A Gentle Introduction to the Art of Mathematics
giam.southernct.edu
giam.southernct.edu
> 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.
Negative numbers are not broken, so they have always been considered whole. For example, Leonhard Euler wrote in his "Vollständige Anleitung zur Algebra" from 1767:
"Alle diese Zahlen, so wohl positive als negative, führen den bekannten Nahmen der gantzen Zahlen, welche also entweder größer oder kleiner sind als nichts. Man nennt dieselbe gantze Zahlen um sie von den gebrochenen, und noch vielerley andern Zahlen, wovon unten gehandelt werden wird, zu unterscheiden."
https://www.math.uni-bielefeld.de/~sieben/Euler_Algebra.ocr....
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...)
No, it is not even a real number, but every whole number surely is a real number.
Yes, it is a "Gaußsche ganze Zahl".
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/
Among your original recommendations, I have heard about "All the Maths you Missed". And I have read several chapters of the one by Kolmogorov et al. It's a fantastic book. It lays the landscape really well, discusses things, and covers the breadth of the field rather than the depth of any particular field. I find the writing style really good. It could be seen as a reference book for people who already know the stuff, or are looking to know about new stuff, but want to have ideas about what those stuff might be.
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.
The overwhelming empirical evidence is that having a course on proofs and basic set theory is much better preparation for advanced mathematics than reading Smulyan’s recreational math books. I guess you’d rather your students read Martin Gardner and then do Fraliegh’s Abstract Algebra book. No one does it that way but go with your so called empirical evidence.
Having a Ph.D. in math ought to have taught you to reason better than to use “actual research” as part of your reasoning when discussing learning topics that are not cutting edge. One doesn’t need to have done research in mathematics to know about Gorenstein rings or projective dimension or other such stuff. It also has nothing to do with teaching basic math.
I could be wrong in my opinion but attack it on its merits without using superfluous things like “actual research” when research has nothing to do with the topic.
As a person interested in self-learning Mathematics, i have read and amassed a lot of "popular mathematics" books by authors like Ian Stewart, W.W.Sawyer, E.T.Bell, George Gamow etc. all of which were great motivators but none of which taught me the basics of "Modern Mathematics" which i could only get from Textbooks. The quality of Textbooks are of course all over the map and so i am always on the lookout for the simplest, clearest and yet rigourous explanations available. The book under discussion seems to check all such boxes for a beginning student.
The language is a baby compared to the applications.
How do you propose one do applications of point set topology without knowing about sets and mappings? Before the applications one must know the language. We don’t teach the quadratic formula and solving simple velocity problems before teaching students how do the basics of manipulating algebraic expressions.
One must first be a baby before being an adult.
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.)
With self study I prefer a bit more breadth to make up for the realities of needing to self study which often ends up with deep but not wide understanding of topics.
Having a brother who had a PHD in complex analysis to bother probably helped with self-learning. That is the only option when you are on-call for decades at a time as higher math courses are/were always in person.
But hopefully someone will figure out a business model to help people who need to grow and adapt.
Thanks again for the suggestions, I have ordered both books to add to my lending library.
It's the ideal book for learning proofs if you are self-learning.
https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...