School of Computer Science THE UNIVERSITY OF BIRMINGHAM CoSy project CogX project

An introduction to toddler theorems: What makes it possible for humans to perceive, play with, think about, and learn from triangles?
Abstract for talk to Language and Cognition seminar Friday 5 Oct 2012 14:15, and to Intelligent Robotics Lab, Monday 8th Oct 3pm
(DRAFT: Liable to change)

Aaron Sloman
School of Computer Science, University of Birmingham.
(Officially retired philosopher in a Computer Science department)

Installed: 5 Oct 2012
Last updated: 5 Oct 2012
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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:
(discoveries about triangles based on playing with diagrams)

or here
(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:
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