Work in progress.
PART-2 will be discussed in my tutorial at Diagrams 2018

A Super-Turing (Multi) Membrane Machine for Geometers
(Also for toddlers, and other intelligent animals)
PART 1: Philosophical and biological background
(DRAFT: Liable to change)

PART 2: Towards a specification for mechanisms, is available at

Aaron Sloman
School of Computer Science, University of Birmingham

Any theory of consciousness that does not include and explain
ancient forms of mathematical consciousness is seriously deficient.

7 Jun 2018

Parts of a paper on deforming triangles have been moved into this paper.

Installed: 7 Jun 2018
Last updated: 7 Jun 2018
This paper is:
A closely related draft, incomplete, paper, extending this one, is on
requirements for a "Super Turing Membrane Machine": available here:

A partial index of discussion notes in this directory is in
This is part of the Turing-inspired Meta-Morphogenesis project
Which is part of the Birmingham Cogaff (Cognition and Affect) project


BACKGROUND PART 1: Different philosophical and scientific goals
Types of (meta-) theory about mathematics In a separate document:
     REFERENCES AND LINKS (also copied below)

Different philosophical and scientific goals

Life is riddled through and through with mathematical structures, mechanisms, competences, and achievements, without which evolution could not have produced the riches it has produced on this planet. That's why I regard evolutionary mechanisms as constituting a Blind Mathematician (as well as being the most creative thing on this planet, as explained in this (draft) paper:

Being a "Blind Watchmaker" in Richard Dawkins' sense is a side effect of this.

If AI researchers wish to produce intelligent organisms they will need to understand the deep, pervasive, and multi-faceted roles of mathematics in the production of all organisms, including reproduction and development, in addition to many and varied uses of mathematical mechanisms and competences in meeting the practical information processing challenges of individual organisms interacting with their physical environments and other organisms (including in some cases intelligent prey or predators).

A consequence is that many forms of (human and non-human) consciousness involve deep mathematical (e.g. topological) competences. The ideas of theorists like Immanuel Kant, von Helmholtz, Jean Piaget, Richard Gregory, Noam Chomsky, Max Clowes, Margaret Boden, Daniel Dennett, James Gibson, David Marr and many others (my sample of names should be regarded as quirky and random), all contribute fragments towards a deep theory of consciousness. And all have errors or omissions, either because they focus on a restricted set of phenomena or because their explanatory accounts are inadequate, or both.

For example, any theory of consciousness that says nothing about mathematical consciousness, e.g. the forms of consciousness involved in ancient mathematical discoveries by Archimedes, Euclid, Zeno and others (including pre-verbal human toddlers), must be an incomplete, and usually also incorrect, theory of consciousness. That rules out most of them!

However, "What is it like to be a mathematician?" or "What is it like to understand a mathematical discover?" are not helpful questions about mathematical consciousness. Compare: What is it like to be a rock? Some of my examples below can be construed as partial answers to "What is it like to be a mathematician?" or "What is it like to make a mathematical discovery?", but the work is still at a stage that's too early for a clear structure to determine the order of presentation.

Types of (meta-) theory about mathematics

In discussing the nature of mathematics and the mechanisms that make mathematical discoveries and their use possible, we need to distinguish the following (the order is temporary, likely to change, and not intended as significant):

In the other document


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