On 08 Jun 2017, at 03:34, Bruce Kellett wrote:
I have just come across this paper from a year or so ago. Rovelli
essentially summarizes many of my own negative feelings about
mathematical platonism.
https://arxiv.org/pdf/1508.00001.pdf
Exercise. Show that the Church-Turing-Post-Kleene thesis refutes
Rovelli's idea that "if it is little, it depends on us". In fact the
question is "who us?", But you don't need mechanism to refute Rovelli.
The CT part of "YD + CT" is enough . Sorry for the exercise, but I am
under the june torture time (exams, corrections).
Mechanism, and Occam, says that the sigma_1 platonism is enough. It is
the Brouwer separable part of mathematic, where classical
mathematicians and intuitionist mathematicians "agree" (extensionally).
The interesting things are invariant for the choice of the phi_i. The
choice of Fortran-phi_i, or the LISP-phi_i, or <whatever>-phi_i will
be locally circumstancial, but the laws of mind and matter are
independent of that base, and their truth-notion are equivalent. This
needs no more than the common arithmetical realisim needed to make
sense of the Church-Turing thesis and of any Turing universal system.
That minimal number realism is conceptually interesting (cf. Number
Theory, the Music of the primes or the crazy partitions of natural
numbers (Ramanujan), etc.), and familiar, to humans since long (taught
in primary school). There are also arithmetic problem in newspapers,
and not much lambda expression problem, nor combinators one.
But the theology (mind and matter) is independent of the basic choice,
the "initial church-turing universal reality.
A measure of the complexity of universal computability in
"provability" scale is the sigma completeness. It is the ability to
search the numbers and find those having a verifiable provable
(decidable) property. A sigma1, or simply sigma formula has the shape
EnP(n) with P decidable. With the Matiyasevich-Robinson-Davis-Putnam
result this amounts to the belief in the existence of solution or non
solution to diophantine polynomial equations!
That part assumed is very small compared to basically all theories,
especially in metaphysics. But it does not depend on us more than it
depends on all universal numbers.
In category theory you can see them as cartesian closed categories
with a natural numbers object, the literature is very rich on this.
We can limit the use of the excluded middle to the "EnP(n)".
That is where I stop to be platonist, and I do not assume the
induction principle at the ontological level. Indeed at that level
"observer/believer"' dream, made by machine having much richer belief,
like including many induction axioms, can be proven to exist (in
infinitely many histories).
Little does not mean it depends on us, except if you meant by us: us
the universal numbers.
Bruno
PS I will comment your other post, at some time, as it is a busy
period of time, but in one word: you don't take the absence of
collapse enough into account.
I want take some time to read the post. You might try to avoid the
patronizing insulting tone which is the tool of those who does not
believe themselves in their own arguments.
Bruce
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