## The Should I Become a Mathematician? Thread

hi Mathwonk

good to know nikuklin flows into Stillwell's Geometry of Surfaces book

i got some interesting notes/quotes for that one and i actually plopped it in book 17 under topology *grin*

Notes:
[This is the book that made me a mathematician.]
[It is an attractive mixture of topology, algebra and a smidgen of analysis.]
[The main theme here is the deep connections with complex function theory.]

------

The preceding book was Stillwell, which because of the comments and the MAA rating, is on my list of old junky books to buy one day...

-----
16 Classical Topology and Combinatorial Group Theory - John Stillwell - [Springer 1980?]
[This book is great! No book on this list coincides with my own mathematical esthetics like this one: I checked this book out this summer while I was doing research on surface topology and read it cover to cover: you'll see how geometry relates to topology relates to group theory. I wish this was my first algebraic topology book, because it's full of exciting theorems about surfaces, three-manifolds, knots, simple loops, geodesics - in other words, it's rippling with geometric/topological content intead of commutative diagrams. Let me also recommend Stillwell's book Geometry of surfaces, along the same lines.]
[an excellent guide]
[Chapter 1 is very intriguing and contains lots of ideas.]
[Chapters 2-5 were a bit slowed down by foundational issues, but now in chapters 6-8 it's all topology all the time.]
[There are many ways to destroy the soul of topology. Stillwell says in the preface: "In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams."]
[Stillwell protects us from such dangers by his emphasis on low dimensions, his insistence on the fundamental group as the best unifying tool, visualisation and illustrations, and his great respect for primary sources. The latter is reflected in frequent references and in the commented, chronological bibliography, which is very useful.]
[MAA - 1 star recommendation] - Topology: Algebraic Topology
-------
-------

a. Henderson is new, what book/s do you recommend before tackling it?

and

b. which book/s by Taimina would you suggest, and what's texts are good before attempting it?

-------
 hi dustbin - I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)? Calculus just blurs into analysis, hardy and rudin are lumped with advanced calculus like courant and kaplan. When it's elementary calculus that's the 'vigour' and when you apply the spiral approach come back to it, with advanced calculus, you add the 'rigour'. Books like Courant and Apostol just start off with a bang with both. And it helps if you started with something like syl thompson or je thompson [calculus made easy/calculus for the practical man] And if you used older terminology, like in the early 50s, rudin could also be classed as 'functions of a real variable' and rosenlicht is no different than rudin. which is well analysis and you could say it turns into real analysis and real variables too... what they are doing is overhauling what the number system is, and shaping your intuition about what functions are and what variables are.... and suppossedly...magically one day you end up with a box of tools where don't fuss with trivial issues... and well one *hopes* that soon after you stop fearing mathematics, can can more often get to the heart of the problem. So, with that emphasis of Nathan Grier Parke [guide to the literature of mathematics Dover 1957] where analysis goes... you can suppossedly save a ton of hours with those 'trivial issues' which went on a century or two before in mathematics... and well with all that analysis you an be 'rigorous' with Fourier Series, and 'rigorous' with probability theory too. And well you can do stretchy rubber sheet geometry too, whoops topology.

Thanks for your insights RJinkies. I appreciate all of your vault notes!

 Quote by RJinkies And well you can do stretchy rubber sheet geometry too, whoops topology.
Lol.

 Quote by RJinkies hi dustbin - I often hear people say that Spivak and Apostol's Calculus texts are basically introductory analysis texts. What is the difference between Spivak/Apostol and books that are specifically titled along the lines of Introductory Analysis or Introduction to Anaysis (such as Rosenlicht)? Calculus just blurs into analysis, hardy and rudin are lumped with advanced calculus like courant and kaplan. When it's elementary calculus that's the 'vigour' and when you apply the spiral approach come back to it, with advanced calculus, you add the 'rigour'. Books like Courant and Apostol just start off with a bang with both. And it helps if you started with something like syl thompson or je thompson [calculus made easy/calculus for the practical man] And if you used older terminology, like in the early 50s, rudin could also be classed as 'functions of a real variable' and rosenlicht is no different than rudin. which is well analysis and you could say it turns into real analysis and real variables too... what they are doing is overhauling what the number system is, and shaping your intuition about what functions are and what variables are.... and suppossedly...magically one day you end up with a box of tools where don't fuss with trivial issues... and well one *hopes* that soon after you stop fearing mathematics, can can more often get to the heart of the problem. So, with that emphasis of Nathan Grier Parke [guide to the literature of mathematics Dover 1957] where analysis goes... you can suppossedly save a ton of hours with those 'trivial issues' which went on a century or two before in mathematics... and well with all that analysis you an be 'rigorous' with Fourier Series, and 'rigorous' with probability theory too. And well you can do stretchy rubber sheet geometry too, whoops topology.
Just to be completely sure, are you saying that it is better to spend your time on Parke's book rather than Spivak & Apostol?

Ohh, okay thanks. And man, that was quite a list!

I'll likely pick Apostol or Spivak and an intro to proof book, I want to make sure I can master my first proof class later on.
 All this talk of old texts has me wondering, have any great mathematicians of today written a text that people will be talking about in 50+ years?
 Silly question - when do you folks read all these books (Like Apostol) if they are not part of the classes you are already taking? Are you doing this while you are taking classes or after you've gotten through the traditional sequence? I do lots of extra reading and studying, but nothing quite this heavy yet. -Dave K
 I just think people collect the books, before or after their classes [if they do the classsic] One just finds the books that speak to you in the library, or if you're lucky, you find someone to talk to or a list somewhere. [a lot easier with the internet in some ways] If you're aware of the curriculum and know what the general syllabus is for the courses, you just go on a lifelong easter egg hunt and find what 'fits' your style. It's one thing to browse and another thing to 'study' the books, but don't ignore the joy of browsing and searching, it's all a part of getting your own unique box of tools. sometimes the curriculum helps and often it hurts... i remember there wasnt any good algebra books at home, but for calculus there was the quirky and likeable Sherman K Stein's book [1969 and then a few 70s editions] and JE Thompson's calculus book from the early 30s. But i probably would hath been better off if i read Stein and Thompson rather than waiting around for an actual class in calculus, in hindsight.... But i was buying Symon and Kleppner for physics without a damn calculus physics problem in my life, and those books 'spoke' to me. Courant i heard about, and didnt see a copy till after i took calculus. Though i saw the creepy gold dustjacket of the first part of the 1963 Courant and John edition, which was definately a 'weird' one..... I think how the curriculum goes against you, is i still think the best ways of learning some things are by taking a course twice, with an easy text and then a harder one after. There is something sorta magical about seeing how clear and straightforward something like Calculus for Electronics can be, and often you get a better working box of tools with that outside of the classroom, than *inside* one with a regular text. Things like the Berkeley Physics Course and Feynman's Lectures didnt take off, though i tend to think of Griffith's books now as a new form of that [now that he's written the other two texts], and surprisingly they are now a solid part of the mainstreain curriculum. ---------- I got a good question.... a. Why did Stewart's calculus textbook take off so successfully? and is anyone out there bold enough to toss some detailed minuses, and detailed pluses to the text? [and like Thomas and Finney the earlier books were better...] b. I thought Flanders book on Calculus was something close to taking off as a popular text in the late 80s - WH Freeman [it's a glossy white one, and the first edition was white and red cloth [he was much more famous for the differential forms calculus book way way earlier] and what were some of the famous calculus textbooks, when Apostol/Spivak and Thomas-Finney werent used in the 50s 60s 70s.... I thought it odd how Thomas and Finney gradually turned into a second year only textbook and dropped for most with first year calculus....
 Recognitions: Gold Member Here's a fun problem to solve that I did a little while back: Let $$x=\frac{1-t^2}{1+t^2}$$ and $$y=\frac{2t}{1+t^2}$$ Show that $x^2+y^2=1$
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Let $t=\tan(x/2)$

 Quote by RJinkies I got a good question.... a. Why did Stewart's calculus textbook take off so successfully? and is anyone out there bold enough to toss some detailed minuses, and detailed pluses to the text? [and like Thomas and Finney the earlier books were better...]
I'd like to know the answer to that too, though I think I can contribute somewhat to the discussion. I think the strongest point of Stewart's book may also be what makes it so disliked by many students - namely, that is extremely concise.

Most students complain that it is "hard to read" and that "it doesn't have enough examples." I think the philosophy of the text is to keep students away from the "plug and chug" method of heading straight for the homework problems, looking for examples that are similar, and re-arranging the necessary formulas.

The explanations are actually very good but have to be read very carefully and "unpacked." Sometimes there are VERY IMPORTANT details that are relegated to a small, fine print marginal sentences. When I took notes out of this book I would often re-write what was contained in a single paragraph to something (for my understanding) that would fill a whole page of notes.

I looked at earlier editions of the book and it seems to have gotten thinner and thinner as the new editions came out. Stewart is putting supplements online, but most of the students I studied with weren't even aware of this (even though it is advertised in the book.)

-Dave K
 That problem is much more fun in reverse! (Finding a rational parametrization of the unit circle, that is.) Give this one a try: parametrize the curve y2=x3+ax+b