How Can I Improve My Proof Writing Without Feedback?

[futurebird]

My adviser asked me to study the first 50 pages of a book so I'm working the exercises. But they are all proofs, so I have no idea if I'm doing them correctly. I can't find any answers online, and even if I did, that would just tell me one way of proving it-- and there are, of course, many.

So, my question for more experienced math folks is how do you work on your own doing proofs when you don't have many ways to get feedback? I can bug my adviser a little but he is on vacation!

I guess I just don't have a lot of confidence in my proofs yet. They seem good enough for me, but I've often turned in a proof and gotten like 3-pages of notes about the mistakes I made back... so I just know there are mistakes. I need to work on being more careful about technical details.

Is it a waste of time to just write out my own proofs if I can't get feedback until fall?

 

[mrandersdk]

It is never a waste to write down your own proofs, but why not just try to post them in this forum, lots of people here will gladly help you when you show you have tried.

 

[ice109]

yea checking your own proofs is kind of like playing chess against yourself. it's tough.

 

[nicksauce]

I second mrandersdk's thought that you should post your proofs on the forum.

 

[n_bourbaki]

As it's your first time writing proofs, what is almost surely true is that you're not putting in enough details. A proof is not just a string of numbers and mathematical squiggles. First, make sure that your proof makes sense as a written piece of English. You can do this by giving the proof to anyone and asking them to read it out loud. The only bits they are allowed to stumble on are the symbols they don't recognise. Ideally, the person should be able to read the proof without having read the statement of the thing you're proving.

If your question is 'prove that every injection from a finite set to itself is a bijection', for example, you DO NOT write

f(x)=y

as your first squiggle. Start with 'Let f be a function from a set X to itself, and suppose that f is injective'. Now ask yourself honestly if that (or something similar) would have been your opening sentence.

Make sure every single implication is justified, writing out the reasons explicitly (and in plain English, or whatever your language of choice is) at each stage. No one should have to struggle to understand what the symbols mean or why you were allowed to manipulate them as you did.But mainly, keep at it - learning to write proofs out for others to follow is very hard to do.

 

[ice109]

As it's your first time writing proofs, what is almost surely true is that you're not putting in enough details. A proof is not just a string of numbers and mathematical squiggles. First, make sure that your proof makes sense as a written piece of English. You can do this by giving the proof to anyone and asking them to read it out loud. The only bits they are allowed to stumble on are the symbols they don't recognise. Ideally, the person should be able to read the proof without having read the statement of the thing you're proving.

If your question is 'prove that every injection from a finite set to itself is a bijection', for example, you DO NOT write

f(x)=y

as your first squiggle. Start with 'Let f be a function from a set X to itself, and suppose that f is injective'. Now ask yourself honestly if that (or something similar) would have been your opening sentence.

Make sure every single implication is justified, writing out the reasons explicitly (and in plain English, or whatever your language of choice is) at each stage. No one should have to struggle to understand what the symbols mean or why you were allowed to manipulate them as you did.But mainly, keep at it - learning to write proofs out for others to follow is very hard to do.

you know i read half of your post then had to take a leak and i thought "hmm i don't agree with that, too much rigmarole obfuscates the math and logic." then i came back and read the bolded part and now I'm sure i wholly disagree. now don't take this personally, I'm not attacking you. i just think that spelling out every detail is not only overly pedantic and lends it self to confusion but is also literally impossible. I've done this before in my own head when imagining myself trying to explain a proof to someone completely clueless; how far back do i have to go to get back to bare set theory. even at the elementary level all it takes is something as simple as one composite idea like compactness or path-connectedness before it's a two hour talk until they can understand. but what i mean by impossible is that if you insist on spelling it all out so that literally no one will struggle you have to go beyond naive set theory and maybe even to beyond axiomatic set theory to propositional calculus and then logic. ok maybe I'm exaggerating but it's not like the paradoxes of material implication are so easily explained away.

some things simply have to be, and should be intuited. a good test of a proof should be not the amount of words but the potency of those words.

so for you question i would simply state any surjective function is bijective on its image, or maybe every 1-1 function is 1-1 and onto on its image. yea that last one is very clear and cogent even if you didn't know the precise meanings of 1-1 and onto.

 

[futurebird]

ice109 I agree with what you are saying, in terms of how I come to understand ideas... but I'm not "advanced" enough yet to ignore the small stuff. I want to know how to do every little pedantic detail of proofs so that I can know that I can do it when I need to. Once I have more confidence, I may feel more comfortable with broader strokes in proofs. But, my background is in the arts, not in science. Working on my masters in math is a huge shift for me. I don't think it' so bad to become a bit of a nit-picker in the short term to avoid big mistakes. I look at the big picture easily, that's just my nature. It's my "safe zone" I think I need to get out of it to learn and grow... you know?

 

[ice109]

ice109 I agree with what you are saying, in terms of how I come to understand ideas... but I'm not "advanced" enough yet to ignore the small stuff. I want to know how to do every little pedantic detail of proofs so that I can know that I can do it when I need to. Once I have more confidence, I may feel more comfortable with broader strokes in proofs. But, my background is in the arts, not in science. Working on my masters in math is a huge shift for me. I don't think it' so bad to become a bit of a nit-picker in the short term to avoid big mistakes. I look at the big picture easily, that's just my nature. It's my "safe zone" I think I need to get out of it to learn and grow... you know?

you got into a math masters program with bachelors in art?

look i'll trade proofs with you. I've been looking for someone to do that with. you send me a proof, i'll check, i send you one and you check it. you tell me whatever book you're working from and i'll get it from the library. I'm completely serious too.

 

[n_bourbaki]

you know i read half of your post then had to take a leak and i thought "hmm i don't agree with that, too much rigmarole obfuscates the math and logic." then i came back and read the bolded part and now I'm sure i wholly disagree.

So you don't think that every implication needs to be justified? Note that I did not specify what was required to make that justification rigorous. That only comes with experience, and depends on whom you are writing the proof for. For one example, if I could assume a lot then I would not even bother mentioning that every compact subset of the real numbers is closed and bounded. But in other situations I might need to mention the theorem by name, and in some examples I might even need to prove it.

Your attempt at answering my 'example' was sufficiently way off the standard required of a proof, that that alone indicates I shouldn't take your opinion seriously.

 

[ice109]

So you don't think that every implication needs to be justified? Note that I did not specify what was required to make that justification rigorous. That only comes with experience, and depends on whom you are writing the proof for.

Your attempt at answering my 'example' was sufficiently way off the standard required of a proof, that that alone indicates I shouldn't take your opinion seriously.

ha and what was so non-standard about my proof? maybe i simply restated the question? hmmm i guess i should've stated that every function is onto on its image, and therefore an injective function is bijective on its image.

anyway, and this is just plain tit for your tat, you should always mention on which topology because we both know that every closed and bounded interval is not compact on the left hand topology

 

[futurebird]

you got into a math masters program with bachelors in art?

look i'll trade proofs with you. I've been looking for someone to do that with. you send me a proof, i'll check, i send you one and you check it. you tell me whatever book you're working from and i'll get it from the library. I'm completely serious too.

I'll take you up on that. I had to take some courses before they would let me in, but I've always loved math and I took math courses even when I was getting my BFA. So, I never fell too far behind. And I had good GRE scores. And I went to the program director and... well... begged.

I said PLEASE give me a chance! They did. Now I have a ton of work to do.

 

[ice109]

are you getting funded?