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Comment by auggierose | Hacker News Reader

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, ...

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> 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").

As a German, this is exactly what whole numbers means - numbers such as -1, 0, 1. Natural numbers are whole numbers > 0.

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.

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....

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.

This comment goes as far back as Euler and 1767:

https://news.ycombinator.com/item?id=39917278

English is being weird by having three terms then?

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

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

Olha, primeira vez que vejo algo em português aqui!

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

Is i a whole number?

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".

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.

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.

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 ))

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.

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.

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