[the answer ether] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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the answer ether
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Posted by: Philip Ronald Dutton
It is rather trivial to understand that we can expect the requirement of running rule 30 out to the millionth step if we want to know the configuration at that point. There are yet any shortcuts. No one "knows" what the answer is unless they run it.
Interestingly, we know there is an answer at that step. We know there is an answer at every step of the CA evolution.
If we know there is an answer then it is as if it has already been computed.
As with all possible computations, the answer exists out there in that ether of answers.
Out there in that answer ether, complexity means nothing. In a place where all answers exist, there is only a sort of simplicity.
Is complexity basically what happens when we cannot get close to the answer? Is complexity simply some sort of approximation? When an answer yields not to exact computation then our best is approximation and also complexity? Is complexity a computation in progress?
Posted by: Jason Cawley
"If we know there is an answer then it is as if it has already been computed."
I rather think not. Something is "as if" something else if there is no practical difference between the first and the second side of that "as if". But there is clearly a practical difference between an uncalculated and a calculated rule 30 center cell - the former I would act as if it were equiprobable white or black, the latter I would act as it it were certainly white (or, as the case happens to be, black).
So there is clearly a practical difference and the "as if" can only make sense in some restricted domain of application. Something is different about one side and the other side. If something else hasn't changed, fine. But that is not normally what one means by such an "as if". The question then becomes, what is this other restricted domain in which the supposedly relevant aspect hasn't changed?
Whether there is an answer, one might say. Equivocation on the word "is" - can be taken in the present sense or in the timeless sense. In the former sense it means, an answer is potentially findable and has been found; in the latter it means, an answer is potentially findable. To equate the two, then, is to adopt the perspective - everything that can be found has been found. Now, this is not actually true for anybody that anyone can point to. But we can consider looking at things that way if it illuminates some practical issue.
What it amounts to is the old idea that when a problem has been reduced to a matter of logic, it has been solved. And NKS forcefully maintains that this simply is not the case. It was an illusion created by the simplicity of the logic we typically used in our models. Enough logic, and involved enough logic, is hard. The epistemological problem is not reducable to getting outside our heads, it can arise inside them. This is not a new observation in NKS, but it makes it forcefully and in somewhat wider contexts than previous thinkers have.
As an example of such a predecessor, consider Peirce.
"If you sit down to solve ten ordinary linear equations between ten unknown quantities, you will receive materials for a commentary upon the infallibility of mathematical processes. For you will almost infallibly get a wrong solution. I take it as a matter of course that you are not an expert professional computer. He will proceed according to a method which will correct his errors if he makes any."
Notice that "a computer" in Peirce's day means a human being whose job is to compute things. Peirce's example of a hard math problem differs somewhat from ours. Mathematica will handle his example in a split second. Whereas it might take rather a long time to compute a million steps of rule 30. His basic point, however, is that we make mistakes in formal matters just as in other matters of reasoning, and that we proceed by trial and error and self correcting methods there, too. He continued
"This calls to mind one of the most wonderful features of reasoning, and one of the most important philosophemes in the doctrine of science, of which however you will search in vain for any mention in any book I can think of, namely, that reasoning tends to correct itself, and the more so the more wisely its plan is laid. Nay, it not only corrects its conclusions, it even corrects its premises."
So the first point is a pragmatic one, that reducing a problem to logic may or may not put us on the road to a robust and self correcting manner of finding a satisfying answer (depending on whether we can do the logic), but it cannot be said, of itself, to provide such an answer.
But the perspective from which the question is posed is not a pragmatic one, but more like a Platonic one. We are invited to distinguish between the existence of an answer and our knowledge of it - or rather, our lack thereof. Now, this is fair enough as far as it goes. One clear meaning it has, is that we can be wrong about such things. The indeterminacy involved has a certain character that marks it as a limit of our knowledge of something independent of us and real.
But trying to take it farther than this, to imagine a "place" "where" answers "are", is a metaphor. Chaitin's omega is in a way an attempt to make that metaphor tractable as an object of mathematical reasoning. An analog of real number oracle constructions but restricted to computables.
Now, one can have a meaning for a term like complexity that restricts to things "beyond omega" as it were - mathematical facts that are just so, mathematical atoms as it were, unrelated to any prior process of calculation and so lacking the minimal discernability of "computable". And one can then observe that the cardinality of the reals allows any number of these. But Chaitin has pointed out, you can't even name most of the reals, so being told you can't compute them all is hardly news.
I don't think this is the natural meaning of the term "complexity", but at bottom that is up to us and conventional. At any rate, it is not what is behind the phenomenon of computational irreducibility, which already arises in formal systems (fully specified and so lacking possible ambiguity about the role of "real continua") that are discrete, countable, and deterministic.
The last bits suggest a mapping from simple to answer, that I agree has been traditional or is an expectation we tend to carry around. But I don't think it actually follows. That is, Dutton suggests that anything still complex is not yet an answer, that answer and reduction to simplicity are coordinate terms. I do not see the necessity for this.
Some questions simply have complicated answers. The answers are no less answers for being complicated. A question with a complicated answer, receiving that answer, is not still on the way to being, but not yet, answered. It is just a question the answer to which happens to be complicated.
There is a connection among complexity, computation, and answer. It is that a question about an irreducible system requires some amount of actual computation to arrive at an answer. That having an answer and having performed a computation are correlates, in such cases.
The reason I would not go beyond this, is that every stage of an ongoing computation can be seen as an answer to some question, just a different one. It does not make sense to say the millioneth step of rule 30 is only an approximation to the center cell value at step a million and ten. If the answer one wants is the value of that center cell on step a million and ten, then yes calculating step a million - or being handed it by somebody who has calculated it already - may be part of the process of determining the answer to this particular question.
Since that process need not be one that gets near and never strays far from the answer, I'd hestitate to call it "approximation" - rather than part of a process or method aimed at an answer. But I'd also refuse to call the complexity at step a million "the computation of the center cell at step a million and ten, in progress".
The complexity is there in the formal behavior of the rule, and not in the question I choose to ask about that behavior. There is a relativity to a question involved, in the idea "in process toward an answer". Which I do not find in the complexity itself. Which is basically indifferent to my choice of question (within limits of course - "is it an elementary CA? Does it take 2 possible values for each cell?" etc are questions that have uncomplex answers, obviously).
I close with my perspective on some philosophic positions and their ability to digest the main results of NKS, at least those about computational irreducibility (PCE is another level).
Any perspective that allows some placement for the idea, "things which we don't know, but that are (or will be) so", and allows that status to purely formal matters, fits with NKS irreducibility. Philosophic positions that do not have a "slot" for that kind of thing (maybe some "subjectivist" positions that make it hard to just not know an objective truth, or overconfident ones wedded to the idea that the formal equals the known), may have problems with NKS style irreducibility. But it seems to me, many of the better positions have no problem.
Pragmatism allows for this, as does the critical position. (Think of Peirce and Popper as examples for those, e.g.). Operationalism is close enough to pragmatism in this respect not to have serious problems with it. I also see no problem for either realism or idealism (of, in Kantian terms, dogmatic varieties), provided they do not try to maintain excessively strong epistemic propositions like "everything logical is knowable". Some positions have maintained that more is knowable than actually is. But that sort of thing has been in formal trouble for a century now, and by now most of them know it.
Even skepticism can handle most of it, though I think the difference between reducible and irreducible formal systems shows that some traditional arguments for skepticism are poorly grounded (since some of them try to map "formal" to "knowable" and "empirical" to "unknowable"). I have other reasons for disliking skepticism, but they aren't really on topic here.
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