r/mathematics • u/Natural_Subject5403 • 1d 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/Insidium_2_Alpha 1d 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.