Bruno Marchal wrote:
On 29 Mar 2015, at 10:04, Bruce Kellett wrote:
OK. If all the connections and inputs remain intact, and the digital
simulation is accurate, I don't see a problem. But I might object if
the doctor plans to replace my brain with an abstract computation in
Platonia
The doctor propose a real physical computer. Either a cheap PC or a more
expensive MAC, but it is done with matter guarantied of stellar origin!
But what sort of program will it be running? A physical simulation, or
some abstract computationalist AI model? See my reply to Brent.
-- because I don't know what such a thing might be,
Nor do I.
and don't believe it actually exists absent some physical instantiation.
Do you thing prime numbers needs physics to exist? If yes, show me what
is wrong in Euclid's proof, which define and prove the mathematical
existence of the prime numbers without assuming anything physical.
I am assuming that Euclid, himself, is physical, and that he devised the
proof -- it did not drop into his lap unsought. In a phrase I have used
before, It did not spring forth fully armed, like Athena from Zeus's
brow. Numbers were a hard-won abstraction from everyday physical
reality. They do not have any independent existence. As someone has
said, you do not come across a number "5" running wild in the undergrowth.
I know that many, if not most, mathematicians report that in their
research it is as though they are exploring a landscape that exists --
they are discovering things that are already there, they are not
constructing them. Hence most mathematicians are realists about
mathematics, which is Platonism.
But I think we need to distinguish two senses in which something can be
said to exist. There is mathematical existence, Exist_{math}, and
physical existence, Exist_{phys}. These are not the same, and are not
even approximately equivalent, although it might seem that way to a
mathematician.
Exist_{math} is the set of all implications of a set of axioms and some
rules of inference. It is not necessary that everything that
exists_{math} can be proved as a theorem withing the system, or that the
completeness and/or consistency of this system can ever be established.
But it is an abstract system, and exist_{math} resides in Platonia,
outside of any physical existence.
Exist_{phys} is the hardware of the universe. It is not defined
axiomatically, but ostensively. You point and say "That is a rock, cat,
or whatever." In more sophisticated laboratory settings, you construct
models to explain atomic spectra, tracks in bubble chambers, and so on.
The scientific realist would claim that the theoretical entities
entailed by his most mature and well-tested scientific theories
"exist_{phys}", and form part of the furniture of the external objective
physical world. The experienced scientist, though, always recognizes
that any such claims of ontology are, at best, provisional, and are
always subject to revision on the advent of new and better date, more
general and sophistical models, and so on.
So there is a very clear difference between the mathematical and
physical worlds. One is axiomatic and subject to proof. Valid proofs are
not open to revision -- they may be abandoned as useless, but once
proved, they remain proved and transfer truth values from the premises
to the conclusions. This is not the case for physics. That is not
axiomatic, it is ultimately based on observation and experiment. Any
theories that might be constructed are always provisional and subject to
revision.
So prime numbers might exist_{math}, but they do not exist_{phys}. If we
keep this distinction clear we will avoid a lot of unnecessary confusion.
Bruce
Likewize, all computations can be proved to exists, and have some
weight, in a theory as weak as Robinson arithmetic.
The doctor will not propose an abstract immaterial brain to you. But the
problem, shown by the UD-Argument, is that you already have an infinity
of abstract immaterial brain in elementary arithmetic, and you can
detect the difference, and that leads to the necessity of justifying the
stability of the physical laws from a measure on all computation,
extending Everett methodology on Arithmetic.
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