Dose Mathematicians understand their books?

[TMSxPhyFor]

That's why you need to read critically and not blindly trust the author. I personally think that anything can be understood infinitely well, everything has an explanation, everything has a reason, etc. Hence, if the author would say that, I wouldn't believe him.

No this why it's an author's problem, not reader's, he should gather this different point of views and light up the differences between them to raise up the reader's awareness, instead of wasting our time!

 

[Obis]

No this why it's an author's problem, not reader's, he should gather this different point of views and light up the differences between them to raise up the reader's awareness, instead of wasting our time!

If you don't understand someone, it's usually your own fault. If you don't like a particular book - just pick a different one.

I'd like to add a perfect quotation about blaming by Epictetus:

"To accuse others for one's own misfortunes is a sign of want of education. To accuse oneself shows that one's education has begun. To accuse neither oneself nor others shows that one's education is complete."

 

[TMSxPhyFor]

If you don't understand someone, it's usually your own fault. If you don't like a particular book - just pick a different one.

Sorry to say that but it is completely misleading, that means that all books & teachers are good, but we are stupid and can't understand them, and you lost in your words two main points:
+Time: why i should spend couple days to find the best book? if you can't write for "humans" so don't.
+The purpose of teaching is passing knowledge that is experience at the fastest way (especially at our days), and to do that you should pass to me all possible view points, otherwise you are subjective, and passing just information as math books do , even so you will be objective, but you will lose totally the point of teaching, so speaking with no offense; formal math books are only suitable to be sent with voyager hoping that Aliens will understand it.

 

[Obis]

Sorry to say that but it is completely misleading, that means that all books & teachers are good, but we are stupid and can't understand them

No, that doesn't mean that all books and teachers are good. But only the fact that YOU don't like a particular book doesn't make that book a bad one.

+Time: why i should spend couple days to find the best book? if you can't write for "humans" so don't.

By saying for humans, you mean for YOU really. So what you are trying to say is, if you can't write for me, then don't write at all! No offence, but that's a terrible attitude. And yes, spending a lot of time choosing a book is crucial.

formal math books are only suitable to be sent with voyager hoping that Aliens will understand it.

Once again, if YOU don't understand it, that doesn't mean that nobody understands it. Some people just might enjoy reading formally written books (my own personal taste asks for a little bit of intuition too though).

No offence, but it seems you base your judgments on your own emotions.

 

[TMSxPhyFor]

By saying for humans, you mean for YOU really. So what you are trying to say is, if you can't write for me, then don't write at all! No offence, but that's a terrible attitude. And yes, spending a lot of time choosing a book is crucial.

If you noticed the title of the post was if mathematicians understand their books, and all of the repliers including yourself stated that they still need intuition, the aim of the question for the first place was to understand if there is some people that can think completely formally without any intuition (as most math books), and it seems that there is non, so what i said has no relation to my emotions.

 

[Obis]

If you noticed the title of the post was if mathematicians understand their books, and all of the repliers including yourself stated that they still need intuition, the aim of the question for the first place was to understand if there is some people that can think completely formally without any intuition (as most math books), and it seems that there is non, so what i said has no relation to my emotions.

Reading formal books is not the same as thinking formally. Even if the book is written completely formally, due to the your ability to convert formality into intuition, you can still read it as an intuitive book.

 

[simplicity123]

You are overstating the problem.

Firstly, Mathematics is hard. Deep Mathematics takes a lot of thought and doing stuff i.e. not passively sitting their like an idiot wanting the knowledge to sink in. Not only that, but not knowing one thing can destroy your understanding of a proof.

Another thing is that thinking Mathematical takes time and I doubt you done any real Mathematics in your Physics degree. Most of it probably like herp derp that is true because some vague intuition or here is some baby proof of this.

If you noticed the title of the post was if mathematicians understand their books, and all of the repliers including yourself stated that they still need intuition, the aim of the question for the first place was to understand if there is some people that can think completely formally without any intuition (as most math books), and it seems that there is non, so what i said has no relation to my emotions.

You suffer from the fact that you studied Mathematical methods in Physics. Everything you learned is probably geared towards being some tool to use in Physics.

Also, the point of proofs is to do them. Not to read them. To answer your question. You get used to formal proofs and most of the time you can see what is going on. I can see what a group is? I can see what a person is claiming.

I can't believe that Helbert was thinking of vector space just as couple formal conditions that objects should satisfy, unless he has a chip instead of a human mind, for example I read a comment says that Grassmann him self who invented the External Algebra wrote couple hundred of pages trying to describe the "physical" intuition he used to build it, can anybody explain for me why other writers of Math and Physical books doesn't use his original ideas? or they are too smart for this? do you think that my IQ is not enough to become a theoretical physicist (as i want) if I can't understand those books as they written?

The point is your aren't a baby anymore. You have to engage your brain and not have people spell everything out to you. Which, is why formal proofs are used most of time. As the Mathematician should be able to work it. A book about Galois theory would expect the person to be able to figure out a proof that uses basic group theory. A book about differential topology would expect the person to be able to do set theory and know how to prove basic facts about topology.

How many hours are you putting in a week to read them? As Mathematics is painfully slow to learn. It takes me an hour to read 4 pages most of the time. Because I'm checking if I can do the proof, checking if I can do what is claimed and then working out examples.

I feel you are just scratching the surface and just trying to follow the reasoning infront of you. But, you need to do it. You need to live it. You can't just pick up an algebra book and read it like a novel. Which, is what you are trying to do. Also, please read if you got the prerequisites as if you are reading books on Differential topology yet you know no topology, then how the hell are you going to understand it? The same thing if I was going to pick up a book on Quantum field theory and knew no calculus.

P.S. There actually emotion when you figure it out. As most of the time you are like this is so clever. Today for example doing Group theory and just figured out how the proof the class equation for finite groups work. It's utter genius and sort of like light turned on and you can see how everything falls into place.

 

[TMSxPhyFor]

You are overstating the problem.

Firstly, Mathematics is hard. Deep Mathematics takes a lot of thought and doing stuff i.e. not passively sitting their like an idiot wanting the knowledge to sink in. Not only that, but not knowing one thing can destroy your understanding of a proof.

Another thing is that thinking Mathematical takes time and I doubt you done any real Mathematics in your Physics degree. Most of it probably like herp derp that is true because some vague intuition or here is some baby proof of this.


You suffer from the fact that you studied Mathematical methods in Physics. Everything you learned is probably geared towards being some tool to use in Physics.

Also, the point of proofs is to do them. Not to read them. To answer your question. You get used to formal proofs and most of the time you can see what is going on. I can see what a group is? I can see what a person is claiming.


The point is your aren't a baby anymore. You have to engage your brain and not have people spell everything out to you. Which, is why formal proofs are used most of time. As the Mathematician should be able to work it. A book about Galois theory would expect the person to be able to figure out a proof that uses basic group theory. A book about differential topology would expect the person to be able to do set theory and know how to prove basic facts about topology.

How many hours are you putting in a week to read them? As Mathematics is painfully slow to learn. It takes me an hour to read 4 pages most of the time. Because I'm checking if I can do the proof, checking if I can do what is claimed and then working out examples.

I feel you are just scratching the surface and just trying to follow the reasoning infront of you. But, you need to do it. You need to live it. You can't just pick up an algebra book and read it like a novel. Which, is what you are trying to do. Also, please read if you got the prerequisites as if you are reading books on Differential topology yet you know no topology, then how the hell are you going to understand it? The same thing if I was going to pick up a book on Quantum field theory and knew no calculus.

P.S. There actually emotion when you figure it out. As most of the time you are like this is so clever. Today for example doing Group theory and just figured out how the proof the class equation for finite groups work. It's utter genius and sort of like light turned on and you can see how everything falls into place.

Dear simplicity123, I don't know from where to start becuase we already discussed most of what you said.

First of all once i spent a complete year trying to visualize linear algebra's highly formal book of 300 pages becuase i hadn't any other choices, even Internet or teacher.
Secondly I'm spending 80% of my time trying understand math, and this what really makes me mad, and pointing that i wanted to became a mathematician at first place, believe me most of what i like is digging as deep as possible, becuase this deepness is the ultimate beauty for me.
Thirdly as I already mentioned is time, what is the logical mystery that forcing me to not read a book of math as a novel? and to spend a day to "reverse engineer" 4 pages of it? even so trying proving theorems by your self is very useful, but it is not what we are talking about here, let me just ask you one question:

Do you really think that when the below two geniuses said those my favorite quotes made a great mistake?

"We invent by Intuition, we prove by logic" , don't be surprised, it's Poincaré
"Any idiot can make simple thing look complex, but you need a genius to make complex thing look simple", yea it's Einstein.

P.S. There actually emotion when you figure it out. As most of the time you are like this is so clever. Today for example doing Group theory and just figured out how the proof the class equation for finite groups work. It's utter genius and sort of like light turned on and you can see how everything falls into place.

I know this feeling, but you know maybe if it was now like 18'th century most probably i will try to proof every single theorem by my self, but at current time, when we supposed to know 10000 times more to be able at least follow (I'm not saying invent) the progress of your field, this becomes physically impossible.

 

[atyy]

micromass: Two math books from hell
https://www.physicsforums.com/blog.php?b=3059 !

I wonder if micromass would agree with Arnold's amusing http://pauli.uni-muenster.de/~munsteg/arnold.html ?

 

[homeomorphic]

That's why you need to read critically and not blindly trust the author. I personally think that anything can be understood infinitely well, everything has an explanation, everything has a reason, etc. Hence, if the author would say that, I wouldn't believe him.

Actually, I don't think everything has a good intuitive explanation. No one seems to understand the classification of finite simple groups. The proof evidently consists of thousands of pages. In principle, maybe it's understandable in the sense that if you had an unlimited amount of time, maybe you could understand it. But I'm not sure about that. It's hard to think of other good examples right now, but there are limits to human understanding. Some things that have been shown to be true might not really be understandable. But in that case, there's no point in learning the proof, unless you are actually in doubt as to the veracity of the statement and wish to check it for yourself. But from a learning point of view, it's a waste of time.

But you have the right attitude, there. Usually my reaction IS to be optimistic, but not unrealistically optimistic about being able to understand things deeply. So, if an author presents things in a way that seems too opaque, my warning signals will flash very strongly and if I can't figure out a better explanation myself, I'll keep looking until I have looked at every available book that covers the material. And if that doesn't work, I'll keep thinking about it. I might set it aside even for years, but I never feel like I am done with my work until I have gotten to the bottom of it. There are many such things I will probably never get to the bottom of, though.

Also, there are times when even I think too formally, and I just don't know that I'm doing it. It happens to me, and I'm sure it happens to everyone else. The risk of that happening is much, much greater if the books you read are too formal.

This is true. However, as I mentioned before, the intuition you found yourself, you've built yourself is even stronger, even more natural to you. The very act of building intuition, building mental models improves the ability to do it, which is the most important thing when learning mathematics, at least in my current point of view.

You can get plenty of practice building your own intuition without spending all your time decoding it from formal proofs. You can prove your own theorems, even. I don't really think it matters whether you come up with it yourself or someone tells you it, as far as understanding the particular thing they are trying to tell you. As far as getting practice, yes, it's better practice to come up with it yourself. But that's a different goal than just trying to learn such and such proof or definition or whatever specific thing it is. Often, you can do a better job than what you are told, but if whoever was telling you did a better job from the beginning, the effect would be the same. Still, of course, you are a different person from them, so you have to think about it critically. Just because they don't force you to decode their formal proof doesn't mean you just sit back and expect to learn it automatically. You still have to mull it over for yourself until it's clear. Maybe they have more intuition built up than you do, so you have to spend more time filling in the gaps that they didn't tell you.

No, that doesn't mean that all books and teachers are good. But only the fact that YOU don't like a particular book doesn't make that book a bad one.

The fact that a particular book didn't take human psychology into account DOES make it a bad book, though.

By saying for humans, you mean for YOU really. So what you are trying to say is, if you can't write for me, then don't write at all! No offence, but that's a terrible attitude. And yes, spending a lot of time choosing a book is crucial.

No, he doesn't mean himself. We evolved out there in the wild. We evolved to do things like hunt, socialize, cooperate, get away from danger, and so on. Writing wasn't even invented until relatively recently in our species history. So, it's no surprise our brain deals better with things that we call "intuitive", like pictures, pushes and pulls, sense of motion and touch. Our brain evolved to process that kind of information, not symbols. It's just foolish not to take full advantage of that and to communicate with that in mind.

Once again, if YOU don't understand it, that doesn't mean that nobody understands it. Some people just might enjoy reading formally written books (my own personal taste asks for a little bit of intuition too though).

It's not just about whether you understand it or not. I can read math better than 99% of the population, including formal math books. But just because I was able to tease the intuition out of it after 100 times as much effort as it COULD have taken doesn't mean that it was well-written. The bottom line is that this type of thing is going to have a negative impact on the efficiency with which people can learn the stuff. And there's a great risk that the quality of understanding will also be compromised as well.

No offence, but it seems you base your judgments on your own emotions.

You too.

Objectively speaking, I have a good reason to believe that the continued over-use of excessively formal books will have a negative impact on how much and how well people will be able to learn, not to mention their enjoyment. I mean, having the experience of reading two books and having one make complete sense and the other wasting your time is pretty good evidence that one book was better than the other at least for you, and the reasons that was the case are reasons that seem to apply to other people, as I mentioned.

Emotion may be part of the point, actually. Why am I a mathematician? Is it because I love boredom? Is it because I am a masochist? No! I want to be entertained when I learn math. That's the reason I do it, and it has the potential to entertain if explained properly. So, regardless of what the "right" way to do it is, my goal is being thwarted.

 

[Obis]

You can get plenty of practice building your own intuition without spending all your time decoding it from formal proofs.

Well, I believe, that at least for me, reading formal proofs and finding intuition in them is the most effective way to build my general ability to understand intuitively. What is more, I only trust my knowledge about a definition when I know the formal definition for it, and when I have built the intuition, a mental model that is based on that formal definition. If I only know the intuition, that has been simply given to me - I won't trust that knowledge, and I wouldn't feel comfortable using that definition.

The fact that a particular book didn't take human psychology into account DOES make it a bad book, though.

Different persons have different psychologies. You can't take into account all the possible psychologies. The same book simply cannot fit every person.

Emotion may be part of the point, actually. Why am I a mathematician? Is it because I love boredom? Is it because I am a masochist? No! I want to be entertained when I learn math. That's the reason I do it, and it has the potential to entertain if explained properly. So, regardless of what the "right" way to do it is, my goal is being thwarted.

It is possible to enjoy reading formal books.

 

[homeomorphic]

Well, I believe, that at least for me, reading formal proofs and finding intuition in them is the most effective way to build my general ability to understand intuitively. What is more, I only trust my knowledge about a definition when I know the formal definition for it, and when I have built the intuition, a mental model that is based on that formal definition. If I only know the intuition, that has been simply given to me - I won't trust that knowledge, and I wouldn't feel comfortable using that definition.

With the example that I cited, Riemannian curvature, I think it is prohibitively difficult to come up with the intuition if all you are given is the definition. I couldn't do it. Do Carmo evidently couldn't do it. But Arnold could do it.

There are two different kinds of books that are "informal" and they both have their place. There are those like visual complex analysis that are NOT rigorous. That book contributed an enormous amount to my education. Without it, I would be half the mathematician I am today. I did very, very well in my advanced undergraduate classes (which, by the way were completely rigorous), and this was partially the result of having read Needham's book. So, you see that's why this is such a big issue to me. Without a book like that I never would have seen how far you can take intuition, and most other people who never see a book like that will also never know how much you can do with it.

The other kind of book IS completely rigorous, but just clarifies everything by motivating it. An example of this sort of book would be Marshall Cohen's Simple Homotopy Theory, which is also one of my favorite books. Note that this book is a definition, theorem, proof-style book. But it manages to convey the intuition. The way in which it is not "formal" has nothing to do with whether it's rigorous and gives precise definitions. It does. What makes this book different from other sources covering this material is the way in which the material is arranged and the particular proofs that have been chosen. You could call it a "formal" book, if you wanted, but the fact is that it is 100 times more intuitive than other books which I am referring to as being overly formal. Formal is not really referring to rigor, but rather just slogging through stuff in such a way that it's nearly impossible to figure out what makes it all work.

Different persons have different psychologies. You can't take into account all the possible psychologies. The same book simply cannot fit every person.

True. You know, I don't care if people want to write some kind of ugly math that I don't like if someone else enjoys it. The problem comes when a textbook is chosen that is harmful to MOST of the students, and they are FORCED to learn from it to an extent. It can be a challenge sometimes to find alternatives. Also, the fact that these atrocious books are chosen for courses is giving them undue credit and publicity. And above all, the most lamentable thing is that often, it's not possible to find a suitable intuitive book on a given subject, even though it is perfectly possible for such a book to exist. That's my biggest complaint. If there always was a good intuitive book out there, I wouldn't care. But the problem is that intuition is given such short shrift that the right book isn't out there. That causes problems. In a way, it's kind of like having a solutions manual. Sometimes figuring out your own intuition is just too hard. This is coming from someone who is exceptionally skilled at doing so. So, it would be nice if someone really racked their brains, and put out the "solutions" to that problem, just in case people get stuck trying to do it on their own.

Emotion may be part of the point, actually. Why am I a mathematician? Is it because I love boredom? Is it because I am a masochist? No! I want to be entertained when I learn math. That's the reason I do it, and it has the potential to entertain if explained properly. So, regardless of what the "right" way to do it is, my goal is being thwarted.

It is possible to enjoy reading formal books.

Enjoyment isn't the only goal, either. Lots of people enjoy do Carmo's Riemannian geometry, although I find it difficult to enjoy. At least certain parts of it. However, I suspect the people who enjoy that book from cover to cover (there may very well be certain sections of it that I would enjoy) don't really understand the subject on a very deep level and are content with that. If they wanted deep intuition, they wouldn't be happy with what they were given there.

 

[Obis]

The other kind of book IS completely rigorous, but just clarifies everything by motivating it. An example of this sort of book would be Marshall Cohen's Simple Homotopy Theory, which is also one of my favorite books. Note that this book is a definition, theorem, proof-style book. But it manages to convey the intuition.

Yes, that's the formalism I was talking about - rigorous and precise, formal on the surface, intuitive deeper.

Formal is not really referring to rigor, but rather just slogging through stuff in such a way that it's nearly impossible to figure out what makes it all work.

If this is what is formal, then I agree that formal books are indeed useless. However, I've seen many cases where people, due to their lack of experience reading rigorous mathematics, have blamed the book for being too formal.

In a way, it's kind of like having a solutions manual. Sometimes figuring out your own intuition is just too hard. This is coming from someone who is exceptionally skilled at doing so. So, it would be nice if someone really racked their brains, and put out the "solutions" to that problem, just in case people get stuck trying to do it on their own.

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".

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.

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.

 

[mathwonk]

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.

 

[Sankaku]

My apologies to the followers, as I know it is not intentionally useless, but it is useless all the same.

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]

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.

:-)