How many pages of math theory can you absorb 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.

 

[tronter]

Crosson, then why did you have trouble with Alg 2 in high school? And how did you become good at problem solving?

 

[Crosson]

Crosson, then why did you have trouble with Alg 2 in high school?

I made bad grades in high school because the teachers and textbooks were/are so bad that I wouldn't want anything to do with them in any lifetime. The math back then was delivered at such a slow pace that I didn't see it going anywhere and I wasn't interested. That changed when I discovered the potential for self-study.

And how did you become good at problem solving?

My advice is to read as many proofs as you can. In the time it takes to write one proof, you might be able to read 10. This way, when you are forced to write a proof of your own, you will have a large enough bag of tricks to try out.

I just want to point out that I am being singled out for questioning, when according to the poll there are several others who claim to read 30+ pages per day.

 

[tronter]

and also read from many sources? what school do you go to? do you do the same thing for physics (e.g. don't do the problems)? Is it all about focus?

 

[mathwonk]

let me offer as a test, the exercises that are included in my notes on linear algebra. if you can do them as you read, then i think you will have absorbed the material.

( a new version of those notes went up on my website a couple days ago.)

let me say however that although you may read the 13 pages of text in those notes in one day, and even learn something, i do not encourage you to try to complete all the exercises in one day.

unless you already know the material from some prior exposure, i think that is quite unrealistic. indeed if you do not already know the material, even completing those exercises at all, over a good number of days, is already impressive.

the reason my notes are so short is that many significant facts are in the exercises. indeed the exercises do not require difficult new ideas, but they do require mastery of the difficult concepts in the text.

hence i believe they measure quite well, and reinforce, understanding those ideas. i plan to teach linear algebra this summer, and will feel successful if we cover those topics in the full 8 week course.

indeed you may note by comparison with a standard book like that of insel, spence, and friedberg, that my 13 pages cover essentially all of their 400 pages.

 

[AzonicZeniths]

I am currently teaching myself calculus and usually every day I cover about 25-30+ pages without reviewing, if I am reviewing that day I usually cover only around 15-20.

 

[mathwonk]

another way to put the fact that my 13 page notes cover the same ground as insel et al's 400 pages, is to say that a person who reads 15 pages of their book a day, should expect to read less than one page of mine a day, maybe a half a page a day

 

[Crosson]

another way to put the fact that my 13 page notes cover the same ground as insel et al's 400 pages, is to say that a person who reads 15 pages of their book a day, should expect to read less than one page of mine a day, maybe a half a page a day

I'm glad you said that, so that I don't have to feel too bad for only getting through the first set of exercises.

I have not been doing very much math lately, so I couldn't work on it for any longer than four hours (then I'm tired). Of course, most of that time was spent typing so that I could "prove" to you guys that I am "learning" what's in there.

Not that I expect anyone to read these solutions carefully, especially because I have not proof read them (I reached my math limit for today and then stopped), and because in many cases I did not get around to including the statement of the problems (just check mathwonks notes, exercise 0 refers to the first unamed exercise and the rest have numbers).

I am nearly finished reading the notes and doing the challenging exercises mentally, but I can only type for so many hours in a day.

 

[mathwonk]

wow! great! i am on vacation, but i Will read your work. anybody else is also encouraged to jump in now that crosson has had the cojones to break the ice on this.

 

[tronter]

how was it?

 

[mathwonk]

what a dork i am. i got back from vacs a couple days ago and totally forgot to read his work. i myself did read and partially absorb about 3-5 pages of looijenga's book on isolated complete intersection singularities today though, [theory of relative differentials, and definition of kodaira - spencer map associated to a family of varieties]. the embarrassing part is realizing i myself wrote a paper on related topics 25 years ago which i have now mostly forgotten. ******!@! man you just have to keep putting one foot in front of the other for your whole life, or you lose your place. [principle of intellectual entropy for you physcics buffs.]

please forgive me. i mean well, but have limited attention span and energy.

i am starting a summer class on that linear algebra stuff on thursday but do not have the moxie to use my own book, assuming few of my students are likely to be able to learn from it.

i say this in case it encourages any of you young students, i am just like you, only older and slower. i know a lot of old stuff, but it cost me many hours to learn it. if i want to learn more, or even stay current on stuff i once knew, i must pay the piper again.

 

[Howers]

I can probably absorb up to 5 pages of a very terse book. Usually, I can understand a chapter completely by 2-3 days (5-10 page chapters). If the book is a little thicker and more chatty, I can easily eat up 10-15 pages. But if a difficult proof arises (ie. Principal Axis theorem or uniform continuity) I can spend 2 hours on half a page and still come out with nothing.

But is this about if I have the entire day to devote to one subject? If so, there is no reason I couldn't do 10-20+ pages, although I doubt it would be full understanding and I think I'd get bored of that.

 

[mathwonk]

look, let me tell you how i evaluate whether i have absorbed or not. i am trying to prove something that requires me to extend the known theory of deformations of isolated singualrities. as long as i cannot do that, i feel i have not understood the existing theory of deformations.

so my measure is not just that i can say, gee i think i understood that. it is that i can use it to prove a new theorem in the subject. to me this is the proof of the pudding.

 

[tronter]

mathwonk do you still take your teacher's advice in writing 3-5 pages when reading a single page?

 

[mathwonk]

well not today, but i was in a doctors office, and thank you for reminding me. actually i usually write far more.

but my purpose in posting tonight is to ask ourselves to raise our sights in respect to measuring what we are learning when we read.

lets ask ourselves, if we can repeat what we have read, if we can give a talk on it, if we can answer questions on it, and maybe if we can extend it, by proving more results.

 

[Topologist]

I think that there are certain "extents" to which you can absorb the material in a mathematics textbook. For instance, on a scale of 1-10, I would say that someone who reads all the definitions, theorems and discussions in a given section of a textbook in a day, and can engage in a 30 minute discussion about the material if asked to do so, is a "5". On the other hand, someone who proves all the theorems in a given section of the book on his own (without looking at the proofs in the book), formulates conjectures and original results in the context of certain definitions, and can lecture the material the following day, is somewhat closer to a "10". Personally, I think (and know) that quite a few mathematicians could easily complete textbooks in a few weeks, but that, in my opinion should not be done when you are still learning "basic mathematics". If someone absorbs a complete first year (Harvard) graduate mathematics education on the scale of "10" (which I described), it is perhaps OK for him/her to learn more advanced branches of mathematics at a quicker place. Basically, it is important to be very experienced in mathematical thinking before learning material quickly (as an example, if someone were learning algebraic geometry, it would be alright for him to read Hartshone's book quickly if he/she had already thoroughly absorbed Eisenbud's commutative algebra, and Harris' algebraic geometry). I think I could absorb approximately 50 pages of mathematics material in a day, but I do not think that I would gain the same excitement equivalent to thoroughly researching the material (formulating conjectures and results, developing intuitions in the area, etc.).