On 3/30/2015 10:42 PM, Bruce Kellett wrote:
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.
I could have written that myself, Bruce. In fact I have. :-)
Brent
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