Also by Sussman - well worth looking into http://mitpress.mit.edu/sites/default/files/titles/content/s...
As I said elsewhere in this thread, this is not like SICP.
Ask HN: Math books like SICP?
31–40 of 92 posts
Re: Ask HN: Math books like SICP?
#32I share exactly the same interest. Can anybody suggest some outline for what topics (more or less) should somebody cover to be on "undergrad-math" level? Unfortunately I have to hold my own tongue this time, because books I found useful are mainly in russian and they surely aren't like SICP. And I still lack the whole understanding of the area anyway.
All the Mathematics You Missed: But Need to Know for Graduate School - http://www.amazon.com/All-Mathematics-You-Missed-Graduate/dp...
Words cannot express how grateful I am. This seems to be the very thing I looked for the long time and missed somehow! Seems not to cover really All the Mathematics I Missed however (nothing on algebra and number theory for example, which is admittedly my weak spot), but I'm already thrilled to start reading.
Re: Ask HN: Math books like SICP?
#33Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fie…
Thanks. Axler's LADR is indeed fantastic, I have already worked through 1/2 of it. I'm really happy to see my opinion seconded. It's really short and precise. Some people seem to prefer Halmos' Finite Dimensional Vector Spaces, but I found the presentation less didactic. Perhaps it's also more of an upper division text, and I'm not there yet. I was looking for a real analysis companion, perhaps baby Rudin. I was also…
I've tried reading Rudin a couple times, but it's a bit of a slog. There are easier analysis texts. Abbott's Understanding Analysis is good, though a bit basic. I'm currently reading Terence Tao's Analysis I, which is very good if you're in the right frame of mind for it. The first 150 pages are spent building the real numbers from scratch, starting with set theory and the Peano axioms. You successively construct the naturals, the integers, the rationals, and then the reals (as equivalence classes of Cauchy sequences of rationals). It's fun to see how the sausage is made, but I can also admit that when I was just starting out in math I might have found this book unbearably tedious.
Re: Ask HN: Math books like SICP?
#34If 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.
Re: Ask HN: Math books like SICP?
#35Well, I don't know how it goes over there, but here in Algeria, Engineers go through two common years (after which they chose a specialty in the third year, and then, in the fourth and fifth year, a specialty of specialty).
All Engineers go through both years, except Computer Science who don't do the common second year and they directly go to Computer Science.
In these two years, everyone goes through this (maybe it'll give you some ideas on what you want to add):
I'll only list the "Maths" we take first and second year:
First year: - Algebra: (a long course, bottom up. From Boole's algebra, to groups, sigma-algebra, yadda yadda), linear algebra(vector spaces, etc)..
- Probabilities and Statistics.
- Analysis: (Taylor series (Lagrange, Laplace, Young, Cauchy, Maclaurin), integrals, differentiations, different series, convergence/divergence kung fu), Riemann overall, proofs, etc.. Functions, multivariable, real and complex, etc.
Second year:
- Analysis I - Numerical Analysis (Equation systems, Gauss-Seidel, different algorithms(also calculating their speeds), Newton-Raphson, extrapolation, interpolation, etc).
- Analysis II - Integrals(up to 3rd - curves, areas/surfaces(Green) and volumes (Ostrogradsky)), Differential equations (Wronskian, etc).
This is the minimum (to be able to function in other modules, and some other modules are needed before you can function in these, so there's sort of bootstrapping of sort).
And then it depends what you take as specialty (if it's something involving Signal Processing, for instance, or Control Systems, you also need to do stuff).
Hope that helps and you can find some things.
PS: None of these are done with computers, so computing stuff with Newton algorithm and operations on big matrices are all done by hand. It takes a lot of time.
PPS: We don't have multiple answer questions. There's a question, and you answer it (and some answers take multiple pages).
Also, most tests are designed in a way that even if you have the answer sheet right next to you, it still takes you more time to copy the answers than the time of the exam itself. i.e: Even if you don't think and only "write", the time-frame is too tight.
Re: Ask HN: Math books like SICP?
#36Sheldon Axler's "Linear Algebra Done Right" has my highest recommendation if you want expertise in linear algebra. As a followup, Paolo Aluffi's "Algebra: Chapter Zero" is the best synthesizing text for abstract algebra for a beginning graduate student. The thing that makes it so amazing is the writing style: it introduces and demystifies category theory, and then discusses groups, rings, modules, linear algebra, fie…
Thanks. Axler's LADR is indeed fantastic, I have already worked through 1/2 of it. I'm really happy to see my opinion seconded. It's really short and precise. Some people seem to prefer Halmos' Finite Dimensional Vector Spaces, but I found the presentation less didactic. Perhaps it's also more of an upper division text, and I'm not there yet. I was looking for a real analysis companion, perhaps baby Rudin. I was also…
As mentioned in another subthread, Pugh's Real Mathematical Analysis is great, and better motivated than baby Rudin.
Wilf - Generatingfunctionology (CRC 3rd ed.; free 2nd ed. pdf[1])
Lovasz - Combinatorial Problems and Exercises (AMS Chelsea)
The classic probability book is Feller (2 vol), but it's absurdly priced. There's also Sidney Resnick's Probability Path and Adventures in Stochastic Processes. Grinstead & Snell - Introduction to Probability Theory is free[2], and there's also Chung's A Course in Probability Theory (Academic Press/Elsevier).
Dover publishes at least three good books on counterexamples and pathological cases: Counterexamples in {Analysis, Probability, Topology}
[1] http://www.math.upenn.edu/~wilf/DownldGF.html
[2] http://www.dartmouth.edu/~chance/teaching_aids/books_article...
Re: Ask HN: Math books like SICP?
#37I dunno about SICP-like, but here's some good book lists
https://github.com/ystael/chicago-ug-math-bib (updated Univ of Chicago bibliography
http://math.ucr.edu/home/baez/books.html
http://www.maths.cam.ac.uk/undergrad/course/schedules.pdf
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and 2 i got from HN and /r/machineLearning
http://www.reddit.com/r/MachineLearning/comments/1jeawf/mach...
https://github.com/vhf/free-programming-books/blob/master/fr...
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finally, the "Maths for PHysics" texts
http://www.scribd.com/doc/156523189/Boas-mathematical-Method...
http://www.goldbart.gatech.edu/PG_MS_MfP.htm
http://www.scribd.com/doc/91670553/Arfken-Math-Physics
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(if i had to recommend only one book, it would be the Boas, or maybe Princeton Companion: http://press.princeton.edu/titles/8350.html
Re: Ask HN: Math books like SICP?
#38Not the same type of book, but you could do a lot worse than reading through Mathematics: Its Content, Methods and Meaning, by M. A. Lavrent’ev, A. D. Aleksandrov, A. N. Kolmogorov. It's an amazing book which gives a mathematical (but not rigorous in the sense of proofs etc.) overview of most of mathematics.
http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do...
Re: Ask HN: Math books like SICP?
#39Earlier quoted context omitted.
My idea was to cover 6 topics, which would be equivalent to a broad freshman and 1/2 sophomore math education: * Set theory * Linear algebra * Geometry * Real analysis * Combinatorics * Probability theory
Eh, I was thinking about something else, actually. What you listed are taught in every CS program, aren't they? It isn't what I imagined when I heard "rigorous" at all. Topology, number theory, abstract algebra (I mean, real one, not CS-course basics), statistics, tensor analysis? Isn't that "undergrad math"? For things like Set theory/combinatorics/logic basics I'd recommend Rosen's "Discrete Math and Applications"[…
Thanks. I have already covered those topics during my CS undergrad. But I'm trying to redo some things at a much higher level. Hence the comparison with SICP, which is far from a regular introductory book. In the same vein, a course with Axler & Rudin is far more advanced than a standard one.
Re: Ask HN: Math books like SICP?
#40The only book on mathematics actually reassembling SICP I found is "What is mathematics?" by Richard Courant. I have read it after reading SICP, largely because I very much liked some of the more mathematical fragments in SICP, and surprisingly it felt like an actual follow up - it spends significant amount of time highlighting high level trends in mathematics, like use of abstraction and generalization, using very concrete examples. Sounds familiar?