Notes For a Joint Meeting of AINC in Computer Science with Developmental
School of Computer Science, University of Birmingham.
Installed: 14 Feb 2011
20 Feb 2011; 23 Feb 2011; 11 May 2011; 4 Mar 2013; 16 Jul 2014
Slides (PDF) for the presentation on this topic are available at
It is not widely known that shortly before he died Jean Piaget and his collaboratorsThe Books
produced a pair of books on Possibility and Necessity, exploring questions about how two
linked sets of abilities develop:
- The ability to think about how things might be, or might have been, different from the
way they are.
- The ability to notice limitations on possibilities, i.e. what is necessary or
I believe Piaget had deep insights into important problems for cognitive science that have
largely gone unnoticed, and are also important for research on intelligent robotics, or
more generally Artificial Intelligence (AI), as well as for studies of animal cognition
and how various animal competences evolved and develop.
These topics are relevant to understanding biological precursors to human mathematical
competences and to resolving debates in philosophy of mathematics, e.g. between those who
regard mathematical knowledge as purely analytic, or logical, and those who, like Immanuel
Kant, regard it as being synthetic, i.e. saying something about reality, despite
expressing necessary truths that cannot be established purely empirically, even though
they may be initially discovered empirically (as happens in children).
It is not possible in one seminar to summarise either book, but I shall try to present an
overview of some of the key themes and will discuss some of the experiments intended to
probe concepts and competences relevant to understanding necessary connections.
Presentation overview:I hope to explain
That kind of computational or "design-based" understanding of the problems can lead to
deeper clearer specifications of what it is that children are failing to grasp at various
stages in the first decade of life, and what sorts of transitions can occur during the
learning. I believe the problems, and the explanations, are far more complex than even
The potential connection between his work and AI was appreciated by
Piaget himself only very shortly before he died, acknowledged at a workshop
on AI and Genetic Epistemology in Geneva 1981
I now think that Piaget's experiments were important and based on some insights that are
not widely shared, but that his theoretical framework for analysing and presenting the
results is flawed because of his "Stage Theory" of development.
In contrast, Annette Karmiloff-Smith's book Beyond Modularity misses out some of
the important topics discussed by Piaget, but has a better theory of development (though
still in need of refinement -- as she is aware).
I discuss Beyond Modularity informally in
Oxford DPhil Thesis 1962,____________________________________________________________________________
Knowing and Understanding: Relations between meaning and truth,
meaning and necessary truth, meaning and synthetic necessary truth,
Necessary', 'A Priori' and 'Analytic', in Analysis vol 26, No 1, pp 12-16 1965.
Interactions between philosophy and AI:
The role of intuition and non-logical reasoning in intelligence,
in Proc 2nd IJCAI, 1971, London, pp. 209--226, (Also published in AI Journal, 1971)
Actual Possibilities, in Principles of Knowledge Representation and Reasoning:
Proc. 5th Int. Conf. (KR `96),
Eds. L.C. Aiello and S.C. Shapiro, Boston, MA, 1996, pp. 627--638,
And more recent papers and presentations on varieties of affordance, perception of motion,
reasoning with diagrams, toddler theorems, philosophy of mathematics, evolution of
mathematics, etc., including:
(Added: 4 Mar 2013)
"Hidden Depths of Triangle Qualia"
(Theorems about triangles, and Implications for Biological Evolution and AI)
Notes for a related talk at AGI 2011 At Google, 3rd--6th August 2011
Reasoning about shapes: curves on a torus.
Biology, Mathematics, Philosophy, and Evolution of Information Processing
Slides from previous related talks are available here.
For more information see presentations on "toddler theorems" and
"Philosophy of mathematics" here:
School of Computer Science
The University of Birmingham