How many pages of math theory can you absorb in one day?

[morphism]

Just thought I'd let you know that quotient spaces are very intuitive, and there's ample motivation behind them, in case you haven't figured this out already. It's all about identifying elements of the underlying set X via an equivalence relation ~. The resulting collection of equivalence classes X/~ is called the quotient space, or identification space. Here, a set is a collection of equivalence classes; it's open if the union of the equivalence classes produces an open set in the original space X.

Of course there's much more to say about this and where it comes from, but this should hopefully help you in knowing what to look for.

 

[andytoh]

Ah, so if Jason, who voted 30+ pages, now realizes that more like 5 pages, I wonder how many others who voted 30+ pages is also around 5 pages in reality...

 

[Lee55]

The question is entirely dependant on mitigating circumstances, so it's hard to judge, but on average I can get through a textbook in a week, I work full time, and I only have those couple of hours before I go to bed, and what I can read as I travel. I try to find time as soon as I get back from work, it depends if there's a math question I'm itching to work out.

 

[JasonRox]

Just thought I'd let you know that quotient spaces are very intuitive, and there's ample motivation behind them, in case you haven't figured this out already. It's all about identifying elements of the underlying set X via an equivalence relation ~. The resulting collection of equivalence classes X/~ is called the quotient space, or identification space. Here, a set is a collection of equivalence classes; it's open if the union of the equivalence classes produces an open set in the original space X.

Of course there's much more to say about this and where it comes from, but this should hopefully help you in knowing what to look for.

Exactly, that's the way I'm feeling now.

But at first, the idea of a quotient map... was just like, uh?

I'm really excited to see where it leads though.

 

[JasonRox]

Ah, so if Jason, who voted 30+ pages, now realizes that more like 5 pages, I wonder how many others who voted 30+ pages is also around 5 pages in reality...

I can do 30 pages.

But that's like Introductory Linear Algebra, or Calculus and things like that.

Later on, things are just different.

 

[JasonRox]

The question is entirely dependant on mitigating circumstances, so it's hard to judge, but on average I can get through a textbook in a week, I work full time, and I only have those couple of hours before I go to bed, and what I can read as I travel. I try to find time as soon as I get back from work, it depends if there's a math question I'm itching to work out.

If you read them that quickly, you should be putting Gauss to shame by now.

 

[hrc969]

Ah, so if Jason, who voted 30+ pages, now realizes that more like 5 pages, I wonder how many others who voted 30+ pages is also around 5 pages in reality...

Well who knows.

I did not vote at all. In my opinion counting number of pages is useless, meaningless, etc. I'll give you an example from Several Complex Variables since that's what I'm studying right now. You can go through two pages of Hormander's and in many cases you can learn more than going through ten of Krantz' book and it will take longer with Hormander's book. The difference is that Hormander's material is far more compressed and if you want to go through everything as thoroughly as you described it is easier to do so with Krantz's book.

So more pages does not necessarily mean you've learned more. Less pages don't mean you've learned less. Yeah, so I think your question is useless. It doesn't matter how many pages I read and it varies far too much. Even within the same subject as I said above it varies. Then it can vary depending on the field. So like Jason Rox said, some one can suck at learning one subject but be good at another.

 

[Gib Z]

Yea i was going to point that out as well. It depends on how hard the book is, how densely the information is packed, how much knowledge or deductions are assumed! 30 pages can be filled with crud, like the Australia textbooks I was talking about. Most arent A4, filled with pretty borders, nice big diagrams and abit of history surrounding the theorem. The proofs will take no calculation too easy to be assumed. This makes it much larger than say, Mathwonks 5 pages of Linear Algebra covering an entire semester!

 

[Lee55]

If you read them that quickly, you should be putting Gauss to shame by now.

I should of mentioned that this expected, as part of my current course. The textbooks are around 100 pages on average, sorry, I was about to go to bed when I wrote the last post.

 

[JasonRox]

I should of mentioned that this expected, as part of my current course. The textbooks are around 100 pages on average, sorry, I was about to go to bed when I wrote the last post.

But can you imagine reading that quickly though!

Apparently Gauss read Gauss's Arithmatica (spelling?) in two to three days!

He died at the age of 20 or so, and only started learning mathematics at like 15! And look at the work he DID! Damn those that ignored him. Ignorance and jealousy does no good!

 

[d_leet]

But can you imagine reading that quickly though!

Apparently Gauss read Gauss's Arithmatica (spelling?) in two to three days!

He died at the age of 20 or so, and only started learning mathematics at like 15! And look at the work he DID! Damn those that ignored him. Ignorance and jealousy does no good!

You don't mean Gauss here, do you? Figuring as Gauss lived to around 80 or so, did you mean Galois?

 

[JasonRox]

You don't mean Gauss here, do you? Figuring as Gauss lived to around 80 or so, did you mean Galois?

Yes, I meant Galois! Ooops, my mistake!

 

[mathwonk]

Lee 55 please read my graduate algebra textbook from my webpage, and let me know next week when you finish, including the exercises.

 

[mathwonk]

according to my best teacher, maurice auslander, a world famous algebraist, he knew only one mathematician (paul cohen, fields medalist) who could learn math by reaDING WITHOUT WRITING 3-5 PAGES PER PAGE read. so anyone who thinks they have learned 30 pages should have written 90 pages-150 pages.

 

[Gib Z]

How large is our handwriting allowed to be?

 

[mathwonk]

as my 8th grade teacher used to say, don't laugh, you only encourage him.(on a somewhat related note, it is sort of amazing how much one used to learn in 8th grade, based on the number of times since then i have heard myself say, "we learned that in 8th grade.")

 

[mathwonk]

hrc969, note that the first short chapter of hormanders book covers all of one variable complex analysis, and more, since it proves the singular integral version of cauchy's theorem.

soon after, as i recall, he solves the mittag leffler problem in several variables, by showing every non negative "divisor" on C^n is the divisor of a global holomorphic function.

i.e. in sheaf language he computes H^1 (O) = {0}. and this is just the beginning. (it was 30 years ago i read this stuff, and since it is not my specialty, I never did finish it all.)

of course note too that when he gets to the commutative algebra chapters, where he is not an expert, he slows down to snail crawl, as if he thinks that mickey mouse stuff is somehow dense or hard for analysts!

so for me personally the number of pages wirtten per page read of hormander, goes from way more than 5 at the beginning, (as high as 10-20 pages per sentence ocasionally, or even per word, for one famous "hence" as I recall) to about Auslanders number nearer the back of the book.

 

[mathwonk]

notice on this forum, when we tell people that a manifold is a top sopace locally homeomorphic to R^n they believe it, but when we tell them to understand math you have to do some work, some people don't want to hear us.

 

[JasonRox]

notice on this forum, when we tell people that a manifold is a top sopace locally homeomorphic to R^n they believe it, but when we tell them to understand math you have to do some work, some people don't want to hear us.

Why do all that work when you can use Wikipedia?

 

[ehrenfest]

A lot of math books include lots of commentary and redundancy and the "motivation" behind the theory in order to guide you through a complicated subject. Rudin's Principles of Mathematical Analysis has none of that. In terms of Rudin's pages, I said 15.

 

[mathwonk]

see how long it takes you to read and understand my 13 page primer of linear algebra on my webpage, which goes from definition of vector spaces through jordan canonical form.

thats right, 13 pages. so one day for some of you 15 page per day guys on that. then spend the next 4 days reading my 53 page notes on riemann roch theorem.

then check back in here with a progress report, and take the quiz.

i would think you would do well to finish those in a semester.

 

[BryanP]

Depends on the math.

Some come really intuitively to me and I can read many pages and understand it (meaning more than 10 pages a day).

 

[Crosson]

see how long it takes you to read and understand my 13 page primer of linear algebra on my webpage, which goes from definition of vector spaces through jordan canonical form.

thats right, 13 pages. so one day for some of you 15 page per day guys on that. then spend the next 4 days reading my 53 page notes on riemann roch theorem.

then check back in here with a progress report, and take the quiz.

I will accept this as a challenge, and within the next week or so I will find a day for each of these sets of notes. It would be nice if I had a copy of the test available in the digital equivalent of a sealed letter on my desk. That way I can read through the notes then put them away and "open" the test and write it that same day.

As for the Linear Algebra, I already have some familiarity with that subject, although your notes look interesting for many reasons, among which are their brevity.

The Reimann-Roch theorem, however, is not something that I have ever encountered (I am a physicist by profession). Therefore I for one will consider it to be a stronger test. Also, 54 pages in one day will exceed the number of pages in the original claim.

Furthermore, the total number of pages will increase by my reviewing / learning for the first time the things that you will in some cases assume as prerequisites. In other words I am asking permission to use the simultaneous use of supplementary readings to help myself learn your notes; in any case I will take the tests with all books closed and away if that is to be set as a condition of the forum challenge.

i would think you would do well to finish those in a semester.

That takes the pressure off somewhat, and would make success feel even better. I just want to say, regardless of any friendly forum challenge, that I really appreciate your contribution to the forums, Mathwonk. And I have a lot of respect for the details of your career; especially that postdoc at Harvard sounds fun :)

 

[daveyinaz]

I'll read the notes as well. Not that you guys know me or anything but still..

 

[tronter]

crosson did you read quickly in undergrad as well? Or only after gettinga PhD?

 

[Crosson]

crosson did you read quickly in undergrad as well? Or only after gettinga PhD?

Yes, I started doing self-study in my sophomore year at university.

Even before that I never felt the need to do exercises. And despite this, in school I always got top grades in math and a lot of respect. This is why I must do everything I can to fight the oppressive idea that exercises are a necessary part of learning mathematics.

Soon someone will say "but if he doesn't do the exercises, then he is only fooling himself in saying that he has learned the material." But this just stubbornly assumes the very proposition I am fighting against! Mathematics for me is more like washing dishes; if someone wants to give me an exam in the form of a sink full of dirty dishes, it is not necessary that I prepare by washing each type of cup and plate repetitively.

I have thought a lot about what it means to learn a part of mathematics. I don't think it is enough just to do well on exercises and exams. Nor is it enough to teach the subject at the university level. Ultimately, I think the necessary and sufficient condition for having learned a part of mathematics is to publish major research in that area. And if that is the standard, then I must I do not know any mathematics, unlike mathwonk. I may have absorbed a lot, and I can recall it quickly to pass any exam or teach a student, but I have no plans of doing major research in these areas.

That is why my original answer to this thread should have been "http://en.wikipedia.org/wiki/Mu_(negative)" ."

 

[uman]

So read Mathwonk's pamphlet in one day and I'm sure he will be able to test your knowledge of it afterwards.

 

[tronter]

Yes, I started doing self-study in my sophomore year at university.

Even before that I never felt the need to do exercises. And despite this, in school I always got top grades in math and a lot of respect. This is why I must do everything I can to fight the oppressive idea that exercises are a necessary part of learning mathematics.

Soon someone will say "but if he doesn't do the exercises, then he is only fooling himself in saying that he has learned the material." But this just stubbornly assumes the very proposition I am fighting against! Mathematics for me is more like washing dishes; if someone wants to give me an exam in the form of a sink full of dirty dishes, it is not necessary that I prepare by washing each type of cup and plate repetitively.

I have thought a lot about what it means to learn a part of mathematics. I don't think it is enough just to do well on exercises and exams. Nor is it enough to teach the subject at the university level. Ultimately, I think the necessary and sufficient condition for having learned a part of mathematics is to publish major research in that area. And if that is the standard, then I must I do not know any mathematics, unlike mathwonk. I may have absorbed a lot, and I can recall it quickly to pass any exam or teach a student, but I have no plans of doing major research in these areas.

That is why my original answer to this thread should have been "http://en.wikipedia.org/wiki/Mu_(negative)" ."

Surely you must find something difficult? Edward Witten did all the exercises in a book when he studied. It builds work ethic right? Then how do you read a math or physics book? Do you read it like a novel, then read another book on the same subject like a novel? Problem solving can only be gained through practice. Or do you do the problems in your head?

 

[Crosson]

Surely you must find something difficult?

Don't get me wrong, math is difficult and it consumes a lot of effort. But I am able to read and comprehend quickly because:

1) I spent a lot of effort studying symbolic logic.

2) In general I am good at learning the rules of language-games, e.g. the modern mathematical style of exposition.

3) I have the sort of memory that can recall important phrases and sentences with high precision.

But problem-solving is a different and related skill from reading and comprehending. All I know with respect to math is that I can solve book problems, because as I said I have not done research in pure math. I am not sure if I could meet my own standard of a worthy publication in those areas.

Do you read it like a novel, then read another book on the same subject like a novel?

Yes and no. I read math like I read a newspaper, but that only means that I read math relatively quickly and I read the newspaper relatively slowly.

As far as novels are concerned, I can't stand to read fiction and I have not done so since the 10th grade.

Problem solving can only be gained through practice.

Either

1) show me a rigorous proof of that claim

2) call me a liar

3) get my university on the phone and have my degrees revoked

or (recommended)

4) accept the fact that I am a counter example to that claim.

 

[uman]

Everyone will believe you when you learn linear algebra from Mathwonk's notes in one day.

 

[tronter]

so you would suggest studying symbolic logic? how did you become good at solving problems?

 

[tronter]

here is a sample random problem: Fix $b >1$, $y > 0$, and prove that there is a unique real $x$ such that $b^x = y$.When you first see this, how do you approach it? What do you think? Could you solve all the problems in Jackson E&M? Do you have a PhD?

 

[Crosson]

here is a sample random problem: Fix $b >1$, $y > 0$, and prove that there is a unique real $x$ such that $b^x = y$.

My first thought is: what do you mean by $b^x$ defined over the reals?

Is it defined as the unique continuous completion of the same function restricted to the rationals?

or

Is it defined by a power series?

or

as the solution of a differential equation

or

as one or another limit...

Either way my first goal would be to prove that the function is monotonically increasing. Then my second goal would be to get the standard algebraic property $b^{x + h} = b^x b^h$ either by working with the Dedekind cuts if we are completing from the rationals or else by direct series manipulation (which involves the binomial theorem).

Then to prove the existence we would use the least upper bound property on the set:

S = {x in Reals | b^x < y }

and go through the usual process that it is empty, bounded above, and that b^(lub(S)) cannot be less than or greater than y (this is where we will use the algebraic property from above). The uniqueness follows from monotonically increasing.

Maybe there is a more interesting proof based on differential equations.

Could you solve all the problems in Jackson E&M?

E&M is one of my favorite subjects! Yes, I think I can solve any problem in Jackson, but I must admit that there are some problems in there that I would never do for fun.

Do you have a PhD?

Not yet, although if I had stuck to one subject I would have.

 

[DeadWolfe]

Edward Witten did all the exercises in a book when he studied.

Source?

 

[tronter]

I emailed him.