Earlier version presented at

http://www.birmingham.ac.uk/schools/mathematics/news-and-events/birmingham-popular-maths-lecture.aspx

Hosted by the School of Mathematics

The University of Birmingham

Contact/Enquiries

**The Association for Science Education (ASE) Conference**

(Key Speakers)

Held its annual conference on Wednesday 8th to Saturday 11th January 2014, at

The University of Birmingham

The Association is a community of teachers, technicians, and other professionals

supporting science education and is the largest subject association in the UK.

It is an independent and open forum for debate and a powerful force to promote

excellence in science teaching and learning,

(Abbreviated link to this page: http://tinyurl.com/CogMisc/math-talk.html )

**_____________________________________________________________________________________**

One of the talks was

Saturday 11th Jan 2014, Aston Webb Building, room WG 5.

Honorary Professor of Artificial Intelligence and Cognitive Science

School of Computer Science

On Saturday 11th Jan 11:30 - 12:30 in lecture room W5 in the

Aston Webb Building.

**Short Abstract**

Since Alan Turing discussed the possibility of intelligent machines in 1950

there have been many outstanding achievements in artificial intelligence,

robotics and computational cognitive science -- including powerful logical

and algebraic theorem provers and proof checkers, programs that model

complex physical processes, and increasingly impressive mobile robots.Yet we don't know how to give a machine spatial reasoning abilities found in

very young children and many other animals, apparently required for the

discovery of the simplest truths and proofs in Euclidean geometry leading

up toEuclid's Elements, one of the greatest books ever written, which

included, for example, proofs about triangles that current computers cannot

follow.Can we understand what's missing in computers, and how it evolved in humans, and

perhaps other animals? Does mathematical spatial reasoning require new forms of

computation?See the longer abstract below.

-----

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.

This is totally different from noticing some regularity in the environment, or

finding by repeating an experiment, that a particular result happens regularly (e.g.

if a spinning coin drops to the floor it doesn't end up balanced on its edge, but lying

flat. What is discovered by mathematical reasoning has nothing to do with high

probabilities. Rather a proof shows that something is **necessarily** the case because

alternatives are **impossible**.

Without help from teachers, biological evolution somehow 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 --

which we also have the ability to discover for ourselves.

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.

[How did the squirrel get up to the bird feeder?]

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 try to analyse some unobvious details of the reasoning

processes, to explain why making such discoveries is not a feature of current robots,

or other Artificial Intelligence, systems.

It is not easy to get machines to do that -- 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

different.

Many current robots perform physical tasks, like walking, juggling, climbing, swimming and

recovering lost balance very impressively, yet they cannot reason about what they have

done or why it works or what else could have been done -- abilities required for discovery

of geometrical constraints on spatial structures and processes implied by Euclid's axioms.

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 planar triangle **must** add up to exactly half a rotation.

A more detailed discussion of some of the issues is available here:

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathsem.html

**From Molecules to Mathematicians:**

How could evolution produce mathematicians

from a cloud of cosmic dust?

A Protoplanetary Dust Cloud?

[NASA artist's impression of a protoplanetary disk, from WikiMedia]

**_____________________________________________________________________________________**

This is part of the Turing-inspired Meta-Morphogenesis project:

http://tinyurl.com/CogMisc/meta-morphogenesis.html

For more on the current state of Artificial Intelligence see http://aitopics.org/

Adam Ford has a large and growing collection of video interviews and recorded workshop and

conference presentations related to AI and other computing advances, by many scientists,

philosophers, artists, and others, here:

http://www.youtube.com/user/TheRationalFuture/

including an interview with Aaron Sloman closely related to the topic of this talk

http://www.youtube.com/watch?v=iuH8dC7Snno

with a transcript here:

http://www.cs.bham.ac.uk/research/projects/cogaff/movies/transcript-interview.html

A more detailed (but still incomplete) discussion of the topics can be found here

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathsem.html

**_____________________________________________________________________________________**

**_____________________________________________________________________________________**

Installed: 26 Feb 2013

Last updated: 1 Apr 2013; 29 Aug 2013 (reformatted); 27 Nov 2013 (Added ASE
Conference); 10 Jan 2014; 17 Jan 2014

Maintained by
Aaron Sloman

School of Computer Science

The University of Birmingham