Re: [more absurd]

[Martin Steffen <msteffen@ifi.uio.no>]

>>>>>   <tomas@tuxteam.de> writes:

    > On Mon, Jul 04, 2022 at 07:10:27AM +0200, Uwe Brauer wrote:

    > [...]

    >> That really su... (My use case only concerned numbers from 0-10).
    >> 
    >> So it boils down to the question: why isn't 0 considered as
    >> natural numbers, as, according to the Peano axioms, it is?

    > I don't know whether you're serious or making fun (Poe's Law and
    > all that), but actually, Peano's axioms couldn't care less: as far
    > as they are concerned, natural numbers could well start at 23 or
    > something.

    > Actually it seems to be some kind of "cultural question" whether
    > mathematicians start counting at 0 or at 1; my observation is that
    > they tend to agree across one faculty at one university.  I know
    > positively one that tends to count from 1 (HU Berlin), another
    > that counts from 0 (Freiburg), both in Germany.


In some sense that's defendable (that what could call natural numbers is
a cultural question or historical, like looking at what Peano did nor
did not define).

On the other hand, one normally does not just deals with the numbers as
such, one does something with it (like comparing them or calculating
with them). If one takes the reservoir of numbers (in decimal notation,
{0,1,2,3 .....} indeed it is irrelelvant where to start, 0,1, or
23. Also if one does nothing else than comparing them (that would be
considering them as "ordinals", one has one single smallest number, and
again it's it's irrelevant if that's ``called'' nor notated $0$, "zero"
or "1", or "23".

Now, if one starts doing simple calculations (addition, multiplication),
the natural numbers including 0 is simply more "elegant" or ``richer''
than without. One has laws like n+0 = n, n*0=0 (one then says, 0 is a
neutral element wrt. +, there is terminology for all than, and it's
simply that N with 0 has nicer ``algebraic'' characteristics than
without. It's quite analogous to the choice between defining lists as to
include the empty list '() as a ``natural'' list, or insist on that
``natural'' lists must have 1 or more elements. 


    > I once asked a maths prof and he said foundational folks (set
    > theorists, math logicians -- that's the typical environment where
    > you'd tend to stumble upon Peano) tend to favour starting at 0.


Foundational folks can elaborate on that analogy between lists and nats,
but as you say, in both cases they favor to include 0 to nats and the
empty list to lists (and there are more examples) and it's favored for
good reasons (at least to them).

best, Martin



    > Historically, Peano himself seems to have been a one-counter:

    >   "Peano's original formulation of the axioms used 1 instead of 0
    > as the "first" natural number,[6] while the axioms in Formulario
    > mathematico include zero."  as quoted in [1].

    > Cheers

    > [1] https://en.wikipedia.org/wiki/Peano_axioms -- t