meekerdb wrote:
On 4/28/2015 10:59 PM, Bruce Kellett wrote:
meekerdb wrote:
On 4/28/2015 9:18 PM, Bruce Kellett wrote:
meekerdb wrote:
On 4/28/2015 7:27 PM, Bruce Kellett wrote:
But you fall foul of the essential slippage here. Arithmetic
certainly supports all calculations -- as I point out elsewhere,
all calculations ultimately reduce to counting. But this does not
mean that all calculations exist in Platonia. The results of all
calculations might exist there, but not the calculations themselves.
That's where I think Bruno's idea diverges. Because "all
computation" can be defined by a UD, and every such definition will
be extensively equivalent per Church-Turing, it must exist in
Platonia. Bruno uses "exist" as in the logical form "There exists
an even prime number."
I think there might be a distinction being drawn between
"computation" and "calculation". I m not quite sure what this
distinction might be, but in the sense that a calculation is an
operation that can be performed with pencil and paper, and takes
physical time in a physical world, I do not see how this can work in
Platonia, which is timeless. There is not an ordered sequence of
steps there, neither spatial nor temporal ordering make any sense in
Platonia. There might be a logical ordering, but all of the steps
exist simultaneously. Platonia is the timeless world of forms, so
although there exists an indefinite number of relations between any
two integers, x and y, none of these paths or relations is ever
actually calculated there. In other words, all valid results 'exist'
in Platonia, but the equivalent computations do not. Computations
(calculations) are left to mere mortals.
I think the idea is that computations are proofs, i.e. proof that
this algorithm with this input produces this output. So they have
Goedel numbers which specify the steps of the proof and so "exist" in
Platonia.
That seems a likely interpretation. Does the Goedel number give the
whole ordered proof?
The number encodes the whole proof in order. But it isn't unique; there
are infinitely many ways to define a Goedel numbering.
So how does my Platonic consciousness know the difference between a
different Goedel numbering of the same proof, or the same Goedel
numbering of a different proof?
Or is there a separate number for each step? One of my problems is
that the notion of a 'step' in a computation or proof is not
well-defined. That seems to depend on the architecture of your
'computer'.
That's where Bruno relies on the equivalence formal digital computation
methods, the Church-Turing thesis: Turing, recursive functions, lambda
calculus,
But that formal equivalence ensures only that for a given algorithm on a
given input, the output is the same. It says nothing about the
intermediate steps. Unless you break your algorithm down to simple
counting or some such.
I am an impoverished physicist and the only computer I can afford is
the simplest Turing machine that I put together out of a few bits of
wood and a couple of pebbles. I have an unlimited supply of paper
tape, so I can do any calculation whatsoever -- it might just take me
a bit longer than it takes other people! Each step on my Turing
machine never gets better than adding a unit to some already
calculated number. So the steps are just counting, or else moving the
tape about. But I can 'prove' any valid relation between numbers that
anyone else can 'prove'. Are my 'steps' rich enough to produce
consciousness and a physical world? After all, each 'step' is a proof
of a valid arithmetical result.
I think it's enough to create a consciousness IN a physical world
created for it to be conscious OF. But that's not exactly a reversal of
physics and psychology - they are on a par and both derivative from
computation.
However, I'm not clear on what the Goedel number of an algorithm that
doesn't halt would be, since it either doesn't have a result or has
an infinite sequence of results, depending on how you look at "results".
Exactly. The nature of a 'computation' in Platonia needs some
clarification. Goedel numbers are, after all, just as timeless and
static as any other number in Platonia.
And even if it is clear that doesn't mean it exists. The question is
whether a value of a variable that satisfies a predicate implies that
the value "exists". I think there are different kinds of existence and
the existence of numbers doesn't entail the existence of electrons.
I think we agreed on that a while ago.
Bruce
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