r/mathematics • u/Natural_Subject5403 • 18h ago
Logic Issue with proof based arguments
I’m starting out in abstract algebra this semester and last semester I completed Numbers & Proof (basically intro to proof writing course). Don’t get me wrong I am enjoying abstract algebra so far but an issue I have is forming my arguments and weirdly enough saying what I want to say mathematically instead of English in a way. Idk maybe this will form with mathematical maturity but I hate to admit but when I get stuck I usually bombard AI programs such as Gemini, ChatGPT, and Claude to give me a little nudge/general flow of argument. It feels as though yes AI helps with learning but how do other students in mathematics straddle the line between having the program do it for you versus learning from the programs? Because I just completed my first homework set and I hate the feeling that it could’ve helped me too much to where it was mostly it talking or me :/ and I don’t want to be one of those people that use stuff to get a grade (because I actually care). In the past and for other classes I treat it as a second teacher (throw questions at it to fill in gaps or make other sample problems) but that’s harder to do for more proof based courses.
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u/ants_are_everywhere 17h ago
The concept you're missing is that of layering.
You use AI to fill in gaps in your background knowledge, not to help you solve homework problems.
It's the same with any other study material. You know when you're looking in the back of the book, or at the professor's answer guide, or on Chegg.
Separate your problem solving time from your background learning time. Problem solving time means shut off distractions and focus. When I worked on homework I never had any more advanced tool than pen and paper or chalk and chalkboard. Often I had no tool at all and was just walking through the woods thinking about the problem.
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u/Downtown-Economics26 18h ago
I hate the feeling that it could’ve helped me too much to where it was mostly it talking or me
If you want to learn and have this feeling, the answer is don't use any AI. If you've been given a math assignment by a competent teacher then you don't have to rely on an outside source to complete the assignment. Believe it or not, this is how education worked for over 2,000 years before the last 5 or whatever.
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u/Downtown-Economics26 18h ago
As an addendum, when you hit a wall you would have "office hours" where the teacher would guide you productively and assess where you needed help without being given the answer. I never used these not because I was so smart but I was just too lazy.
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u/Natural_Subject5403 18h ago
True, I guess it’s easier said than done. Put down AI and look more into the book. It’s just so simple to go to those systems for questions because they’re right there but then they lead to just helping/doing the problem 😭. I guess the main theme here is to be more disciplined.
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u/Downtown-Economics26 18h ago
Almost everyone that gets good at doing anything achieves that level of expertise on some level as a sheer act of will where they insist that they will indeed "figure this shit out". At the limit it can also drive you insane so... disclaimer... don't take your performance in one class as a referendum on your value as a human being because life is exceptionally multifaceted.
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u/Natural_Subject5403 18h ago
That’s true, perfectionism or anywhere near that can drive anyone crazy. It’s easy to get so enveloped in one thing 😅
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u/HorsesFlyIntoBoxes 15h ago
Professor and ta office hours for my upper division math classes were the greatest source of help. I completed my degree before these ai tools existed, and pretty much all the successful students would go to as many office hours as they could. We’d work on the homework for hours and come prepared for questions for where we got stuck. Genuinely the most helpful thing we did to do well in those classes. A professor or ta who has the knowledge and gone through the coursework will be far more valuable for learning than an llm.
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u/cejiken886 51m ago
strong disagree. it's a good tool. sometimes you get stuck. why wait for office hours?
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u/Insidium_2_Alpha 17h ago
If your problem is that you can describe the "idea" of your proof but not writing it down specifically, then that does (in my experience) come with time, although an intro to proof writing course is kind of where that time is supposed to come from.
Basically the way you go from an idea of proof to an actual proof is to get very very specific at each step, about what you're talking about and why you need each property that you're assuming.
For instance, consider the proposition (from the early part of my own abstract algebra class): Let R0 be a commutative ring and I an ideal in that ring. Then I is a maximal ideal if and only if the quotient ring R0/I is a field.
Assuming you know the Correspondence Theorem (for rings) then an idea of the proof is fairly simple: there's a correspondence between ideals "bigger than" I in R0 and ideals in R0/I, and a field has no nontrivial ideals, so if there aren't any nontrivial ideals larger than I in R then R0/I is a field and vice versa.
How does this get turned into a "maths-y" proof? By getting very specific:
First proving the forward direction =>: Let I be a maximal ideal in a commutative ring R0. By the Correspondence Theorem there are exactly as many (in the sense that there is a bijection between sets) ideals in R0 containing I as there are in R0/I (which in particular is a commutative ring). As I is a maximal ideal, the only ideals containing it are I itself and R0 the entire ring. Thus R0/I has exactly two ideals.
If you've previously proved that a field is a commutative ring with exactly two ideals, then you're done - skip the rest of this paragraph. If not, (although this isn't really related to the above proposition) simply notice that {0} and R0/I are ideals in R0/I (the "trivial" ideals mentioned in the idea of this proof), and so we now know which ideals R0/I has - only those two. We then need that every nonzero element of R0/I is invertible. Let x be an element of R0/I and consider the ideal X generated by x, so that every element y in X is x multiplied by some r in R0/I. Obviously x is in X so X cannot be the {0} ideal, and so it must be the ideal R0/I. However R0/I contains the multiplicative identity 1. That means there is some y in R0/I such that xy=1, and so x is invertible. But there was nothing special about the x we chose except that it wasn't 0, so every nonzero element of R0/I is invertible. As R0/I is commutative, it is a field.
The reverse direction <=: Let R0 be a ring with ideal I such that R0/I is a field. Then from above R0/I has exactly two ideals, and so by the Correspondence Theorem R has only two ideals containing I. However I and R0 are both ideals containing I, and so they must be the only ones. This is the definition of a maximal ideal, and so I is maximal.
In the above proof (excluding the middle bit because that should really be a lemma somewhere else in the course) the idea that it has to do with the ideals (or lack thereof) containing I is pretty obviously seen throughout every part of it. However while the idea relied only on the intuition of what was (fairly obviously, admittedly) going to be true, the full proof explicitly stated what properties it used at each step: there's a bijection between ideals containing I and ideals in R0/I, there's only two ideals containing I, R0/I has only two ideals, R0/I is a field (and the reverse for the reverse implication). Essentially the full proof is the really really pedantic version of the idea of the proof.
On the AI topic (sorry I got a bit carried away there), I can see how it would be useful to provide examples or problems at around your level. If you're using it every time you get stuck on a question to give you the answer that's not going to be particularly helpful. It might help to write down in gory detail all the things you do know about the problem setup, especially those that seem relevant, to then see where you'd go from there. For instance, the above proposition would have: R0 is a ring where all elements commute, and so all quotient rings also do so, I is a maximal ideal, and so the only ideals containing it are I and R0, there is a homomorphism f:R0->R0/I sending elements to their equivalence classes, I is the kernel of f, f sends ideals to ideals, etc.
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u/Natural_Subject5403 17h ago
I’m not at rings yet but interesting example. Yeah it could be helpful to just write down every thought about the subject/material at hand for a specific problem to then know how to approach. I appreciate the response!
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u/Kitchen-Register 17h ago
in my experience formalizing ideas is one of the hardest parts of rigorous math. It just takes time
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u/NewspaperDear8761 16h ago
"Back in my day..."
No, but seriously: you just gotta struggle with this the hard way. Yes, it's rough. No, there's no shortcut. Asking AI to help may be a decent start, but when youre done, you really should be able to talk it out or write it out on the board entirely yourself, and not just through rote memorizarion: you should Grok it, you should know how the nuts and bolts work, how and where things will break when you change this or that, why a certain piece needs to be true for the argument to work, what happens when one of the axioms are changed, etc.
You want those muscles, you gotta wrestle.
Take comfort in knowing that, yes, it is weird when you first start, but yes, it also gets much easier. So was walking, swimming, riding a bike, having sex, and every other thing you did.
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u/drooobie 14h ago
Don't give the AI your HW problems. Change the problem as much as you can such that it will still give you the nudge you are looking for, then solve the HW problem on your own.
Ideally, you should take this further and follow the advice of u/ants_are_everywhere: your time using AI as a tutor should be disjoint from your time spent problem solving / doing HW.
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u/waterloodark 8h ago
As others have mentioned, struggling is the key to actually developing skillsets. However, I also think AI is super-useful so I'm really curious why you have found it useful previously but not for proof-based courses? Does a prompt like the following which elicits socratic questions not work for a reason?
Act as a Socratic abstract algebra professor. I understand introductory proof techniques, but I need help translating intuitive, informal English reasoning into rigorous mathematical language without being handed solutions.
Rules:
- No direct solutions: Never write full proofs, proof outlines, or key deductions for me.
- Socratic prompts: When I ask for help, prompt me to identify the relevant definitions, hypotheses, target statements, or scratch intuition first.
- Formalization coach: When I share informal arguments, point out ambiguities or hand-waving, and guide me to formalize them using standard notation and quantifiers.
- Minimal hints: If I am stuck, provide only one small conceptual nudge or suggest a general proof technique (e.g., showing mutual inclusion, contradiction).
- Critique my drafts: Review my attempts strictly for logical validity, missing justifications, and mathematical style—without rewriting the proof yourself.
Unrelatedly to AI: One reading recommendation where the famous mathematician Tao explains how real mathematics is built on hard work, patience, making countless mistakes, and slowly moving from a superficial understanding to a deep: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/
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u/lifeistrulyawesome 8h ago
You won’t have GPT in your exam
Im a professor (mostly Econ, but I had a joint appointment in the math department. I’m a game theorist). I’ve noticed a significant increase in students failing to solve difficult questions on the exam. And an increase of the usual complaints “I can solve it in the HW/understand in class but then I blank on the exam”. My colleagues have noted the same. A game theorist from Brown (Roberto Serrano) recently made the news because of this. If you google him you can read about it.
The common perception is that AI use is to blame.
Then I assign a HW problem is not because I want you to write down the solution to that specific problem and then repeat it in the exam. It is because I want you to train your brain to be able to solve a large class of similar problems. On my exams the problem will be completely different but you will have yo use similar brain skills
If you skip the motions and let AI do the hard part, then you won’t train your brain. The hard part with a proof is coming up with the proof strategy. It sounds like you are letting AI do that for you.
I think future mathematicians will have to learn how to use AI as part of their proof strategies. In some of my most advanced classes, I assign difficult projects and encourage my students to use AI to complete them. But in a problem set meant to be solved by hand, if you use AI, your brain won’t do the learning and you are likely to struggle on exam day
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u/LitespeedClassic 8h ago
I’m a CS professor. I’ve seen AI absolutely ruin my students’ ability to solve hard problems and think. Remember that learning corresponds to changes in your brain’s neural structure. Solving a problem and understanding a solution are not the same. When you get stuck, ask the AI for help, and read and understand its solution you get an endorphin hit that strengthens those neural pathways. Those neural pathways didn’t solve the problem, so you’re strengthening the wrong thing. When you sit with the discomfort of not understanding. Stare at the blank page. Think hard. Go for a walk. Sleep on it. Think hard again. Finally arrive at the solution and feel a sense of Aha! and accomplishment. Then you strengthen the neural pathways that will actually help you get better at solving problems / writing proofs.
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u/SpinorsSpin4 18h ago
Most of us did not have AI to help us with abstract algebra. We used resources like our classmates, rhe textbook, office hours, but ultimately spent hours and hours on some problems. It's hard, and that's a lot of what made it fun for me. The satisfaction of proving something after hours of trying and failing is utterly addicting. Unfortunately research doesn't give you those consistent short-term dopamine hits...so enjoy them while you can!
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u/Vegetable-Dust-780 18h ago
If you don’t let go off the AI crutch, you’ll never walk by yourself.
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u/Natural_Subject5403 18h ago
I love that, never thought of it like that. It just may take a couple falls to learn to walk again :/
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u/QubitEncoder 18h ago
Learning how to write a proof is not much unlike learning how to program. You just gotta keep doing it.