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January 09, 2005
A Mathematician Plays the Market....and Loses.
After losing his shirt on WorldCom stock, mathematics Professor John Allen Paulos took a sceptical tour through the multitude of pseudo-mathematical, statistical, and outright mystical theories of the financial markets, concluding from their average ineffectiveness and the constant hidden role of faith-based psychology in their advocates' behaviour, that the movements of the market may after all be beyond our 'complexity horizon'. In his final chapter he makes passing reference to Chaitin.
Effective scientific or mathematical modelling requires the construction of a set of instructions that accurately predict the appropriate outcomes whilst being simpler, more tractable, or quicker to compute (these three being arguably more or less identical) than the system in the entirety of its workings.
The relationship between system and model is what Chaitin calls program-size complexity, the bit-size of an algorithm as compared to that which it is designed to compute. Paulos' conclusions suggest that the only feasible way to 'model' the market would be to create an atom-perfect simulacra of it; like Chaitin's Ω, it cannot be explained (= predicted) in any terms simpler than the sheer enumeration of all its interactions in their entirety. (We would concur here with Ray B's hypothesis that "Ω...indexes the uncomputability of the real...the objective randomness of Capital's non-computable dysfunctions...").
Explanation and its cognate, prediction, rely on the principle of sufficient reason, which bespeaks a faith in the greater simplicity of the essential as compared to the real - a faith which founders in situations, paradigmatically weather-systems, where what is essential is an irreducible mass of real interactions.
We can suggest a direct connection here to Kittler's argument in "There is no Software". Phenomena such as weather systems, argues Kittler, cannot be modelled by discrete computers anything like fast enough to predict their movement - the massive density of interactions makes this technically impossible (this is a strong claim, of course, stronger than merely saying we do not yet possess the computational power. It needs further examination.) The only way, suggests Kittler, that our digital computers could begin to approach the task would be to reverse the fundamental microelectronic engineering trend of the last century - towards discrete components designed so that their operations can be isolated from crosstalk and noise - and build computers that actually take advantage of this excess connectivity.
The Hilbertian paradigm of the formal system, having already been irretrievably damaged by Gödel, transformed into an engineering rather than a syntactic problem by Turing, now displays in a more crushing sense its constitutive insufficiency. Given Kittler's strong hypothesis of discrete-uncomputability, it would not matter at all how much we tinkered with ideas like connectionism, distributed computing, or any imaginable elaboration of the Turing paradigm.
If you think about it, the obvious corollary to the universe being conceived as a giant information-processing device is that digital computers are bound to be a relatively computationally-poor subcomponent of that device. But on the other hand the pragmatic problem with Kittler's "noise engine" is one of throwing out the computational baby with the discrete bathwater; for how could such an aformal system, open to contingent interference, be programmable in any useful way?
The computational corollorary to Chaitin's vision of a "quasi-empirical" mathematics may be that the quest for ever-growing computational power necessitates a fraught trade-off between the instrumental use of bounded, discrete computational systems and their apparently necessary escape into unbounded, nondiscrete materiality and contingent utilisation. The boundary between a system's internal model of the world, and that world's concrete impingement on the system's physical instantiation has never seemed so close to collapse.
We could venture a speculation that is is on this collapsing boundary that synthetic intelligence is likely to be forged, as an as-yet obscure function of the interplay between matter-as-computational substrate and matter-as-uncomputable-turbulence. We may need to accept that beyond certain unrealistically simplistic cases, we cannot usefully define formal systems without reference to their aformal environments. AI may need to refocus from bounded discrete information spaces to messy margins of interpenetration between the real and the virtual...and stop hoping to create intelligence in a box.
The strong discrete-uncomputability hypothesis suggests that the incapacity of formal reason (understood as discrete computation) is a fundamental consequence of its conditions of possibility, the virtual separation from matter and the machinery of representation and modelling. The labyrinthine convolutions of matter necessary to instantiate apparently substrate-indifferent reasoning (Kittler's "software") are the very conditions of such reasoning's inherent limitation.
Godel would not, of course, have raised an eyebrow...
Posted by undercurrent at January 9, 2005 04:55 PM