I'm slowly reading through Feynman's Lectures on Physics and I find myself wondering, is there an analogous book (or books) for math?
Mathematical equivalent of Feynman's Lectures on Physics?
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11By this, do you mean a good approach to physics given through sweeping motions, appeals to intuition, and a lack of focus on historical precedence in favor of developing patterns for modern thought? If that's the case, I recommend Feynman's Lectures on Physics. Or do you mean for calculus books, which are sort of the bread and butter of math? If that's the case, then no. I don't. Really though, what do you mean? – 2011-09-06
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2Although I'm sure some readers will be familiar with Feynman's Lectures, perhaps you could elaborate on what it is about them that you're searching for. Easy to follow? Witty? Deep?? – 2011-09-06
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3By this, I mean a good approach to mathematics given through sweeping motions, appeals to intuition and a lack of focus on historical precedence in favor of developing patterns of modern thought. :-) Seriously, I thought that was very well-said. – 2011-09-06
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0What strikes me about Lectures on Physics is its conceptual orientation. It is easy to read in that I don't need a pen and paper at hand to work through details, but I find it necessary to regularly stop and contemplate his ideas and examples. Content-wise, it is not at the level of a calculus book. My impression is that it covers much of what would constitute an undergraduate education in physics. – 2011-09-06
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2There will be much praise for the people who can write a three-volume series of books which covers ‘an undergraduate education’ in mathematics, let alone ‘modern mathematics’. (The same goes for physics really. A lot has happened in the 50-or-so years since the Feynman lectures, but undergraduate mathematics barely even reaches the early 20th century!) – 2011-09-06
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4I didn't mean to imply The Lectures were equivalent to an undergraduate education in physics, but rather, it covered a similar amount of material, had a similar breadth. This impression might be quite wrong. Obviously it isn't a replacement for such an education, nor does it cover the material at the same level of depth. Of course, I'd be happy to find a book that covered modern mathematics through, say, 1950. – 2011-09-06
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0@Zhen Lin "..but undergraduate mathematics barely even reaches the early 20th century!" I think its not meant to be! It is supposed to build strong conceptual abilities and problem solving skills rather than feed us with as much mathematics as we can take in 3 or 4 years.Is'nt it? – 2011-09-06
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0@Dinesh: That's a very applied point of view. I think mathematics is a subject worthy of study in its own right, and I feel that at the end of a good undergraduate course students should have a decent chance of understanding the latest research. This, mutatis mutandis, was the motivation behind Feynman's lectures. – 2011-09-06
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0@Zhen Lin : I certainly think that mathematics is worthy of study in its own right. However, I think it is wildly unrealistic for a student with an undergraduate education to understand most of the latest research. I'm a professor and have been doing this a long time, but despite going to many a colloquia I rarely understand the latest research in areas other than those in which I specialize. – 2011-09-06
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0@Adam: I understand this as well, having finished three years of full-time mathematics education but still finding most things impenetrable. But I think this is something that could be changed—which is what I meant by ‘should’—after all, four years is a long time and one imagines that it should be possible to develop at least one specialism in that time. – 2011-09-06
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1@Zhen Lin : What evidence do you have that this could be changed? Math is huge and it takes an enormous amount of time to get to most of its frontiers. Do you have some kind of magic wand that will make it easier? In a related question, why the focus on understanding the "latest research"? Who cares when a result was proved, as long as it is beautiful and gives you pleasure? When I want to learn math for fun (as opposed to doing research), I usually spend my time reading papers from the '50's and '60's... – 2011-09-06
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7http://www.physicsforums.com/archive/index.php/t-199223.html has a detailed discussion on the same topic – 2011-09-06
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0@Adam: I largely agree, but (a) I am planning to go into research, and (b) the field I'm interested in—topos theory—didn't even really _exist_ until the 1960s. So _I_ necessarily have to catch up to the latest mathematics. As for evidence, no, I don't have much evidence to suggest that it is possible, beyond anecdotes of what _some_ undergraduates have achieved with their own efforts. (Some of the top users here on MSE and on MathOverflow are still undergraduates!) – 2011-09-06
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1@Zhen: Dear Zhen, This discussion is a litle off-topic, but let me just highlight your use of the expression *their own efforts*. Speaking as someone who has taught undergraduates and graduate students, some of them very strong, for many years, I can say that it is not realistic, as an institutional goal, to reach the frontiers of research in mathematics in an undergraduate education. (I presume the same is true of physics, byt the way.) However, nothing stops you from trying to learn more mathematics yourself. The number of students who are ready and able to teach themselves mathematics – 2011-09-06
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0... at a significantly faster rate than a standard undergraduate curriculum is small (too small to design a university curriculum around, I think), but I agree that those who are in this group should be encouraged. Regards, – 2011-09-06
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0+1 to Matt E. One should point out to Zhen Lin that the top undergrad users on MSE and MathOverflow (eg Qiachu and Akhil) are not at all normal undergraduates. At this point, they are basically graduate students in all but name. Certainly when I was their age I was not nearly as advanced, and when I started graduate school (about 15 years ago) I was completely ignorant of "current research". And most of my peers (at a top 5 department) were similar. – 2011-09-06
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0I think that the most significant features of Feynman's treatment of the subject are his unconventional approach (RPF's analyses are often unique, where other texts are similar to each other) and clarity of explanation from first principles. I am very fond of mathematics, and am likewise looking for a mathematical work with a similar level of non-conventionality in topic choice in addition to treatment. – 2011-09-07
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0As an aside, if you like Feynman's "Lectures on Physics", I'd also highly recommend QED (again on Physics) which is a small but remarkable book. On a different subject is his "Lectures on Computation", which is another very unconventional treatment, also highly recommended. – 2011-09-07