Bruno Marchal wrote:

On 29 Apr 2009, at 23:30, Jesse Mazer wrote:
But I'm not convinced that the basic Olympia machine he describes doesn't 
already have a complex causal structure--the causal structure would be in the 
way different troughs influence each other via the pipe system he describes, 
not in the motion of the armature. 
>But Maudlin succeed in showing that in its particular running history,  *that* 
>causal structure is physically inert. Or it has mysterious influence not 
>related to the computation. 

Maudlin only showed that *if* you define "causal structure" in terms of 
counterfactuals, then the machinery that ensures the proper counterfactuals 
might be physically inert. But if you reread my post at you 
can see that I was trying to come up with a definition of the "causal 
structure" of a set of events that did *not* depend on counterfactuals...look 
at these two paragraphs from that post, particular the first sentence of the 
first paragraph and the last sentence of the second paragraph:

>It seems to me that there might be ways of defining "causal structure" which 
>don't depend on counterfactuals, though. One idea I had is that for any system 
>which changes state in a lawlike way over time, all facts about events in the 
>system's history can be represented as a collection of propositions, and then 
>causal structure might be understood in terms of logical relations between 
>propositions, given knowledge of the laws governing the system. As an example, 
>if the system was a cellular automaton, one might have a collection of 
>propositions like "cell 156 is colored black at time-step 36", and if you know 
>the rules for how the cells are updated on each time-step, then knowing some 
>subsets of propositions would allow you to deduce others (for example, if you 
>have a set of propositions that tell you the states of all the cells 
>surrounding cell 71 at time-step 106, in most cellular automata that would 
>allow you to figure out the state of cell 71 at the subsequent time-step 107). 
>If the laws of physics in our universe are deterministic than you should in 
>principle be able to represent all facts about the state of the universe at 
>all times as a giant (probably infinite) set of propositions as well, and 
>given knowledge of the laws, knowing certain subsets of these propositions 
>would allow you to deduce others.

>"Causal structure" could then be defined in terms of what logical relations 
>hold between the propositions, given knowledge of the laws governing the 
>system. Perhaps in one system you might find a set of four propositions A, B, 
>C, D such that if you know the system's laws, you can see that A&B imply C, 
>and D implies A, but no other proposition or group of propositions in this set 
>of four are sufficient to deduce any of the others in this set. Then in 
>another system you might find a set of four propositions X, Y, Z and W such 
>that W&Z imply Y, and X implies W, but those are the only deductions you can 
>make from within this set. In this case you can say these two different sets 
>of four propositions represent instantiations of the same causal structure, 
>since if you map W to A, Z to B, Y to C, and D to X then you can see an 
>isomorphism in the logical relations. That's obviously a very simple causal 
>structure involving only 4 events, but one might define much more complex 
>causal structures and then check if there was any subset of events in a 
>system's history that matched that structure. And the propositions could be 
>restricted to ones concerning events that actually did occur in the system's 
>history, with no counterfactual propositions about what would have happened if 
>the system's initial state had been different.

For a Turing machine running a particular program the propositions might be 
things like "at time-step 35 the Turing machine's read/write head moved to 
memory cell #82" and "at time-step 35 the Turing machine had internal state S3" 
and "at time-step 35 memory cell #82 held the digit 1". I'm not sure whether 
the general rules for how the Turing machine's internal state changes from one 
step to the next should also be included among the propositions, my guess is 
you'd probably need to do so in order to ensure that different computations had 
different "causal structures" according to the type of definition, 
you might have a proposition expressing a rule like "if the Turing machine is 
in internal state S3 and its read/write head detects the digit 1, it changes 
the digit in that cell to a 0 and moves 2 cells to the left, also changing its 
internal state to S5." Then this set of four propositions would be sufficient 
to deduce some other propositions about the history of this computation, like 
"at time-step 36 the Turing machine's read/write head moved to memory cell #80" 
and "at time-step 36 the Turing machine had internal state S5."

So if we define causal structure in terms of relationships between propositions 
concerning the history of the Turing machine in this way, then look at 
propositions concerning the history of the Olympia machine described by Maudlin 
when it was emulating that Turing machine program, it's not clear to me whether 
it would be possible to map propositions about the original Turing machine to 
propositions about Olympia in such a way that you'd be able to show their 
causal structures were isomorphic (I think it is clear that such a mapping 
would be impossible in the case of your MGA 1 though, so if we identify OMs 
with causal structures this would suggest that the brain which functioned via 
random cosmic rays correcting errors would not have the same inner experience 
as the brain which was functioning correctly and did not require these cosmic 
rays). But either way, what is clear is that the presence or absence of inert 
machinery designed to guarantee the correct counterfactuals would not affect 
the answer, since we'd only be looking at propositions about events that 
actually occurred in the course of the Olympia machine's operation. If it 
turned out there was an isomorphism between these propositions and the 
propositions about the operation of the original Turing machine, then that 
would show Maudlin was too quick to dismiss the original Olympia machine (the 
one lacking the counterfactual machinery) as giving rise to phenomenal 
experience (even though the armature behaves in a monotonous way, the way the 
troughs influence each other via pipes might be enough to ensure that the 
causal structure associated with Olympia's operation does depend on what 
program is being emulated). If there wasn't such an isomorphism, then there 
still wouldn't be an isomorphism even with the counterfactual machinery added, 
so that could make it more clear why the Olympia machine was not really 
"instantiating" the same computation as the original Turing machine.

One interesting thing about defining causal structure this way is that we could 
talk about causal structures being contained in pure mathematical structures 
like the set of true propositions about arithmetic. A Platonist should believe 
that if you take the set of all well-formed formulas concerning numbers and 
arithmetical operations (as well as logical symbols like 'there exists' and 
'for all'), then there is a particular infinite set of WFFs which represents 
all true propositions about arithmetic, even if Godel showed that this infinite 
set cannot be generated by any finite set of initial propositions taken as 
axioms (and it also cannot be generated by a computable infinite set of axioms, 
I think). If you take any finite subset of true propositions (P1, P2, P3, ..., 
PN), then these propositions will be logically interrelated in some particular 
way--it might be that if you start out taking P2 and P3 as axioms you can 
deduce P5 from this but you can't deduce P4, for example. I imagine 
representing each proposition as a dot in a diagram, and then arrows would show 
which individual dots or collections of dots in this finite set can be used to 
deduce other dots in the same finite set. This diagram would define a unique 
"causal structure" for this set of propositions, and then if you have a set of 
propositions about something different from arithmetic, like the history of a 
particular Turing machine computation, you could see whether there was a subset 
with an isomorphic pattern of logical implications (and thus the same 'causal 
structure' according to my definition). And even within arithmetic you might 
have two different subsets of propositions (P1, P2, ..., PN) and (p1, p2, ..., 
pN) which could be mapped to one another in such a way that the implications 
within each set were isomorphic to the implications within the other, in which 
case they would be two different "instantiations" of the same causal structure 
within the Platonic set of all true propositions about arithmetic. 

Maybe you could even make a TOE based on the idea that all that really "exists" 
is this infinite set of propositions about arithmetic, and that this infinite 
set defines a unique measure on all finite causal structures, based on how easy 
it is to find multiple "instantiations" of each finite causal structure within 
the infinite set of true propositions. I don't suppose this has any resemblance 
to your approach? I suppose the answer is probably "no" since I'm suggesting 
some kind of absolute measure on all causal structures, and if you identify 
particular causal structures with OMs that would correspond to the ASSA, but 
you have said that your approach only uses the RSSA. Anyway I have no idea how 
you'd actually "count" the number of appearances of a given causal structure in 
the infinite set of propositions about arithmetic, so the idea of getting a 
measure on causal structures this way is very vague...but if there's one thing 
this list is good for it's vague speculations! ;)

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