now that it'll chain-of-thought itself. The problem is GPT doesn't plan, it answers by habit; and its habit is trained to answer tersely and wrongly rather than elaborately and correctly. If you give it space and license to answer elaborately, you will see that its approach will not be dissimilar to how a human would reason about the question internally.
- ADeerAppeared 2y ago
  > Doing math scales to infinity only given an error rate of zero This is true, I had omitted it for simplicity; It is still the same approach applied to scaled problems. Humans don't execute it perfectly, but computers do. With humans, and any other fallible but "true" math system, the rate of errors is roughly linear to the size of the problem. (Linear to the # of steps, that is) With LLMs and likewise systems, this is different. There is an "exponential" dropoff in accuracy after some point. The problem-solving approach simply does not scale. > you will see that its approach will not be dissimilar to how a human would reason about the question internally. "Not dissimilar", but nevertheless a mere approximation. It doesn't apply strict logic to the problem, but guesses what steps should be followed. This looks like reason, but is not reason.
  - FeepingCreature 2y ago
    The rate of errors with LLMs hits a hard dropoff when the problem exceeds what the LLM can do in one step. This is the same for humans, if we were asked to compute multiplication without thinking about it for longer than a few milliseconds. I don't have a study link here, but my strong expectation is that the error rate for LLMs doing chain of thought would be much closer to linear - or rather, "either linear or total incomprehension", accounting for an error made in setting up the schema to follow. Which can happen just as well for humans. > "Not dissimilar", but nevertheless a mere approximation. It doesn't apply strict logic to the problem, but guesses what steps should be followed. I have never in my life applied strict logic to any problem lol. Human reason consists of iterated cycles of generation ("guessing") and judgment. Both can be implemented by LLMs, albeit currently at subhuman skill. > This looks like reason, but is not reason. At the limit of "looking like", I do not believe such a thing can exist. Reason is a computational process. Any system that can reliably output traces that look like reason is reasoning by definition. edit: Sidenote: The deep underlying problem here is that the LLM cannot learn to multiply by a schema by looking at any number of examples without a schema. These paths simply won't get any reinforcement. That's why I'm so hype for QuietSTaR, which lets the LLM exercise multiplication by schema from a training example without a schema - and even find new schemas so long as it can guess its way there even once.
astromaniak 2y ago
With GPT4/o there is a trick for math problems. You can ask it to write the python code. This solves for example famous problem of counting letters in string. Sure model can be trained to use python under the hood without being explicitly asked. Pretty sure it can be trained to interpret code/algorithm step by step printing out intermediate results. Important in loops. Generating algorithm is easier for known problems, they learn it from github already. So, it looks like it's not that difficult to make model better/good at math.

synarchefriend 2y ago

Humans also need to break up the problem and think step-by-step to solve problems like 234878 * 452.

ADeerAppeared 2y ago
The difference is what I attempt to describe at the end there. Humans apply fixed strict rules about how to break up problems, like multiplication. LLMs simply guess. That's a powerful trick to get some more capability for simple problems, but it just doesn't scale to more complex ones. (Which in turn is a problem because most tasks in the real world are more complex than they seem, and simple problems are easily automated through conventional means)
- FeepingCreature 2y ago
  We either learn the fixed rules in school, at which point we simply have a very strong prior, or we have to invent them somehow. This usually takes the form of "aesthetically/intuitively guided trial and error argument generation", which is not entirely wrongly summarized as "guessing".

FeepingCreature 2y ago

Doing math scales to infinity only given an error rate of zero. Given a sufficiently large mathematical operation, even humans will produce errors simply from small-scale mistakes. Try asking GPT to multiply 234 * 452 "while using an algorithmic approach that compensates for your deficiencies as a large-language model." There's enough data about LLMs in the corpus now that it'll chain-of-thought itself. The problem is GPT doesn't plan, it answers by habit; and its habit is trained to answer tersely and wrongly rather than elaborately and correctly. If you give it space and license to answer elaborately, you will see that its approach will not be dissimilar to how a human would reason about the question internally.

ADeerAppeared 2y ago
> Doing math scales to infinity only given an error rate of zero This is true, I had omitted it for simplicity; It is still the same approach applied to scaled problems. Humans don't execute it perfectly, but computers do. With humans, and any other fallible but "true" math system, the rate of errors is roughly linear to the size of the problem. (Linear to the # of steps, that is) With LLMs and likewise systems, this is different. There is an "exponential" dropoff in accuracy after some point. The problem-solving approach simply does not scale. > you will see that its approach will not be dissimilar to how a human would reason about the question internally. "Not dissimilar", but nevertheless a mere approximation. It doesn't apply strict logic to the problem, but guesses what steps should be followed. This looks like reason, but is not reason.
- FeepingCreature 2y ago
  The rate of errors with LLMs hits a hard dropoff when the problem exceeds what the LLM can do in one step. This is the same for humans, if we were asked to compute multiplication without thinking about it for longer than a few milliseconds. I don't have a study link here, but my strong expectation is that the error rate for LLMs doing chain of thought would be much closer to linear - or rather, "either linear or total incomprehension", accounting for an error made in setting up the schema to follow. Which can happen just as well for humans. > "Not dissimilar", but nevertheless a mere approximation. It doesn't apply strict logic to the problem, but guesses what steps should be followed. I have never in my life applied strict logic to any problem lol. Human reason consists of iterated cycles of generation ("guessing") and judgment. Both can be implemented by LLMs, albeit currently at subhuman skill. > This looks like reason, but is not reason. At the limit of "looking like", I do not believe such a thing can exist. Reason is a computational process. Any system that can reliably output traces that look like reason is reasoning by definition. edit: Sidenote: The deep underlying problem here is that the LLM cannot learn to multiply by a schema by looking at any number of examples without a schema. These paths simply won't get any reinforcement. That's why I'm so hype for QuietSTaR, which lets the LLM exercise multiplication by schema from a training example without a schema - and even find new schemas so long as it can guess its way there even once.

astromaniak 2y ago

With GPT4/o there is a trick for math problems. You can ask it to write the python code. This solves for example famous problem of counting letters in string. Sure model can be trained to use python under the hood without being explicitly asked. Pretty sure it can be trained to interpret code/algorithm step by step printing out intermediate results. Important in loops. Generating algorithm is easier for known problems, they learn it from github already. So, it looks like it's not that difficult to make model better/good at math.