School of Computer Science THE UNIVERSITY OF BIRMINGHAM CoSy project

Could a child robot grow up to be a mathematician and philosopher?

Aaron Sloman

An updated version of the presentation, still changing, can be found here (PDF)

Abstract for talk at Liverpool seminar in Thinking about Mathematics and Science series.
21st January 2008

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

     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.


Some of these ideas are explored in online papers and presentations:
    Natural and artificial meta-configured altricial
        information-processing systems (PDF)
        (Chappell and Sloman, IJUC, 2007)
    Predicting Affordance Changes (Discussion paper HTML)
    Why robot designers need to be philosophers (PDF presentation)
    Two Notions Contrasted: 'Logical Geography' and 'Logical
    Topography' (Variations on a theme by Gilbert Ryle: The logical
    topography of 'Logical Geography')
        (Discussion paper HTML)
        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).