Hi, I've got a Masters in Physics, but always feel that I'm missing out slightly by not knowing and appreciating the beauty that there is in all the maths I never had cause to study in my physics degree. Is there a book (or set of books) that does for maths what the Feynman Lectures do for physics? As in books that take relative novices and then give them a good understanding of the subject. I think Feynman only assumed a knowledge of as much calculus as you could fit on a postcard and then took them through the first two years of a university degree, which might be a greater challenge for a book on maths. Anyway, does anyone have any recommendations? Thanks.
Maybe What is Mathematics? by Courant and Robbins, or Matters Mathematical by Herstein and Kaplansky (you can get this for $6 from abebooks.com).
Here is a link to What is Mathematics http://www.amazon.com/Mathematics-E...5192/ref=pd_sim_b_title_2/002-2775545-7002437
I'll try and dig up my old copy of "The Mathematics of The Strapless Evening Gown" or some such and send a link. It was published both as a whimsical work, but with a mathematical perspective as a small publication rebuttal, or at least companion, to "A Stress Analysis of the Strapless Evening Gown" Granted it's only a single topic but in math, not unlike pharmacology, one takes solace where one finds it....
The thing is the soul of mathematics is about problem solving. So any book that dosen't have problems for the reader to solve will have missed a large chunk of mathematics. Although Feyman dosen't have any problems in his book. I wonder why?
From what I've seen, no one in mathematics has summed up so eloquently yet simply the essence of some aspect of maths, as Feynman did for quantum theory. Feynman is renowned for being a genius of the first class, who behaved just like a normal bloke off the street. He played poker (and was exceptionally good at it), he liked practical jokes and traveling abroad. That sort of made him for accessible to the lay man, and though I can think of people who are studying mathematics yet are just average people who do average things, none of them are genius as the rank of Feynman. The entire world got a bit dumber when he died. I could be wrong, and just have not heard about an equivalent though.
Because I could just as easily say "theory building is the soul of mathematics, so any book that doesn't build up in this way will have missed a large chunk of mathematics". As to why Feynman didn't include any exercises, maybe he thought students should spend more time learning the content then learning how to jump through hoops.
True. There was a great line from Hans Bethe in his obituary for Feynman which read "It was said that there were only two ways to solve the most difficult problems in Physics. One was to use Mathematics. The other was to ask Feynman." You forgot to mention his great skills as a bongo player.
Eh. I would strongly advise AGAINST reading a calculus textbook as a survey of mathematics. I would recommend this book followed by some of the things here. Tim Gowers is great. (I do like Ian Stewart, too.)
[EDIT: just noticed that Xevarion just beat me to the draw in recommending PCM!] Not AFAIK, but you will probably want to read the eagerly awaited Princeton Campanion to Mathematics (PCM); see also Tim Gowers's blog entry on PCM and Terry Tao's draft articles from PCM.
Feynman didn't because he had TAs who did it for him ;). There's a special problem book written to accompany his lectures, made of problems thought up partially by himself and partially by his TAs. I don't think it's in print anymore, but universities someone have it in stock. -------- Assaf Physically Incorrect
There are many areas of applied mathematics that dosen't involve theory building but just solving applied problems. It borders on engineering.
Mention an odd error In a half dozen threads similar to this one, including at least two recent threads, I recommended the book by Kac and Ulam, Mathematics and Logic, Dover reprint. The recommendation stands, but although this is tedious, I should add a correction every time I mention this book. Late in the book (p. 118-119 in the Dover reprint) the discussion of the braid group is flawed by what might be a misprint followed by what certainly seems to be a very odd brain blip. In the braid group on four strands, the relations satisfied by the generators [itex]A_1, \, A_2, \, A_3[/itex] should be [tex] A_1 \, A_2 \, A_1 = A_2 \, A_1 \, A_2, \; A_2 \, A_3 \, A_2 = A_3 \, A_2 \, A_3, \; A_1 \, A_3 = A_3 \, A_1, \; A_1^2 = A_2^2 = A_3^2 = I [/tex] This is one instance where the WP article, in this case Braid group, is correct (in the version cited, at least), but a widely read book by two leading mathematicians contains inexplicable errors! But I must unfortunately add that nonetheless WP is inherently unstable and unreliable and cannot be safely used as a reference except by experts who may be able to spot vandalism (e.g. maliciously flipping a sign), honest misstatements, and misleading portrayal of a fringe POV as "mainstream" [sic], where laypersons probably would be misled, perhaps seriously.
i'm in the same boat. i hear What is Mathematics, and Mathematics: Its Content, Method, and Meaning are good for undergrad math.
Mathematics, especially if one is looking for particular curiosities, is too vast even in its most concise form. And it is true to a certain degree(though more than anyone would agree to) that none of the greats of mathematics were geniuses in the classical sense. They were phenomenal theoreticians, and excelled in contributing towards the body of human knowledge; not 'mere' prodigy. Back to the topic though- in mathematical physics I particularly enjoyed Applied Mathematical Methods in Theoretical Physics by M. Masujima. I also think Walter Rudin's Analysis series- Principles of Mathematical Analysis, Real and Complex Analysis, Functional Analysis are the de facto standard for training the hand at pure mathematically minded analysis. Yoshida's Functional Analysis book is also strongly recommended. Rudin does a remarkable job at guiding the calculus student through the transition into the more rigorous flavour. In number theory my interests are very much localised to certain areas but Apostol gives a good account of the analytic bit. As for curiosities and more references try Penrose's Road to Reality and Havil's Gamma.