Space of Possible Theory Languages Gottfried Wilhelm von Leibniz (1646-1716), on reasoning, 1677 "All our reasoning is nothing but the joining and substituting of characters, whether these characters be words or symbols or pictures, ... if we could find characters or signs appropriate for expressing all out thoughts as definitely and as exactly as arithmetic expresses numbers or geometric analysis expresses lines, we could in all subjects in so far as they are amenable to reasoning accomplish what is done in Arithmetic and Geometry. For all inquiries which depend on reasoning would be performed by the transposition of characters and by a kind of calculus, which would immediately facilitate the discovery of beautiful results ..."      The position put forward by Leibniz in 1677 suggests that reasoning can be appropriately described by what we would today refer to as a formal language. With the advent of computing, and more particularly, the development of mathematical languages intended to formally describe computing; some have suggested that the formal languages that are being developed to describe computing are in fact the languages that Leibniz was looking for to describe reasoning.      Is 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?      The 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.      The 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.      Contrast this with what can be said when the language adopted is a specific formal mathematical languages. None of these ideas could be expressed.      All 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.      Formal 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.

Introduction - Table of Contents

© Charles F. Schmidt