Keith Devlin argues that the history of logic is a search for the rules of thought. In a lively and lucid style he discusses the work of Plato, Aristotle, Zeno of Elea and the Stoics, through to Ockham and Duns the Scot, then onto Liebniz, Euclid, Boole, Alan Turing and others, and eventually to the ideas of Noam Chomsky on language.
His conclusion is that all of this work depends on a separation of syntax from semantics. It uses methods that involve the manipulation of blind symbols that can be interpreted in a number of different ways, and it is this that has given rise to their success.
But Devlin is dissatisfied with these approaches. None of them captures something essential to cognition: the meaning. They track merely formal properties, like hollow boxes being moved around.
This is the main problem facing artificial intelligence. Programs are discussed that appear to be intelligent and show understanding, for example the Eliza psychotherapy program, but which on closer inspection deliver mere empty illusions. As a last resort, it is argued that cognition, at a fundamental level, cannot be reduced to rules. But one cannot help asking, "do logical rules necessarily flatten out meaning?", and, "what does it really mean to talk about cognition without them?''. To throw out the idea of a purely logical or mathematical science is to discard the central tenets of rationalism set up by Descartes.
Devlin argues for more intuitive concepts within a science of the mind. He talks about a soft mathematics made up partly of these intuitive concepts with some maths joined on, as a tool, rather than an all-embracing theory.
One might imagine that aesthetic concepts, such as classical and romantic, are not easily reduced to mathematics. Soft mathematics is a good start in trying to understand them. But in the end you need rigorous mathematical concepts, or concepts developed from some formal system or rule-based game to give tough predictions and a complete understanding.
However, this does not mean that content need be lost. Form can generate content. How it does so is the crux of the problem of consciousness. Douglas Hofstadter's ideas, which he presented in 1976 in Godel Escher Bach, might provide an answer. Hofstadter's idea was that Godel's theorem - or a meta-mathematical idea like it - might be the archetype of how new levels of explanation arise. In particular it could point to the model that explains consciousness. The way the Godel formula both floats on the workings of its formal system, yet is sealed off from it, might parallel the mind's relationship to the brain.
In elucidating Hofstadter's ideas, one might envisage a picture in which not only is physical science mathematical but a distinct science of mental states is mathematical as well. Then the relationship between the two separate discourses is not a relationship in mathematics, but about it - ie, the field of meta-maths, which is the realm of Godel's theorem and the like. A science of mental states may have to involve a mathematics all-encompassing and tough enough to fit in with such a picture about how new levels of explanation arise.
Furthermore, these ideas may suggest how structure and syntax can generate meaning. There is an intimate link between the interpretation we are justified in giving to a formal system and the Godel formulae that are specified as "true". Given a number of as-yet-tenuous assumptions, we can sketch a scenario in which the mind effectively models the structure of the Godel argument, using self-reference and mapping between levels. This could generate Godel formulae that effectively constrain what interpretations are logically possible. Such a constraint on interpretation may be what "fixing meaning" constitutes.
In a discussion about logic and cognition, "mistakes" may be important in pinpointing the mind. Logic only captures how we think when we think correctly. However we do not always do so. Blind laws of nature neither work correctly nor incorrectly. But for the mind there is an internal canon for the justification of rationality. There is something real for someone to think or see straight.
Hofstadter's Fluid Concepts and Creative Analogies picks up another idea that is central to his thinking: that developing analogies and variations on a theme is the core to creativity and cognition. Here Hofstadter and his team, now at Indiana University, summarise their 14 years' work on the computer modelling of this ability.
Consider for example his copycat program. This deals with a simple universe of letter strings. The program is shown a letter string being changed, eg ABC goes to ABD. It is then asked to do the same thing to XYZ (with the proviso that Z's successor is not A, as in a circular alphabet). A number of alternatives are possible for example XY, XYD, XYY, WYZ. Choosing which one seems best is almost a matter of aesthetic vision.
Hofstadter is always aware of the aesthetic validity of different approaches and research agendas. It is exciting to follow him as he takes us through the different layers of this thinking as his aesthetic antennae feel for a deeper understanding. In each case he clearly explains the development of his ideas.
Here is a meaty approach that gives us programs that are creative. Hofstadter points out that they capture something of the "halo of meaning", "slippability" and "connectedness of concepts", not unlike the human case. In this they go beyond other programs that merely ape creativity. However Hofstadter's work, though concrete in its realisation, and aimed with an eye to great generality, still needs to forge predictions in neurophysiology.
I look forward to him linking this work to his ideas on Godel and the mind; to take on issues such as self-reference, recursion, mapping between levels, tangled hierarchies and meta-languages. Indeed, his ideas about Godel and the mind become very pertinent when one realises that they set constraints on the way research can approach consciousness. They point out a problem with his recent hard AI approach. It may unknowingly bypass consciousness. This is because one cannot tackle consciousness from below.
In fact there may only be two research routes to grasping consciousness. Either we develop models that start from a first-person perspective, or we use a third-person approach but equipped with the array of weapons borrowed from an understanding of the Godel argument. Here are two books about the mind that are clear, lively and stimulating.
Paul Caro is honorary research associate, department of mathematics, University College London.
Fluid Concepts and Creative Analogues: Computer Models and the Fundamental Mechanisms of Thought
Author - Douglas Hofstadter
ISBN - 0 713 99155 0
Publisher - Allen Lane The Penguin Press
Price - £.50
Pages - 518