> It has been said that “God invented the integers, all else is the work of Man.” This is a mistranslation. The term “integers” should actually be “whole numbers.” The concepts of zero and negative values seem (to many people) to be unnatural constructs.
It is not a mistranslation. In German "ganze zahlen" actually means "integers". It is just if you translate it word for word, "ganze" -> "whole", "zahl" -> "number", that you get whole numbers, which would be a mistranslation, because in English whole numbers mean 0, 1, 2, ...
According to Wikipedia [1], the term is ambiguous. Its talk page [2] has plenty discussion about it though.
Inteiros - (...-2, -1, 0, 1, 2...)
Naturais - (1, 2, 3...)
E.g - it's always been true, that given a set of axioms A, B, and C, the claim X is true. Humans discover these conclusions when they explore implications of taking different assumptions to be true.
Peano would go further: God invented 0 and +1, and all the rest are the work of people.
Certainly, nature follows a very similar pattern. For limited values of infinity ))
But in practice there are a bunch of propositions/proofs where 1 is treated as a number just like any other.
and then there's Grant Sanderson
It needs to be suitable for someone studying alone.
I happened to pick up a copy of Fearless Symmetry at a garage sale recently. Was wondering if it was worthwhile (kind of hard to go wrong for $1), thanks for the recommendation.
What are some other resources for learning more Math that are very approachable when studying alone?
I found Strang's youtube lectures the best for linear algebra along with 'vector maths for 3d graphic'. Khan's linear algebra course seems like more a collection of practical things you can do with linear algebra rather than a coherent cource but it's still a great resource.
Khan academy's multi-variable calculus is good enough to give you a reasonable grasp, but I'd call it a good complement rather than sole resource.
I've tended to go through the cycle of watching the lectures, then doing the exercises, going forward, going back after a few months to ensure I understand the whys and not just the hows, and then writing my own notes. Machine Learning et al is a great practical application of all this maths.
I am not really looking for the basics. I am looking for advanced material that can be handled without the help of a teacher. The book should be written in that manner.
As I have a good background, I can read AI papers and get the math directly or study some and get that.
But whenever I start to study something, more often than not, the book is written in a dry manner and cannot hold my interests.
* For an introduction (and a reference) to various areas of Modern Mathematics that one didn't even know existed, The Princeton Companion to Mathematics and The Princeton Companion to Applied Mathematics are a must.
* All the Math You Missed: (But Need to Know for Graduate School) by Thomas Garrity - A survey and a good adjunct to a textbook.
* Mathematics: Its Content, Methods and Meaning by Kolmogorov et al. - Classic text from the great Russian Mathematicians.
* Methods of Mathematics Applied to Calculus, Probability, and Statistics by Richard Hamming - Unique text from the great Richard Hamming (also checkout his other books).
There is plenty more of course, specifically; checkout "Dover Publications" texts, many of which are classics and affordable.
I already know the basics, and I want to learn more.
I cam handle hard, but the textbooks need to be written in a manner that doesn't require the help of a teacher or a classroom.
Personally, I think if you want an introduction to the "art" of mathematics, it would be a lot better to pick up a more idiosyncratic book that doesn't aim to cover the basics of the standard curriculum in a textbook-style way, which in my opinion is rather tedious. That could either be a more high-level book like Ian Stewart's "From Here to Infinity" or one of Raymond Smullyan's fun texts on logic.
Or for a more basic book, something like "The Mathematical Universe" by William W. Dunham is a much more interesting introduction to the "art" of mathematics than a textbook-style intro.
Smullyan’s books are great but one isn’t going to go from Smullyan to abstract algebra, point set topology, or real analysis.
So I think the idea behind the title is get students to see this as the gateway to the good stuff as opposed to a lot of proof texts which might be seen as irrelevant.
For those interested: https://webpages.csus.edu/jay.cummings/Books.html
Right now I have a student working on this material and we're using "How to Prove It: A Structured Approach" by Daniel Velleman, which so far I'm finding decent. Some others I've seen (but that I haven't looked at in as much detail) are "Proofs: A Long-Form Mathematics Textbook" by Jay Cummings and "Book of Proof" by Richard Hammack.
While I was motivated, I used one of the typical college books. For me Abstract Algebra is what opened a lot of doors for me... but I am simply using applied math.
That moving away from proofs being magical across sub-topics is what I would like to share with some co-workers who are unwilling to buy a textbook and answer key.
As I didn't even mind Spivik for calc, my radar is way off for making suggestions to most people.
It's the ideal book for learning proofs if you are self-learning.
https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...
