Re: Proof that counting is valid?

Michael Press wrote:
In article <XEP_g.114183$aJ.14139@attbi_s21>,
Stephen Montgomery-Smith <stephen@xxxxxxxxxxxxxxxxx> wrote:


bogus61847688@xxxxxxxxxxxxxx wrote:

I remember, many years ago, having seen a Sesame Street episode in
which Bert tries to show Ernie that four cookies are always four
cookies, no matter how they are arranged.

I have never actually seen a proof that "counting works": that is, that
no matter in what order you count a set of objects, the answer will
always be the same. I know that this is (to most of us) "self-evident",
yet mathematicians seem to feel that it is not a waste of time to prove
the "self-evident".

For the sake of completeness, can I see a mathematical proof that
"counting works"?

My feeling is that this is really a "primitive" mathematical fact. "Primitive" here meaning that it really cannot be proved from or broken into other facts.

I guess that you could prove this sort of thing using set theory or the Peano axioms. But really these axiom systems are built with the idea that this kind of fact should be true, not vice versa. Indeed even to formally describe what you mean by an axiom system, you already need an elementary concept of counting (and indeed even of set).

I think that you should regard your question as unanswerable using present day philosophy. You probably need to wait a few hundred years, and then maybe someone will have a good answer. So for now, simply regard it as something obviously true and completely self-evident, and leave it at that.


I had the impression that all this was sorted out long ago. I mean, what is Foundations all about if counting is a primitive?

I read a lot about foundations. While it does give a great foundation for doing mathematics, I really don't think it can be considered as capturing such a fundamental philosophical concept like "four."

Stephen
.

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