Fearless Symmetry and Elliptic Tales are fascinating, accessible and fun introductions to the art of mathematics.
A Gentle Introduction to the Art of Mathematics
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Re: A Gentle Introduction to the Art of Mathematics
#12Earlier quoted context omitted.
> 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
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").
Okay, but what about in the 1880s?
Re: A Gentle Introduction to the Art of Mathematics
#13At 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"…
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.
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.
Re: A Gentle Introduction to the Art of Mathematics
#14It's been hugged to death perhaps. The site is also hosted on github here:
Re: A Gentle Introduction to the Art of Mathematics
#15At 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"…
> 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
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.
Re: A Gentle Introduction to the Art of Mathematics
#16Would 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.
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…
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.
Re: A Gentle Introduction to the Art of Mathematics
#17At 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"…
> 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
English is being weird by having three terms then?
Inteiros - (...-2, -1, 0, 1, 2...)
Naturais - (1, 2, 3...)
Re: A Gentle Introduction to the Art of Mathematics
#18Would 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.
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…
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.
Re: A Gentle Introduction to the Art of Mathematics
#19It's been hugged to death perhaps. The site is also hosted on github here: https://osj1961.github.io/giam/
In fact, this seems to link to an even newer version of the text.
Re: A Gentle Introduction to the Art of Mathematics
#20Earlier quoted context omitted.
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…
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.
Meh, well different strokes for different folks I guess. I got into higher math while reading Ian Stewart's book in grade school but I guess some people are going to want to go the standard way.
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.