291–300 of 323 posts
Some that stand out "Concrete Mathematics: A Foundation for Computer Science" by Knuth, Graham, and Patashnik - solid foundation in mathematical concepts and techniques, and it helped me develop a deeper understanding of mathematical notation and problem-solving. "Introduction to the Theory of Computation" by Michael Sipser - introduced me to the theoretical foundations of computer science, and it helped me develop a…
Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.) Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.
I read the first chapter of “Mathematics; Its Content, Form, and Meaning” (or something close to that) and it explained the entirety of my high school mathematics. If I had been given that to read back then I might have gone into mathematics in college. Instead I got burned out and quit doing any math for a decade or more. Sigh.
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History of mathematics as well, it will give you a very subtle entry into the minds of mathematicians and the motivation behind their theorems. This will surely make you more appreciative of subjects and concepts you are learning.
Absolutely, and there are some really interesting personalities in the history of mathematics. Newton and Galois come to mind.
See https://www.newyorker.com/magazine/2005/02/28/time-bandits-2
“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on every…
As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.
Sussman (who wrote the famous SICP book) wrote another book structure and interpretation of classical mechanics. Tough book to go through. But they start with the same premise: mathematical notation is confusing (and hand wavy at times). A better symbolic notation should reveal enough details to be able to code up the mathematics in a program. I found this approach to be bang on target, but could never get enough time to actually go through the book.
And I realised why the 'let us build it up from scratch' books work. They force you to think about the function signatures and shape of objects passed to each function. This approach reveals gaps in our understanding much better. For example, F=ma is looks like an algebraic statement, hiding the fact that `a` on the right is about time evolution of the system through the derivative.
Steven Strogatz made a funny quote in his infinite powers book. (I'm paraphrasing), if Newton was doing this in today's era he might create a flipbook animation to make this point and not symbols.
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As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.
Mathematics only clicked and became fun for me when I started using Wolfram Mathematica, because I could fairly easily mess around with the formulas I saw in books until I understood the types and arguments and what is an index vs a reference to some unnamed convention of the field.
By and large mathematics education has missed the point of invention of computers. It is only occasionally used to make a point. We should be teaching mathematics with a programming first approach: get your function code to compile, ponder on its signature, write some test cases to really understand what is going on.
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These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is: * Basic Mathematics by Lang. Covers basic algebra and geometry at high school level. Then one of these two, depending on your interests, or both: * Vector Calculus, Linea…
These are great. Thanks for sharing. Do you happen to have any on probability or statistics and or both?
If you are trying to do inference on large datasets, I think Andrew Gelman's books are superb and will teach you invaluable skills using multi-level Stan models, i.e. generative models and Bayesian inference.
If you want to study basic probability theory and stochastic processes, I really like Elementary Probability Theory by Chung. His more advanced book is really famous, but has a measure-theoretic approach and that's not very practical unless you are interested in developing theories a bit further.
Taleb is really fond of Probability, Random Variables and Stochastic Processes by Papoulis. But I find the typesetting in later editions to be really disorganized and confusing. Take a look. It's a good alternative to the first Chung book.
To add one more thing: the "thing" that has helped me most lately isn't a specific book or video or anything, but rather simply committing to spending 1 hour every day on math. I even set up a Google Calendar task to remind me of this every. single. day. And so far this year I haven't missed a day yet. Now what constitutes that hour can vary. It can be watching math videos, it can be solving problems on paper, and I…
Yeah, I've been doing the same thing, but with the rule that I have to do "a math problem". Right now I'm going through a stochastic processes book a bit at a time that way and really enjoying it.
https://www.goodreads.com/book/show/714583.The_Man_Who_Loved...
* Algorithms to Live By, by Brian Christian & Tom Griffiths (not really a Math book, mostly computer science, but still has some math algorithms and their implementations to real life)
https://www.goodreads.com/book/show/25666050-algorithms-to-l...
For sure I got significantly(?) better with classics like Spivak, Apostol, Rudin. "Real and Complex Analysis" by Rudin, and the two books both named "Calculus" from Spivak and Apostol. But also from Apostol his more concise and far-reaching "Mathematical Analysis". And from Spivak his small gem "Calculus On Manifolds" made quite a dent on me. Other than more "classic math" books, I also wanted to mention two outliers…
> * Geometric Algebra for Physicists, by Doran and Lasenby. I found the power and elegance of geometric algebra mesmerizing, and even if this book is also about physics and there may be more appropriate math-only books about geometric algebra, this is the one that made it for me. I've tried to read several of them, and, sadly, I feel most geometric algebra books fail at explaining it. It's a shame as it's part of wha…
In addition to "Geometric Algebra for Physicists" (whose first two chapters I'd recommend to get a nice overview), I found Hestenes' "New Foundations for Classical Mechanics" to be very good and readable.
Also, there are many good resources in https://bivector.net/ , including videos, papers, presentations and programs.
Finally, an interesting paper (that got me kickstarted) is "Imaginary Numbers Are Not Real—The Geometric Algebra of Spacetime" by Gull, Lasenby and Doran.