20 Apr 2009

Argument from Noise, 4. Three Arguments, 4.1 Argument from Representation, in Schonbein, Cognition and the Power of Continuous Dynamical Systems

[Nick Bostrom & Anders Sandberg argue for digital computation instead of analog for simulating human cognition. They base their contention in part on the "argument from noise." The entries in this series summarize Schonbein's defense of that argument.]

Whit Schonbein

Cognition and the Power
of Continuous Dynamical Systems

4. Three Arguments against AANNs

Schonbein will give three arguments against « the empirical utility of analog systems for understanding information processing in physical systems. »

4.1 The Argument from Representation

Hadler writes that no biological neural network or digital system can handle 'weights' (numerical values?) of infinite precision:
certainly, no one has discovered any means whereby, in a finite span of time, a finite precision substrate can operationally represent values requiring an infinite sequence of numerals to represent. (qtd 62a)
From this Schonbein concludes:
Such weights ought to be rejected on the grounds that the physical medium that is to realize them electrochemical potentials, spiking frequencies, etc. provides only a finite number of potential representational states, and therefore cannot represent values that require an infinite series of numerical to encode. (62b)
Schonbein schematizes this argument:
1. No finite physical device has infinite representational capacity.
2. An infinite-precision value requires infinite representational capacity to represent that value.
3. Therefore, no physical device uses infinite-precision values in performing computations. (62c)

This argument is based on the assumption that such a value as say pi would require an infinite number of numerals; but no system has infinite representational capacity.

Schonbein explains that there is one sound interpretation for this argument. Consider that in digital computers, we assign signals a 1 or 0 value. High voltage is 1. Low is 0. Because we may represent all decimal numerals in binary, we may arrange the wire so to represent numerical values. So '000' would be 0. '001' would be 1. '010' would be 2, and so on. Clearly we will always have physical limitations as to how many wires we have. Or if we keep reusing the same wires, and represent the infinitely many values successively over time, we still will have temporal limitations.

One response might be that we assert without evidence that a physical device has infinite representational capacity.

Schonbein sees a more fundamental problem with this argument.

We must distinguish
a) using physical states to represent numbers (as in the case of digital numerical representation). Here we interpret a physical state as some numerical value. And,
b) the task of accurately modeling the behavior of physical systems using mathematical formalisms. In this case, we intend the variables in our model to represent physical magnitudes. (63bc)

So when we use physical states to represent numbers, we are modeling the physical process to suit our mathematics. However, when we are constructing models of physical processes, we want the mathematical model to represent the physical process.

[We have a device that takes-in water from the gutter pipe. Whatever amount of water it takes in, it gives-out the square of it. But because it comes from natural sources, the amount of water that comes in will vary continuous through infinitely many variables. But we can mathematically represent its computational function as f (x) = x^2. Now we want to have a meter that determines the exact amount that flows out. No matter what physical device we use, it will not be able to present us with infinitely many decimal places of precision.]

This argument is reinforced by a common understanding of representation. Something is a representation when it 'stands in' for what it represents "when that thing is absent" (63c) For example, we are looking at the computer screen while reaching for a glass of water. We dodge the flower vase, because in our mind there is a representation of it, even though we do not see it in our field of vision.
Similarly, it makes sense to conceive of the states of our homemade computer as standing in for the presence of numbers when performing some calculation, and to invoke the content of those representations in accounting for its behavior. (63d)
[Now consider if we could meter the water going in and out of the square-function device. The water is coming-in from the rain drainage. We would not say that the water 'stands-in' for its metered numerical value. Rather, we would say that it has such a value, which we might not be able to represent precisely. So] we would not consider a naturally-occurring physical-system's states as standing-in for the variables of its model. For, those numerical values are irrelevant for explaining the system's behavior. So for example, we might say that an automobile accelerates at some certain rate because its motor systems are producing so many Newtons of force. However, we would not say that the car accelerates at that rate because its engine is representing that much force.
a modeled physical system does not manifest its behavior because it tokens states that have values of the model as their representational contents. (63-64)
Hence, this argument has the representation-relation backwards. We construct computers that will perform mathematical operations and display the results according to a digital scheme whose limitations disallow absolute precision. However, when we model something like the brain's processing of continuously-varying signals, we do not need to structure our model so that it suits our digital numeral system. We merely need to accurately model its physical processes. Our brains do not have digital readout counters on our foreheads.
The homemade computer was designed to represent numbers according to our binary representational scheme, and realizing such a scheme places constraints on the organization of the physical system. In contrast, the possibility of a physical system realizing some process that requires a certain type of model to adequately capture does not depend on the features of the representational system of the model, because it is the model that must answer to the physical process. (64b)
Analog artificial neural networks (AANNs) use real-valued (continuously variable) weights (numerical values). That does not mean they are physically impossible because we cannot translate those values into digital numerals.

As a result, the argument from representation fails to discount their potential utility for the proponent of dynamical systems theory. (64c)

Schonbein, Whit. "Cognition and the Power of Continuous Dynamical Systems." Mind and Machines, Springer, (2005) 15: pp. 57-71.
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