Jason Resch wrote:
On Thu, Jan 22, 2015 at 12:48 AM, Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
John Clark wrote:
On 18 January 2015 at 18:27, Jason Resch <[email protected]
<mailto:[email protected]> <mailto:[email protected]
<mailto:[email protected]>>> wrote:
> Do you believe that *one and only one* of the following
statements is true?
the 10^(10^(10^100))th decimal digit of pi is 0
the 10^(10^(10^100))th decimal digit of pi is 1
the 10^(10^(10^100))th decimal digit of pi is 2
the 10^(10^(10^100))th decimal digit of pi is 3
the 10^(10^(10^100))th decimal digit of pi is 4
the 10^(10^(10^100))th decimal digit of pi is 5
the 10^(10^(10^100))th decimal digit of pi is 6
the 10^(10^(10^100))th decimal digit of pi is 7
the 10^(10^(10^100))th decimal digit of pi is 8
the 10^(10^(10^100))th decimal digit of pi is 9
Either you answer yes, or no to that question. If you answer
yes, I don't see how you can escape mathematical realism.
Seth Lloyd has estimated that the maximum number of
computations that could be performed in the visible universe is
about 10^121 operations on 10^90 bits, if this is insufficient
to find your number is it meaningful to say pi has a
10^(10^(10^100))th decimal digit? I don't know, it depend on if
mathematics gave rise to physics or physics gave rise to
mathematics.
Realist and constructivist approaches to mathematics do not cover
all the possibilities. You can believe that one of the above
statements is true without knowing which is true. It is logically
necessary that one of the statements is true, given the meanings of
the terms involved. This does not entail mathematical realism.
So one of them is true, but can you (or anyone in this universe) prove:
the 10^(10^(10^100))th decimal digit of pi is 0 ?
the 10^(10^(10^100))th decimal digit of pi is 1 ?
the 10^(10^(10^100))th decimal digit of pi is 2 ?
the 10^(10^(10^100))th decimal digit of pi is 3 ?
the 10^(10^(10^100))th decimal digit of pi is 4 ?
the 10^(10^(10^100))th decimal digit of pi is 5 ?
the 10^(10^(10^100))th decimal digit of pi is 6 ?
the 10^(10^(10^100))th decimal digit of pi is 7 ?
the 10^(10^(10^100))th decimal digit of pi is 8 ?
the 10^(10^(10^100))th decimal digit of pi is 9 ?
If you answer no to all 10 of those questions, then none of those
statements is provable by any entity operating within this universe, yet
we know one of the statements is true. So Pi is a mathematical object
with properties that don't depend on the physical existence of
conceptions/proofs realized by entities or processes operating
physically. It follows then that if these properties don't depend on
physical processes of this universe, that even if this universe did not
exist at all, those properties would not be affected. And from that it
follows that mathematical properties and truth statements concerning
them have an existence independent of physics, hence mathematical realism.
Jason
No, your conclusion does not follow -- unless you have some non-standard
meaning of mathematical realism. It is all quite simple, really. We have
a certain set of axioms. Those axioms encompass the definition of pi
which can be shown to be a transcendental number. These are simple
consequences of the axioms. Nothing at all need exist, or in any sense
be /real/, for all of this to be true.
Bruce
--
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/d/optout.
