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A Gentle Introduction to the Art of Mathematics

giam.southernct.edu

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Re: A Gentle Introduction to the Art of Mathematics

#21

Post by Someone »

auggierose wrote: post #15
tromp wrote: post #5

Earlier 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, 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.

> In German, it is not and never was ambiguous

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.

Re: A Gentle Introduction to the Art of Mathematics

#22

Post by skhunted »

vouaobrasil wrote: post #20
skhunted wrote: post #18

Earlier quoted context omitted.

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 s…

The language comes first then the applications. There’s a reason the order of topics evolved the way it has.

Re: A Gentle Introduction to the Art of Mathematics

#23

Post by vouaobrasil »

skhunted wrote: post #22
vouaobrasil wrote: post #20

Earlier quoted context omitted.

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 s…

The language comes first then the applications. There’s a reason the order of topics evolved the way it has.

Well, I'm just going by empirical evidence: what has worked for me and what has worked for many of my fellow colleauges that have done actual research in mathematics.

Re: A Gentle Introduction to the Art of Mathematics

#24

Post by weinzierl »

jacobolus wrote: post #12
zelphirkalt wrote: post #7

Earlier quoted context omitted.

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?

I think, historically the term "Ganze Zahl" (a whole or entire = integer number) was always used in contrast to "Gebrochene Zahl", meaning broken or fractured number.

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....

Re: A Gentle Introduction to the Art of Mathematics

#25

Post by beryilma »

vouaobrasil wrote: post #10
xk_id wrote: post #8

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.

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…

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

Re: A Gentle Introduction to the Art of Mathematics

#26

Post by skhunted »

vouaobrasil wrote: post #23
skhunted wrote: post #22

Earlier quoted context omitted.

The language comes first then the applications. There’s a reason the order of topics evolved the way it has.

Well, I'm just going by empirical evidence: what has worked for me and what has worked for many of my fellow colleauges that have done actual research in mathematics.

Before one can do advanced mathematics (I have too have done research in math) one needs to learn the basics. Reading Smullyan did not in any way help you learn advanced mathematics. It may have helped you to get motivated to learn math and want to learn advanced math but it didn’t help you accomplish this. There’s a reason just about every mathematics department teaches classes on how to do proofs and on basic set theory but almost none of them teach from Smullyan’s book.

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.

Re: A Gentle Introduction to the Art of Mathematics

#27

Post by n4r9 »

tromp wrote: post #5
auggierose wrote: post #4

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"…

> 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

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

Re: A Gentle Introduction to the Art of Mathematics

#28

Post by nicf »

xk_id wrote: post #8

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.

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.

Re: A Gentle Introduction to the Art of Mathematics

#30

Post by lupire »

auggierose wrote: post #4

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"…

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.

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