On 8/2/2010 11:14 AM, Mark Buda wrote:
Brent Meeker<meeke...@dslextreme.com>  writes:

On 8/1/2010 3:42 PM, Quentin Anciaux wrote:

     The only problem is if numbers were a human invention... other
     humans could come with a prime number that is even and not
     2... There would exists a biggest number, 1+1=2 could be false
     somewhere sometime (even by following the rules that makes 1+1=2
     true always)...

They can and do.  In modulo two arithmetic 1+1=0.  You can invent all
kinds of number systems or other logics and axiomatic systems.
No. You can define your terms, and you can use your terms, but you can't
redefine your terms while you're using them and end up with a valid
argument. When Quentin says 1+1=2 always, he has a meaning behind those
symbols.

But the meaning isn't mathematical - it's the idea of putting pebbles together and counting them. He abstracts away the pebbles and supposes that he has discovered a Platonic realm in which the numbers "exist" without anything to count or succeed. But I think that meaning (i.e. reference) only comes from action, or at least potential action, within an environment. It's fine to abstract away particulars for purposes of inference - but to say that discovers new "existences" seems to me simply inventing a new kind of "existence" that could as well be called "non-existence" or "imaginary existence".

He's talking about the idea in his mind underlying the
utterance "1+1=2" being true always. You can't take a different idea
that happens to be expressed using the same symbols and then assert that
that has any bearing on the truth of Quentin's original idea.

But his original idea existed in his brain - at least that's the physicalist theory.

You could do that if he were writing a formal mathematical proof,
because then you would be explicitly bound by the same
symbol-manipulating rules he is.

So what you said above is perfectly true, but doesn't make your case
that numbers are a human invention. The symbols and words we use to talk
about numbers are a human invention. Not the numbers.

My point was that since we can invent other mathematical structures - including number systems. Why should we suppose the natural numbers exist and the others don't. Or do you contend that all mathematical systems exist and are discovered, not invented. In which case what distinguishes them from all Sherlock Holmes stories - were they to discovered?

Brent

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