Abstract for talk to Language and Cognition seminar Friday 5 Oct 2012 14:15, and to Intelligent Robotics Lab, Monday 8th Oct 3pm

School of Computer Science, University of Birmingham.

(Officially retired philosopher in a Computer Science department)

This paper is

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/irlab-lang-cog-abstract.html

Also accessible as http://tinyurl.com/CogMisc/irlab-lang-cog-abstract.html

A PDF version may be added later.

A partial index of discussion notes is in http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html

It is obvious to anyone who has had personal experience of doing mathematics, e.g. finding a proof of a theorem in Euclidean geometry, or a proof of a theorem about numbers (as opposed to merely memorising theorems -- as required in bad mathematics teaching), that humans can make discoveries about structures, relationships, and processes that are not empirical, even if the discovery is triggered by experience, and also not trivial, like definitional truths (e.g. No odd number is divisible by 2). Anyone who does not recognise this is invited to play with the examples here:

http://tinyurl.com/CogMisc/triangle-theorem.html (discoveries about triangles based on playing with diagrams) or here http://tinyurl.com/CogMisc/toddler-theorems.html#primes (discoveries about numbers based on playing with blocks)

Immanuel Kant (around 1781) referred to the knowledge acquired in such discoveries as 'synthetic apriori knowledge' of 'synthetic necessary truths'. Of course, he knew that new-born babies do not have such knowledge, so for him 'apriori' did not mean 'innate'[*]. Most learners nowadays acquire such knowledge only with the help and stimulation provided by teachers, but in the millenia before production of Euclid's elements there must have been people who made such discoveries without having mathematics teachers. I suspect that the required human mathematical abilities are based on the mechanisms produced by biological evolution that enable humans and other animals to discover what J.J.Gibson called 'affordances' of many kinds, in structured environments. (I shall ignore his seriously mistaken ideas about 'direct' perception.) We can generalise Gibson's notion of 'affordance' to cover a wide range of possibilities for change and constraints on change. Some of them can only be discovered empirically, and many of those are merely statistical correlations. But a subset have the features Kant described, e.g. affordances related to interactions between rigid objects, such as levers and gear wheels. I suspect that in the first few years of life young children make many such proto-mathematical discoveries about possibilities and constraints on possibilities in the environment, which enable them to learn by working things out, instead of learning only by trial and error, or by being told, or by imitation. The discoveries in question, i.e. hundreds of "toddler theorems" seem to be closely related to what Annette Karmiloff-Smith called "Representational redescription" in "Beyond Modlarity"(1992). Many of them are made before children have developed the meta-semantic and meta-cognitive capabilities required to be aware of what they have learnt or to be able to communicate it to others. So discovering that such learning is going on is a task with severe methodological problems, especially as the processes seem to be highly idiosyncratic and unpredictable, ruling out standard experimental and statistical methods. I'll present some examples of ways of investigating toddler theorems and invite suggestions for more systematic research. I think Piaget had some important insights about these phenomena, but lacked a theoretical framework for thinking about mechanisms, though shortly before his death, at a workshop in Geneva, he acknowledged that knowing about Artificial Intelligence would have helped. [*] Some theoretical ideas about the nature-nurture issues related to toddler theorems are presented in this paper published in an obscure journal in 2007: http://tinyurl.com/BhamCosy/#tr0609 Jackie Chappell and Aaron Sloman Natural and artificial meta-configured altricial information-processing systems

Maintained by
Aaron Sloman

School of Computer Science

The University of Birmingham