A Gentle Introduction to the Art of Mathematics

https://giam.southernct.edu/GIAM/

by Keegs • 2 years ago

314 69 2 years ago

giam.southernct.edu

Keegs• 2 years ago

The quality of writing in this book is fantastic. If you're studying math or have a recreational interest I really recommend it.

dr_kiszonka• 2 years ago

Thanks for sharing here! Do you know of any textbooks in this spirit that cover other topics in math?

__rito__7• 2 years ago

What other resources would you recommended for learning Math that is a bit more advanced?

It needs to be suitable for someone studying alone.

rramadass4• 2 years ago

Seems very student-friendly and accessible which is good. A quick browse shows that it covers Set Theory, Logic and Proofs i.e. the absolute basics for "Modern Mathematics". One of the biggest difficulties for the student i.e "Mathematical Notations" (which are merely a shorthand language) are explained from the ground up which is great.

auggierose4• 2 years ago

At the very start of the book:

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, ...

tromp• 2 years ago

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.

[1] https://en.wikipedia.org/wiki/Whole_number

[2] https://en.wikipedia.org/wiki/Talk:Whole_number

zelphirkalt• 2 years ago

In German though "Ganze Zahlen" is not ambiguous. At least in my entire life I have not seen any other understanding of it and every child at school learns about them and that they include 0 and negative numbers, in contrast to natural numbers ("Natürliche Zahlen").

auggierose4• 2 years ago

In German, it is not and never was ambiguous, so it is not a mistranslation. What the meaning of "Whole number" in English is, is not really relevant, except for explaining the mistake in the book.

zorked• 2 years ago

English is being weird by having three terms then?

Inteiros - (...-2, -1, 0, 1, 2...)

Naturais - (1, 2, 3...)

n4r9• 2 years ago

Huh, I never knew there was an ambiguity. To me it makes total sense to include negative integers. "Whole" means "without a fractional part".

falcor84• 2 years ago

I can't seem to find a reference for it now, but a math professor once told us that some early cultures didn't even have a proper word/symbol for the number "one", and the act of counting didn't kick in until there were two items.

jacobolus2• 2 years ago

Greek geometers (e.g. in Euclid's Elements, Book 7) define a "unit" to be 1 and "numbers" (arithmoi) to start from 2. https://mathcs.clarku.edu/~djoyce/java/elements/bookVII/book...

But in practice there are a bunch of propositions/proofs where 1 is treated as a number just like any other.

lupire4• 2 years ago

Mistranslation or not, conceptually "natural numbers Z+" or "whole numbers Z*" are a more fitting claim.

Peano would go further: God invented 0 and +1, and all the rest are the work of people.

dotancohen• 2 years ago

In software development it is often said that there are only three relevant numbers: 0, 1 and infinity. Everything else is contrived.

Certainly, nature follows a very similar pattern. For limited values of infinity ))

yboris• 2 years ago

I find that philosophically, "all math is discovered" is a better way to think of mathematics.

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.

xk_id• 2 years ago

Would be nice if we could see experienced mathematicians endorse this, please. It would help to decide whether to put time into it. At first glance it looks really nice.

vouaobrasil3• 2 years ago

I'm a mathematician (PhD in numnber theory...) and I took a look. It's a basic textbook about some concepts like functions, relations, and basic proof techniques. It seems okay. I was expecting something different from the title, though. It's not too different from many basic books introducing such concepts but obviously a lot of effort was put into it.

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.

seanhunter• 2 years ago

The purpose of the book as stated in the preface is to be used for a course in proof writing ie it is a sort of bridge to higher maths which the author defines as "defining axiomatic systems and proving statements within them" vs "elementary maths" which he defines as "solving problems".

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.

skhunted4• 2 years ago

I’m ABD in math for discolosure purposes. I strongly disagree with your recommendations if the purpose is to get an introduction to higher math. The book in question is much, much better for introducing one to higher math than any of the books you recommended.

Smullyan’s books are great but one isn’t going to go from Smullyan to abstract algebra, point set topology, or real analysis.

beryilma• 2 years ago

One the same subject and as accessible, I love the two books from Jay Cummings: Proofs and Real Analysis. Each just $16 on Amazon. It is a joy to read these books and try some of the exercises. I wish PDF versions were also available...

For those interested: https://webpages.csus.edu/jay.cummings/Books.html

nicf2• 2 years ago

I'm a mathematician, currently working as a private tutor for adults who want to learn proof-based math. I had a quick glance through this book and it seems to me like a pretty nice version of this "intro to proofs" sort of book. This is a topic that's done well in a lot of different books, though, so if you really want to dig into this topic I'd maybe recommend looking at a couple different ones and finding the one that agrees with you the most.

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.

nyrikki3• 2 years ago

As someone who tutors adults, can you suggest a more digestible book for abstract algebra?

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.

__rito__7• 2 years ago

I have read many parts of Cummings'es book, and I can vouch for its quality.

It's the ideal book for learning proofs if you are self-learning.

sn92• 2 years ago

The 3rd edition of Velleman's book has an online supplement that uses Lean to work through the book so you can get feedback about your proofs.

mcguire• 2 years ago

Putting on my Computer Scientist hat (and getting all of the tentacles out of my face so I can see), GIAM seems to be a good, if a little verbose[1], introduction to formal logic and basic set theory. Those are fundamental to CS and this is a heck of a lot better than many of the intros to those topics that I have seen in curricula without a dedicated class.

thewanderer1983• 2 years ago

Fearless Symmetry and Elliptic Tales are fascinating, accessible and fun introductions to the art of mathematics.

UncleOxidant• 2 years ago

Fearless Symmetry

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.

Koshkin• 2 years ago

But (as usual) reviews on Amazon offer a wide spectrum of opinion on these.

magpi3• 2 years ago

It's been hugged to death perhaps. The site is also hosted on github here:

https://osj1961.github.io/giam/

leadingthenet• 2 years ago

In fact, this seems to link to an even newer version of the text.

drBonkers• 2 years ago

Are there prints for sale?

mindcrime• 2 years ago

GIAM is now available in hardcopy as a printed-on-demand paperback from CreateSpace. Please rest assured that GIAM will remain available for free download from this site.

https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...

danielvaughn• 2 years ago

I was prepared to hate this book as much as I've hated any other "introduction" to math, since I've found that mathematicians generally suck at introducing or explaining things. But the first little section that breaks down set notation into plain english was fantastic. That's exactly the kind of thing I need personally, as notation is extremely off-putting and confusing for me.

ChainOfFools2• 2 years ago

I've found that mathematicians generally suck at introducing or explaining things.

and then there's Grant Sanderson

wyager• 2 years ago

I don't think it's much of an exaggeration to describe Sanderson as a contemporary Feynman, at least from the pedagogy side of things.

VinLucero• 2 years ago

Been looking for this!

__rito__7• 2 years ago

This looks good. And I have already covered these things.

What are some other resources for learning more Math that are very approachable when studying alone?

tarkin2• 2 years ago

Khan academy will teach you all of pre-k, high-school, pre-calc, calculus and infinite series pretty well. mathsisfun.com is a great resource if that's ever lacking. High school physics on khan is good to give you a practical application of the concepts if they seem too abstract.

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.

__rito__7• 2 years ago

I have a Bachelor in Physics, and a Master’s in CS.

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.

WillAdams• 2 years ago

My context is CNC programming, but I've found the Make: Geometry, Make: Trigonometry, and Make: Calculus books excellent.

__rito__7• 2 years ago

Thanks.

rramadass4• 2 years ago

"Approachability" might vary somewhat but you might find the following useful;

* 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.

__rito__7• 2 years ago

Yeah, thanks for the recommendation.

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.

A Gentle Introduction to the Art of Mathematics

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