The Should I Become a Mathematician? Thread

mathwonk

my perception of a calculus book is partly influenced by when it came out relative to my math education. kleppner i believe was a harvard physics prof who wrote his book after i had taken a spivak style course from john tate at harvard (as spivak also may have), so did not interest me at the time.

gootman is one of my favorite books for struggling students and i have a copy signed by gootman, my long time colleague and a master teacher and analyst.

i liked lang's calculus books and learned how easy and simple riemann integration is from them.


i loved comenetz's book, and wrote the initial rave review of that book. unfortunately i gave away my review copy as a prize to a good student. I attach my (edited) review, no longer available on the publisher's website: (see below)
unfortunately for the buyer, the price has increased from under $40 to over $125. Perhaps that is one reason my review has been removed, since it originally contained a grateful comment about the price.


i loved the first edition of edwards and penney, two wonderful scholars and teachers and friends of mine, but to my taste the book did not improve through several editions apparently designed to enlarge its audience at the behest of the publisher. it seemed to serve as the model for stewart's book.

schaum's outline series was wonderful in the old days, extensive and good problems, plus brief and useful theoretical summaries; but more recently when i tried to use it in a course, it seemed greatly reduced in quality and usefulness somehow, no longer worth it.

the elementary error in cruse and granberg is the fact that the fermat criterion for a tangent line is not that the polynomial which vanishes where the line meets the curve should have only one root, but that it should have a double root at the given point.

this is easy to check for polynomials where one can always divide (x-a) out from
[f(x)-f(a)]/(x-a) because of the first forced root, and after doing so, simply set x=a to see if there is a second root. the result is as usual that the slope of the tangent line to y = x^n at x=a is na^(n-1).


in fact i have experimented using this method to teach derivatives to undergrads, for polynomials. of course more analysis is required for transcendental functions like sin, e^x.


i wrote out this result in complete detail for the author and publishers when they commissioned me to review the book prior to publication, but they ignored it. perhaps the authors did not understand it either, but i suspected at the time, the book was already ready to go to print and thy did not care to know its flaws.

i have written this method up completely with examples in the class notes attached to post #6 of this thread:

http://www.physicsforums.com/showthread.php?t=441018

Attached Files

comenetz review 2 edited.pdf (78.7 KB, 14 views)

dustbin

In reponse to RJinkies partI above:

My calculus course is using Briggs this semester. I think it is a pretty good book... but I feel the exercises are too easy. The explanations are good, though. Definitely better than what I've read of Stewart. Actually, my favorite "popular" calc book is Thomas, I think. There are tons of exercises (100+ per section typically); some of which I've found are also in Apostol and Spivak (decent selection of proof problems). However, certainly not as good as Apostol, Spivak, Courant...

I read a while ago a suggestion for Calculus by Kitchen (forget first name) from mathwonk... I happened to see it in a university library today. Looked like a nice book that covers a lot of material most other books do not.

tinylights

Hi, I wanted to introduce myself. :)

I have recently discovered that math is my calling, and am studying it at a small 2-year college before transferring out next Fall to pursue my BS. I'm taking Calc 1 right now with a Stewart textbook (though due to the earnest recommendations for it all over this site I have ordered Spivak's Calculus as well) and am doing well, though there is a definite change in difficulty level between Pre-Calculus math and Calculus. It's actually quite exciting to me because I remember finding myself so bored in other classes when I could easily predict where my teachers were going with every idea, and the course I am in now is a lot closer to my pace.

Out of curiosity, does anyone know what the best colleges/universities in Florida are for a solid math education? I live nearby UCF so it is my most likely option, but I want to consider others so as to avoid my grad school speaking at me in a new language. And I've heard of a lot of people having issues with UCF's massive enrollment, primarily that of never getting a chance to connect with your professors.

Secondly, I've looked at a lot of grad school programs and they recommend acquiring reading fluency of mathematical texts in French, German or Russian. Which one(s) are most useful to learn, in your experience?

mathwonk

I "read" French, German, and Russian, well enough to pass a grad school math proficiency test, but only French well enough to actually read a math paper fairly easily.

As far as Russian goes, so few English speakers read it that most big Russian journals are routinely translated into English.

I staggered through a few sections of Riemann's papers in German but even those are at last available in English.

I always thought I could read Serre's clear papers in French, but boy the English version of Algebraic groups and class fields is much easier to get something out of.


So while it is recommended to learn these languages, at least french, and less so german, most of us get by quite well in english, occasionally having to struggle through an original language with a dictionary. but even to do that you need to know the basics of the language.

i.e. learn what languages you can, but be aware that you will be able to read almost everything written fairly recently in english. original languages are needed especially for reading some important works from the 19th century and early 20th cent.

e.g. with my weak german, i still have not read the great paper on linear series on algebraic curves, treated purely algebraically, including an early algebraic proof of the riemann roch theorem, by brill and noether.

it was kind of entertaining trying to struggle through a russian textbook on vector spaces (vyektornye prostranstva) when i kept running across the same words (ochevidno shto and silno) over and over, which turned out to mean "obviously" and "clearly"!

symbolipoint

MATHWONK,

you described your career progress a few times, but not remembering exactly, could you tell us: Did you study anything (Mathematics) while you were a meat-lugger, not in school? Or did you just work your labor job without studying your subject?

mathwonk

thats a little like asking country joe mcdonald what he remembers about the 60's, and he answers "nothing".

those were stressful days. suffice it to say i did very little math then. my progress went backwards for a while. i would say i lost about 10 years of my math career in there somewhere.

this is not a thread for discussing politics, but that was a great distraction. those were years when we were fighting fruitlessly in vietnam, nixon was president and using the resources of the government for spying on his "enemies", mayor daley's cops in chicago beat peaceful demonstrators as well as news reporters at the democratic convention, my friends were dying in the war, police and covert government agents were harassing demonstrators, tapping phones, arresting people, hundreds of thousands of citizens were marching in washington dc, there were demonstrations for grape pickers and other farm workers. mlk jr and bobby kennedy were assassinated. it was hard to focus on just preparing for a narrow scientific career. so although i occasionally still tried to read some pages of grothendieck, at age 27 i knew significantly less math than i did at 24 or 25. the one advance i made in those years was by assisting/grading in honors calculus, i had to read spivak's calculus book, and learned a lot of calc i should have known much earlier. reading grothendieck was of no help at all, sort of like telling an average 10 year old he needs to read dieudonne'. you learn by reading important basic material in a good concrete treatment and computing examples, not by reading fancy abstract stuff that is over your head. i also graded algebraic topology and learned some of that, as well as the new york times every day for a while. i still remember reading in 1970 that ARCO oil company had paid zero income tax for several years running on a profit of over 300 million dollars, which was a lot in those days, because of favorable tax codes that allowed oil companies to get a credit for the fact that they are gradually depleting the publicly owned oil supply, called the "oil depletion allowance". i.e. they got a tax break to compensate them for the fact that their free ride was slowly running out because of their own exploitation of public resources. this may still be in place. If we want to discuss these issues we should take it elsewhere though. but these issues distracted me from math for a few years. [/QUOTE]

converting1

has here been any older mathematicians (30+) who've made any impact on mathematics (if so who)? Reading up on mathematicians it seems as though everyone makes great work in their early twenties then just die down

micromass

http://mathoverflow.net/questions/35...-learners-list

converting1

Guess it's never too late then! Thought I had little time left seeing as I'm 17,

thanks

homeomorphic

30 isn't that old. Actually, very few mathematicians today even get to the point where they can make any significant contributions UNTIL they are about that age. The average PhD age is like 27 or 28, and my impression is that postdocs were this extra thing that they had to stick in because a PhD isn't really enough to become a mathematician anymore. So, by the time you are done just getting started, you're that old.

eliya

30 isn't that old to start or to finish. A lot of mathematicians ``made impact" beyond their 30's. Andrew Wiles, for instance, missed the Fields Medal by a few months.

As a general rule though, don't think about making an impact. Every mathematician who's active and writing papers is changing mathematics, of course, to different extents. To paraphrase Robion Kirby, don't worry about the significance of your mathematical results, worry about being the best mathematician you can be, and the rest will follow.

chiro

Take a look at George Polya, who started late relative to a lot of others (consider also that mathematics has exploded since 100 years ago) and didn't start studying mathematics:

http://www.nasonline.org/publication...lya-george.pdf

Born in 1987, got the doctorate in 1912 so got the doctorate at the age of 25 (but please put that into context for mathematics especially probability at that time, and I am not denigrating Polya when I make these statements).

mathwonk

whether or not you will do important math is not determined by your age, surname, gender, or anything else. It is based on your desire. go for it.

dkotschessaa

Just turned 36. Still an undergrad. Not giving up. :)

Mariogs379

@mathwonk,

regardless of what I do re: staying in NYC vs. Brandeis program, I'm gonna take some math in the spring semester. Seems like it makes sense, for continuity's sake, to take real analysis II.

Was also thinking Algebra I. Thoughts?

giorgiv19

Well, I have read some of the posts about textbook recommendations and want to offer an insight of my own:
Normal calculus textbooks? Don't bother. Don't read them, they do more damage, than good. The best thing to do is pick up a Russian Analysis textbook, like Fihtengolz, Zorich or Kydriatsev. They all come in 3 volumes.
Also no textbook is good without exercises. For this the best one by far is Demidoviche's "A Collection of Problems in Analysis".
The other essential thing for mathematics is linear algebra and analytic geometry. Serge Lange has very good book in linear algebra.
But the most important thing is not just studying at a university. You should look for open seminars. These seminars will give you much greater knowledge, than any course ever would.

mathwonk

thank you for these views which differ from many usually found here, and supplement them nicely!