## The Should I Become a Mathematician? Thread

freshman calc: Courant/Hardy/Landau - Foundations of Analysis

how long were people using Hardy? I got the impression both Courant and Hardy did well into the early and mid 60s, though i think hardy faded a bit quicker. Esp with so many people trying to replace both with all the newer 60s texts.

What did you think of Hardy and Landau?

Hardy seems like a pretty rough ride for anyone taking math after the Space Race.
I think anyone reading it would go through it at a glacial pace, and i wonder if anyone finished the damn thing...

Landau looks cool, totally minimalist, and puzzling as Babylonian cuniform....

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sophomore calculus: i forget what book,

oh damn, that's the best part...

Was it more of Courant? and was there another vector book?

were any of these possibly on the reading list, or recommended by the teachers?

1955 AE Taylor - Ginn
1957 Apostol [I'd think you'd remember that one]
1959 Nickerson Spencer and Steenrod - van Nostrand
1964 Protter and Morrey - Addison-Wesley [all these would probably be after you took your degree/classes]
1965 Buck - McGraw-Hill [actually that's probably the second edition, there was probably a first edition 1957-1963ish]
1967? Spivak - WA Benjamin?
1968 Loomis and Sternberg - Addison-Wesley- free pdfs at his website
1970 Rossi - Addison-Wesley [oh oh another Brandeis person]

[I'm not sure if missed anyone from 1955-1980s there, but if there's any famous forgotten text from the 50s 60s 70s, tell me someone]
[oh hell tell me about the terrible ones too!]

my feeling there wasnt really anything out in the 70s... just Thomas and Finney clones and 15% of the books just mentioned...

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I get the feeling that Apostol and Buck soaked up most of the sales at the high end, and Thomas and Finney for the rest]

when did the first Spivak come out? wasnt that like in 1967 I assume you read it after your degree, and the other book he did i think was 1965 on manifolds..
[or did you zoom through it after your degree and before grad school]

I always found it interesting where i'd struggle with a mainstream book and then eons later, find it more approachable [or find the easy and hard books on the same subject more approachable]

I used to think that you liked Loomis before, but it was more 'something you went through' but wouldnt really recommend... [when you clarified things a while later]

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sophomore algebra - linear algebra and matrix theory - nering
fundamental concepts of higher algebra - aa albert

What did you think of Nering?

I assume that was a fixed up edition of albert's 1930's abstract algebra books
[Modern Abstract Algebra - Chicago 1937]
[Introduction to Algebraic Theories - Chicago 1941 - more an introduction to the other book]

Linear Algebra didnt really seem to take off till the 50s/60s, or bits of it in a Calculus III part of the text....
[or they dropped it being called Theory of Equations like using that famous Uspensky book and made it way easier and modern looking in the mid 60s]
[maybe it was all the mainframes doing Linear Programming that got it popular in the schools]

1952 Perlis - Theory of Matrices - Addison-Wesley
1952 Stoll - LInear Algebra and Matrix Theory - McGraw-Hill
1964 Bickley-Thompson - Matrices and their Meaning - van Nostrand 1964

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- complex variables, text by Henri Cartan

so the pures went cartan and the applied went to churchill? [or did anyone do the easiest thing and read churchill first?]

Kaplan did a big Addison-Wesley on Complex too in 1953...

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advanced calculus: official text: calculus of several variables by wendell fleming, but the lectures followed more closely the book Foundations of modern analysis by Jean Dieudonne

Did you take adv calculus at two different times, or was fleming out that early?

[I got the impression that Courant and Spivak and Fleming were the best of the texts from the good ole days from you]

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senior: real analysis taught by lynn loomis, no textbook, it covered abstract measure theory as in the book of Halmos

Halmos came out in 1950 and probably the closest in style is Bruckner.

I remember seeing a strange set of analysis books at Simon Fraser, they used Goldberg [Wiley 1976] and Bruckner [Prentice-Hall 1996]

Goldberg looked stiff, but i heard it's pretty traditional and a touch gentler as far as dry analysis books go, but it's sure a rare one, musta been popular in the mid 70s and with the MAA and got tossed into obscurity when Rudin got pushed more and more...

[I still find Binmore or Colin Clark [The Theoretical Side of Calculus] as the two easier books out there]

and didnt Marsden write a pretty gentle and wordy Analysis text? It seemed the book to read before tackling Hardy]

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algebraic topology taught by raoul bott, text: algebraic topology by spanier. most people today recommend the book by allen hatcher.

How did you find Bott's texts? [Bott and Tu]

Spanier..... well i was going to say, amazon, but i peeked and it's from the chicago list of books...

[Spanier is the maximally unreadable book on algebraic topology. It's bursting with an unbelievable amount of material, all stated in the greatest possible generality and naturality, with the least possible motivation and explanation. But it's awe-inspiring, and every so often forms a useful reference. I'm glad I have it, but most people regret ever opening it.]

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I want to remind that i did not learn much from this somewhat harsh and user unfriendly first exposure to mathematics

people say that Caltech's course probably 'teaches' more, but if you throw teaching out the window, Harvard is the most difficult one...

I found these notes 'somewhere' and it had to deal with Rudin's textbook ...

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[Harvard 55ab takes about 50 hrs a week of study]
[Thoughts on the flaws of Harvard 55]
[After having chosen Caltech over Princeton and Harvard to pursue a math major, I feel strongly that the math department's main feeder course here - Math 5 - is by far the strongest of the various courses at top universities which are taken by the strongest math students. It's main virtue is that it is long enough (a year) to do something serious, and that it does it in a thorough methodical way, building up steadily to huge, important theorems that you actually understand fully by the time you get to them.]
[I know that the 'stronger than the others' claim is true for sure in comparison to Princeton, since I actually took their math major feeder courses when I was a high school senior. (Problems there: teaching quality haphazard, too-advanced material rushed through so that even the brightest students are lost, though Jordan Ellenberg's Math 214 was a well-known and beautiful exception - but he's not there anymore.) And yes, I think Math 5 here is stronger even than Harvard's Math 55. While Harvard's famous course covers a lot of esoteric and advanced topics, it does so with very little unity and requires overwhelming amounts of outsdie reading so that even the best students miss 30% or so of the ideas.]
[After a year and a half at Caltech, I knew everything that a Math 55 graduate knew, but various comments I've heard make it pretty clear that most of them come out with a "scattered" feeling - they've been exposed to a lot but don't have a particularly unified picture. Math 5 keeps to a more manageable area and explores it more deeply, and so one comes away with some very tangible and coherent knowledge.]
[Those are my feelings on the subject.]

and...

[Caltech Math108a - used Rudin and Carothers and Elias Stein Complex Book - 2 real+1complex]
[the combo of the three is better than Harvard 55]
[Loomis and Sternberg's book used to be used for Harvard 55ab]

and

[I think this book is inappropriate for use as an undergraduate textbook. Its use at the introductory graduate level is defensible, but I see no reason to choose this book when better ones are available. Apostol's Analysis book is at a similar level but has much richer discussion and is more comprehensive. For a book slightly more elementary than that, I would recommend Taylor and Mann. Like I said above--as a sequel to this or similar books, I think the Rudin "Real and Complex Analysis" book is absolutely wonderful. This book does have one purpose for which I found it to be very well-suited: it is useful to work through, perhaps only once, to review the subject and solidify your understanding of the material. But its value as such does not warrant purchasing it at the obscene price.]

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- almost all algebra books seem to fail this test to me

[high school or abstract?]

a. Gallian
b. Fraleigh
c. Beachy and Blair
d. Allenby
e. Saracino
f. Pinter
g. Childs

those 7 i think are the easiest ones on my list, and the first two are probably 'well-known'

how did you find Paul Cohn's books [1970s-1990s]

[i think one of his introductory books was fixed up considerably with the newer editions]

not sure what to think though, since it's not used that much in any of the syllabuses out there [or anymore]

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- along with books like rudin's analysis.
- Many people recommend Dummitt and Foote and it does have many good qualities but I have several criticisms of it.

What texts are you somewhat [or completely] sour on?

It's rare to actually hear people criticize a popular book, or classic [in whole or part]

Heck, the first time i saw Apostol's texts i said, man, none of this is really necessary... but i was impressed at how huge the books were, and thought man it would be one hell of a school that used these as 60 weeks of 'an introduction to calculus'....

but i'm sure if one tackled a mini calculus course or had a book to read in parallel, it would be much better. But as a first and only textbook, oh i shuddered, but i definately spent a good 30 minutes at it in the 1980s saying, wow this is surreal, it's the hardest calculus book i seen.

much later on, i added it to my 'shopping list'

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I added three books to the list too..

Nering is a new one...
Mackey's complex text
and Arbarello....

your stories definately do get better the more we hear them mathwonk!
much appreciated
 Recognitions: Homework Help Science Advisor well heres one more story about my sophomore calc book and why i don't remember the name. after getting a D- in freshman honors calc part 2 from john tate (a course i had only attended once a month, during my slow decline before eventually getting kicked out for a year), when i returned in the fall i had to take non honors sophomore calc, taught as it happens also by tate. tate was a great prof, but in the non honors course he had to use the book chosen by the departmental calculus committee instead of picking his own. So it was one of those routine mediocre books they use at places that are not harvard, reasonable but not too challenging (Taylor?). the course was ridiculously easy in comparison to the previous year's course, and although i did not work or attend much and seldom handed in hw, i was still passing as i recall. one day in discussing the implicit function theorem in class on a day when i was there, tate read disgustedly from the book's treatment: "the proof of this result is beyond the scope of this book". He slammed the book on the desk and said loudly "well it's not beyond the scope of this course!" and went over to the board. Then he stopped, looked back at the offending book lying on the desk, strode quickly back, grabbed the book and slammed it into the trash can with both hands. Then at the end of the class, he went back, calmly retrieved the book from the trash and assigned homework from it.

 Quote by mathwonk well heres one more story about my sophomore calc book and why i don't remember the name. after getting a D- in freshman honors calc part 2 from john tate (a course i had only attended once a month, during my slow decline before eventually getting kicked out for a year), when i returned in the fall i had to take non honors sophomore calc, taught as it happens also by tate. tate was a great prof, but in the non honors course he had to use the book chosen by the departmental calculus committee instead of picking his own. So it was one of those routine mediocre books they use at places that are not harvard, reasonable but not too challenging (Taylor?). the course was ridiculously easy in comparison to the previous year's course, and although i did not work or attend much and seldom handed in hw, i was still passing as i recall. one day in discussing the implicit function theorem in class on a day when i was there, tate read disgustedly from the book's treatment: "the proof of this result is beyond the scope of this book". He slammed the book on the desk and said loudly "well it's not beyond the scope of this course!" and went over to the board. Then he stopped, looked back at the offending book lying on the desk, strode quickly back, grabbed the book and slammed it into the trash can with both hands. Then at the end of the class, he went back, calmly retrieved the book from the trash and assigned homework from it.
This needs to be said.
 Recognitions: Homework Help Science Advisor another reason for not remembering the name of the sophomore calc book may be that i did not own a copy and just borrowed one to read the day before the test. i thought that was cool, then.
 Recognitions: Homework Help Science Advisor for me it was not so much the book, as when i started to take learning seriously, but some books like spivak went out of their way to reach me before i knew how to study. i.e. no book is too hard for a serous student, but some books reach out to the clueless.
 hi Dowland hi mathwonk best write up on Euler's book is http://plus.maths.org/content/eulers-elements-algebra ------ Euler's Elements of Algebra Leonhard Euler, edited by Chris Sangwin paperback - 276 pages (2006) Tarquin Books $22 [The style is engaging; the structure and language is clear, and the explanations logical. The approach is surprisingly modern and does not suffer either from being nearly 250 years old, or from being an edited version of a "charming" English translation from the 19th century. In fact, this English text comes from an 1822 English translation of a French translation of the original German. That such writing can still be called clear and readable is something of a miracle, and testament to Euler's original clarity and readability. This edition has excised various later accretions such as editors' footnotes and introductions, as well as an entire chapter added by Lagrange, material which may be reproduced if a reprint of Part II of Euler's work is ever attempted.] [For me, the mystery of this old school textbook, which doesn't hold your hand and so seems to lead you rapidly through a ton of material, is that so much is conveyed in a spare, clean style. In fact, I expect that more material is covered than in more modern textbooks which spend an age going over and over material, and yet books like Elements seem less hurried than modern ones.] [For example, Euler's definition of the integers seems to exclude zero. Later, he gives good reason to suppose that there is an infinity of numbers between two integers, but he couldn't know of the different "sizes" of those infinities which Georg Cantor discovered, and which a brief note might bring alive. He also anticipates the great utility of imaginary numbers. An index would also increase the usability of the book, especially for those interested in the history and development of mathematical concepts.] [Overall, the book is to be highly recommended. The broad range of elementary topics means the book can and should be referred to often. The structure, readability, and standard of explanations lead to a rapid and rewarding learning experience, while the elegance of the prose is frankly a joy to read. The book soothes ageless anxiety caused by learning the mysteries of logarithms and imaginary numbers and yet does not shy away from addressing practical problems, even how to calculate interest — a footnote on the dangers of credit cards would go well here.] ------ I'm not yet sold on it, anyone wanna twist my arm? A few years ago Springer in 3 vols did his calculus text, finally translated in English, seemed interesting enough off amazon for me to dump it in my 'neat' list.... people seemed to like it browsing at what was essentially the first textbook on calculus... [hold on let me drag it out] 32 Foundations of Differential Calculus - Leonhard Euler - Springer -$70 [The First calculus texts] [more intuition than formalism] 33 Introduction to Analysis of the Infinite: Book I (Books 1 + 2) - Leonard Euler - Springer - $105 34 Introduction to Analysis of the Infinite: Book II - Leonard Euler - Springer -$90 If you got \$275 kicking around... but it's probably a better and weirder read than new copies of Stewart or Thomas and Finney.