IN CLARIFICATION OF THE CTMU AND ITS APPLICATIONS IN NOESIS
by Chris Langan
The following enumerated remarks are in response to those of Chris Cole and George Dicks in Noesis 57 (Sections 1 and II resp.).
SECTION 1
1) The CTMU is a theory of reality as mind, not "the universe as a computer". The latter analogy was given advisedly as a conceptual aid and as a means by which to apply mechanistic and computational formalisms to the model. It does not encapsulate the whole theory.
2) The components of the CTMU are not all new. The idea as a whole is new. Technically, there are no new ideas, just new combinations of old components. The CTMU is more deeply rooted in analytical philosophy and the foundations of mathematics than in practical computer science, which nonetheless plays a large part in it.
3) I know of the late physicist Richard Feynman. But who is Ed Fredkin, and where can his ideas be sampled? If Chris means to give him credit for the CTMU, where are the applications? Given the current mania for computer simulation of natural processes, it would be incredible if no writers or scientists had latched onto the general notion of a computational universe. However, we don't give those who make mathematical conjectures automatic credit for proving (hem as theorems, we don't give Michelson, Morley, or Lorentz credit for Special Relativity, and we don't give Jules Verne credit for the deep-sea submarine. Credit is reserved for those who display greater understanding of the ideas in question--for example, by application to problems like Newcomb's paradox, quantum wave-function collapse, and logical and empirical induction. Those are the rules; I'm only following them.
4) Anyone who thinks I'm an "incautious thinker" has a responsibility to point out some specific flaw in my reasoning. No matter how "general" my reasoning is, it is either right or wrong, and if it is wrong, then it contains an identifiable flaw. This group has had a standing invitation to locate such a flaw, but has not done so. For example, if my reasoning is too general to solve the (very general, but very important) problems I claim to have solved with it, such a fact is demonstrable; my claim could be falsified.
5) The CTMU is a theory of the codeterminate relationship between reality and the intellect. The empirical evidence offered so far includes the many experiments now confirming EPR-Bell quantum non-locality and Heisenberg uncertainty. These are hard, physical facts which were fundamentally unexplained prior to the CTMU. The logical evidence presented in favor of the CTMU has been nothing short of overwhelming. Resolutions of paradoxes are evidence for logical theories as surely as predictions and explanations of anomalies are evidence for physical theories. Because physical theories necessarily have logical structures, logic has priority. A single unresolved paradox of the general form x = -x invalidates the entire logical system containing it, and thus every semantical theory formulated within that system; it renders determinations of truth and falsity impossible for every statement the system and theory contain. In principle, I can thus claim every confirming instance of every logically consistent scientific theory as evidence of the CTMU, making it the best-confirmed theory in history. In point of fact. this is precisely what I've already done.
6) Even if CTMU were a definition rather than a theory, definitions are necessary components of theories. There is an inclusory relation, not a total distinction, between the two. In fact, the CTMU can be characterized as a THEORY of how the mind DEFINES and IS DEFINED by the universe. If you must, re-read Noesis 46 and 47.
7) That some details of CTMU were omitted in my presentation was quite deliberate. I used only what I had to use to solve the problems I attacked. The size and scheduling of Noesis imposed severe restrictions on me; even so, I was simultaneously accused of being overly general and excessively technical. This is a paradox that even I find hard to resolve. 8) I have a razor-fine dissection of the original marble problem (#26, Trial Test B, Insight 10) in the form of a dialogue so clear and amusing that even an average high schooler could understand it. However, the current editor-despite what I'm sure are his many virtues-shows an apparent preference for scatological humor over this kind of material, and threatens to "edit like crazy" anything not completely to his liking. That naturally precludes future extensive discussion of the marble problem, towards which he professes antipathy. This is unfortunate, since the dialogue is illuminating on Bayesian paradox in general. Had anyone expressed any real encouragement, I'd have printed and mailed it myself. But supporters are apparently few. So for now, I'll just reiterate my former assertion: there is one and only one way to solve the problem as presented. In a nutshell, the reasons are:
a) Observation is all you have to go on, and the l<=(r)<=10 marbles you observe display just one color. The number of possible colors is critical information, so you can't just guess it or make it up. It has to come from observation. The way you refer to the electromagnetic distinctions on which color is defined is also important, but only as an upper limit on what may be observed. I.e., if you can discern only n colors, you can observe no more than n. Letting your chromatic sensitivity or vocabulary play a greater pan than this implies that you are controlling the sample by observing it, a ridiculous assertion. You are not a "demon"... or are you?
b) Since you can't make up the number of colors, you're entitled to just two, the one you've observed and a generalized complement to symmetrize (remove imbalances in) the initial information necessary for Bayes' rule to work. This is the direct negation of the observed color: white -> nonwhite. Failure to add this complement imbalances the calculation and makes whiteness "axiomatic" for marbles on the basis of just one observation ... an absurdity, given that the formulation requests a "probability" and thus implies an exhaustive and disjunctive set of at least two nonzero colors. The idea of "color" does not exist for monochromatic observers.
c) You must not only symmetrize color (i.e., white-nonwhite), but also the possible distributions in terms of color alone. Otherwise you're imparting phony confirmation to information you can't legitimately claim ... the information embodied in any asymmetry among allowed colors or initial distributions. Thus, the white nonwhite ratio n/(10 - n) is equally probable for all n from 1 to 10.
Let me explain. The method by which marbles were colored and put into the box can be known to you only through the actual distribution. Unequal probabilities among possible (initial) distributions would imply that you have a priori information on this method and so on the actual distribution itself. But the problem formulation gives you no such information. If such information emerges over repealed experiments, the problem changes and Bayesian inference is relativized to new data. But until then, you're more limited.
These conditions--that only two colors may be used, and that all distributions in these two colors are initially assumed to be equiprobable--plug directly into Bayes's rule and yield a probability of .67 that all ten marbles in the box are white, given that sampling with replacement has produced ten white marbles in a row. This is a very specific determination which should be easy for all to follow. There is nothing "ambiguous" about it. But just to make sure, let's run the computation one more time.
BAYES'S RULE: p(a|b) = [p(b|ai)p(ai]/[p(b|ai)p(ai + ... + p(b|an)p(an]
where p(a|b) means "the probability of a, given b", and the ai are exhaustive independent alternatives defining a random variable A.
Let A = {a0 or ... or a10} be the actual distribution of marbles in the box, and let an be the possible distribution with n white and (10 - n) nonwhite marbles. The symmetry condition on possible distributions means that p(a1) = ... = p(a10) = .1 (since at least one white marble has been observed, p(a0) = 0). Let b represent the evidential data, a sample-with-replacement of 10 while marbles in a row.
Then
The original marble problem has now been solved more explicitly than any other problem on any Ronald Hoeflin test. So, unlike most of them, it can now be cautiously used to measure intelligence.
SECTION II
8) Regarding George Dicks's criticisms, I'm forced to reiterate a crucial distinction that makes all the difference in the world. He has solved Newcomb's PROBLEM; I resolved Newcomb's PARADOX. George is one of many who have solved the problem in essentially the same way. I didn't solve it at all except as a "side effect" of resolving the paradox between (he arguments that George and Chris employed. This can only be done in one way, as defined with great precision in Noesis 44. The Resolution of Newcomb's Paradox represents the first and only time that this has been done in print. While I've already congratulated George on writing a fine paper with some original features, his entire "rebuttal" is virtually irrelevant in view of the foregoing distinction. But I'm going to go over a couple of things I'm afraid he and Chris might have missed in Noesis 44-49, just to make sure that nobody backs himself into a corner he can't pitch his tent in.
9) Computative regression is the essence of Newcomb's paradox. The regression is defined by the conventional formulation which George changed to get his own version of the problem. Trying to avoid a computative regression in resolving Newcomb's paradox is like trying to avoid the idea of viruses while searching for an AIDS vaccine. It can't be done with any hope of success.
10) The arguments used to solve Newcomb's problem fall into two general classes, mathematical expectation and temporal directionality. The former class is probabilistic and statistical; the latter is intuitive. Both classes involve subjective utility or self-interest, and both claim an empirical basis. Chris Cole argues from physical intuition grounded in his experience of reality. George Dicks argues from numerical data. But it's easy to see that Bayes's rule, the proper use of which I outlined above, can easily be adopted to Newcomb's problem in lieu of the method he employed. The way I've used it to avoid "false axioms" in the marble problem applies as well to Newcomb's problem, and yields a numerical determination comparable to his. The point is, CTMU principles cover all numerical methods in this class while depending on none. The CTMU is determinant over all of them in kind.
11) "Empiricism works" is a statement with which I fully agree. However, it does not work by itself. Empirical induction is to theorization what observation is to recognition; one is of no use without the other. The CTMU is primarily a theory of induction. To claim that it was used to resolve Newcomb's paradox at the expense of empiricism is a non sequitur. The CTMD resolves the paradox because, unlike certainty theory and standard numerical statistics, it accounts for the means by which empirical data--like the perfect success of Newcomb's Demon and quantum nonlocality--can violate intuitive (and observationally-confirmed) assumptions about lime and space. These means involve the relationship of mind to reality, not just mind or reality alone.
"Anything less is just theology" is a statement which I find impenetrable. Your past experiences and disappointments with what has traditionally passed for theology have nothing whatever to do with the CTMU. which realizes Kant's dream of a "theology" derivable from the union of mind and reality. If you mean to challenge that, you'd better gear up for a whole new game. CTMU "theology" owes nothing to the history of religion, and whatever arguments you use against it had better not either. That includes any idea you might have that the theological implications of the CTMU arc "unverifiable" or mere "wishful thinking".
12) The Resolution of Newcomb's Paradox allows for any arrangement of chooser and predictor. These designations are meaningful only relative to each other. That particular arrangement in which the predictor dominates the chooser happens to be set by the canonical formulation. In Noesis 45, the theory of metagames was incorporated into the CTMU. This theory, by merging the best interests of the players, allows them to cooperate in nondominative situations where "choosers" and "predictors" are indistinguishable. This idea was further developed in Noesis 48. The bottom line is, if neither player is "predictive" relative to the other, then there exists a jointly accessible algorithm which lets them cooperate to achieve mutual benefits (the theory of metagames). This prevents what George calls a "chaotic exchange of move and counter-move," provided both players know of it. But if one is dominant, as the formulation stipulates, then the situation is identical to that of The Resolution. These possibilities are exhaustive and mutually exclusive, so no further explanation is necessary. George's attention to specific arguments is commendable, but unnecessary resolve the original version of Newcomb's paradox. It can't be used to justify a claim of superiority or priority, particularly when it completely ignores the theory of metagames.
If, despite the ubiquity of computative regressions in every aspect of our mental lives, you doubt their possibility, you still can't use your doubt as an objection to the CTMU or its resolution of Newcomb's paradox. If you deny it, then you also deny the validity of mathematical achievements like nonstandard analysis, which itself uses an "incredible" model-a nonstandard universe containing "infinitesimal" (nonzero but subfinite) quantities-to resolve inconsistencies in the infinitesimal calculus. The CTMU is on even firmer ground, and you will ultimately find it harder to avoid than death and taxation. It is a fait accompli.
My remarks are numbered. If you disagree with anything I've said, please refer to it by number and line. The exchange has thus far been circular, and I can't waste time on circular objections. You supposedly have I.Q.'s in the vicinity of 180 and vocabularies to match. If you find these comments "stultifying", "unreadable", or anything but crystal clear, don't blame me. I'm just the one who tentatively bought your PR.
This is not to say that I don't hold your intellects in high regard despite your lack of encouragement. I only wish thai some of you could forget about defending your special views long enough to recognize that you have witnessed a remarkable development. I expect only what is reasonable and seek nothing that is not mine by right. One supposed benefit if a high IQ is knowing when to cut your losses and reinvest in new concepts, and you've all had an adequate prospectus since early 1990.
Chris Langan
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