reasoning a kind of computing and therefore something that is
best described using the mathematics of computing? This hypothesis
continues to be controversial and we have only begun to examine
some of the implications of the hypothesis. But what are the
alternatives to this hypothesis?
figure to the right is an attempt to give you a visual image
of the space of alternatives. The main break in this space is
between informal and formal languages. Natural language is the
best example of an all-purpose informal language that can be
used to describe aspects of our world. But other examples can
be given. Musical notation is also an informal language as is
the set of conventions used in a road map.
significance of this distinction between formal and informal
languages lies in the degree to which the choice of language
commits us to certain assumptions about the things that we are
using the language to describe. In our everyday use of language
we can describe reasoning as being: 'correct or incorrect', 'clever
or stupid', 'rational or irrational', 'intuitive or analytic',
'black and white or gray', 'sophisticated or naive', 'linear
or nonlinear' or even 'geometric or non-geometric', or 'green
or purple'. That is, there is no constraint on what words and
combination of words that I might choose to describe what I find
important about thinking.
this with what can be said when the language adopted is a specific
formal mathematical languages. None of these ideas could be expressed.
attempts at describing aspects of our world begin with informal
description. It wasn't until the mid-20th century that specific
formal languages were proposed for use in describing human reasoning.
Mathematical learning theory and connectionist models were the
first to be proposed. These models were all probabilistic. Some
were formulated as continuous systems and others as discrete
systems. And some were linear and others nonlinear.
theories of computation (or machines) are a small subset of this
space of possibilities. Computational theories of cognition assume
that the mathematical language that will be appropriate to describe
human cognition will be found somewhere in the space of mathematical
languages that are developed to describe machines. This space
is shown as the red circle and in expanded as a gray circle to
provide more detail. Note that the mathematical objects typically
referred to as Machines may be finite or infinite and deterministic
or non deterministic.