Hi Jason,
On 06 Feb 2012, at 14:51, Jason Resch wrote:
On Sun, Feb 5, 2012 at 12:23 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 05 Feb 2012, at 17:14, Craig Weinberg wrote:
Talk with them, meaning internal dialogue?
Public dialog. Like in Boolos 79 and Boolos 93. But the earlier form
of the dialog is Gödel 1931.
Solovay 1976 shows that the propositional part of the dialog, with
the
modal Bp, is formalized soundly and completely by G and G*. It is
the
embryo of the mathematics of incompleteness, including the directly
accessible and the indirectly accessible parts, and the
explanation of
the why we feel it is the other way around, etc.
When you talk with them, do they answer the same way to the same
question every time?
The conversation is made in Platonia, and is not entangled to our
history, except for period where I implement it on some machines.
Even in that case, they didn't dispose on short and long term
memories, except for their intrinsic basic arithmetical experiences
(which bifurcate up to you and me).
Bruno,
Would you say this is the source of all mathematical truth?
Interview / study of platonic objects and machines?
I don't think so. We can only explain why we believe in the natural
numbers, by having some model for "we". With comp "we" is modeled by
natural numbers, (and captured as such by the "doctor" on its hard
disk), so I have to postulate the numbers at the start (or other
finite equivalent things). Also, we cannot logically derive the laws
of addition and multiplication from simpler logical theory. We can
only start explanation by agreeing (implicitly) on some system which
is at least Turing universal.
I am not sure if analysis is ontological, nor if that question is
interesting. What is sure is that analysis and higher order logical
tools are a necessity for the numbers to "accelerate" the
understanding of themselves.
I am agnostic on some possible platonism extending arithmetic. With
comp, this should be absolutely undecidable, because for arithmetical
being (of complexity p), bigger arithmetical being (of complexity q
bigger than p) can behave analytically.
With comp, the source of all mathematics is the natural imagination of
the universal numbers. It obeys laws, and that is why there is
metamathematics (mathematical logic) and category theory, up to, with
comp, the theology of numbers.
And the source of physics is the same, but taking the global first
person relative self-indetermination into account.
Global means that the indeterminacy bears on the UD-computations (or
the theorem of RA and their proofs). the state is relative to its
infinities of UM, and other quasi UM machine, implementation/
incarnation/interpretations.
Bruno
http://iridia.ulb.ac.be/~marchal/
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