
Natural Numbers (Was: Fermath's sake)
Posted:
Nov 11, 1997 6:39 PM


First a comment on subject headings. Later on when I look through my files (or the archives, I'm never going to know that "Fermath's sake" contains a discussion of what counts as a natural number. Hence my change of the topic heading. If the old topic is listed behind the first such change, threads should still be visible, shouldn't they?
Now for the topic: what counts as a natural number, taking a more historical perspective.
The Greeks start with 2, as Sam Kutler observed.
Dedekind starts with 1, as does Peano. Since they were responsible for formalizing the natural numbers, we might follow their lead and do likewise.
John Conway notes that it is more "natural" to start with 0, and that he has done so since the 60s. Phil Parker says the same, noting its connection with SMSG textbooks (also the 60s). This probably coincides with New Math trends generally, where set theory was taken as the foundation for arithmetic.
My conjecture, therefore, is that it was the set theoretic cardinality approach to natural numbers (Cantor? Zermelo? von Neumann?) that switched (many of us) to taking the natural numbers as starting with 0: it's the cardinality of the empty set.
Footnote: Dedekind takes a set theoretic approach in 1889, but seems not to consider the empty set. When did this first enter set theory? Is it already in Cantor?
On 11 Nov 97 John Conway wrote:
> On Tue, 11 Nov 1997, Michael Button wrote: > > > >Usual notation is : > > > > > >Natural numbers = {0,1,2,3,......} > > > > > >positive integers = {1,2,3,......} > > > > This does not agree with what I learned as an undergraduate or as a > > graduate, so I looked it up in some of my old texts, and the definitions I > > learned are > > > > Natural numbers = {1,2,3,...} > > > > Whole numbers = {0,1,2,3,...} > > > > Thus I would have said (and this is what I teach) that the set of natural > > numbers is the same as the set of positive integers. Is this nonstandard? > > Does anyone know the history behind the naming of these sets? > > > > Michael Button > > > > The older nomenclature was that "natural number" meant > "positive integer". This was unfortunate, because the set > of nonnegative integers is really much more `natural' in > the sense that it has simpler properties. So starting in > about the 1960s lots of people (including me) started to > use "natural number" in the inclusive sense. After all, for > the positive integers we have the much better term "positive integer". > > John Conway >

