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Ask HN: Math books like SICP?

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Re: Ask HN: Math books like SICP?

#61

Post by pmiller2 »

nextos wrote: post #17
zodiac wrote: post #9

Math is a much, much broader field than the part of Computer Science than SICP covers, so it's really hard to search for "consistent and broad knowledge throughout all areas". Do you really need to learn analytic number theory? category theory? axiomatic set theory? tensor calculus? My suggestion would be to narrow it down to a specific field you're interested in like abstract algebra and ask for suggestions about th…

I was planning to structure my learning around linear algebra and real analysis, sort of like Harvard's Math 55 does. Apart from that I wanted to expand on basic set theory, geometry, combinatorics and probability theory to have a well-rounded basic education. My ultimate goal is to be able to digest advanced probability and statistics books.

Set theory: You should definitely look at Naive Set Theory by Halmos. Despite the name, it's totally rigorous development of ZF set theory with discussion on the meaning and equivalent statements of AC.

Re: Ask HN: Math books like SICP?

#62

Post by nextos »

pmiller2 wrote: post #61
nextos wrote: post #17

Earlier quoted context omitted.

I was planning to structure my learning around linear algebra and real analysis, sort of like Harvard's Math 55 does. Apart from that I wanted to expand on basic set theory, geometry, combinatorics and probability theory to have a well-rounded basic education. My ultimate goal is to be able to digest advanced probability and statistics books.

Set theory: You should definitely look at Naive Set Theory by Halmos. Despite the name, it's totally rigorous development of ZF set theory with discussion on the meaning and equivalent statements of AC.

What about Lawvere's Conceptual Mathematics or Sets for Mathematics?

Re: Ask HN: Math books like SICP?

#63

Post by davidjhall »

I'm surprised no one has mention Mathematics for the Millions by Lancelot Hogben. http://www.amazon.com/Mathematics-Million-Master-Magic-Numbe...

He takes the approach of starting right at the beginning of human history when we first came to look at the stars and seasons and try to understand why they work, and then builds the math knowledge with each lesson. Excellent book!

Re: Ask HN: Math books like SICP?

#64

Post by lambdaphage »

cabacon wrote: post #6
gms wrote: post #2

It's difficult to find what you are looking for, since maths is a much older field. However, Spivak's 'Calculus' is pretty close: http://www.amazon.com/gp/aw/d/0914098918 .

I clicked on this topic in order to make this suggestion. To expand on it a little bit, I was a math major and read Spivak's 'Calculus' after I had already taken real analysis. I found it delightful - it really approaches the topics from first principles and unlike many calculus textbooks actually goes through the effort of presenting proofs of the theorems. Highly recommended. As some recreational reading, less suit…

I appreciated DFW's book for even attempting to do a popular treatment of what we would now call the history of real analysis. But there were some serious technical problems with it: http://www.ams.org/notices/200406/rev-harris.pdf

Re: Ask HN: Math books like SICP?

#65

Post by mturmon »

midas007 wrote: post #58

Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik https://en.wikipedia.org/wiki/Concrete_Mathematics Also Knuth's TAOCP series are very formal and dense, in a good way.

Concrete Mathematics is a great book, really good content and techniques about computing things, and a lot of depth and useful history and side information.

Re: Ask HN: Math books like SICP?

#66

Post by karamazov »

I'm a big fan of SICP. I'd describe it as "hard-core": requiring hard work, but extremely rewarding for the people who take the time and effort to go through it.

I'd recommend the following three books to similarly cover the three main areas of higher mathematics (analysis, algebra, and topology):

1. Principles of Mathematical Analysis by Rudin (http://www.amazon.com/Principles-Mathematical-Analysis-Inter...)

2. Algebra by Artin (http://www.amazon.com/Algebra-2nd-Featured-Titles-Abstract/d...)

3. Topology by Munkres (http://www.amazon.com/Topology-2nd-James-Munkres/dp/01318162...)

Working through all three (including the exercises!) will give you a solid understanding of the basis of modern mathematics. If you don't have experience with proofwriting, you might find them difficult at first - the activity is very different from performing calculations or solving equations. It's also best to have someone trained in mathematics talk to you about the proofs, until you develop a feel for the needed level of logical rigor.

Rudin, in particular, leaves a lot of work to the reader; going through that book is the most intellectually difficult work I've ever done. If you find it hard-going (which is completely natural), you might want to try Artin first, especially since you have some background in algebra.

(Incidentally, these are the three books used to teach analysis, algebra, and topology to MIT mathematics majors. You can look up the assignments and exams for the three courses - 18.100B, 18.701, and 18.901 - for a good list of exercises to work through.)

Re: Ask HN: Math books like SICP?

#67

Post by zbinga »

reikonomusha wrote: post #30
agumonkey wrote: post #20

Not Math, but Physics/Mechanics, by Sussman again. Title ? SICM hehe https://mitpress.mit.edu/sites/default/files/titles/content/...

SICM is not like SICP at all. SICM assumes you're pretty well versed in classical mechanics and already have an understanding of both Lagrangian and Hamiltonian mechanics. Don't be fooled by the title and authors of the book!

Heh. I bought SICM expecting it would be "SICP for physics". I started reading and felt out of my depth after just a few pages. But it seemed interesting, so I got a Lagrangian and Hamiltonian mechanics book and started reading that. That was 6 months ago, I'm about halfway through and I'm really enjoying it!

Learning Lagrangian mechanics when all your life you only ever heard about the Newtonian approach is a lot like stumbling upon Scheme after years of programming in C.

I'm looking forward to cracking the cover of SICM once more this summer!

Re: Ask HN: Math books like SICP?

#68

Post by zbinga »

It's not at all like SICP (I've never met a math book that was similar), but it's a wonderful introduction and I don't see it mentioned in this thread yet: Needham's "Visual Complex Analysis".

Re: Ask HN: Math books like SICP?

#69

Post by nextos »

tel wrote: post #51
decasteve wrote: post #34

If you've already covered Vector Spaces, Groups, and Rings in your studies, I would suggest checking out Bill Lawvere's "Conceptual Mathematics: A First Introduction to Categories". i.e. An introduction to category theory through the category of sets. His "Sets for Mathematics" is also good but it's a more concise presentation of the topic but a less than gentle introduction than the former book.

I was going to recommend this one. Conceptual Mathematics is wonderful in that it provides a narrative of intuition instead of just the proofs. It's kind of like seeing How It Is Made for algebra.

Do Conceptual Mathematics and Sets for Mathematics cover the same ground?

Re: Ask HN: Math books like SICP?

#70

Post by jvreeland »

dominotw wrote: post #29
jvreeland wrote: post #4

It's not nearly as easy to work through as SICP is but Principles of Mathematical Analysis by Walter Rudin (sometimes referred to as little Rudin) is a great place to start if you're interested in analysis. It's a hard book, but it's pretty much the standard for undergrad analysis.

Baby Rudin still gives me nightmares.

Real Analysis by Royden is the source of my nightmares.
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