This is the original abstract
Some old problems going back to Immanuel Kant (and earlier) about
the nature of mathematical knowledge can be addressed in a new
way by asking what sorts of developmental changes in a human child
make it possible for the child to become a mathematician.
This is not the question many developmental psychologists attempt to
answer by doing experiments to find out at what ages children
demonstrate various abilities e.g. distinguishing a group of three
items from a group of four items.
Rather, we need to understand how information-processing
architectures can develop (including the forms of representation
used, and the algorithms and other mechanisms) that make it possible
not only to acquire empirical information about the environment and
the agent, but also to acquire non-empirical information, for
example:
o counting a set of objects in two different orders must give
the same result (under certain conditions);
o some collections of objects can be arranged in a rectangular
array of K rows and N columns where both K and N > 1, while
others cannot (e.g. a group of 7 objects cannot);
o optical flow caused entirely by your own sideways motion is
greater for nearer objects than for more distant objects;
o when manipulating two straight rigid rods (where 'straightness'
refers to a collection of visual properties and a set of
affordances) it is possible to have at most one point where they
cross over each other, whereas with a straight rod and a rigid
wire circle it is possible to get one or two cross-over points,
but not three;
o if you go round an unchanging building and record the order in
which features are perceived, then if you go round the building
in the opposite direction the same features will be perceived in
the reverse order;
o if one of a pair of rigid meshed gear wheels each on a fixed
axle is rotated the other will rotate in the opposite direction.
Some of what needs to be explained is how the learner's ontology
grows (e.g. discovering notions like 'counting', 'straight',
'order'), in such a way that new empirical and non-empirical
discoveries can be made that are expressed in the expanded ontology.
I shall try to show how these ideas can provide support for the
claim that many mathematicians and scientists have made, that
visualisation capabilities are important in some kinds of
mathematical reasoning, in contrast with those who claim that only
logical reasoning can be mathematically valid.
Some aspects of the architecture that make these mathematical
discoveries possible depend on self-monitoring capabilities that
also underlie the ability to do philosophy, e.g. being able to
notice that a rigid circular object can occupy an elliptical region
of the visual field, even though the object still looks circular.
Although demonstrating all this in a working robot that models the
way a human child develops mathematical and philosophical abilities
will require significant advances in Artificial Intelligence, I
think I can specify some features of the design required.
There are also implications for biology, because the notion of an
information-processing architecture that grows itself as a result of
creative and playful exploration of the environment and itself can
change our ideas about nature-nurture tradeoffs and interactions.
No claim is made or implied that every mathematician in the universe
has to be a human-like mathematician. Some could use only
logic-engines, for example.
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Some of these ideas are explored in online papers and presentations:
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#tr0609
Natural and artificial meta-configured altricial
information-processing systems (PDF)
(Chappell and Sloman, IJUC, 2007)
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#dp0702
Predicting Affordance Changes (Discussion paper HTML)
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#bielefeld
Why robot designers need to be philosophers (PDF presentation)
http://www.cs.bham.ac.uk/research/projects/cosy/papers/#dp0703
Two Notions Contrasted: 'Logical Geography' and 'Logical
Topography' (Variations on a theme by Gilbert Ryle: The logical
topography of 'Logical Geography')
(Discussion paper HTML)
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/wonac
Evolution of two ways of understanding causation: Humean and
Kantian (PDF presentation) (With Jackie Chappell.)
Some relevant empirical research can be found in
Eleanor J. Gibson, Anne D. Pick, 2000,
An Ecological Approach to Perceptual Learning and Development,
Oxford University Press, New York,
Last updated 4 Feb 2008
Trial presentation in Birmingham
4pm Jan 10th
(filling a gap caused by visiting speaker cancellation).