Does Incomputable Mean Not Engineerable?

Self-referential systems have some remarkable properties. The processes of life and mind are not only self-referential, but self-reference turns out to be a crucial property of both. However, they are difficult to understand. From a given starting point, both endogenous systems (self-referential natural systems) and impredicative systems (self-referential formal systems) have infinitely many logically consistent consequences. Both are incomputable; neither halts after a finite number of steps. Therefore, neither can produce an exact prediction of the behavior of the other in finitely many steps. Despite the fact that all engineering decisions are based on incomplete information, this inherent inability of an impredicative model to produce exact predictions of an endogenous system is troubling to some engineers. Nevertheless, self-reference leads to a more general, but no less rational, form of modeling than that provided by traditional reductionism. Although the mathematics of self-reference is unfamiliar to engineers, its power is dramatic. For example, it resolves the apparent paradox of how a brain/mind possessing freewill can operate in a deterministic Universe. INTRODUCTION WHY SHOULD WE CARE? Engineers seek to develop artificial systems that exhibit behaviors similar to the cognitive processes observed in biological brains. This is a worthy goal, and reaching it will be one of the crowning achievements of 21 Century technology. By no stretch of the imagination has the goal been reached yet. Indeed, the astoundingly simple nervous system of the nematode (all 300 neurons) is completely mapped out. Nevertheless, despite years of research, nobody has been able to devise an algorithm that behaves anything like a nematode (Chomsky 1993). Is it really such a stretch to suppose that the reason that the goal has not been met is that cognitive behavior is beyond the scope of the traditional reductionistic methods used by engineers? The answer to this question is yes, and this leads to another question. What other rational methods might we invoke? IMPREDICATIVE MATHEMATICAL OBJECTS What is really being said by the remarkably tricky proposition, φ(x) = φ(x)? It appears to say nothing and everything simultaneously. On the one hand, it says practically nothing. As a definition of φ(x), it does not inform us how φ(x) differs from any other entity, such as ψ(x). This one fact alone seems to illustrate the futility of