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
http://www.mail-archive.com/everything-list.domain.name.hidden/msg16244.html 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 04 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 above...so, 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 00 and moves 02 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 01 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! ;)
Jesse
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Received on Thu Apr 30 2009 - 12:29:45 PDT