On Jan 21, 2014, at 12:47 AM, meekerdb <[email protected]> wrote:
On 1/20/2014 7:20 PM, LizR wrote:
On 21 January 2014 14:25, meekerdb <[email protected]> wrote:
On 1/20/2014 5:00 PM, LizR wrote:
On 21 January 2014 06:42, meekerdb <[email protected]> wrote:
On 1/20/2014 1:11 AM, LizR wrote:
On 20 January 2014 18:51, meekerdb <[email protected]> wrote:
You seem not to appreciate that this dissipates the one essential
advantage of mathematical monism:
we understand
mathematics (because, I say, we invent it). But if it's a mere
human invention trying to model the Platonic ding and sich then
PA may not be the real arithmetic. And there will have to be
some magic math stuff that makes the real arithmetic really real.
Surely the real test is whether it works better than any other
theory. (The phrase "unreasonable effectiveness" appears to
indicate that it does.)
Would it work any less well if there were a biggest number?
I don't know. I would imagine so, because that would be a theory
with an ad hoc extra clause with no obvious justification, so
every calculation would have to carry extra baggage around. If I
raise a number to the power of 100, say, I have to check first
that the result isn't going to exceed the biggest number, and take
appropriate action - whatever that is - if it will... what would
be the point of that?
Just make it an axiom that the biggest number is bigger than any
number you calculate. In other words just prohibit using those
"..." and "so forth" in your theorems.
So you are saying "there's a biggest number, but we don't know what
it is. But it's big. Really big. You may think it's a long way down
the road to the chemist, but that's peanuts copared to this number.
You just can't imagine how vastly, mind-boggling huge it is..."
Or words to that effect (with thanks to the late and occasionally
great Douglas Adams).
Why would you want to do that? It seems like an unnecessary extra
axiom that doesn't have any purpose or utility.
It prevents the paradoxes of undeciability, Cantor
diagonalization, and it corresponds more directly with how we
actually use arithmetic.
I'm not sure it helps. What you may gain from avoiding paradoxes makes
many of our accepted proofs false. E.g. Euclids proof of infinite
primes. Or Euler's identity. Most of math would be ruined. A circle's
circumference would not even be pi*diameter.
Would this biggest number be different for different beings in
different universes? What is it contingent on?
Jason
Brent
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