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.