Lecture 18: Turing Reductionism, Token Physicalism: The Computational System Assumption (Turvey, 2019, Lectures on Perception)

We have encountered the idea in Lectures 6 and 16 that perception might be formalised as a form of computation, and that this might be the step that enables the Cartesian programme to succeed. This Lecture spends time on work on the underpinnings of this hypothesis, namely work on what kind of thing mathematics and computation are and can do. 

The potential value of mathematics to questions about perception is the hypothesis that mathematics can be fully derived from logic; given some necessary principles, mathematics will be a set of necessary truths. This is the kind of thing you need to ground cognition and perception, to avoid loans of intelligence. So is mathematics this kind of system?

The brief answer is 'no'. Russell and Whitehead proposed a system in which this was true, but it hinged on the answer to three questions. Is mathematics complete (can every statement be proved or disproved)? Is mathematics consistent (if you follow the rules do you only get valid statements)? Is mathematics decidable (is there a definite method that will tell you whether a statement is true or false)?

Turvey reviews in some detail how Gödel addressed the first two questions and finds that any consistent system shows incompleteness. This is a hard knock on the ability of mathematics to be an adequate basis of perception, although people vary in how much they worry about this one.  

The one that definitely matters for perception is decidability. Turvey discusses this in the context of scene analysis (think of theories such as Marr's approach) that identifies basic scene elements and then follows steps to recover what the scene is. The question is whether this kind of problem is decidable, can it be done? 

The first piece of the puzzle is Turing, and the Turing machine. This is an input/output device with a read/write mechanism and a finite set of symbols it can manipulate. This is the basis of any computational approach, effectively: symbol manipulation. Scene analysis has been proposed to use symbols such as geons or Marr's system. In modern terms, the decidability becomes a question of whether a given problem is P or NP complete; can it be computed in polynomial time, or does it require nondeterministic polynomial time. 

Turvey then reviews work that demonstrates a problem such as scene analysis is NP complete. This is a problem! It means that perception cannot be a process of computational inference, because it takes too long. Perception must be direct, and based in lawful information, which we shall review soon.