> 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.
I’m not sure about that ”and never was”. Mathematicians used to have fairly loose definitions for all kinds of things and different fields sometimes have incompatible definitions for a term, so I think it’s not impossible “ganzzahlig” was used ambiguously for ℤ or ℕ (in- or excluding zero) for a while in some corners.
When was the phrase “natural number” even invented? The best I can find is https://jeff560.tripod.com/n.html (via https://mathoverflow.net/questions/379699/origin-of-phrase-n...) which says “Chuquet (1484) used the term progression naturelle for the sequence 1, 2, 3, 4, etc.”
There may well have been a time where “ganzzahlig” existed but “Natürliche Zahl” didn’t yet.
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
I'm a big fan of Sanderson.
But I don't think we can compare Sanderson to Feynman on raw intelligence or contributions to the state of the art in math/science.
[0] paraphrasing from memory, might have been a different supervisory figure.
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.
https://press.princeton.edu/books/hardcover/9780691118802/th...
https://press.princeton.edu/books/hardcover/9780691150390/th...
PS: In an earlier HN thread, somebody had highly recommended the 4-vol Foundations of Applied Mathematics developed for Brigham Young University’s Applied and Computational Mathematics degree program for beginning graduate and advanced undergraduate students. I have not browsed/read them yet but they are on my "future acquisition and study" list. They seem great and well worth looking into - https://foundations-of-applied-mathematics.github.io/
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.
My problem with the book we are discussing is that it seems rather prosaic -- it doesn't really give a sense of the true reason to practice math: the asking of interesting questions and creating new universes. It's just the same old stuff that we're taught because it's a convention.
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.
The exercises for each chapter are split into several sections each section covering a different aspect of the chapter's material. Sometimes there is a section of exercises applying the material to some interesting area.
For example, the chapter on groups of permutations has 6 pages of text, then 5 pages of exercises divided into 9 sections. Those sections are: computing elements in S6 (5 problems), examples of groups of permutations (4 problems), groups of permutations in R (4 problems), a cyclic group of permutations (4 problems), a subgroup of SR (4 problems), symmetries of geometric figures (4 problems), symmetries of polynomials (4 problems), properties of permutations of a set A (4 problems), and algebra of kinship structures which consists of 9 problems covering how anthropologists have applied groups of permutations to describe kinship systems in primitive societies.
There are answers in the back for a decent number of the exercises.
It's a Dover republication so is not too hard on the wallet. List price is $30 at Dover but its around $20 on Amazon.
The combination of short chapters and lots of exercises make it easier than most textbooks to fit into a busy adult schedule.
In general, I think self-studying proof-based math can certainly be done if someone's motivated enough, but it's pretty hard and takes a lot of work, especially if you're still getting used to the skill of reading and writing proofs. It's very valuable to be able to have a person available to evaluate the proofs you're writing, and I've definitely seen a few people who came to me thinking they'd mastered proof-writing on their own and were kind of mistaken about that. (I've definitely also seen people who really did learn this skill pretty well without help! It varies a lot.)
It's the ideal book for learning proofs if you are self-learning.
https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...
