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. ------------------------------------------------------------------- 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).