CHAPTER XVII
Church, Turing, Tarski,
and Others
Formal and Informal SystemsWE HAVE COME to the point where we can develop one of the main theses
of this book: that every aspect of thinking can be viewed as a high-level
description of a system which, on a low level, is governed by simple, even
formal, rules. The "system", of course, is a brain-unless one is speaking of
thought processes flowing in another medium, such as a computer's cir-
cuits. The image is that of a formal system underlying an "informal
system"-a system which can, for instance, make puns, discover number
patterns, forget names, make awful blunders in chess, and so forth. This is
what one sees from the outside: its informal, overt, software level. By
contrast, it has a formal, hidden, hardware level (or "substrate") which is a
formidably complex mechanism that makes transitions from state to state
according to definite rules physically embodied in it, and according to the
input of signals which impinge on it.
A vision of the brain such as this has many philosophical and other
consequences, needless to say. I shall try to spell some of them out in this
Chapter. Among other things, this vision seems to imply that, at bottom,
the brain is some sort of a "mathematical" object. Actually, that is at best a
very awkward way to look at the brain. The reason is that, even if a brain is,
in a technical and abstract sense, some sort of formal system, it remains true
that mathematicians only work with simple and elegant systems, systems in
which everything is extremely clearly defined-and the brain is a far cry
from that, with its ten billion or more semi-independent neurons, quasi-
randomly connected up to each other. So mathematicians would never
study a real brain's networks. And if you define "mathematics" as what
mathematicians enjoy doing, then the properties of brains are not mathe-
matical.
The only way to understand such a complex system as a brain is by
chunking it on higher and higher levels, and thereby losing some precision
at each step. What emerges at the top level is the "informal system" which
obeys so many rules of such complexity that we do not yet have the
vocabulary to think about it. And that is what Artificial Intelligence re-
search is hoping to find. It has quite a different flavor from mathematics
research. Nevertheless, there is a loose connection to mathematics: AI
people often come from a strong mathematics background, andChurch, Turing, Tarski, and Others 559