6 ms·
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
by bananaflag 15d ago
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.
- contubernio 15d agoAs 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.
- sigbottle 15d agoWell, 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.
- CrazyStat 15d agoIm glad you brought up abstract algebra—that was the one class in my math undergrad that I never developed an intuition for. I learned to do the proofs by pushing symbols around and putting bars on top of them but I never felt like I understood what was happening.
- lupire 15d agoThat'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.
- fidotron 15d agoSurely 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.
- watwut 15d agoIt 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.
- fidotron 15d agoThe parent comment literally said "You cannot do proof without intuition". Therefore, according to that logic, an entity producing proofs must have intuition. Edit to add: the parent commenter has now confirmed my interpretation of their statement.
- bunderbunder 15d agoYour unstated major premise here is that their intent was to make a universal statement about how proofs work and not just talking to humans about how they teach humans. That premise seems unlikely to be correct.
- fidotron 15d agoWhy? The entire subject of conversation is triggered by things which are not humans producing proofs. If it's possible for a machine to produce a proof without intuition then clearly a human could also do it too. (And in fact I'd argue I've seen many people like that, simply very good at pattern matching over memorised items).
- 15d ago
- zmgsabst 15d agoTo 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.
- lupire 15d agoEngineering is a religion based on faith. Math is the god you follow :-)
- zmgsabst 15d agoNot really: engineering is explicitly empirical compared to other fields — and mathematics serves as ontology for that experience. There’s not faith involved.
- oliculipolicula 14d agoThere could be meta-faith, if you insert engineering into Mathematics, physics, chemistry, astronomy, march in one front [lockstep]. Whichever lags behind is drawn after. Whichever hastens ahead helps on the others... --Karl Schwarzschild The problem or nonproblem before (elite) software engineers were pointed at rather bespoke conjectures, depending on one's specific denomination, was that the frontier mathematicians got too far ahead of the others to effectively drag them along (hence Tao's recent fundraising attempt using his one-off compressed-sensing work) There is also the Experience<->Understanding "wave equation" if you will, codified by the popular engineers' joke about how mistakes/bugs mediate the two Imho what academia+industry really need are GLM-wielding plumbers cheap but capable enough to find these abstraction leaks between silos. One taxes these plumbers so brutally that their clients can get by on basic tokens (morally speaking, so as not to drive demand in the farflung silos of billions bottles and babes) There were interdisciplinarian buzzwords but these did not live outside the grant proposal, and probably won't survive better under the reign of Pangram
- bunderbunder 15d agoAs 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.
- lupire 15d agoLove 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?
- deleted 15d ago[deleted]
- bananaflag 15d ago> 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.
- adastra22 15d agoIf what you teach is proofs, then wheat you will filter for are students who live proofs.
- bunderbunder 15d agoAnd if your job is to train people to become mathematicians, that is absolutely what you should be doing.
- adastra22 15d agoThe idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
- Aerroon 15d ago>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.
- bell-cot 15d agoAnalogous 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.
- jvvw 15d agoI 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?
- bradleyjg 15d agoAn aside, but tests should be that difficult. Otherwise you aren’t getting any signal at the high end. Even in the face of grade inflation that signal can be translated into grad school recommendations (in the uk case, getting all 10 might even garner a “really not bad.”)
- senderista 15d agoSame 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).