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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”

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 3y 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?
    - weinzierl 3y ago
      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.pdf https://www.math.uni-bielefeld.de/~sieben/Euler_Algebra.ocr....
- auggierose 3y 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.
  - Someone 3y ago
    > 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 https://jeff560.tripod.com/n.html (via https://mathoverflow.net/questions/379699/origin-of-phrase-natural-number 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.
    - auggierose 3y ago
      This comment goes as far back as Euler and 1767: https://news.ycombinator.com/item?id=39917278 https://news.ycombinator.com/item?id=39917278
- zorked 3y ago
  English is being weird by having three terms then? Inteiros - (...-2, -1, 0, 1, 2...) Naturais - (1, 2, 3...)
  - noobr 3y ago
    Olha, primeira vez que vejo algo em português aqui!
- n4r9 3y 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".
  - lupire 3y ago
    Is i a whole number?
    - auggierose 3y ago
      Yes and No. No, it is not even a real number, but every whole number surely is a real number. Yes, it is a "Gaußsche ganze Zahl".
    - deleted 3y ago
      [deleted]
falcor84 3y 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.
- jacobolus 3y 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/bookVII.html 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.
  - lupire 3y ago
    log(1)=0, so it's unavoidable to consider 1 a something at least slightly different from the other numbers, as much as 0 is.
lupire 3y 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 3y 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 3y 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.

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 3y 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?
  - weinzierl 3y ago
    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.pdf https://www.math.uni-bielefeld.de/~sieben/Euler_Algebra.ocr....
auggierose 3y 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.
- Someone 3y ago
  > 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 https://jeff560.tripod.com/n.html (via https://mathoverflow.net/questions/379699/origin-of-phrase-natural-number 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.
  - auggierose 3y ago
    This comment goes as far back as Euler and 1767: https://news.ycombinator.com/item?id=39917278 https://news.ycombinator.com/item?id=39917278
zorked 3y ago
English is being weird by having three terms then? Inteiros - (...-2, -1, 0, 1, 2...) Naturais - (1, 2, 3...)
- noobr 3y ago
  Olha, primeira vez que vejo algo em português aqui!
n4r9 3y 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".
- lupire 3y ago
  Is i a whole number?
  - auggierose 3y ago
    Yes and No. No, it is not even a real number, but every whole number surely is a real number. Yes, it is a "Gaußsche ganze Zahl".
  - deleted 3y ago
    [deleted]

falcor84 3y 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.

jacobolus 3y 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/bookVII.html 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.
- lupire 3y ago
  log(1)=0, so it's unavoidable to consider 1 a something at least slightly different from the other numbers, as much as 0 is.

lupire 3y 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 3y 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 3y 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.