# The Should I Become a Mathematician? Thread

by mathwonk
Tags: mathematician
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 Wow. You would do a superb PhD if you have the inclination, but as you are already earning a living that would be a sacrifice. You have the innate power and creativity of a PhD level mathematician. This is unusual with only a BS.
When I thought I was going to look for a programming job, my plan was to go back to school and learn more math.

But since I'm actually employed as a mathematician (and have become fairly good at self-study), I don't feel as much need. OTOH, my employer will pay for some full-time schooling (both the classes, and giving me my full pay!), so I really ought to take advantage of it. My buddies keep trying to tell me to go and get a masters in logic.
 HW Helper Sci Advisor P: 9,371 Some recommended undergraduate books for future mathematicians. Introductory calculus. 1. Calculus (ISBN: 0521867444) Spivak, Michael Bookseller: Blackwell Online (Oxford, OX, United Kingdom) Price: US$53.66 Shipping within United Kingdom:FREE Book Description: Cambridge University Press, 2006. Hardback. Book Condition: Brand New. 3Rev ed. *** CONDITION NEW COPY *** TITLE SHIPPED FROM UK *** Pages: 672, Spivak's celebrated textbook is widely held as one of the finest introductions to mathematical analysis. His aim is to present calculus as the first real encounter with mathematics: it is the place to learn how logical reasoning combined with fundamental concepts can be developed into a rigorous mathematical theory rather than a bunch of tools and techniques learned by rote. Since analysis is a subject students traditionally find difficult to grasp, Spivak provides leisurely explanations, a profusion of examples, a wide range of exercises and plenty of illustrations in an easy-going approach that enlightens difficult concepts and rewards effort. Calculus will continue to be regarded as a modern classic, ideal for honours students and mathematics majors, who seek an alternative to doorstop textbooks on calculus, and the more formidable introductions to real analysis. Preface; Part I. Prologue: 1. Basic properties of mumbers; 2. Numbers of various sorts; Part II. Foundations: 3. Functions; 4. Graphs; 5. Limits; 6. Continuous functions; 7. Three hard theorems; 8. Least upper bounds; Part III. Derivatives and Integrals: 9. Derivatives; 10. Differentiation; 11. Significance of the derivative; 12. Inverse functions; 13. Integrals; 14. The fundamental theorem of calculus; 15. The trigonometric functions; 16. Pi is irrational; 17. Planetary motion; 18. The logarithm and exponential functions; 19. Integration in elementary terms; Part IV. Infinite Sequences and Infinite Series: 20. Approximation by polynomial functions; 21. e is transcendental; 22. Infinite sequences; 23. Infinite series; 24. Uniform convergence and power series; 25. Complex numbers; 26. Complex functions; 27. Complex power series; Part V. Epilogue: 28. Fields; 29. Construction of the real numbers; 30. Uniqueness of the real numbers; Suggested reading; Answers (to selected problems); Glossary of symbols; Index. Bookseller Inventory # 0521867444 2a. Calculus. Volume I. One-Variable Calculus, with an Introduction to Linear Algebra. Second Edition Apostol, Tom M Bookseller: Paper Moon Books (Portland, OR, U.S.A.) Price: US$ 20.00 Shipping within U.S.A.:US$4.50 Book Description: New York John Wiley & Sons, Inc. 1967., 1967. Fine. 666pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068435 2b. Calculus. Volume II. Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probabil Apostol, Tom M Bookseller: Paper Moon Books (Portland, OR, U.S.A.) Price: US$ 20.00 Shipping within U.S.A.:US$4.50 Book Description: New York John Wiley & Sons, Inc. 1969., 1969. Fine. 673pp. Clean and bright book. No previous owner's markings. 2nd.Edition. Binding is Hardback. Bookseller Inventory # 068436 3a. Introduction to Calculus and Analysis (Volume I) Courant, Richard; Fritz John Bookseller: Harvest Book Company (Fort Washington, PA, U.S.A.) Price: US$ 9.95 Shipping within U.S.A.:US$3.95 Book Description: Interscience Publishers/ New York 1965, 1965. First American Edition, 1st Printing Hardback in Decorated Boards. 661p. Very good condition. Very good dust jacket with one small closed tear and sunned jacket spine. Satisfaction Guaranteed. Bookseller Inventory # 515288 3a, alt. Introduction to Calculus and Analysis Volume 1 (ISBN: 0470178604) Richard Courant Bookseller: Frugal Media Corporation (Austin, TX, U.S.A.) Price: US$ 10.00 Shipping within U.S.A.:US$3.70 Book Description: Wiley, John Sons. Hardcover. Book Condition: VERY GOOD. USED Ships within 12 hours. Bookseller Inventory # 873302 3b. Differential and Integral Calculus Volume 2 R. Courant Bookseller: Pioneer Book (Provo, UT, U.S.A.) Price: US$ 13.50 Shipping within U.S.A.:US$3.50 Book Description: Interscience Publishers, 1947. rebound Hard Cover Good. Bookseller Inventory # 481571 4. ANALYSIS 1 Lang, Serge Bookseller: The Book Cellar, LLC (Nashua, NH, U.S.A.) Price: US$ 39.99 Shipping within U.S.A.:US$4.00 Book Description: Addison-Wesley 1968., 1968. Fine in Good dust jacket; Light shelf wear to book. Heavy wear to DJ. 460 pages. Binding is Hardcover. Bookseller Inventory # 374309 5. Calculus of One Variable Joseph W. Kitchen, Jr. Bookseller: Antiquarian Books of Boston (Winthrop, MA, U.S.A.) Price: US$ 150.00 [sorry] Shipping within U.S.A.:US$3.50 Book Description: Addison-Wesley Publishing, Reading, Mass., 1968. Hard Cover. Book Condition: Very Good. No Jacket. 8vo. xiii, 785 pages. Tightly bound and clean. No writing in book. The book also deals with plane analytic geometry and infinite series. Bookseller Inventory # 7620 also Honours Calculus*(ISBN: 0965521117)$24. from the author. Helson, Henry http://members.aol.com/hhelson/ Calculus of several variables. 6. CALCULUS ON MANIFOLDS: A MODERN APPROACH TO CLASSICAL THEOREMS OF ADVANCED CALCULUS Spivak, Michael Bookseller: BRIDGEWAY ACADEMIC BOOKSTORE, ABA (TAOS, NM, U.S.A.) Price: US$25.00 Shipping within U.S.A.:US$ 6.50 Book Description: W. A. Benjamin, NY, 1965. PAPERBACK COPY. Book Condition: Very Good. VERY GOOD CONDITION, PAPERBACK, 146pp. Bookseller Inventory # 001874 7. Mathematical Analysis: A Modern Approach to Advanced Calculus Apostol, T. M. Bookseller: Textsellers.com (Hampton, NH, U.S.A.) Price: US$12.50 Shipping within U.S.A.:US$ 3.50 Book Description: Addison Wesley, 1957. Book Condition: Good. Dust Jacket Condition: Fair. 8vo - over 7¾" - 9¾" tall. Hardcover, 559 pp. Notes, jacket has edge chips. Bookseller Inventory # 011916 8. Functions of Several Variables. Fleming, Wendell H. Bookseller: Significant Books (Cincinnati, OH, U.S.A.) Price: US$12.00 Shipping within U.S.A.:US$ 3.50 Book Description: 337 pp. Addison Wesley (1965) (Hardback) Good condition, ExLib. Glue Spot on cover. Bookseller Inventory # MATH10273 9. Advanced Calculus Loomis and Sternberg free download from Sternberg’s website. Linear Algebra: 10. Linear Algebra : A Geometric Approach (ISBN: 071674337X) Malcolm Adams, Ted Shifrin Bookseller: www.EMbookstore.com (Flushing, NY, U.S.A.) Price: US$67.98 Shipping within U.S.A.:US$ 3.25 Book Description: W. H. Freeman; (August 24, 2001), 2001. Book Condition: New. Free Delivery Confirmation!! Brand New Hardcover, US Edition, Quality Paper Printed in USA. Bookseller Inventory # 071674337X-2 11. Linear Algebra. Hoffman, Kenneth, & Ray Kunze Bookseller: Zubal Books (Cleveland, OH, U.S.A.) Price: US$11.46 Shipping within U.S.A.:US$ 6.50 Book Description: Englewood Cliffs: Prentice-Hall 1965, 1965. 1st edition, fourth printing (1965) 332 pp., hardback, wear to spine & covers, previous owner's name to front free endpaper else textually clean & tight. Bookseller Inventory # ZB471098 Ordinary Differential Equations 12. Ordinary Differential Equations (ISBN: 0262510189) V. I. Arnold Bookseller: A1Books (Netcong, NJ, U.S.A.) Price: US$28.77 Shipping within U.S.A.:US$ 4.95 Book Description: Brand new item. Over 3.5 million customers served. Order now. Selling online since 1995. Few left in stock - order soon. Code: M20060602184422T0262510189. SKU: 0262510189-11-MIT. Bookseller Inventory # 0262510189-11-MIT 13. Lectures on Ordinary Differential Equations. Hurewicz, Witold. Bookseller: Significant Books (Cincinnati, OH, U.S.A.) Price: US$7.00 Shipping within U.S.A.:US$ 3.50 Book Description: Book Condition: Good condition, no dj. 122 pp. Wiley (1958 ) Hardback. Bookseller Inventory # MATH12978 Topology 14. First Concepts of Topology Chinn, W. G. & Steenrod, N.e. Bookseller: aridium internet bookstore (Cranbrook, BC, Canada) Price: US$8.32 Shipping within Canada:US$ 8.95 Book Description: SInger, 1966. Trade Paperback. Book Condition: Very Good. First Printing. Usual library markings in and out. non-circulating. very light use, clean crisp pages. edge rub/wear. A solid copy. Ex-Library. Bookseller Inventory # 010917 15. Differential Topology: First Steps Wallace, Andrew Bookseller: Books on the Web (Winnipeg, MB, Canada) Price: US$30.25 Shipping within Canada:US$ 5.50 Book Description: NY: W.A. Benjamin, 1968, 1968. paper bound, 1st edition, illustrated in colour, 130pp including bibliography and index. As new. Bookseller Inventory # 16779 16. An Introduction to Algebraic Topology Wallace, Andrew H. Bookseller: BOOKS - D & B Russell (Shreveport, LA, U.S.A.) Price: US$12.00 Shipping within U.S.A.:US$ 4.00 Book Description: Pergamon Press, New York, 1963. Book Condition: Very Good hard cover/ no dust. Octavo, 198 pp., Last name of prior owner inside front cover. One of a series of the International Series of Momographs in Pure and Applied Mathematics. Bookseller Inventory # 013208 Abstract Algebra. 17. Algebra (ISBN: 0130047635) Artin, Michael Bookseller: DotCom Liquidators / DC 1 (Fort Worth, TX, U.S.A.) Price: US$44.50 Shipping within U.S.A.: US$ 3.50 Book Description: Bookseller Inventory # NA/DC8/T999/*114552 Abstract Analysis 18. Foundations of Modern Analysis. Pure and Applied Math., Vol. 10 Dieudonne, J. Bookseller: Zubal Books (Cleveland, OH, U.S.A.) Price: US$9.49 Shipping within U.S.A.:US$ 6.50 Book Description: Academic 1960, 1960. 361 pp., hardback, ex library, else text and binding clean, tight and bright. Bookseller Inventory # ZB472982
 HW Helper Sci Advisor P: 9,371 notice there is a dearth of books listed for elementary diff eq since few of them inspire much admiration among people. on the other hand i have found some amazing bargains for you, including courant, apostol, hurewicz, hoffman/kunze, and dieudonne, at prices about 1/5 to 1/10 those often seen. sorry about kitchen. its a nice book but at that price it is absurd to buy it, given that copies of fleming, dieudonne, courant, etc... exist for so much less. almost any one of these books will give you an enormous amount of education. i have also shortchanged complex analysis, but you will find another example on henry helson's website. he is a student of loomis i believe, and former berkeley professor who writes excellent books and publishes and sells them himself at reasonable rates, with some written by others. he has a linear algebra book too but i have not seen it.
 HW Helper Sci Advisor P: 9,371 here are two more really good, really cheap books: Elementary Theory of Analytic Functions of One or Several Complex Variables Henri Cartan Format: Paperback Pub. Date: July*1995 B&N Price: $13.95 Member Price:$12.55 Usually ships within 2-3 days also: Differential Forms Henri Cartan Format: Paperback Pub. Date: July*2006 NEW FROM B&N List Price: $12.95 B&N Price:$11.65*(Save*10%) Member Price: \$10.48
 P: 1,240 For any mathematicians (pure or applied) did you guys intern anywhere during your summers? I am trying to find places where an applied mathematics major could go and intern during the summer (freshman). Maybe I could go abroad? Typically, does a math major do research over the summer or intern if he opting for a pHd? Does it have to be necessarily math related? Also, for an applied mathematician, what would you say is the most important area to know? Would it be ODE's / PDE's. I might be interested in going into quantitative finance, or something like biological math. This summer, I want to try to focus on learning a range of math rather than a depth of math (i.e. only studying Apostol, but not studying other areas of math like probability theory). Sure, I may not be a scholar in the end in any of the particular fields, but I can always go ahead and brush up later when the time calls for it (i.e. if I do a pHd). I find that the internet offers me the most versatility in learning different fields of math.
 HW Helper Sci Advisor P: 9,371 I did not intern myself. Today there are several programs for math types in summer funded by VIGRE grants from NSF. Some schools also offer sumemr research opportunities but these are often voluntary activites by faculty, hence may fall short of volunteers. I.e. we are asked to do it for free, and that is something hard to sustain for long.
 HW Helper Sci Advisor P: 9,371 hurkyl, i do not see how you can resist getting paid plus free tuition to study something interesting. how can you lose? it also adds to your resume for pay increases, new job opportunities, etc. i say grab it. you will do it easily. you are really strong mathematically. I am sure of this.
 HW Helper Sci Advisor P: 9,371 Here are some foundational graduate books for future professionals. * means an especially high level recommended book. Grad math books: Algebra: 1. *Lang, Algebra, 2. Jacobson, Basic algebra 1,2. 3. Dummit - Foote, Algebra 4. Hungerford, Algebra Reals 5. Measure and Integral: An Introduction to Real Analysis Richard L. Wheeden, Antoni Zygmund 6. Royden, Real Analysis 7. Rudin, Real and complex analysis 8. * Functional Analysis, Riesz - Nagy Complex 9. Ahlfors, Complex analyhsis 10. Conway, complex analysis 11. *Hille, Complex Analysis 12. Complex Analysis in One Variable, R. Narasimhan, Topology 13. Fulton, Algebraic topology 14. *Spanier, algebraic topology 15. Hatcher, algebraic topology. 16. Vick, Homology theory.
 HW Helper Sci Advisor P: 9,371 A remark for graduate students that they do not always seem to understand: Your instructor in a basic graduate course is often an expert in the field, at least on a level with many authors, although perhaps not all, of basic books. Hence it is not to be expected that the instructor will slavishly plow through a standard book on the topic, but may well merely present the material as best suits him or her. Do not be automatically disappointed if your instructor lectures from his/her own notes as they are often actually superior to what is found in many books. At the elast the lecturer will probably select from the best presentations available for each topic. This is a plus for the student. I am having difficulty citing here standard books for each subject, since at this level the presentation given in class is normally better than that found in any one book, for one thing as it is more up to date, being given by a practicing professional. I.e. at this level the best instruction is often obtained in person rather than from books.
 HW Helper Sci Advisor P: 9,371 Another remark: You will notice that all the books I have cited are theoretical ones, on specific bodies of theory, rather than being say problem books. This is the way I was taught, proving theorems. We were expected to find and work problems on our own. But in Russia e.g., there is a wonderful tradition of problem solving and problem teaching. This type of activity was what brought me to math in high school but was slighted in my college instruction. Nonetheless it is gresat fun, and leads well toward the experience of doing a PhD and solving open problems. Thus it would be good to list some books of problems, but I will have to do some research to find them.
 P: 61 Hey mathwonk, would you mind doing a little comparison between pure math and applied math? As in the types of classes you'll take in each major, their differences, what you can do with each degree, etc I am starting my undergraduate studies in September. While I don't have to decide on my major until I am done by first year, I'm still kind of curious as to how it all works. Awesome thread, by the way. I've been reading it since you've started it.
 HW Helper Sci Advisor P: 9,371 Well I really don't know squat about applied math, but i gather you should go heavy on the ode, partial de, and numerical analysis courses. Thanks for the feedback. I have been dominating the discussion but I want to explicitly solicit reports from other math people on their experiences in school, getting ready, what helped, what was a problem, what led to productive results at work, etc,... Perhaps Matt could shed some light on his journey to a math PhD, and Hurkyl on his path to gainful employment, and J77 on his life as an applied math guy. Also physics guys like Zapper and others could help us with input on what math you really need if you might want to get into physics, or mathematical physics. My friends in physics have emphasized group representations, but that was a long time ago. More recently it has been Riemann surfaces and algebraic geometry.
 HW Helper Sci Advisor P: 9,371 My older son was a math major with numerical emphasis at Stanford, and now does web based internet stuff for Arriba. He likes it. He also needs some business skill, as in a company you have to manage people who work for you, motivate, sell, service, hire and fire, and educate customers and clients.
 HW Helper Sci Advisor P: 9,371 My wife was also a math major and is now a pediatrician. Math is not her main resource but all dosages require mathematics to scale them to suit each child by weight. I may be trivializing her math usage, but math majors can do a lot of things because they can reason and calculate well. She also needs to manage people and service customers. Besides her ability to deal with all people she meets, her main skill that impresses me is her terrific diagnostic ability. She actually saves lives when she detects a serious infection by its outward signs. This is deductive ability appield to real life emergencies.
 HW Helper Sci Advisor P: 9,371 One thing i can guarantee, everyone needs to take linear algebra, pure applied, whatever. The thing that is so frustrating about the AP courses in high school is their focus on calculus instead of linear algebra. I.e. linear algebra is easier than calculus, more important for more people than calculus, and even a prererquisite for understanding calculus. So it sems odd to make calculus the focus of high school AP courses instead of linear algebra. Unfortunately no one listens to math professors when planning math education curricula.