Mar

12

I can't recommend this book highly enough. It is quite possibly my favorite of all time. Today I read Chapter 13, "Metafont, Metamathematics, and Metaphysics: Comments on Donald Knuth's Article 'The Concept of a Meta-Font,'" subtitled "The Mathematization of Categories and Metamathematics."

The chapter makes a case for the absurdity of the notion that creative processes can be mechanized via parameterization. He invokes Godel's Incompleteness Theorem, which states that no non-trivial formal system can demonstrate the truth of all possible assertions, because a set of those assertions is self-referential with respect to the system in question. The canonical example is, "System X is not powerful enough to demonstrate the truth of Sentence S." If the statement is true, it is false, and if it is false, it is true. This contradiction lies at the boundary of all formal systems.

This is the genesis of ever-changing cycles. The market is a system of systems, which continually generates new states, without ever exhausting the set of all possible states. This "essential incompleteness" is the core property of the family of sets called "productive" in mathematics.

(It is tempting to think that all possible states could be examined via a generative process, but there are two problems with this: in a world of zero-bounded asset prices, there is an infinite set of possibilities. Further, even if it were possible to generate a complete set, it would be impractical to attempt to analyze them all, and then the question would become one of screening which states were "relevant." But Godel's Theorem, stated another way, says that you cannot satisfy both of these goals simultaneously. Further, relevance would almost certainly have to be defined a posteriori.)

But this is not to say that pattern recognition and the search for superior risk-adjusted profits is an impossible one. It merely means that minimizing your degrees of freedom (parameter "knobs") will allow you to find relevance more reliably. And those degrees of freedom should be chosen to find the most matches, so that we are nearing the (unreachable) goal of completeness.

But how to choose our parameters? Statistics are of great use here. Analyzing predictive significance of past states is sufficient if the generative process can be (a priori) known, assumed, or believed. An obvious example would be equity markets. If you believe that the circumstances that led to a 1,000,000% price increase in the past century will continue to exist, then your job is to find past states that have predicted higher prices. This is true for any process or market, and is the key underlying factor in trading and investing success: you must have a model of how your market behaves before you can try to take advantage of said behavior via acting on state information. It is worth noting here that the derivatives expert and cohorts' central argument is that the nature of the generative process can not be known, and that therefore statistical moments and properties are just crutches for those who are fooled.

Finding these past states is of course the real pursuit. Hofstadter argues that it is essential to understand that creative processes do not just follow the "letter of the law," but also the "spirit of the law." This suggests a wide world of potential parameters and approaches that are outside the mainstream.

Hofstadter suggests that in generating states it is more effective to consider conceptual roles that must be satisfied, rather than specific, concrete conditions. These roles are quite abstract, and can be partially or wholly fulfilled. Further, these roles are modular and can exist in more than one simultaneous state (or process). The example here is that of serifs (the "feet" on the bottom of fonts), where larger serifs on the letter "a" most likely mean larger serifs on the letters in the font.

It is easy to draw an analogy to volatility in asset markets. If equities suddenly experience a jump in intraday activity, it is at least worth examining if that is predictive of a similar jump in other markets, as it would affect the possible states of those markets.

I could go on (I probably already have), but I think the "meat" of the question I'd like to ask is, "What behaviors can be examined for predictive significance that are not normally parameterized?" It's there that edge lies.


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