Studying Dose Mathematicians understand their books?

[TMSxPhyFor]

Hi all

this is maybe a strange question, but it really bothers me, I can't believe that there is human beings that can understand a book on Topology or differential Geometry for example that written in formal way starting with lemmas and dry theorems from the first page, and expecting one to be able understand what is tensor and covariant space and homomorphism, exterior algebra...etc

I spent my last weeks going though tens of books, papers, physics books with graphics that some how trying to build "intuition", even so I'm still hardly following what they saying and still can't completely visualize that, this really makes me mad becuase i feel my self as a complete idiot when I'm trying to understand how people was getting this ideas without having Internet.

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?

 

[homeomorphic]

Hilbert talked about viewing math as a complete formal thing, but actually he was really big on intuition (refer to his book, Geometry and the Imagination).

To some extent, you should be able to come up with the intuition for yourself, but the way math books are usually written is usually just wrong in my opinion. Too formal, not enough motivation, and calculation or brute force logical deduction is often favored over concepts. Sometimes, it seems like they have this bizarre religion that tells them to be really rigidly formal. It's really mind-boggling. When you figure out the way they obscure things, you realize you could have learned the subject 100 times faster with better understanding if they had just told you certain things from the start. It's not always like that, but too often.

Try reading Visual Complex Analysis or Geometry and the Imagination. Also, check out John Baez's website.

It's not for us to say whether you're "good enough" to do physics or not. But, the reality is you will have to deal with a lot of overly formal writing, whether you approve of it or not. You don't have to like it, but you have to be able to live with it at times. You can seek out the best books and writers, though, wherever possible, and try to come up with your own ideas.

So, yes, those overly formal books are doing it wrong to some extent, but you have to be able to deal with it to some extent in order to survive. When I decided to study math, I just thought I didn't want to let those overly formal and boring mathematical macho guys bully me out of being a mathematician. It would have been easier to just give up, but I didn't want to.

 

[TMSxPhyFor]

But, the reality is you will have to deal with a lot of overly formal writing, whether you approve of it or not. You don't have to like it, but you have to be able to live with it at times.

Yes I agree with you, but in the formal writing using symbolism, if authors don't explain the worlds behind those symbols, how I can understand it? it's really strange how such smart writers don't catch such a basic idea, otherwise as i said before, I'm not smart enough for that.

When you figure out the way they obscure things, you realize you could have learned the subject 100 times faster with better understanding if they had just told you certain things from the start.

Exactly! I used to spend weeks trying to understand some basic ideas (which later you wondering how you couldn't get it much faster), after that the book becomes (almost) a matter of story reading, so I'm not alone in this? :)

By the way John Baez's website very is interesting, thanks .

 

[Number Nine]

Yes I agree with you, but in the formal writing using symbolism, if authors don't explain the worlds behind those symbols, how I can understand it? it's really strange how such smart writers don't catch such a basic idea, otherwise as i said before, I'm not smart enough for that.

The symbols convey words and meaning. Yes, many texts are dry and unmotivated, but the reason we developed mathematical notation in the first place is because ordinary language is often insufficient. What is your mathematical background? Maybe you just aren't ready to be studying topology and differential geometry.

 

[Obis]

There are hundreds of books written for every major topic in mathematics, all they differ in one way or another. I think it's extremely important to find a book that suits you - that you find interesting, that you find understandable. However, you should not avoid formal treatment, because it's the formal treatment of mathematics that guarantees objectivity. Intuitive remarks are very helpful, however, too much of them could be dangerous. Everyone's intuitive understanding can be slightly different, hence the author does not present his own intuitive understanding, but simply presents everything in a pretty formal way, so it's up to you to build an intuitive understanding out of all the formalities. This is an ability that develops, it's like learning a new language. Even though it might seem extremely difficult to find the intuitive sense out of all the formal symbolism, I find that a lot of times there's simple ideas, which could be explained to someone without any background in mathematics, that are presented in a rigorous, formal fashion. Don't be scared of it, you'll get used to it (and you'll even start to like it, I believe). Be patient.

 

[TMSxPhyFor]

but the reason we developed mathematical notation in the first place is because ordinary language is often insufficient.

As per my knowledge, human language is much reacher than mathematical symbolism that used at least at first order logic, this why basically we can't build a human quality language translation machine, and symbols used to reduce the amount of words, and becuase they built basically to reflect the pure logic laws when we construct them alike human language.

What is your mathematical background? Maybe you just aren't ready to be studying topology and differential geometry.

I don't know how to explain my mathematical background, but I'm at 3rd year in theoretical physics, but anyway i didn't find any book of pure mathematics that explains what is differential manifold as wikipedia dose, for example.

You see, actually I like the formal and step by step logical contraction that used in math books more than than the fuzzy way used in physics books (actuality i wanted to study math some time ago :) but the luck of intuitional motivation when new concepts are explained , blows up things.

 

[homeomorphic]

Yes I agree with you, but in the formal writing using symbolism, if authors don't explain the worlds behind those symbols, how I can understand it? it's really strange how such smart writers don't catch such a basic idea, otherwise as i said before, I'm not smart enough for that.

It's a cultural thing, I think. They just copy what they see other people doing. Even mathematicians and scientists, though they like to think of themselves as independent-minded, can act like sheep sometimes and just do what the herd is doing.

Exactly! I used to spend weeks trying to understand some basic ideas (which later you wondering how you couldn't get it much faster), after that the book becomes (almost) a matter of story reading, so I'm not alone in this? :)

No, you're not alone. Some people will conclude that maybe they are wrong, even though it seems much easier to do things in a more reasonable way. But you have to have the guts to say, no, formal and unmotivated is boring and not a good way to learn. But, you shouldn't be too rigid. The intuition doesn't always have to be spelled out in every single case. You have to learn to see the intuition behind formal proofs, sometimes.

In some ways, Munkres topology book could be called formal, but I did okay with it. I might argue with the particulars of the way the subject was laid out, but I was able to learn from it without too much trouble. Partly, having background knowledge, partly help from the professor.

Yes, many texts are dry and unmotivated, but the reason we developed mathematical notation in the first place is because ordinary language is often insufficient.

Often, when you find a dry and unmotivated book, you can find one that gets by perfectly well WITHOUT being dry and unmotivated. Of course, sometimes, maybe the book is assuming you have background knowledge, so that you don't need as much motivation. Also, you have to learn to see the intuition that is just beneath the surface, if you know where to look. That takes some practice. But let's face it, some books are just bad books.

Unfortunately, in many subjects the right book doesn't yet exist, but it very well COULD exist if someone were to write it.

 

[homeomorphic]

However, you should not avoid formal treatment, because it's the formal treatment of mathematics that guarantees objectivity.

You shouldn't avoid formal proofs, yes, but you should avoid formal treatments, meaning those that rely EXCLUSIVELY on formal proofs and nothing else with no intuition or motivation. When I say formal, that's what I mean.

Intuitive remarks are very helpful, however, too much of them could be dangerous.

I suppose it is possible to over-do it, but that's hardly ever an issue. Mainly, what could be dangerous is not too much intuition, but not enough rigor.

 

[TMSxPhyFor]

Intuitive remarks are very helpful, however, too much of them could be dangerous. Everyone's intuitive understanding can be slightly different, hence the author does not present his own intuitive understanding, but simply presents everything in a pretty formal way, so it's up to you to build an intuitive understanding out of all the formalities.

This why i mentioned the example of Grassmann, why we can't simply learn our selfs from the intuition that had been used by the inventor of the concept it self? how it can be wrong if he discovered something depending on it? we actually by knowing their own way of thinking we can teach our selfs too.

Another example, most books that derive Dirac equation for relativistic particles uses extremely ad-hoc way (which mathematically some how not rigous) and for me looked to be something like astrology more than physics, until i found a wonderful book which states that they will use Dirac's own way of discussing that, and it comes to be really insightful, genius, clear approach, the question is why i had to lose so much time to get such an intuitive and original explanation? just a total waste of time...

 

[TMSxPhyFor]

I suppose it is possible to over-do it, but that's hardly ever an issue. Mainly, what could be dangerous is not too much intuition, but not enough rigor.

I totally agree with you!

 

[Obis]

This why i mentioned the example of Grassmann, why we can't simply learn our selfs from the intuition that had been used by the inventor of the concept it self? how it can be wrong if he discovered something depending on it? we actually by knowing their own way of thinking we can teach our selfs too.

Because his own intuitive understanding is too subjective, and it simply might not suit you. The brain structure of yours is different, the mathematical background of yours is different, hence the same intuition might just not work. Intuitive explanations can be easily misinterpreted, they are vague, just as our everyday language.

 

[homeomorphic]

It's very problematic to say that intuition is too subjective. If we insist on not talking about intuition, then we insist on not talking about math, as far as I am concerned. The real objects of investigation ARE the intuitive things. That's what you're trying to study. The rest is just there to check to make sure you get it right. So, being formal all the time is just silly. It's like musicians trying to claim that they only care about music theory and they don't give two hoots about playing music because it's beneath them. It's just acting rough and tough for no good reason. No, the music is the whole point. It's not that there's anything wrong with music theory.

 

[Obis]

It's very problematic to say that "The real objects of investigation ARE the intuitive things" either. The same formal argument can be understood intuitively by two different persons completely differently. In fact, I think it is crucial for a mathematician to be able to extract intuition from formal argument, and, conversely, to convert his own intuition into formal argument.

Yes, being formal all the time is quite silly, however, I think the treatment should be 80%-90% formal and 10% intuitive, however, the formalism should increase only gradually. The way I see mathematics should be thought is going from games and 100% intuition continuously towards formalism.

 

[TMSxPhyFor]

It's very problematic to say that intuition is too subjective. If we insist on not talking about intuition, then we insist on not talking about math, as far as I am concerned. The real objects of investigation ARE the intuitive things. That's what you're trying to study. The rest is just there to check to make sure you get it right. So, being formal all the time is just silly. It's like musicians trying to claim that they only care about music theory and they don't give two hoots about playing music because it's beneath them. It's just acting rough and tough for no good reason. No, the music is the whole point. It's not that there's anything wrong with music theory.

In supporting what you said, there can't be math at first place without physics, real word is what motivates us invent geometry, numbers, ect... so it's totally unnatural to work with pure formalism, and most of math objects has really physical roots.
And however intuition is subjective, it is more important than any formalism, becuase the last one is straight forward logical deduction on mathematical objects, but why we invented those objects at first place? it's physics!

I think it is crucial for a mathematician to be able to extract intuition from formal argument, and, conversely, to convert his own intuition into formal argument.

with the same success, i can rebuild all this mathematics by my self (if i smart enough), what for the book then if it is just a giant puzzle?

 

[Obis]

I don't agree that mathematics is just a tool for physicists. Since you're studying physics, it's natural that that's the way you see it (or want to see it) though. "Real world" is a complicated concept. The world that we perceive as objective reality around us is actually a creation of our own brain (at least partially), we understand what our brain understands, even though I agree that there exists some objective reality that "sends" us information, however, what we understand is the information already processed and structured.

 

[TMSxPhyFor]

I don't agree that mathematics is just a tool for physicists. Since you're studying physics, it's natural that that's the way you see it (or want to see it) though. "Real world" is a complicated concept. The world that we perceive as objective reality around us is actually a creation of our own brain (at least partially), we understand what our brain understands, even though I agree that there exists some objective reality that "sends" us information, however, what we understand is the information already processed and structured.

I didn't say that it's just a tool, I said that it's roots are physical, I agree that math has it's own abstract depth that beyond physical reality, Mathematicians takes physical objects and abstracting there concepts, but those objects or information doesn't came to us processed and structured, we just "see" them like this due to our senses limitation, if we was able to see on molecular level, i hardly can believe that we was able to invent a mathematical object like vector.

 

[homeomorphic]

It's very problematic to say that "The real objects of investigation ARE the intuitive things" either.

Well, we will just have to disagree on this point. You could call it a subjective judgment, but like I said, it's like if you try to say that making actual sounds isn't the main point of music. If you like music theory, rather than music, you are entitled to that opinion, but I am going to consider it to be in poor taste. It's just obvious that sound is the main point. If you try to say, yes, it's nice to hear sound, but we can't have anyone playing actual music because that's too subjective, so you have to get a sound-proof room so that we don't have everyone arguing over how it should sound, that's a step ahead of the guys who think music is just about writing notes on a page, but has nothing to do with sound, but it's still not quite right. The fact that people have different intuitions is analogous to the fact that different pianists will have a different interpretation of each piece they play. The fact that they don't agree on the interpretation doesn't stop them from discussing their interpretation or performing it in front of others. It's rather irrelevant that they have a different interpretation. So what? Just because I tell you my intuitive argument doesn't mean you have to have the same one. It just gives you the idea. Maybe if I don't tell you, you're in danger of thinking too formally. If I want to play some piece on the piano, my piano teacher tells me how he interprets it to some extent. He doesn't leave the whole task up to me. If I interpret it with no help, it's just not going to be good enough. I'm trying to learn. I'm not an expert at it already. Math is analogous to that. If you don't have any guidance about the intuition, even people who THINK they can figure out all the intuition for themselves will just end up missing a hell of a lot of it. That's the end result of downplaying intuition. People will just end up not learning stuff as well and it will be much less interesting. Intuition is my strong point, and I can't figure out all the intuition myself. Usually, I do, after I have expended considerably more effort than would have been required if someone had explained it properly from the start, but sometimes, I miss it. Sometimes, even I am guilty of thinking too formally because I'm not aware that there's a better way. So, if even Mr. intuition himself can't do it, I think it will take a very gifted person, indeed, to not need considerable guidance as far as intuition goes, and even then, it's probably not the most efficient way for them to learn.

In fact, I think it is crucial for a mathematician to be able to extract intuition from formal argument, and, conversely, to convert his own intuition into formal argument.

True. But it's also crucial to know as much math as you can cram into your head. That means, you want to learn as quickly as possible, which means you need the most efficient way of learning, which, in turn, means we can't be wasting ALL our time converting proofs into intuition. Yes, you should have some practice with that. But it's generally not the way to go. Actually, I typically would rather come up with my own proofs, rather than read someone else's if it isn't well-motivated because it is so much harder to decode it than understand the idea directly that it's just easier to come up with the idea yourself. You don't want to be banging your head against the wall, trying to decode cryptic proofs all day. There are other things that need to be done in math.

Yes, being formal all the time is quite silly, however, I think the treatment should be 80%-90% formal and 10% intuitive, however, the formalism should increase only gradually. The way I see mathematics should be thought is going from games and 100% intuition continuously towards formalism.

Well, I don't think I would attempt to put any percentages on it. I would rather say you just play it by ear. If such and such proof is easy to interpret or gets cumbersome to talk about, you just do the formal proof. Otherwise, it should be intuitive, by default, either as a prelude to the formal proof or mixed in with it.

 

[homeomorphic]

Also, you can't separate the formal proof from the intuitive proof, completely. Some formal proofs are actually much more intuitive than other formal proofs. Actually, MOST of the intuition can be conveyed in formal proofs, if it's the right proof. What happens is that a lot of times, you get the wrong proof. So, it's not always working formally that is to blame, but more often than not, it's bad taste in proofs. Again, subjective. But subjectivity matters. You can't run away from it because the fact is that we are not ONLY concerned with truth, we are also concerned with learning and that involves the psychology of learning which gets subjective. You can't just run away from the problem because it's subjective. That's just like throwing your hands up and saying it's too hard to deal with, so we'll just live with the problem instead of trying to cope with it effectively.

Different people may have different formal proofs of the same fact. Does that mean that we should only state theorems because the path to getting them is too subjective?

 

[mathwonk]

it is not easy to transform the geometry of the universe into a sequence of symbols on paper. why should it easier to reconstruct that geometry from those symbols? it also helps to talk to human beings who understand it. that's why there are colleges and professors and not just books to learn from.

 

[Obis]

True. But it's also crucial to know as much math as you can cram into your head. That means, you want to learn as quickly as possible, which means you need the most efficient way of learning, which, in turn, means we can't be wasting ALL our time converting proofs into intuition.

I don't think it's crucial to know as much math as possible. I think it's the quality that matters the most, not quantity. The quality of learning, at least to me, is much better if I find the intuition myself reading a pretty formal proof, not reading someone's interpretation.
Even though I agree that a few intuitive sentences before a proof saves a lot of time.

I feel pretty strange though, if you would defend formalism, I would probably be defending intuition. Both have their own advantages and disadvantages, the perfection lies somewhere in between.

 

[homeomorphic]

I don't think it's crucial to know as much math as possible. I think it's the quality that matters the most, not quantity.

That's probably because you don't really know what it's like to do math research. Say you want to do string theory. Then, you're going to have to learn a lot of math. A ridiculous amount of math, in fact. So, if you insist on taking too much time, you're just not going to be able to do it. Period. If you are an undergrad, you have no clue. If you are a grad student who's been in the world of mathematicians, you have a clue how much math is out there. You know how much your adviser knows, for example. By the way, my adviser is famous in his area. Why was he so successful? In part, because he knows a ridiculous amount of math! It's quite obvious. You also go to math talks and you see the speakers know 100 times more math than you do. That starts to clue you in on the reality of it. Of course, some people have more breadth than others and if it's your style to be more thorough and know less stuff, there's a place for that, too. However, if you want quality of knowledge, you NEED more intuition. Also, when you do research, you don't want to worry about having too much knowledge because you want to go ahead and tackle the problems. Actually, I put quality over quantity, too, and that's why I can't learn enough math and maybe my career is doomed as a result.

The quality of learning, at least to me, is much better if I find the intuition myself reading a pretty formal proof, not reading someone's interpretation.

So you would like to think. If that's really true in general, then you must be a true genius. It may be true for SOME subjects. But I suspect it's just that you haven't studied difficult enough subjects, yet. Or perhaps, you are not aiming for as deep an understanding as what I aim for. I like to feel as if I could have invented the subject, myself. If you can reach that level of understanding from the typical books alone, then you are a great genius. I have some level of skepticism to that. I don't think you are quite aware of the difficulties I am talking about. Without concrete examples of how certain books have made things much harder than they need to be, you are unable to grasp what I am getting at. You can get that by reading enough of Baez's articles, for example, alongside more typical treatments of the material and see for yourself how much difference his exposition makes. It's not the kind of thing you just figure out for yourself, no matter how smart you think you are, except MAYBE if you're Riemann or Gauss or something.

I feel pretty strange though, if you would defend formalism, I would probably be defending intuition. Both have their own advantages and disadvantages, the perfection lies somewhere in between.

No, it's just that you need both. It's not one at the cost of the other.

 

[Sankaku]

No, it's just that you need both. It's not one at the cost of the other.

I just wanted to say that I totally agree with you. In support of this position, here is a great quote from the preface of Robert Ash's graduate algebra book:

Many mathematicians believe that formalism aids understanding, but I believe that
when one is learning a subject, formalism often prevents understanding. The most important
skill is the ability to think intuitively. This is true even in a highly abstract field
such as homological algebra. My writing style reflects this view.

...snip...

Another goal is to help the student reach an advanced level as quickly and efficiently as
possible. When I omit a lengthy formal argument, it is because I judge that the increase
in algebraic skills is insufficient to justify the time and effort involved in going through
the formal proof. In all cases, I give explicit references where the details can be found.
One can argue that learning to write formal proofs is an essential part of the student’s
mathematical training. I agree, but the ability to think intuitively is fundamental and
must come first. I would add that the way things are today, there is absolutely no danger
that the student will be insufficiently exposed to formalism and abstraction. In fact there
is quite a bit of it in this book, although not 100 percent.

...snip...

I never use the phrase “it is easy to see” under any circumstances.

I particularly liked the last sentence!

It is available for free here:
http://www.math.uiuc.edu/~r-ash/
And from Amazon here:
http://www.amazon.com/dp/0486453561/?tag=pfamazon01-20

 

[Nano-Passion]

Because his own intuitive understanding is too subjective, and it simply might not suit you. The brain structure of yours is different, the mathematical background of yours is different, hence the same intuition might just not work.

I don't understand your reasoning. Because an intuition is subjective it should not be included at all? So instead of helping out a certain percentage of people you would rather help out none since not everyone will be akin to the intuition? Edit: I am not speaking of substituting formalism for intuition, but nevertheless it would be better for some intuition to work hand in hand with formalism.

Intuitive explanations can be easily misinterpreted, they are vague, just as our everyday language.

Don't agree here. I see a bit too much subjectivity on this issue. I bet neuroscientists/psychologists look at these statements and scoff. In defense of that statement, I would argue that intuitive explanations can only add to one's understanding, not subtract. Whether or not it will help every single person that reads the arguments is an unnecessary point, probability posits that something is better than nothing.

 

[Obis]

I don't understand your reasoning. Because an intuition is subjective it should not be included at all?

I didn't say that it should not be included at all.

 

[Nano-Passion]

I didn't say that it should not be included at all.

But you seem to be against intuition and for the use of more formalism.

 

[Obis]

That's probably because you don't really know what it's like to do math research.

You're right, I don't. However, I do understand that there's a huge amount of mathematics that has been produced. But in my point of view, knowing more math doesn't mean to be a better mathematician. Everything can be understood infinitely well and infinitely deeply, it's not discrete - either you understand or not, it's a continuous scale, that has no upper bound. What I prefer and enjoy more is not learning a lot of different material, but learning the material which I think is extremely important as well as I can.

I'm not against intuition. However, I prefer the intuition I've built myself instead of the one that has been given to me. It's somehow similar to the difference between reading a solved problem, and trying to solve it on your own. Even an unsuccessful attempt is useful.

 

[homeomorphic]

You're right, I don't. However, I do understand that there's a huge amount of mathematics that has been produced. But in my point of view, knowing more math doesn't mean to be a better mathematician.

The point is that there are now enormous prerequisites to being able to do good research in probably most areas. There are certain areas where you might be able to get by without knowing that much. But more knowledge means more discovering power. You can't work on something you don't even understand in the first place. I know an enormous amount of math, yet my adviser was telling me, "you can't take forever to learn the basics."

So, after years of study, I just know the basics of the field. Having a vast amount of knowledge is a necessary, but not sufficient condition to be a good mathematician.

Everything can be understood infinitely well and infinitely deeply, it's not discrete - either you understand or not, it's a continuous scale, that has no upper bound. What I prefer and enjoy more is not learning a lot of different material, but learning the material which I think is extremely important as well as I can.

There is a certain pitfall, here, aside from the considerable prerequisites to working in most fields. Math is very interconnected. Knowing about many branches of math and the connections between them is crucial. One of the professors I talk to says, I'm telling you, everything you saw in your graduate classes is something you'll need. Maybe that's a slight exaggestion, but it's an awful lot of math. It's probably good to focus on the most important things, but how do you even know what they are. If you are going to do analysis, maybe Tychonoff's theorem is important. If you are going to do topology, maybe you care more about paracompactness.

I'm not against intuition. However, I prefer the intuition I've built myself instead of the one that has been given to me. It's somehow similar to the difference between reading a solved problem, and trying to solve it on your own. Even an unsuccessful attempt is useful.

That is okay up to a point. But, I think you'll find you are not always up to the task of doing it all by yourself, especially when you get to more advanced material. It keeps getting harder and harder, each year. As I look back throughout my education, it's easier and easier. I think it doesn't matter that much if you came up with it yourself or were given it. You can always modify what you are given.

If two mathematicians are trying to do some math together, they are probably going to be talking to each other in intuitive terms. It's much more direct. Converting it into formalism is putting in an extra middle man that is unnecessary, artificial, and will just make things harder. It's just putting in an extra obstacle for no reason.

Fields Medalist and renowned topologist Bill Thurston says something very revealing to this effect in the following essay:

http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf

"When a significant theorem is proved, it often happens that the solution can be communicated in a matter of minutes from one person to another in the subfield. The same proof would be communicated and generally understood in an hour talk to members of the subfield. It would be the subject of a 15 or 20 page paper, which could be read and understood in a few hours or perhaps days by members of the subfield."

If you read further, you find that Thurston's explanation for this is very much in line with what I am trying to say.

So, now we're beginning to see the extreme magnitude of the inefficiency that is being introduced by trying to put this middle man of formal proof in the way of direct communication of the ideas. It's like trying to communicate to a friend by speaking in some code that they have to decipher. Why not just tell them directly? Don't be so caught up in the idea that you have to practice to be good at deciphering the code. For now, it may be working, but in the long run, you'll be shooting yourself in the foot.

Mathwonk is right. Math is a lot more social than people think. It's better not to rely too much on books. Talking to people is often an MUCH better way to learn, by orders of magnitude.

 

[Obis]

That is okay up to a point. But, I think you'll find you are not always up to the task of doing it all by yourself, especially when you get to more advanced material. It keeps getting harder and harder, each year.

Yes, it is getting harder, but the ability to extract intuition also becomes stronger and stronger. Once again, it is similar to solving mathematical problems - yeh, the problems get harder, but the ability to solve them is also improving, especially if you are solving problems by yourself, and not just reading the solutions.

If two mathematicians are trying to do some math together, they are probably going to be talking to each other in intuitive terms. It's much more direct. Converting it into formalism is putting in an extra middle man that is unnecessary, artificial, and will just make things harder. It's just putting in an extra obstacle for no reason.

I agree that intuitive arguments are much more direct, alive, interesting and clear. However, sometimes they are just not sufficient. This is especially true, for example, for definitions. Basically every definition has an infinite amount of intuitive explanations, however, there exist various extreme cases, for which the intuitive explanation is not sufficient - it simply can't tell whether some object is an example of a definition or not, while the formal definition is "complete" - there's no vagueness.

Once again, I am not against intuition. In fact, it's the intuition in mathematics that I find beautiful and it's the reason I am actually studying it. However, it has weak points, and the formalism in mathematics is simply necessary.

 

[homeomorphic]

Yes, it is getting harder, but the ability to extract intuition also becomes stronger and stronger. Once again, it is similar to solving mathematical problems - yeh, the problems get harder, but the ability to solve them is also improving, especially if you are solving problems by yourself, and not just reading the solutions.

Here's my concern. Someone will hit you with some horrific definition, like, say, the Riemannian curvature tensor. If you read do Carmo's Riemannian geometry book, he even talks about how his own definition looks unnatural and removed from the intuition. However, I disagree completely with do Carmo. It is, in fact, do Carmo's rigid and overly formal approach that makes the definition look so unnatural. Where would you find the cure to do Carmo's overly formal nonsense? Look at Baez's discussion of the meaning of the Einstein equation, The Road to Reality, and, the best source of all, one of the appendices of Arnold's mathematical methods of classical mechanics. Without these references, if someone is given do Carmo or many other differential geometry books, they may very well conclude that do Carmo is right. They may think that the definition is inherently unintuitive because they are not aware of the alternatives. That is the biggest danger with your approach. You can't always know what you're missing out on because if you're missing out on it, you don't know that you're missing out.

I agree that intuitive arguments are much more direct, alive, interesting and clear. However, sometimes they are just not sufficient. This is especially true, for example, for definitions. Basically every definition has an infinite amount of intuitive explanations, however, there exist various extreme cases, for which the intuitive explanation is not sufficient - it simply can't tell whether some object is an example of a definition or not, while the formal definition is "complete" - there's no vagueness.

Usually, when the intuition breaks down, you just come up with different intuition to handle it. But, yes, the formal proof is there to check it. I never said you shouldn't verify it formally. You should ideally verify everything formally. But you are trying to learn, so the things you are going to remember and take away from the experience are the intuitive things. If it's not intuitive, how are you going to remember it 5 years from now? That's my point. That's why I say intuition is the most important thing to learn. You also have to learn the skills of making things rigorous. But when you say you want to learn some subject, like topology, mainly what you are trying to learn is the intuition because, psychologically speaking, that's the only thing you are going to be able to learn, anyway. The rest you will just forget very quickly. It won't stick. You also need to have some practice converting the intuition into proofs, so that you can prove theorems, too, though. But that's what problems are for. So, you see, reading proofs can be inefficient because your goal in reading them is to learn the intuition. You are trying to decode something that someone could have told you directly. There's essentially no difference between reading a formal proof and two mathematicians trying to speak to each other formally. We established that, at least in many cases, this would be an inefficient way to communicate the ideas.

 

[Sankaku]

There is an interesting article discussing similar material from a Category Theory point of view on Eugenia Cheng's website:

http://cheng.staff.shef.ac.uk/morality/

 

[kjohnson]

This is an interesting conversation that seems to touch on the philosophy of:

Is mathematics a tool created used to describe the universe?
or
Is mathematics the "language" of the universe (i.e. it would exist even without the humans who use it)?​

This is probably better off for another thread, but still an interesting question. And the direction I see this thread headed in...

 

[TMSxPhyFor]

Is mathematics a tool created used to describe the universe?
or
Is mathematics the "language" of the universe (i.e. it would exist even without the humans who use it)?​

Oh please no! this is an endless discussion I hope it will not turn to this point even so I agree it's quite interesting!

 

[TMSxPhyFor]

There is an interesting article discussing similar material from a Category Theory point of view on Eugenia Cheng's website:

http://cheng.staff.shef.ac.uk/morality/

As i know it was back to 30s, and then again in 60's when the borders between mathematics and Meta-Mathematics, or analytical philosophy became totally destroyed, by introducing incompleteness theorems by Goudel and non standard analysis by Robinson.

 

[kjohnson]

Oh please no! this is an endless discussion I hope it will not turn to this point even so I agree it's quite interesting!

Haha, yeah it is a discussion that could go on forever. I'm not trying to start it, just trying to point out that it is headed that direction.

 

[Obis]

Here's my concern. Someone will hit you with some horrific definition, like, say, the Riemannian curvature tensor. If you read do Carmo's Riemannian geometry book, he even talks about how his own definition looks unnatural and removed from the intuition. However, I disagree completely with do Carmo. It is, in fact, do Carmo's rigid and overly formal approach that makes the definition look so unnatural. Where would you find the cure to do Carmo's overly formal nonsense? Look at Baez's discussion of the meaning of the Einstein equation, The Road to Reality, and, the best source of all, one of the appendices of Arnold's mathematical methods of classical mechanics. Without these references, if someone is given do Carmo or many other differential geometry books, they may very well conclude that do Carmo is right. They may think that the definition is inherently unintuitive because they are not aware of the alternatives. That is the biggest danger with your approach. You can't always know what you're missing out on because if you're missing out on it, you don't know that you're missing out.

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.

But you are trying to learn, so the things you are going to remember and take away from the experience are the intuitive things. If it's not intuitive, how are you going to remember it 5 years from now? That's my point.

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.

 

[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...