Comment by ForgotMyUUID
1 day ago
I’m reminded of that famous debate between Poincaré and Hilbert at the International Congress of Mathematicians in Paris in 1900. It was then that everyone decided to follow Hilbert’s path, and proof came to be valued more than intuition. I think modern math at school and at applied university kind of lost this intuitive part.
I try to teach my students that mathematics is, first and foremost, a very precise language of communication. It’s sometimes amusing to ask those who don’t like math to do without it entirely, just to see how much harder it becomes to describe the things around them.
Second thing I tell them, formulas are the essence of mechanisms in their purest form. And in this form, they’re much easier to grasp and mentally manipulate. It always amused me, after taking a mechanics course, to imagine that for any formula, you could visualize a mechanism or process that implements it.
And third thing, I suppose, the ability to verify one’s own statements as proof. Although, of course, mathematicians would probably tear me apart here for my heresy:sorry, I’m not a mathematician, but an engineer. You can make mistakes by using incorrect assumptions, but at some point, analysis itself will show you that you were mistaken. There’s a wonderful book, How to Prove It by Daniel Velleman, which provides an introduction to proof for the uninitiated like me. I really enjoyed it.
As a math prof, I care(d) much more about proof than intuition, not because proof is more important, but exactly because intuition is (I'm a bit Chesterton-ish here haha). You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself. Whereas if you emphasize intuition, students won't have any idea of what a rigorous proof should be.
As someone who mostly only applies math, that strikes me as a peculiarly academic take. Intuition is more important for me because it’s what enables me to know what methods are most applicable to whatever practical problem I’m trying to solve. The proof’s purpose is to verify my intuition. It’s just a means to an end. I only take the time to do my own when I can’t confirm what I need from a textbook or paper.
> As someone who mostly only applies math, that strikes me as a peculiarly academic take.
Yeah I was talking strictly about preparing students to become pure mathematicians. No opinion here on other goals.
5 replies →
Love is more important than breathing. It is and it isn't.
What good is an end you can't reach, or worse, you can reach but it's wrong?
>You cannot do proof without intuition hence, if you emphasize proof, intuition will take care of itself.
This isn't always the case. Our algebra (or analysis) course focused a lot on proofs for the exam. The result was that a lot of people learned the proofs by heart.
I think intuition is hard to test in a way that feels 'fair'.
You can do it - I doubt you could have got a first when I was at Oxford just by learning and understanding the material, but you should probably have been able to get an upper second. The final part of every question virtually always involved insight, but you'd obviously then have to prove what that insight helped you understand.
If you give people questions like those, there is the risk of complaints about the university not having been taught the material for the exams I guess, or you might find that nobody can answer those harder intuition parts. Certainly most students at Oxford couldn't answer that many of them - you needed to answer about three 'final' parts out of about ten questions say in each three hour exam to get a first and perhaps about 20 percent of students got firsts?
1 reply →
Same here, but I didn't memorize the proofs, I tried to internalize their logic, so I could reconstruct them on demand by just thinking systematically. It did work for me pretty well on my real analysis final exam IIRC (27 years later).
Analogous to the Archimedean Property - there is no approach to teaching mathematics so intrinsically good that it cannot be done poorly enough to yield arbitrarily bad results.
Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that, and rightly so, certainly not in the way Poincaré was on about.
Maybe it's been done, but I'd like to see an LLM recreate Euclid from questioning without having seen it during training.
> Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition . . . and honestly I don't think many people believe that
Yes, I believe that, it's part of what I was implying (I believe the LLM weights have some internal representation of math in the same way brains do that allow them to produce proofs)
5 replies →
I think it's actually the opposite. Intuition is the only thing LLMs can do, which is why they are prone to hallucinating when they can't validate their intuition against reality.
There is an idea that human intuition, expertise, and critical thinking are largely pattern recognition. When you encounter a situation, your brain gives you a plausible starting point, based on what it has experienced before. You then continue with explicit reasoning, which is slow and inefficient, and try to validate your ideas. The more relevant the patterns you have learned are to the situation, the more likely you reach a useful conclusion.
LLMs are largely the same, except that they cannot learn from experience in normal usage. And except that they experience the world only through symbolic data, while the human brain has access to plenty of sensory data.
My hunch (or intuition, hah!) is that intuition is an instinctive mental shortcut required to navigate large problem spaces that can’t entirely fit into our heads.
Maybe LLMs do not need intuition because they can scale their “cognitive capacity” with hardware and brute force their way through these problem spaces.
4 replies →
>Surely this implies these LLM generated proofs require the LLMs to have mathematical intuition
Well yeah they do, obviously.
That has been done, like back in the 1960’s.
LLMs have LLM intuition, not human intuition. (See the movie Her.)
LLM cannot reinvent Euclid from scratch, but a larger system including LLM might.
It does not imply that. He is talking about how people do math. Intuition is what you use when deciding what to try and how to think about things.
Proof is the rigorous outcome.
LLM running probabilistic loop is different kind of process.
6 replies →
As a math professor, I care much more about the key idea, heuristics, and motivation than the proof. With the others in place the proof is clear, something an AI or a student can do.
Well, it's knowing when to push and when to not. You probably have an intuition for, I don't know, abstract algebra objects (I don't know your field of specialty :P), without needing to symbolically manipulate all of it, but you developed a deep intuition for them through many proofs and attempts at proofs with them.
1 reply →
That's leaning into engineering, away from math. Heuristics aren't always accurate. Math history before proof is the history of delusion. Idea, heuristics, and motivation aren't nearly enough for correctness outside of a sandbox.
To agree:
In my experience, proof is the gym reps that allows you to harness strong intuition elsewhere.
In practice as an engineer, intuition is far more useful, eg, being able to “feel” when something is off in our reasoning — but proofs are where I train those same sensibilities on “harder” problems, (eg) details about how to model identity, equality, and equivalence in a formal model.
Engineering is a religion based on faith. Math is the god you follow :-)
2 replies →
Similarly, programming is also a precise language of communication. Initially, we focused on direct machine behavior but every abstraction above the hardware (including assembly) has been to make that behavior legible to humans.
Developed notations and shared procedural abstractions have made thinking about computation more intentionally human and source control has established a protocol for conversing with other humans in the language of a program and changes to that program.
The moment just now feels like a neglecting of the idea of communication being central. If the program is a compile target but not sufficiently legible or if the conversation moves too quickly for us to keep up then we retain the effects of computation but loose its meaning as communication. We loose the understanding and the ability to develop and evolve further shared abstractions.
Open source programs could be more like motivated explanations of computation. For open source to survive, maybe we should start to make the distinction between free product distribution and programming as communication and community building.
This feels a bit too optimistic regarding programming. Or perhaps betrays an imprecise definition of programming. To me, programming is about expressing algorithms. But algorithms may be totally illegible and still work fine.
Math on the other hand is exclusively about being understood. It is ideas from math that made algorithms legible and thus made the act of programming an act of communication. If by programming you mean using notation and ideas that were borrowed from mathematics to specify algorithms, then for sure it’s communication, but only inasmuch as it was math first. If you mean only specifying algorithms, then no communication need take place; the executor of the algorithm will deterministically execute it irrespective of its ability to communicate.
Programming was never about communication. It was always about making the machine do the thing we want. Back in the day, a good game programmer knew which time intervals had writable video memory and which CPU cycles drew which scanlines, and spent more time rearranging the code to hit these timings than to write the actual algorithm. Later programmers (I hesitate to call them good) learned everything there is to learn about OS internals and wrote theoretically nonsensical and invalid code that still worked thanks to those internals, to save CPU cycles and especially memory use. And the next generation of programmers took the principles of late binding and abstraction to the logical extreme and created architectures that cannot be described in words anymore, only in diagrams - but are crazy good for code reuse, traceability and A/B testing.
> Open source programs could be more like motivated explanations of computation.
It is already that. Every time a method/function is created, a structure is defined, a variable is added, a file is created or renamed,… It’s all for the purpose of human communication. The computer only need binary in a single file.
But people feels like they should be able to jumpninto curl code without any understanding of networking, or linux code with no knowlede of computer architecture. Few code are meant for total beginners.
> There’s a wonderful book, How to Prove It by Daniel Velleman
the name sounds familiar but i don't think i have read that one, i did enjoy "introduction to mathematical reasoning" by eccles.
personally my relationship with mathematical proofs has been complicated. it took some work to understand basic proofs (dedekind cuts, ideas vs. instructions with mathematical notation), but all of the theory of computation proofs, which supposedly are difficult for many, were completely intuitively easy for me.
i think mathematicians are facing a similar confusion as computer programmers. the medium used to require precise thinking and the simple act of reading, writing and composing it was a mechanism for thinking and learning. in the llm era, the question is: should there be a new mechanism and if so, what should it look like?
People often hate math because it was not explained to them correctly, usually by people who are good mathematicians but know close to nothing about teaching.
It was so infuriating to see everyone in the class absolutely fail on a specific subject and the "teacher" assumed that everyone must be stupid then. No self reflection, no questioning himself why he is not getting gaussian distribution in marks, just straight Fs.
> usually by people who are good mathematicians but know close to nothing about teaching.
I higly doubt that. Maybe in university level courses. Most people’s only experience with mathematics is an elementary or high school teacher who were probably themselves at best mediocre at the subject. Simply because of selection factors. Those who are good at math are encouraged to go into STEM. There will be of course exceptions everywhere, but that is not what “usually” happens.
And thats just about being good at maths the school subject, which is distinct from being “ good mathematicians” the science / research topic. Mathematicians are few and far between, simply because it is a specialist subject. There just aren’t enough of them to go around for them to be the formative experience around math for most people.
I think the problem is partly circular. Most people do not like maths. This includes most primary school teachers - in places I know primary school teachers are not subject specialists so just reflect the population of those with the required level of education in terms of their attitude to maths.
If you do not enjoy a subject, any subject, you cannot make it fun for those you teach. In the case of maths specifically its pretty bad: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician%...
My daughter hated maths when I took her out of school at the age of nine. A few years later she was very good at it and enjoying maths and STEM subjects. When she went to a sixth form college[1] she liked it well enough to pick it as one of her A levels[2].
[1] https://en.wikipedia.org/wiki/Sixth_form_college
[2] https://en.wikipedia.org/wiki/A-level
[dead]
Another response to math that makes me sad: "I must be too stupid to understand this," "my brain is too small for this," etc. Different people say it for different reasons, but it's almost always in response to a hand-wavey explanation that doesn't makes sense to anyone not already in the know. Math is so much more about humility and skepticism than it is prodigy.
This is tricky, because, in fact, hard math having an intelligence floor is one of the nastier realities of the human condition. Anyone who is even quite intelligent but has really pursued the rigorous stuff, unless they are in fact a prodigy, eventually realizes they have an abstraction ceiling (and this term is a common one thrown around in people studying mathematics, because intelligence denial is so obviously false when you do hit your abstraction ceiling).
Most people are correct that they lack the intelligence / mind for a lot of hard math (even epsilon-delta proofs are enough to eliminate the majority of the population, no matter how good a teacher you are, and these are just basic undergrad calc).
And yeah, sure, people have different kinds of intelligence and such, but there is still a g-factor, and people of low intelligence almost universally can't do hard math, whereas most people who can do e.g. advanced undergrad math can generally do almost all other advanced undergrad fields reasonably well. The world isn't fair here.
25 replies →
> People often hate math because it was not explained to them correctly
Spoiler: this is also why mathematicians hate vibe-math. AIs are outright terrible explainers even when they do have a watertight logical argument—and honestly, this is the load-bearing seam.
It goes beyond "proof vs. exposition": the logical derivations AI comes up with fail to even qualify as human-directed proof because of how terrible they are (far below even the most novice mathematician doing their roughest work) at the exposition part.
> AIs are outright terrible explainers even when they do have a watertight logical argument
I think this only applies to cutting edge mathematics (novel proofs of hard problems). I have seen it reported more than once that such AI proofs are cumbersome to follow.
But in my experience, when it comes to explaining well-established math that is already in the training data, AIs can be very good teachers (at least with recent models). Especially if you use it along with a textbook and ask it about anything that might not be explained well in the textbook.
> AIs are outright terrible explainers
Gemini's explanations are very good.
a good teacher remembers the journey, not just the destination.
socratic method exists. almost none follows it.
>socratic method exists. almost none follows it.
I have a hatred for people who think they can use this method.
If used incorrectly which it is a great percentage of the time it confuses the student. The person employing the socratic method must actually know the answer and where the student is in their mind. Failure on either account makes it pointless.
Ask anyone unfortunate enough to ask for help on IRC
2 replies →
That only works if the one you're trying to guide can figure it out mostly on their own and is interested in cooperating. Aka does not work for anything below university level.
"How To Prove It" is used to initiate people. It was required reading for an introductory class on formal mathematics at university.
Intuition happens naturally and people are prone to inducting the wrong conclusions. Proofs provide a framework for rigorously analyzing drawn conclusions such that it can be used to build intuition in others. If math is about sharing the insights gained in a particular class of problems, proofs are the means to getting there.
> a very precise language of communication
I find that difficult to match to my own experience, in that there is seemingly endless domain specific notation that heavily obscures communication
[flagged]