Ah, ok. Well, as your friend checked my proof, what I was/am working
on is an effective theory.
Bruno Marchal wrote:
On 02 Jan 2011, at 18:01, Brian Tenneson wrote:
Bruno Marchal wrote:
On 02 Jan 2011, at 11:31, silky wrote:
On Sun, Jan 2, 2011 at 8:31 PM, Brian
Tenneson <[email protected]> wrote:
In the case of a TOE, the model IS
reality.
Okay, I won't reply further, this has become irrelevant noise.
I suspect the traditional confusion between "model" in the sense of
physicists (where model = a theory, like a toy model), and model in the
sense of the logician, where model = the reality studied (like a woman
serving as model for a painter, or the mathematical structure (N, +, x)
for PA or RA).
Logicians and physicists use the word "model" in the complete opposite
sense, and this leads often to complete deaf dialog.
This makes even more problem with computationalism, where an observer
accept that some "theories/brains/finite-describable-objects" fits the
reality. When you say "yes" to the doctor, it is because you believe
that the artificial brain does capture (locally, with respect to your
current environment) the real thing (your conscious you). In that case
*you* are a fixed point where a model-theory correspond to a
model-reality, a bit like in Brouwer fixed point theorem, where a map
of a territory is shown to have a point on it matching the real point
in the territory, provided the map is not ripped in two disconnected
parts, but only transformed continuously. The point is that in some
contexts some overlap can exist between a theory and its (or one of
its) model, between description and realities, like with the painting
of a painting of a pipe (cf Magritte).
Things get confusing also if, like Brian, (but also logicians in some
circumstances) people makes a model (a "reality") into a (non
effective) theory. This can be justified for some technical reason,
when working on super non effective structure, but is really out of
topic, imo.
Bruno
What makes a theory effective?
That its proofs are checkable. That its set of theorems is
recursively enumerable.
I'm going to be less precise
given that my audience has changed in a way I do not know.
We argue in an interdisciplinary field.
Given a couple of
assumptions, which are essentially that (1) reality is independent of
humans (which will imply that a model (in the logical sense) can be a
TOE as defined in this thread) and
I don't see this. I prefer to use "theory" for something
finitely presentable (finite or recursively enumerable).
(2) a model every model can
be embedded within endows that model with a universality that makes it
a candidate for being reality. This is then a brief description of
reality, though I couldn't hope to give all the details about reality.
In that case the model (N, +, x) is the "TOE" that your are
searching. It is rather a ROE (realm of everything), and the embedding
relation is simulation (emulation or partial emulation). My point is
that we have no choice in the matter once we assume that brains work
like a digital machine at some level of description.
I am also working on the
hypothesis that a TOE can be given in an finite/infinite presentation
such as found in ZF with axioms and axiom schemata.
Question: what is the theory with no assumptions? I know that in
logic, the consequent closure of the empty set of statements is the set
of tautologies, which is not really what I'd call an effective theory.
The set of (classical tautologies) is effective. But the empty
theory as all models, the structure of which depending of the
meta-theory. It is the trivial theory satisfied by all structures.
But what about if we remove
all assumptions? Sounds like chaos to me. This is connected to all
this as I can explain.
In fact, I can prove (1) on the grounds that there is no largest
number. It took me a while to find this argument.
"1)" follows from comp, which assumes arithmetical realism (used
in "there is no largest number").
Bruno
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