On 1/20/2014 7:20 PM, LizR wrote:
On 21 January 2014 14:25, meekerdb <[email protected] <mailto:[email protected]>>
wrote:
On 1/20/2014 5:00 PM, LizR wrote:
On 21 January 2014 06:42, meekerdb <[email protected]
<mailto:[email protected]>> wrote:
On 1/20/2014 1:11 AM, LizR wrote:
On 20 January 2014 18:51, meekerdb <[email protected]
<mailto:[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.
Brent
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/groups/opt_out.
