Studying Dose Mathematicians understand their books?

[homeomorphic]

this thread, possibly well intended, is absurdly unfocused. it is a little like bill cosby's album, "why is there air?" With all respect, I suggest deleting or closing it, as a stimulus to people to try harder to post meaningful threads. My apologies to the followers, as I know it is not intentionally useless, but it is useless all the same. Or maybe it belongs in a section devoted to rambling nonsense, rather than academic guidance. there is no guidance to be found here, nor any genuinely sought.

Well, it looks like Obis and I are reaching an agreement, and we're almost done, so it seems as though it will take care of itself. I don't think the discussion will last more than one or two posts.

I don't think that the discussion is useless. However, I do think that it is of LIMITED use because we are talking in very imprecise terms. That's why it has taken so long for us to discuss it and understand what was trying to be said.

The impulse not to discuss such things is, I think, a harmful one. Such a taboo may be part of what has landed us in the less than desirable state of affairs with regards to pedagogy that we see today. These things ARE important and should not be ignored. However, it is of limited use, since it is too removed from the actual practice of learning math, which involves a lot of specifics that we couldn't possibly begin to touch on here. But I do think it can help nudge people in the right direction, form opinions, and so on. It can be sort of a hint.

Another limitation of this thread is its long-windedness, and that is regretable, but I was not able to express myself more concisely enough and get the point across.

I've enjoyed it, though.

 

[homeomorphic]

But you can't read a solution manual only and learn something. Similarly, I think a mathematician should read a rigorous text, try to find the intuition himself, and only after he tried for a while and failed, he could check for the solution in the "solution manual".

You can treat the interpretation of each proof as an exercise. However, you only have to do a certain number of exercises to get enough practice. So, maybe you are given 200 exercises. 100 exercises give you plenty of practice to master the skills you are seeking. But maybe you still want to know the answers to the other 100 exercises. So, the smart thing to do is not to do all the exercises to save time. This is a bad analogy, but I think this conveys the idea. Part of your goal is to learn to read proofs and part of it is just to learn the subject. You shouldn't overemphasize learning to read proofs if the cost in terms of time is too great. Often, I find the subject matter SO interesting that it can be extremely frustrating not to be able to advance quickly in understanding the main ideas when I've already practiced reading proofs up the wazoo. I don't think you ever reach a point where reading very formal stuff is as easy as processing the intuitive arguments.

Edit: Actually, I think a mathematician should only read a rigorous text on subjects that are of the largest importance to him. Reading not-so-important subjects, it would be better to save time and read more intuitive books, since yes, it's much faster to read an intuitive book.

Ok, looks like we agree to some extent. All I am trying to say is that you want to be flexible. Sometimes, you might have to just entirely skip a proof to save time. Maybe you will come back to it later if you care about it enough. Sometimes, maybe you go through the proof just to practice your skills, to find out roughly what might be involved, even if it isn't really enlightening. Sometimes, you come up with your own intuition from a formal proof. Sometimes, you might just want an intuitive proof because you don't have time to get all the details. Sometimes, you ask the experts. How do you know when do you choose each of these options? There's no easy answer. What I'm just trying to point out is that all of these options. They all have their advantages and disadvantages. Just do what works.

I should also mention that mathematicians will go through different stages in their education, so their approach will vary over time. They will also change their approach depending on what subject they are studying and what their goal in learning it is.

If they wanted deep intuition, they wouldn't be happy with what they were given there.

Maybe, but in my opinion, this statement is an overgeneralization.

I don't think so. If they understood it deeply, they probably did enough work to have written their own book on the subject, fairly independently of the original.

 

[TMSxPhyFor]

this thread, possibly well intended, is absurdly unfocused. it is a little like bill cosby's album, "why is there air?" With all respect, I suggest deleting or closing it,

from the beginning, my question was crystal clear, and i tried a lot to avoid a side discussions, so I suggest closing it, becuase I got my answer long time ago, and which is:

No, none of the mathematicians can read and understand purely formal math books without any other help/sources unless they will spend 100 times more time that it should be spent on it trying to "guess", "build", "refract" or "reverse engineer" the hidden intuition between the lines.

Anyway, they still and will continue to write formal books, maybe they simply idealists...

Thank you for every person that shared his opinion.

 

[homeomorphic]

No, none of the mathematicians can read and understand purely formal math books without any other help/sources unless they will spend 100 times more time that it should be spent on it trying to "guess", "build", "refract" or "reverse engineer" the hidden intuition between the lines.

Ah, but they can read it and "understand" it, in the sense that they know what is being said on a superficial level. They can follow the logic perfectly well and even "know" the subject when they are done (on a superficial level, which is enough for some of them). But following the logic doesn't always mean they know why it works. Often, I suspect, they just don't care about understanding why it works.

Some mathematicians are much more interested in solving problems than they are in understanding theory. There's some wiggle room here. Not all of us can or should think in the same way, and doing problems often contributes to your understanding of the theory.

 

[Number Nine]

Ah, but they can read it and "understand" it, in the sense that they know what is being said on a superficial level. They can follow the logic perfectly well and even "know" the subject when they are done (on a superficial level, which is enough for some of them). But following the logic doesn't always mean they know why it works. Often, I suspect, they just don't care about understanding why it works.

Some mathematicians are much more interested in solving problems than they are in understanding theory. There's some wiggle room here. Not all of us can or should think in the same way, and doing problems often contributes to your understanding of the theory.

I would argue that "understanding" has as much, if not more, to do with the content of the logic than the motivation. A proof can be enlightening even if presented in formal logic, and the content of the proof can foster understanding of the subject. For example, many elementary proofs in number theory can be motivated perfectly well and explained very clearly, but very often they don't provide any real insight into why something is true. A few lines of complex analysis, even if densely and formally presented, will usually be more informative.

 

[Nano-Passion]

this thread, possibly well intended, is absurdly unfocused. it is a little like bill cosby's album, "why is there air?" With all respect, I suggest deleting or closing it, as a stimulus to people to try harder to post meaningful threads. My apologies to the followers, as I know it is not intentionally useless, but it is useless all the same. Or maybe it belongs in a section devoted to rambling nonsense, rather than academic guidance. there is no guidance to be found here, nor any genuinely sought.

I would contend that the debate between homeomorphic and others is co-essential to whether mathematicians actually understand their books. I've followed every post in this thread and have followed the thread of argument, you might see a connection if you do the same. Sometimes it doesn't do justice if you read a couple of posts and take things out of context.

Oh great guru, you know best for everyone what conversation is useful? Now that you have pronounced your unerring judgement, we shall all bow down and cease to be interested...

 

[homeomorphic]

I would argue that "understanding" has as much, if not more, to do with the content of the logic than the motivation. A proof can be enlightening even if presented in formal logic, and the content of the proof can foster understanding of the subject. For example, many elementary proofs in number theory can be motivated perfectly well and explained very clearly, but very often they don't provide any real insight into why something is true. A few lines of complex analysis, even if densely and formally presented, will usually be more informative.

I talked about that above a bit. For example, I mentioned Cohen's book. Indeed, the content of the logic is mostly what makes the book great. But the some of the same theorems could have been proven with much less illuminating methods. It's not just a single proof, but the way the whole subject is laid out. So, you can distinguish between the content of the logic and the content of the subject. If a proof doesn't give any insight into why it works, I always, by default, suspect that it is a bad proof, for which a more insightful proof can be found. 95% of the time, this suspicion turns out to be correct, if not more, in my experience. It's possible, though unfortunate, that, in some cases, there may be no such enlightening proof waiting to be found. But it's not at all easy to tell when that is.

 

[Obis]

No, none of the mathematicians can read and understand purely formal math books without any other help/sources unless they will spend 100 times more time that it should be spent on it trying to "guess", "build", "refract" or "reverse engineer" the hidden intuition between the lines.

I don't want to go to the beginning, however.. Yes, mathematicians can read and understand purely formal math books. Trying to "guess", "build", "refract" or "reverse engineer" the hidden intuition between the lines is an enjoying thing to do for some people (including me), and it's useful, since it develops very abstract, general mental abilities.

In short:

1. Intuitive mathematics is beautiful, insightful, natural, relatively easy to understand, it is practically never forgotten.
2. Formal, rigorous mathematics is precise, concentrated, also beautiful in a slightly different sense.

It is better to understand only intuitively, than to understand only formally. However, the true understanding is when you understand it both formally and intuitively, and these two understandings are coupled to each other.

 

[Sankaku]

It is better to understand only intuitively, than to understand only formally. However, the true understanding is when you understand it both formally and intuitively, and these two understandings are coupled to each other.

I would say that is a nice summary for the thread.

:-)