Departmental Seminar, School of Computer Science
The University of Birmingham

Why is it so hard to get machines to reason like our ancestors
who produced Euclidean Geometry?

Speaker: Aaron Sloman

4pm Thursday 20th June 2013,

Official abstract for the talk:

This informal, extended abstract may be modified from time to time.
Last modified: 18 Jun 2013; 7 Dec 2014


Extended Abstract

The ideas presented here cannot be presented in a clear and simple cumulative fashion:
there's too much mutual dependence between sub-topics that do not naturally form
a linear order. So I have to hope for readers who are persistent and patient, as
well as highly critical.

The universe seems to include (potentially) infinitely many domains, of many different
sorts, with many different degrees and kinds of complexity. The domains instantiated on
this planet seem to have increased steadily in size and complexity and diversity since
life began, partly because of the increasing diversity and complexity of forms of life,
and their interactions.

A domain is a set of possible structures and possible processes involving those structures.
For quick introductions to many domains see Vi Hart's amazing high speed online math-doodles.

The fact that a specific type of structure or process is possible in a domain does not
imply that instances of that specific type will ever exist. For example, it is likely that
there are hugely many 500-word possible (grammatical, meaningful, sentences of English
that will never be uttered, and many possible machines and buildings that will never be
constructed. The domain of shape changes of a 5 metre long piece of string lying on a flat
surface may include process-types that will never be instantiated. It's even more likely
that the domain of possible shape transitions of a climbing plant, such as ivy, is so vast
that despite the many millions of actual instances of ivy growing, only a small subset of
the possibilities will ever be realised. (To some extent this will depend on how demanding
the criteria are for saying that two slightly different instances are of the same type.)

Both the instances included in many of the domains, and the domains themselves have become
bigger, more complex, and more varied over time -- for example, as more species of organisms
have evolved and, the size and complexity of organisms and their capabilities have increased.

As organisms acquired more complex sensors, effectors, and control mechanisms, including
mechanisms that change their behaviour through learning, and mechanisms that take account
of information processing and control processes in other organisms, increasing varieties
of domains of information processing have developed including domains of information about
information, both in individual meta-cognition, and in cognition concerned with
information-processing (sensing, learning, wanting, planning, choosing, acting on
intentions, hypothesising, reasoning, etc.) in others.

A major driver of this process has been biological evolution, perhaps the richest positive
feedback system known.

Each domain comprises a set of possible structures and processes (transformations), in
which structures change.

Some domains are very simple, (e.g. the domain of possible ways a coin tossed once can
land: heads up or tails up). Others are more complex, including

Organisms need repertoires of behaviour related to their structures, their capabilities,
their varying needs and varying circumstances. Domains of possible behaviours for
organisms become increasingly complex and varied as evolution produces more complex and
varied organisms that interact with other organisms (as prey, as predators, as mates, as
offspring, as parents, as collaborators, as competitors, as symbionts) as well as with
changing physical circumstances produced by weather, earthquakes, volcanoes, floods,
migration, etc.

Some domains of possible behaviours are produced (very slowly) by biological evolution.
Later, evolution produced organisms that can discover domains relevant to their
capabilities and circumstances and produce appropriate reactions to particular
circumstances, for instance, using learnt reflexes and responses to triggers of various
sorts. (Varieties of online intelligence.)

Later, as the challenges and opportunities became more complex, organisms evolved, and
learnt/developed more sophisticated ways of selecting behaviours by reasoning about sets
of possibilities. For this they required information about the domains of possibilities
and constraints on possibilities. This required evolution to produce new abilities to
acquire knowledge about how to use information about domains, for example in constructing
plans, finding explanations for perceived situations or events, etc. (Varieties of offline

The elements of some domains can be physically instantiated, whereas others cannot, though
for different reasons, e.g. the domain of impossible 3-D objects depictable in 2-D line
drawings, as in Escher's pictures -- impossible to instantiate because spatial constraints
would be violated -- or the domain of possible re-orderings of the set of natural numbers
-- impossible to instantiate if the universe is finite.) A major challenge for biology,
psychology, neuroscience, AI/Robotics and philosophy is to identify the various kinds of
domain, with various origins, that have proved relevant in one way or another to
individuals, to species, to social groups, to ecosystems, to nations, ..., and to explain
how individuals can come to know about and use information about such domains, including,
in some cases creating new domains with greater power (e.g. adding metrics to pre-existing
spatial domains, inventing calculus, inventing programmable looms, etc.).

It seems that without a certain type of richness in the domain of possible chemical
structures and processes there would not have been life as we know it (Ref T. Ganti).

Each form of life has a domain of possible sensory-motor interactions with each of the
types of environment in which that life is possible. More complex forms of interaction
became possible when evolution produced organisms that could explore and learn about
domains and deploy that knowledge, e.g. in planning action sequences, or building maps of
extended spatial structures on the basis of sequences of sensory motor interactions. Using
such domain knowledge enabled more intelligent animals to detect and reason about positive
and negative affordances for action. (Ref: J.J. Gibson).

In humans some of that led to meta-knowledge about such competences and eventually to the
production of Euclid's Elements, one of the greatest achievements of biological evolution.
Many other forms of mathematical knowledge grew out of later explorations of domains, and
then meta-domains of many kinds.

One of the key features of such knowledge is that it concerns grasping some set of
possibilities and then discovering constraints on those possibilities, e.g.
learning that some extensions of a set of possibilities can be described or depicted but
are not included in the set, e.g. the penrose triangle or a set of 3 objects combined with
a non-overlapping set of 2 objects forming a set of 4 objects.

This is totally different from and more fundamental than discovery of probabilities
through empirical observations, the current focus of huge amounts of research in
AI/Robotics and neuroscience (much of it misguided in my opinion).

In the last few decades there have been tremendous advances in AI theorem proving
techniques, and we now have programs that can find and prove theorems that would defeat
most humans, including a package that will sell you a new, unique, non-trivial theorem
named after you (REF). But it has proved extremely difficult to get computers to engage in
the kinds of reasoning even a human toddler can do and some other animals seem able to do
that made the development of Euclidean geometry possible. This talk will present some
examples and discuss possible ways of making progress, with potential implications for
developmental psychology, neuroscience, theories of animal cognition, and philosophy of
mathematics, as well as AI and Robotics.

This abstract is available at:

A partial index of discussion notes in this directory is in

Examples of simple forms of geometrical reasoning that seem hard to model on
computers are in:
Hidden Depths of Triangle Qualia

This is part of the Meta-Morphogenesis project:

The ideas here are closely related to some of the ideas developed by Annette Karmiloff-Smith
in her 1992 book: Beyond Modularity, briefly reviewed here:

A partial survey of types of "toddler theorems" is here (contributions welcome):

To be modified and extended.

A video interview introducing some of these topics:
Euclidean geometry is one of the greatest products of human minds, brought together in
Euclid's Elements over two millennia ago.

However, at some distant earlier time there were no geometry textbooks and no teachers.
So, long before Euclid, our ancestors, perhaps while building huts, temples and pyramids,
measuring fields, making tools or weapons, or reasoning about routes, must have noticed
facts about spatial structures and processes that are both useful (like facts about
physics, geography, biology, and human languages), but are also demonstrable by reasoning
with logic and diagrams. Mathematicians do not have to keep checking that their
discoveries remain true at high altitudes, or in cold weather, or on surfaces with unusual
materials or colours -- because they can prove things.

Without teachers to help, biological evolution must somehow have produced
information-processing mechanisms that allowed ancient humans to develop the concepts,
notice the relationships and discover the proofs that their descendants are taught at
school, but which we have the ability to discover for ourselves, as our ancestors did.

This suggests that normal human children have the potential to make those discoveries
themselves, under appropriate conditions. I suspect there are also deep connections with
competences that have evolved in other intelligent species that understand spatial
structures, relationships and processes -- such as nest-building birds, squirrels that
steal nuts from bird feeders, elephants that manipulate water, mud, sand and foliage with
their trunks, and apes coping with many complex structures as they move through and feed
in tree-tops.

Can we replicate evolution's achievements, and create robots that start off with
competences of young children and later, as they develop, make simple discoveries in
Euclidean geometry? I'll explain why that's hard to do -- but perhaps not impossible.
There have been great advances getting computers to reason logically, algebraically and
arithmetically, but the kinds of reasoning in Euclid, e.g. using diagrams, are very

I'll discuss some of the problems and possible ways forward. Perhaps, someone now studying
geometry and computing at school will one day design the first baby robot that grows up to
be a self-taught robot geometer, and, like some of our ancestors, discovers for itself why
the angles of a triangle must add up to exactly half a rotation.

Other related materials

Installed: 11 Jun 2013
Last updated: 12 Jun 2013

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