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A little background:

Almost two months ago I started to seriously self-study mathematics and so I searched the web for the best first book to expose myself. I found the following invaluable resources:

http://www.stumblingrobot.com/best-math-books/

https://hbpms.blogspot.com/

and based on the above websites I decided to go with Velleman's How to Prove It. This was the first time that I was seeing proofs.

My problem:

From my experience on this website it seems that my proofs are too pedantic or wordy. But in the above mentioned book it seems that the author emphasizes such proofs. So I am really confused!

Here are some of the examples that I was told my proofs were too pedantic:

Suppose $\{A_i | i ∈ I\}$ is an indexed family of sets and $I \neq \emptyset$. Prove that $\bigcap_{i\in I}A_i\in\bigcap_{i\in I}\mathscr P(A_i)$.

Suppose $A$, $B$, and $C$ are sets. Prove that $C\subseteq A\Delta B$ iff $C\subseteq A\cup B$ and $A\cap B\cap C=\emptyset$.

Prove that for any family of sets $\mathcal F$, $\bigcup!\mathcal F=\bigcup\mathcal F$ iff $\mathcal F$ is pairwise disjoint.

In the second question the answer by halrankard really opened my eyes to a whole new world. That I should try to work at the level of sets. From then on I tried to do exactly that but sometimes I really have a hard time doing it or simply I cannot see it.

In the third question as in many others the answer by Brian M. Scott helped me to see how I was wordy about a certain problem but in general whenever I try to prove statements from the above mentioned book my proofs automatically become too pedantic. I simply do not know which parts of my proofs are redundant.

How can I fix this problem? Is it too soon to fix this problem? Does everybody experience such a problem when they are at the beginning of the road?

Thanks for your attention.

Edit:

I was going to accept the answer by Mike but since the answer by CogitoErgoCogitoSum was controversial I decided to put a bounty on my question to see more perspectives.

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    $\begingroup$ I think it's only natural to have some "teething" problems when it comes to starting to write formal proofs, just like with any skill. But keep going at it! $\endgroup$ Jul 28, 2020 at 20:41
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    $\begingroup$ Writing proofs is something that takes a considerable amount of time, practise and mostly experience. It's not something that comes over night. It also highly depends on your targeted audience and your "mathematical maturity" in the respective area. $\endgroup$
    – Zest
    Jul 28, 2020 at 20:42
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    $\begingroup$ @Riemann'sPointyNose Isn't this "teething" a natural part of mathematics in general? $\endgroup$ Jul 28, 2020 at 20:45
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    $\begingroup$ In my experience, which is pretty extensive, it’s a fairly common problem. It’s by no means the worst problem to have: first, it’s far better than the opposite problem of not saying enough, and secondly, students who have it generally have a pretty good understanding of the material and are able to recognize when they don’t understand something. At this point I’d say that you should simply keep working at it; I’ve not been paying close enough attention to be sure, but I think that you’ve been improving. $\endgroup$ Jul 28, 2020 at 20:45
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    $\begingroup$ @KhashayarBaghizadeh Indeed for sure yes :) learning Mathematics takes time, but it's definitely worth it. If not for practical reasons - just for the enjoyment of it $\endgroup$ Jul 28, 2020 at 22:15

4 Answers 4

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I've looked at your three proofs, but I only analyzed the first one very closely (since the answerers of the last two provided detailed comments on your proofs). I've added an answer to the first question.

You'll notice that in my answer I use a very common lemma: $B \subseteq \cap_{i \in I} A_i$ iff $\forall i \in I \ B \subseteq A_i.$

Everyone learns these kinds of lemmas eventually, usually from reading proofs that use them. Sometimes one discovers them on one's own, but this usually ultimately depends on inspiration from encountering broadly similar arguments in other people's proofs first.

I'm not very familiar with Velleman's book, but from looking at some of it casually, it seems that most of the arguments presented go back to the level of elements rather than using any kind of higher-level lemmas on sets. So you can hardly be blamed for reproducing the same style of proof the author uses.

Your proofs will naturally become more sophisticated when you start reading more sophisticated mathematics. In the meantime, you're doing the right thing by breaking things down so that you understand every detail of a proof. That's the main thing.

Another way you can improve your proofs is by selecting textbooks or problem books with full solutions. That way you can compare your solution with the book's. You seem to have the discipline to do things on your own before looking at a solution, so this is likely to be a help to you, not a hindrance.

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  • $\begingroup$ Thank you. :$)$ $\endgroup$ Jul 29, 2020 at 23:56
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    $\begingroup$ I would write proofs in the way that makes sense to you. I wouldn't worry about it if other people find your proofs too wordy. Eventually you will reach a level of mathematical sophistication where you find your proofs to be too wordy. At that point, you will naturally start being less wordy. $\endgroup$ Jul 30, 2020 at 0:19
  • $\begingroup$ I agree with that. $\endgroup$
    – Mike
    Jul 30, 2020 at 4:59
  • $\begingroup$ @DanVelleman Your book is awesome. :$)$ $\endgroup$ Jul 30, 2020 at 9:32
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Here are my two cents. I've only read the second proof you linked to, and only the first direction of it, and I think it is not too pedantic at all.

The thing it, it all depends on how far gone you are in your mathematical studies and how used to you are, and how comfortable you're with, rigorous proofs. Since you are just beginning studying rigorous mathematics - and self-studying at that - I think it is actually crucial you begin by writing such "pedantic" proofs. Then, once you grow familiar with such proofs and become more confident, you can start writing more 'casual' proofs, because you'd develop an instinct that the proof is really correct and that you could make it completely rigorous if you needed to.

This doesn't stop at your level: the more you study, the less rigorous proofs becomes. And in fact, people often make mistakes, thinking something is obvious and that they could prove it completely rigorously if they wanted to, which then later turns out to be false. But that's just the way these things work, and you have to at least become a little more relaxed at proof-writing, or else you'd never have the time to prove anything more "serious", involving more complex mathematics.

One thing which is perhaps confusing about this is the feeling that you can always be more pedantic. And it's more or less true. I suggest reading about formal languages and formal proofs, if you find this point interesting/confusing. But in my opinion (and most people's) the level of rigor in the proofs you linked to (at least what I read) is enough. Why? Because, usually, when you write at such a level of rigor, even in the beginning of your studies, you don't make mistakes, thinking something is clear while it is in fact incorrect. That is, since your proofs are fairly rigorous (even if they could be even more rigorous), there aren't really too many "subtle" points you might be missing.

This is at least how I see it.

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  • $\begingroup$ Well, I do think there's an upper limit on pedantry: If your proof is computer-verifiable, you cannot make it even more pedantic, as computers are the biggest pedants imaginable. $\endgroup$
    – celtschk
    Aug 7, 2020 at 9:23
  • $\begingroup$ @eltschk Yeah, I agree. This is what I wrote "and it's more or less true" ;-) $\endgroup$
    – Cronus
    Aug 7, 2020 at 13:26
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Sharing: I've been given math classes for a long time, so what I'll say is only my own experience.

That "redundancy" you speaks of is as relative as it can possibly be. I'm researching for this exact situation because my work now have to do with the elaboration of materials for students to work basically alone, for themselves, at home, with minimal interaction with tutors. And the answer so far is: there's no perfect way to elaborate any materials like that. There's no perfection in writing something that it will, 100% sure, be satisfactory to every reader. It may be never enough. In this case, I'm just like you. I have this need for detail, give the maximum number of information that is possible to make the reader understand every little thing about the demonstration. What isn't written anywhere is that when we "give" someone an answer for a problem, we give "our" experience with that exact problem, the way "we" saw it. And what the reader will do with that answer is HIS problem, isn't part of the solution that we proposed. Many times my friends ask me: there's any way to resume it, to make it simple? My answer is always: if you understand the answer, you can write it with your own words, put anything that you judge necessary and take off everything you judge that is too much.

But keep one thing in mind: you maybe never be able to know exactly what the person - the one that is asking the question - have on their baggage. So, always put the maximum information that you can, for the good sake of the answer. And when and if someone says it's pedantic, just ignore that, because he probably understood your answer and, in this situation, he fells able to rewrite more efficiently. It means that you have succeeded in this answer, it means a full victory.

That's my point of view, obviously... I remember using a textbook where the autor used to say "the proof is obvious" and skip to the next part, and it always made me feel so stupid for not seeing the "obvious" part. Today I make my own way through the "obvious" and rewrite anything that I judge that need more explanation, and my friends always want my notebooks for further instruction. I'm at some point that I can never throw away an old notebook because there's always someone that needs my notes. And they are always welcome, because math isn't easy anyway, and if I can help, I fell completely honored.

As for my students, I always ask for them to put the maximum information possible, and if they don't know exactly how to say something, that they should describe it with his own words, the best way they can, because with this material and if the answer went through a good place, we can use it and discuss in the class. I've been paying my price. My students sometimes put some long answers, some big text trying to proof their points. So the work reviewing my students classwork is huge... But I like it, and most of the time they understood the basics of it, so I have work to do at the class with them, time to polish the knowledge they have and make them feel comfortable with themselves when writing in math.

That's it. Sorry for the long outburst.

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  • $\begingroup$ Thank you. It was a beautiful answer. $\endgroup$ Aug 5, 2020 at 8:35
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No such thing as too pedantic. People are just illiterate and dont want to have to read. If it cant be squeezed into a 140 character tweet their attention span wont cut it. Id propose that this is most peoples motivation. Especially when, such as on a site like this, answering questions is a gamed proposition. People are quick to get through to the next one.

You want to be as rigorous as possible, and I dont see a distinction between rigor and pedantry, especially in proof-writing.

Isaac Newton, the Bernoullis, Gauss, the list goes on... these men get a lot of credit for a lot of things throughout history. But if you look at their work, especially the further back you go, the less their proofs withstand scrutiny against modern day rigor expectations. How many times have we as a community been forced to doubt and re-write anew proofs that were once thought valid?

And how many years will pass before the works written by us today no longer meet the standards of rigor? It makes sense to me to be as sure of our logic as possible by including as much detail as possible.

Its easy to overlook innocuous details that could invalidate a proof if they arent even written; the peer reviewer simply glosses over it accepting the statements on intuition and such. But if every bit of minutia is included, even if one person doesnt notice the leap in logic, surely another would; they'd be able to point out some limitation in a theorem, etc., but only because it was explicitly stated. The leap from A to C through an unmentioned B might seem obvious, even intuitive... but if you point out that B explicitly, it might just dawn on someone that that's a bad move. Just as an example.

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    $\begingroup$ Can the person who downvoted this explain themselves? This is a perfectly legitimate answer. $\endgroup$
    – Abhi
    Jul 30, 2020 at 0:44
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    $\begingroup$ @AbhijeetVats I downvoted because I strongly disagree with the answer, though I agree that it is an answer - although I think editing out the statement "People are just illiterate and dont want to have to read" would make the answer more appropriate to the site. Personally, I think proofwriting should mostly serve the reader and communicate an idea clearly - which often means eliding some details or sometimes avoiding technical language. One could write all their proofs in Coq or another proof assistant, assuring some mechanical rigor - but too much would be lost in doing so, I think. $\endgroup$ Jul 30, 2020 at 0:49
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    $\begingroup$ The writer is only responsible for the quality of his own work. The reader isnt obligated to read it, but he sure cant criticize it if he isnt going to. There is a certain intellectual honesty implied here, isnt there? Those who dont want to read the work are intellectually lazy. Any "dumbed down" statements are for people who really dont want to think too deeply about the subject: they are in no position to be reading proofs. They are there only for the final theorem. Its my assertion that EVERY statement in a proof/argument demands justification. Just me, I guess, but I call that rigor. $\endgroup$ Jul 30, 2020 at 1:09
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    $\begingroup$ On the other hand, there is a point to be made about clear proofwriting. It is entirely possible to be rigorous and clear at the same time. You wouldn't want to read a piece of work that is a messy hodgepodge of mathematical statements. It'd be hard to parse through it and certainly, a waste of the reader's time to do so. This isn't intellectual laziness. It is a demand for clear writing in the text. If the author has delivered on that, then they can be as rigorous as they want and I would be the one having to contend with the difficulty of the text. $\endgroup$
    – Abhi
    Jul 31, 2020 at 16:56
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    $\begingroup$ "How many times have we as a community been forced to doubt and re-write anew proofs that were once thought valid?" -- Not all that often, actually. Euler's solution of the Basel problem is a perfect example in my mind. Sure it's not rigorous, but it has "the right idea". Demanding everything be perfectly rigorous from the get-go is just not reasonable when non-trivial research is involved. Usually you have a vague notion and fill in the details until you're satisfied everything's been done. It's very easy to fail to see the forest for the trees if too much detail is included. $\endgroup$ Aug 1, 2020 at 0:00

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