On 22 Jan 2015, at 07:48, Bruce Kellett wrote:
John Clark wrote:
On 18 January 2015 at 18:27, Jason Resch <[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.
Well, the disjunction of the statement above would be non constructive
(but in this case, it is constructive has pi is algorithmic).
Intuitionists accept the truth of the proposition "the
10^(10^(10^100))th decimal digit of pi exists and is equal to 0, or 1,
or 2, ..., or 9. They are realist on this disjunction.
And their are realist on the double negation of a disjunction which
would be non constructive, like, with omega being Chaitin number, or
the halting oracle made into a real number, etc.)
~ ~ (the 10^(10^(10^100))th decimal digit of omega is 0 V the
10^(10^(10^100))th decimal digit of omega is 1)
(in binary to be less long!).
On the arithmetical reality intuitionism and classical logic does not
differ much, and admit many two ways cross roads.
"It is logically necessary that one of the statements is true, given
the meanings of the terms involved" Is classical realism. It does not
entail intuitionist realism, but they are equivalent on the base
ontological level.
What happens, *in* computationalism, is that the intuitionist capture
better the constructive act of its mind when conceiving the numbers.
It is the first person view of the mathematician.
For a theory of everything, theology for short, we needs all the
points of view and their relations. The knowable gives a particular
intuitionist logic, and the observable gives a particular quantum logic.
The points of view are giving by intensional variants of Gödel's
beweisbar predicate:
p
beweisbar('p')
beweisbar('p') & p
beweisbar('p') & ~beweisbar('~p')
beweisbar('p') & ~beweisbar('~p') & p
With p restricted on the sigma_1 sentences.
The logic of the second is not entirely captured by the machine, so
its truth logic (G*) differ from its justifiable logic (G). The same
happens for two more views: "beweisbar('p') & ~beweisbar('~p')", and
"beweisbar('p') & ~beweisbar('~p') & p" (Z1*/Z1*; X1*/X1). The logic
of beweisbar('p') & p is given by the modal logic S4Grz1, which does
not split. It is the one giving a canonical intuitionist logic to each
"first person machine".
Bruno
Bruno
Bruce
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