231–240 of 323 posts
Meta comment: might be good to add the level of mathematical maturity needed to enjoy the book.
Along the lines of fascicules de résultats, I find talks are a good way to get a coup d'oeil for a field: people giving a talk tend to take a direct approach to what they want to introduce, hitting only the salient points. But that yields enough keywords to then consult any relevant texts.
* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world. * Calculus by Spivak. This was used in my intro calculus course in university. It…
* The Natural Number Game https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g...
Earlier quoted context omitted.
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.
I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing. I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop. Programming languages are designed to describe a specific computat…
No, we think that because proofs and programs are isomorphic[1]. It's not a mistake: traditional mathematical notation provably is a terse badly implemented programming language. Actually it's worse than that, because oftentimes it doesn't even parse. Now I'm not going to say I can't on some level see the appeal. After all I think Perl is a lot of fun to code in.
Naturally, its adherents are practiced at making a virtue out of its defects. Who wants to admit they dedicated considerable brainpower to doing something in a fundamentally suboptimal way? That doesn't really matter though. As Mathematica and other tooling shows, the formalists have already won and now it's just a matter of mopping up the stragglers, or waiting for them to age out. This isn't terribly surprising to those who know the basics of the history of mathematics. It took something on the order of two centuries before Recorde's innovation of the equal sign was generally accepted.
[1] https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspon...
I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in underst…
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…
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 what kindled my interest in pure mathematics and I still feel I'm nowhere nearer understanding it despite working through several other mathematics textbooks, including just plain algebra. But, it did spark my interest and now I've moved on to other interesting topics, though Geometric Algebra is still my white whale.
Nonlinear Dynamics and Chaos by Strogatz Chaos theory and deterministic systems are a fascinating vantage point for thinking about the dynamics of large computer systems. Thinking of them as stochastic systems is sometimes useful, but most of the systems are actually just operating in unstable periodic processes which are much closer to being a chaotic system rather than a stochastic system. This influences how I thi…
Earlier quoted context omitted.
I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?
I’m reading and like Thomas Garrity’s “All the mathematics you missed (but need for graduate school)” which is this but for people who did a bachelors degree but missed certain areas (or forgot them). https://www.amazon.co.uk/All-Math-You-Missed-Graduate/dp/100...
https://www.amazon.co.uk/Princeton-Companion-Mathematics-Tim...