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Comment by manfromchina1 | Hacker News Reader

I personally don't get this attitude. For one thing, nobody on earth can give you a 500-word summary of quantum physics in a way that lets you internalize even a tiny bit of it. That's not how you learn anything.

Here's how I do it: I open Baby Rudin (3rd ed.), second chapter, and read the main text - absolute brutality. I unpack almost every sentence with Claude/GPT until I finally get what's going on. No ELI5 nonsense, just examples and counterexamples galore while absorbing the techniques and the way of thinking in analysis/topology. How do I know I've learned the material? By solving every single problem in that chapter. Here's the thing, though: the problems in Rudin can be brutal and decoupled from what's in the text, so if you can handle them, you've definitely mastered the material. No 500-word summary of analysis here.

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Baby Rudin is a special textbook: it comes from a time when mathematics textbooks were not at all judged by their pedagogical value, but instead by their aesthetic appeal to mathematicians (a la Bourbaki).

If you actually want to learn analysis, there are a nearly infinite number of friendlier resources (e.g., Understanding Analysis by Abbott).

Learning to unpack difficult text on one’s own is a valuable skill. Research papers often require a similar amount of suffering, and at the frontier of knowledge, despite all the advances we’ve seen, LLMs seem to have absolutely no understanding or intuition. They are much better at things that have been expounded at length by humans before.

I would argue that if you’re going to use an LLM to make Rudin easier to understand, you are not learning how to absorb difficult material, nor are you learning analysis efficiently.

> I would argue that if you’re going to use an LLM to make Rudin easier to understand, you are not learning how to absorb difficult material, nor are you learning analysis efficiently.

I would argue that if he can solve the problems on his own, without LLM assistance, than he has mastered it.

That may be, but my point stands. It is neither learning to absorb difficult material nor efficient.

The same could be said for someone taking a university course where Rudin is the textbook, and they rely on the professor/TA to help them understand the book.

This is needlessly nitpicky. The stark reality is that the majority of undergrad math majors do not achieve what this person has merely be reading Rudin (assuming he is actually solving most/all the problems).

Your argument sound remarkably familiar. Humor me:

"For this invention will produce forgetfulness in the minds of those who learn to use it, because they will not practice their memory.“

-Socrates, on writing. From Phaedrus

Secondary source: https://www.historyofinformation.com/detail.php?id=3439

I don't think so. There are many objections to methods of making learning "easier" and the invalidity of one does not imply the invalidity of all of them.

We've also seen the effects of LLMs on learning, at least when used instinctively by students. An entire generation of students seems to have atrophied their critical thinking skills this way. Talk to 21-year-olds today and you'll see what I mean.

>There are many objections to methods of making learning "easier" and the invalidity of one does not imply the invalidity of all of them.

I am pointing out a pattern of technological hubris stretching thousands of years.

>An entire generation of students seems to have atrophied their critical thinking skills this way. Talk to 21-year-olds today and you'll see what I mean.

Ah, yes, the old "kids these days" routine... quite fond of it myself sometimes. But I find myself benefitted when I take care to learn from the young.

ChatGPT also offered this:

"For the correct analogy for the mind is not a vessel that needs filling, but wood that needs igniting - no more - and then it motivates one towards originality and instils the desire for truth. Suppose someone were to go and ask his neighbours for fire and find a substantial blaze there, and just stay there continually warming himself: that is no different from someone who goes to someone else to get some of his rationality, and fails to realize that he ought to ignite his innate flame, his own intellect, but is happy to sit entranced by the lecture, and the words trigger only associative thinking and bring, as it were, only a flush to his cheeks and a glow to his limbs; but he has not dispelled or dispersed, in the warm light of philosophy, the internal dank gloom of his mind."

-Plutarch, on listening https://books.google.com/books?id=0U-hsAonP1AC&lpg=PA50&dq=p...

I didn't read the whole passage, but it seems to be talking about the same thing.

You didn’t read it but I should?

Hey @turzmo I found your 21 year old and I agree with you!

woosh

Is that meaningfully different from the study methods of the past? That doesn't require an LLM, and using one risks correctness in exchange for speed. You might not even get that speed if you're stuck in the weeds often enough.

A decade ago, a google search for study guides written by another professor would have been slightly slower. A decade before that, you'd be even slower fumbling through several books. Every single word could at least be trusted. You don't get that from an LLM.

> Is that meaningfully different from the study methods of the past?

The fundamental service a teacher provides is personalized feedback, quickly identifying where you are stuck and focusing the explanations and exercises on that area, drastically increasing the speed and quality of learning versus the self-supervised route.

The lack of this closed loop effectively killed the high hopes that were placed in e-learning and MOOCs 15-20 years ago, TV learning in the 1960s and many other failed revolutions, seems every generation has its own version.

It appears to me LLMs have a real potential to close this loop and become the failed educational revolution of our own generation.

> personalized feedback, quickly identifying where you are stuck and focusing the explanations and exercises on that area,

This has been a huge blocker when I tried to study advanced math myself. Many of the exercise books don't have worked out answers, so often you're either stuck or you have to hunt a variety of sources online for solutions and advice. It kills flow.

If you have an actual mathematics professor to ask questions to then sure do that, but most people don't have that luxury. Also the error rate of frontier LLMs on textbook highschool/college level mathematics is going to be extremely low.

It is also significantly more engaging and fun.

>> the error rate of frontier LLMs on textbook highschool/college level mathematics is going to be extremely low.

The scientific basis for this statement is unclear.

AIME and HMMT are high school math olympiad competitions now used to benchmark AI. Frontier models score 95-99%

https://benchlm.ai/blog/posts/aime-hmmt-competition-math

They score 99% but then can’t explain _why_ their solution works with any accuracy. Not as useful for learning as one might think.

are you referring to this section? > Verification is different from generation: Models scoring 98 can solve problems but can't always explain why their approach works at the level a human mathematician would. It doesn't say it can't explain why with any accuracy, it just says it can't *always* explain at the level of a mathematician, but most of us don't have such a mathematician at our beck and call to answer our questions anyways (thinking of the perspective of a self-learner outside of formal education)

It’s not just math. Anecdotally, LLMs struggle the same way with software engineering where the code they write is correct (compiles and passes tests), but reasoning is wrong often enough to eliminate most trust in these models’ ability to explain codebases or even features they themselves produce. It’s not about the _level_ of the supposed intelligence where a model struggles to summarize things succinctly or simply enough (responding to the “pocket mathematician” comment) or can’t grasp certain concepts at all (if so, how tf is it able to apply them?). It’s that by their design LLMs have no concept of truth and no concept of causality. They guess with every single inference and it’s very hard as a user to understand which guesses are more or less certain, since, you know, confidence ratings aren’t part of these models’ design either.

>The scientific basis for this statement is unclear.

Is it? Actually this seems like one of the MOST clear aspects of LLM performance we can measure.

>most people don't have that luxury.

I wonder how long before LLMs need to cash in and even simple prompts are beyond the non-existent budget of a student. Which means it will need to be subsidized by education... and we're back in the teacher loop once again.

Or perhaps libraries. That would be neat.

>It is also significantly more engaging and fun.

That won't last long either. I remember when phone apps were the "engaging and fun ways to learn". Half life of 2 years, and we're already seeing people lose the fun factor.

When studying a mathematics text, it is good practice to question what you are reading, trying to prove things to yourself etc., which IMO makes the problem of trusted sources much less than it is in things which you are not able to verify e.g. historical accounts.

I certainly would only trust my textbook as the authoritative source, but i can see that in the absence of an expert teacher it's nice to have something that can critique a proof. Imo the fact that it's hard to verify that your own proof is correct is one of the main barriers in self-studying math, especially if one is at a level where one is not completely fluent in applying the various techniques. This also applies to judging answers to open-ended questions in any other field.

I'm a private math tutor specializing in exactly this sort of material, and I agree with this very strongly. Knowing what "counts" as a proof is one of the most common gaps I see in students who come to me after self-studying, and most students do need some back-and-forth with an expert to really get that skill down. I imagine that LLM's could be very helpful for this if they were used judiciously!

There's been research done (I know, big words -- don't have any links readily available), where it was pointing to the fact that what goes easily in (into the brain) also goes easily out, and conversely, what the brain requires effort and literal calories and energy to understand, stays there longer (investment must be rewarded / recouped somehow).

So while there's no doubt that facilitating learning is a net-positive, at some point it becomes a net-positive, I suppose -- your brain just chucks it out faster because it knows you can re-obtain the same information again since it worked so easily the first time. It doesn't know the difference between easy and hard, all it knows is how much effort it takes and how much reward (hormones) was generated for it all (to cement the habit/result).

Spaced repetition could address this, no? Also, the right example or metaphor can make a difficult concept both easier to understand and stickier in my mind which I think is different than something easy to learn.

I agree regarding metaphors, I myself aggressively utilise analogies with both myself and others, as a "digestive", but I feel like with LLMs we're treading on thin ice as it's not straightforward to classify a particular use case as one scaffolding learning with a metaphor, or just having the opposite effect where your brain "sails" in a faulty sea of "learning" while in reality there's no real work being done, not of the kind the brain needs to do in order to re-order all the synapses and own networks that ends being "knowledge" eventually.

Spaced repetition is a good way (perhaps even the optimal way) to get large amount of trivial memorization into your brain, but it not a good way to understand a complex subjects.

Memorizing one word in a foreign language is not that complex, nor hard. The tricky part about learning a language is that you have to memorize thousands of words, and the trickier part is that you have to retain most of those words over a long period of time. Spaced repetition helps by finding an optimal schedule to for the exact same activity as you would otherwise.

I also think that this broad claim (LLMs are good for learning) is also a testable hypothesis, and given the size of the AI industry I think it is reasonable to assume that AI companies would be eager to proof that LLMs are good for learning (and then pushing their own proprietary implementations into school systems and make more money).

The absence of evidence here feels very much like evidence of absence. At least my cynical view of capitalism tells me that if there was a good way to use LLMs to learn stuff, we would have research showing it, and AI companies would be waving that research all over our faces.

LLM have only been useful for ~12 months (arguably <6 months). Too early to throw the baby out with the bathwater...

LLM‘s have been useful for ~12 months for the past 3 years. I’m not buying this any more: https://hn.algolia.com/?dateEnd=1753056000&dateRange=custom&...

They released air tools but you’d rather use a hammer to drive nails. It reflects more on /you/ than on the technology.

What are you referring to with "they released air tools"? I am not sure I am able to interpret your argument in any meaningful way.

Your obvious use of an LLM to write this comment isn’t exactly confidence-inspiring that it’s helpful for learning to do things on your own. Seems like it’s evidence that LLM’s quickly become a crutch for even trivial efforts like posting internet comments.

Feel free to share your chat that has explanations for Baby Rudin.

I’d like to see what you consider helpful in this context.

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