Why don't most textbooks help the readers to get abstract concepts?

  • Thread starter Calmarius
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I was always good at maths, just because primary/high school math was simple enough to find concrete examples for the abstract concepts, and that helped me a lot on exams.

Since then I tried to grasp more advanced concepts. But I always faced with pure overformalized, overgeneralized stuff with n's, m's, sets and spaces. And the only numbers in the whole paper were page numbers.

Conversely sometimes I can find an informal writings about the topic (probably written by a non-mathematician), that uses simple examples to demonstrate the concept, and it makes the whole topic easier to comprehend when I return to the previous formal thing.

As far as I know the human brain is very good at classifying and building models as long as it saw enough examples. But from my experience it isn't always trivial to create good examples when you are given only a definition.

I don't find proofs hard: just use the rules you know, a computer can do it nowadays. But definitions are magic: why do you define it that way? What does it express? Textbooks often don't answer these questions, but proceed assuming you already know everything about it inside out... And this is the point where I usually get lost, and I cannot proceed for weeks/months, eventually I give up and resign that I'm too stupid to comprehend it.

I understand that formalism is important in mathematics, but I still wonder why don't textbooks help the reader to get the point?

This often makes me research why mathematics hard...

I've recently found these lines [crackpot link deleted]

I really hope this isn't the truth...
 
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  • #2
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If you're reading a book that only uses "sets and n's and m's and stuff", then it probably isn't intended for a beginner (and if you say "I don't find proofs hard: just use the rules you know, a computer can do it nowadays.", then you are most definitely a beginner). There's no reason to expect an advanced book on, say, analysis to waste time motivating the concept of a set. Start with something easier. It's true that many textbooks are unmotivated and poorly written, but you don't seem to be reading books at your level.

If you look at the CV (curriculum vitae, or resumé) of a professional
mathematician, you will see paper after paper listed, each with a title that only a
specialist can understand
This really is a bizarre statement. The purpose of a CV, among other things, is to list the academic achievements of the author, including the papers that they've written. It would be silly to expect the title of an article in a journal of topology to carefully define all of the standard terminology used in topology.
 
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D H
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I've recently found these lines [crackpot link deleted]

I really hope this isn't the truth...
I edited your post, Calmarius. The lines you found are from a crackpot site. We do not want this site to promote crackpot notions, and one of the best ways to promote things nowadays is with links and quotes from those links.

With regard to your complaint, it's important to realize that mathematics education has two very distinct goals:
- To teach non-mathematicians the limited, concrete tools they need to succeed in their world, and
- To teach future mathematicians to think like mathematicians.

Mathematicians think in the abstract. The very, very abstract. It's how huge advances have been made. Their job is not to make things accessible to non-mathematicians. The same applies to *any* technical discipline. Every discipline has its jargon, its presumed level of knowledge. Writing a technical paper is hard enough as it is. Writing one so that the general public could possibly understand it would require turning each paper into a multi-volume book.
 
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Stephen Tashi
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I don't find proofs hard: just use the rules you know, a computer can do it nowadays.
If you don't find proofs hard then you don't have much experience in reading proofs! There are attempts to get computers to prove theorems, but currently these fall far short of enabling computers to prove most theorems that interest people.

But definitions are magic: why do you define it that way? What does it express? Textbooks often don't answer these questions, but proceed assuming you already know everything about it inside out... And this is the point where I usually get lost, and I cannot proceed for weeks/months, eventually I give up and resign that I'm too stupid to comprehend it.
You limit your progress in math if you take a passive approach. You should try to make up your own examples - and that requires that you read definitions carefully in a legalistic manner rather than phrasing them "in your own words". If you take a passive approach then you will forever be searching for the right book or article to learn a topic.

I agree that it is delightful to find a book or article that explains a puzzling mathematical concept in an intuitive and concrete manner. But what is intuitive to one person may not be intuitive to another, so people's reaction to such articles is highly personal. The only common ground in mathematics is the formal part of it.
 
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arildno
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Intuitions develop and are not static.
If a carpenter was to tell me, in painstaking detail how to make the "good choices", rather than discussing them with a peer, his lines of conversation would be totally different. As they should be.
 
  • #6
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That's what teachers are for. It's very common these days to find questions in online math forums from people reading textbooks all by themselves. But that's a very hard way to go. In any upper-division or grad school math class, you want to have a good teacher and you want to form a study group with your peers. That's the only way to get the "meaning" of the symbols.

A good teacher makes all the difference. It's a rare individual who can pick up an advanced math book and just read it. All you get that way is the definition-theorem-proof presentation. You need a teacher to provide insight.

By the way, that's why online learning is doomed. You cannot replace the role of a gifted teacher.

Of course many (most?) teachers aren't much better than the textbook. But once in a while you have a teacher who inspires you and who explains the hidden meaning of the symbols, and that's when you have a profound educational experience.
 
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A good teacher makes all the difference. It's a rare individual who can pick up an advanced math book and just read it. All you get that way is the definition-theorem-proof presentation. You need a teacher to provide insight.

...

Of course many (most?) teachers aren't much better than the textbook. But once in a while you have a teacher who inspires you and who explains the hidden meaning of the symbols, and that's when you have a profound educational experience.
So the textbooks don't do much along the lines of giving intuition and concrete examples, and many teachers don't either? In that case, shouldn't there be more textbooks along the lines of what the OP is asking about?

I don't think it's fair to ask "most textbooks" to be like this, but there should be--and I believe there are, to various degrees--some (which is why this thread might be better if it was in the textbook section asking what books do this, instead of why most don't...). Then again, maybe the internet is now the best place for this, with mathematicians like Tim Gowers and our own mathwonk helping people see "the big picture", while the textbooks take care of the n's and m's.
 
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I edited your post, Calmarius. The lines you found are from a crackpot site.
I haven't checked whether the site is crackpot or not... But I experienced that kind arrogance from math pros at the college. (I stop here, because the remaining would be just a rant...)

Their job is not to make things accessible to non-mathematicians.
That's sad. Whose responsibility to make it accessible then?

I see this attitude on Wikipedia where technical articles should be written in an accessible manner... Google often ranks it as the first search result, but I find the information there on technical topics the most cryptic ever... I don't question it's accurately but it's Chinese for me.(oops rant again)

If you don't find proofs hard...
You don't need the intuitive idea and understanding of the concepts to check whether a proof is correct. Just know the definitions, their properties and axioms and theorems deduced from them, don't you? Of course, intuition is good heuristic when proving something.

If you can make all this knowledge computer edible, a computer can prove theorems.

There are attempts to get computers to prove theorems, but currently these fall far short of enabling computers to prove most theorems that interest people.
As far as I know the four color theorem was proven by computer.

You should try to make up your own examples...
My room is full of sheets of paper scribbled full of symbols. So I'm trying... But when after hours or days of trying I finally realize what's that definition is about, I will be a bit angry: why don't the textbook told me what's the basic idea behind the definition, why do I need to decypher it myself?

I understand why formalism is important, but sometimes, just a single plain English sentence greatly helps.

For example: if a function's graph breaks somewhere, it's not differentiable at that point.

Another scary example: every programmer understands what a loop is, how to use, and when to use it. But would you get the point of it from this formal definition of the loop? (from the Hungarian lecture notes of the Introduction to the programming course at ELTE)
 
  • #9
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As far as I know the four color theorem was proven by computer.
The FCT was proven by mathematicians Kenneth Appel and Wolfgang Haken. Appel died just this past April. Haken is still alive.

They used conventional math to reduce the problem to a large number of special cases; and then they used a computer to analyze those cases.

At the time it was quite a shock to traditional mathematicians. In 1977 computers were not commonplace in society, let alone math departments.

Math is a human activity. Computers, by themselves, are unlikely to ever prove anything of interest. I say this knowing that a lot of smart people are currently trying to program computers to do exactly that.

I apologize if I took your remark out of context. I wasn't following the specific discussion. But I wanted to clearly assert that math is done by humans and not by machines. I admit this is a philosophical point. I know this forum doesn't do philosophy.

But isn't this thread about wishing for textbooks that put the formulas into a larger context? To do that, sometimes we must veer into philosophy.
 
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That's sad. Whose responsibility to make it accessible then?
Teachers. Textbook writers. Nobody is responsible for making science and technology accessible for those who don't want to take advantage of those fantastic resources.

I see this attitude on Wikipedia where technical articles should be written in an accessible manner... Google often ranks it as the first search result, but I find the information there on technical topics the most cryptic ever.
Encyclopedias are rarely a good place to learn about complex technical subjects. They can be a good resource if one only needs a refresher, but a place to start learning? No. With regard to wikipedia articles, most are quite accurate, but many are terribly written. You paid *nothing*, why do you expect anything in return?


You don't need the intuitive idea and understanding of the concepts to check whether a proof is correct. Just know the definitions, their properties and axioms and theorems deduced from them, don't you?
No. It's not that simple. It's not nearly that simple.

If you can make all this knowledge computer edible, a computer can prove theorems. As far as I know the four color theorem was proven by computer.
Once again, it's not nearly that simple.

Yes, some theorems painstakingly reduced to logic statements can be checked by computer. Note that I said some rather than all; there's an important theorem that says this is impossible. Yes, some theorems can be proven with the aid of a computer. The four color map theorem is a famous example. This turned out to be a very tedious divide and conquer kind of problem. Humans had to write the software to divide the problem up, to make the compute solve each little bit of the problem. The computer did not do this all by its lonesome.

Another scary example: every programmer understands what a loop is, how to use, and when to use it. But would you get the point of it from this formal definition of the loop? (from the Hungarian lecture notes of the Introduction to the programming course at ELTE)
Whoever told you that this is from the Hungarian lecture notes of the Introduction to the programming course at ELTE is selling you a bunch of baloney.

You are the one who said "if you can make all this knowledge computer edible, a computer can prove theorems." That is what this is about. It's called "formal methods". When you absolutely need to know whether a program will work exactly as the requirements say it should work, what you do is to make the knowledge and the code "computer edible." Everything has to be reduced to pure logic. As you can see, that is no mean feat. Then, maybe, those formal methods can determine if the program will work. Maybe. Sometimes the formal methods magic works, sometimes it doesn't.
 

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