School of Computer Science THE UNIVERSITY OF BIRMINGHAM CogX project

Discussion of some themes in Piaget's two books on
Possibility and Necessity

These notes are available here:
(A PDF version, generated automatically, may be added later.)

Notes For a Joint Meeting of AINC in Computer Science with Developmental Psychology Seminar
Aaron Sloman
School of Computer Science, University of Birmingham.

Installed: 14 Feb 2011
Last updated:
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 collaborators
produced a pair of books on Possibility and Necessity, exploring questions about how two
linked sets of abilities develop:
  1. The ability to think about how things might be, or might have been, different from the
    way they are.
  2. The ability to notice limitations on possibilities, i.e. what is necessary or
The Books Comment
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

  1. The relevance of Piaget's work to the problems of designing intelligent machines that
    learn the things humans learn.
    (Most researchers in both Developmental Psychology and AI/Robotics have failed to notice
    or have ignored most of the problems Piaget identified.)

  2. How a deep understanding of AI, and especially the variety of problems and techniques
    involved in producing machines that can learn and think about the problems Piaget
    explored, could have helped Piaget describe and study those problems with more clarity and
    depth, especially regarding the forms of representation required, the ontologies required,
    the information processing mechanisms required and the information processing
    architectures that can combine those mechanisms in a working system -- especially
    architectures that grow themselves.


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
Piaget thought.

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

Importance of Piaget's ideas

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


Related papers and presentations

These are topics I have been working on and writing about for many years, though
without knowing about Piaget's last two books until 2010. A sample of related work:
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
Toddler Theorems
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:


Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham