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A Gentle Introduction to the Art of Mathematics

314 points by Keegs 3y ago

Keegs 3y 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 2y ago
  Thanks for sharing here! Do you know of any textbooks in this spirit that cover other topics in math?
- __rito__ 2y 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.
rramadass 3y 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.
auggierose 3y 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 3y 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 https://en.wikipedia.org/wiki/Whole_number [2] https://en.wikipedia.org/wiki/Talk:Whole_number https://en.wikipedia.org/wiki/Talk:Whole_number
  - zelphirkalt 3y 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").
    - freilanzer 3y ago
      As a German, this is exactly what whole numbers means - numbers such as -1, 0, 1. Natural numbers are whole numbers > 0.
      - ahf8Aithaex7Nai 2y ago
        Every mathematical introductory text that excludes zero from the natural numbers contains an unwieldy additional notation like N_0 in the very next sentence. Why the hell should the number that is both used in the first Peano axiom and is the additive neutral element of the natural numbers not be included in the natural numbers? I've never understood why so many people insist on this, relying on pseudo-anthropological reasoning or something. Zero is at least two and a half thousand years old. You could just as easily claim that the natural numbers end with the number 10 because humans don't have any more fingers.
    - jacobolus 3y ago
      Okay, but what about in the 1880s?
      - 2y ago
xk_id 3y 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.
- vouaobrasil 3y 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 3y 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.
  - skhunted 3y 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.
thewanderer1983 3y ago
Fearless Symmetry and Elliptic Tales are fascinating, accessible and fun introductions to the art of mathematics.
- UncleOxidant 2y 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 2y ago
  But (as usual) reviews on Amazon offer a wide spectrum of opinion on these.
magpi3 3y ago
It's been hugged to death perhaps. The site is also hosted on github here: https://osj1961.github.io/giam/ https://osj1961.github.io/giam/
- leadingthenet 3y ago
  In fact, this seems to link to an even newer version of the text.
drBonkers 2y ago
Are there prints for sale?
- mindcrime 2y 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-version/dp/1494943557/ https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...
danielvaughn 2y 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.
- ChainOfFools 2y ago
  > I've found that mathematicians generally suck at introducing or explaining things. and then there's Grant Sanderson
  - wyager 2y 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.
    - ChainOfFools 2y ago
      And Feynman himself was characterized by his advisor in a recommendation letter as "another Dirac, but this time human."[0] [0] paraphrasing from memory, might have been a different supervisory figure.
    - Nifty3929 2y ago
      Perhaps on pedagogy, or at least nice, intuitive, layperson (but technically accurate) explanations of things. And plenty of fully-rigorous explanations of some things too. 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.
VinLucero 2y ago
Been looking for this!
__rito__ 2y 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 2y 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__ 2y 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 2y ago
  My context is CNC programming, but I've found the Make: Geometry, Make: Trigonometry, and Make: Calculus books excellent.
  - __rito__ 2y ago
deleted 2y ago
[deleted]

314 points by Keegs 3y 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 2y ago
Thanks for sharing here! Do you know of any textbooks in this spirit that cover other topics in math?
__rito__ 2y 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.

rramadass 3y 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.

auggierose 3y 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 3y 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 https://en.wikipedia.org/wiki/Whole_number [2] https://en.wikipedia.org/wiki/Talk:Whole_number https://en.wikipedia.org/wiki/Talk:Whole_number
- zelphirkalt 3y 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").
  - freilanzer 3y ago
    As a German, this is exactly what whole numbers means - numbers such as -1, 0, 1. Natural numbers are whole numbers > 0.
    - ahf8Aithaex7Nai 2y ago
      Every mathematical introductory text that excludes zero from the natural numbers contains an unwieldy additional notation like N_0 in the very next sentence. Why the hell should the number that is both used in the first Peano axiom and is the additive neutral element of the natural numbers not be included in the natural numbers? I've never understood why so many people insist on this, relying on pseudo-anthropological reasoning or something. Zero is at least two and a half thousand years old. You could just as easily claim that the natural numbers end with the number 10 because humans don't have any more fingers.
  - jacobolus 3y ago
    Okay, but what about in the 1880s?
    - 2y ago

xk_id 3y 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.

vouaobrasil 3y 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 3y 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.
- skhunted 3y 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.

thewanderer1983 3y ago

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

UncleOxidant 2y 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 2y ago
But (as usual) reviews on Amazon offer a wide spectrum of opinion on these.

magpi3 3y ago

It's been hugged to death perhaps. The site is also hosted on github here: https://osj1961.github.io/giam/ https://osj1961.github.io/giam/

leadingthenet 3y ago
In fact, this seems to link to an even newer version of the text.

drBonkers 2y ago

Are there prints for sale?

mindcrime 2y 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-version/dp/1494943557/ https://www.amazon.com/Gentle-Introduction-Art-Mathematics-v...

danielvaughn 2y 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.

ChainOfFools 2y ago
> I've found that mathematicians generally suck at introducing or explaining things. and then there's Grant Sanderson
- wyager 2y 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.
  - ChainOfFools 2y ago
    And Feynman himself was characterized by his advisor in a recommendation letter as "another Dirac, but this time human."[0] [0] paraphrasing from memory, might have been a different supervisory figure.
  - Nifty3929 2y ago
    Perhaps on pedagogy, or at least nice, intuitive, layperson (but technically accurate) explanations of things. And plenty of fully-rigorous explanations of some things too. 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.

VinLucero 2y ago

Been looking for this!

__rito__ 2y 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 2y 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__ 2y 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 2y ago
My context is CNC programming, but I've found the Make: Geometry, Make: Trigonometry, and Make: Calculus books excellent.
- __rito__ 2y ago

deleted 2y ago

[deleted]