On Jan 19, 2014, at 11:51 PM, meekerdb <meeke...@verizon.net> wrote:
On 1/19/2014 3:41 PM, Jason Resch wrote:
On Jan 19, 2014, at 3:31 PM, meekerdb <meeke...@verizon.net> wrote:
On 1/19/2014 9:45 AM, Bruno Marchal wrote:
But why should that imply *existence*.
It does not. Unless we believe in the axioms, which is the case
for elementary arithmetic.
But what does "believe in the axioms" mean. Do we really believe
we can *always* add one more? I find it doubtful. It's just a
good model for most countable things. So I can believe the axioms
imply the theorems and that "17 is prime" is a theorem, but I
don't think that commits me to any existence in the normal sense
of "THAT exists".
Axioms are a human invention which only approach the truth that was
already there. Our picking some axioms to believe in changes
nothing.
You seem not to appreciate that this dissipates the one essential
advantage of mathematical monism: we understand mathematics
(because, I say, we invent it).
I would say we understand some things about sine mathematical objects
because we can simulate them in our mind (or on computers) and learn
their properties.
For example, the general shape of the Mandelbrot set.
But if it's a mere human invention trying to model the Platonic ding
and sich then PA may not be the real arithmetic.
It certainly isn't, as it leaves inaccessible an infinite number of
true statements. However, when it can prove something, like that a
certain Turing machine halts at a certain step, we find it is in
agreement with other axiomatic systems and reality.
And there will have to be some magic math stuff that makes
the real arithmetic really real.
I thought we were talking about truth not reality, though perhaps the
two are related. The Sanskrit word for truth also means that which
exists.
So what makes 2+2=4 true? I don't know, I suppose you might say
"magic".
Jason
Brent
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