School of Computer Science THE UNIVERSITY OF BIRMINGHAM Ghost Machine

Tutorial at Diagrams 2018 -- Monday June 18th: 14:00-15:30

Were "Super-Turing" diagrammatic reasoning competences
ancient products of biological evolution?

Alternative, shorter title:
Evolved "diagrammatic" spatial intelligence.

PRESENTER Aaron Sloman
School of Computer Science, University of Birmingham

(DRAFT: Liable to change)

Last updated: 16 May 2018


A summary overview can be found here:

Keywords/phrases for tutorial and its implications:
-- Evolution of ancient spatial mathematical competences using "diagrams in minds";
-- Evolved biological construction kits: concrete and abstract, fundamental and derived;
-- Biological foundations for mathematics -- Evolution: the blind mathematician
-- Forms of representation for intelligent machines
-- Examples of toddler and non-human mathematical competences.
-- Serious gaps in current computational, neural, psychological and philosophical theories of spatial
    reasoning and mathematical knowledge.
    (But Immanuel Kant had major insights over 200 years ago.)

Depending on who comes, and their backgrounds, I'll present a subset of these topics (definitely including the roles of spatial cognition in mathematical reasoning):

Background: The Turing-inspired Meta-Morphogenesis project
Varied examples of natural intelligence will be used to introduce aspects of the Turing-inspired Meta-Morphogenesis project, investigating changes in information processing functions and mechanisms since the earliest proto-life forms, with aim of identifying and filling major gaps in current understanding of how minds work and how different kinds of minds vary.

Evidence is sparse, so the project requires intelligent guess-work: expanding available evidence by interpolating possible functions and mechanisms.

Evolving requirements for biological information processing
Changing life-forms include: changing needs, physical forms, control processes, kinds of terrain, opportunities, dangers, types of resource and types of information relevant to survival and reproduction.

Figure: Evolutionary Steps in Information processing

Evolution to Betty crow

As a result, requirements for information-processing also change: organisms need increasingly complex and varied forms of information, and increasingly complex and varied mechanisms for acquiring, manipulating, storing, and using information, including, for example, spatial information for controlling actions and for reasoning about structures and processes, on various spatial and temporal scales. Current diagrammatic reasoning abilities build on this, but mechanisms are mostly unknown, and not yet in AI. The importance of statistics-based probabilistic learning is currently vastly over-rated.

Evolved cognition is increasingly disembodied
Because of the complexity and variety of kinds of information, uses of information, the diversity of needs and preferences, and the increasing spatial and temporal regions to be considered in making decisions and forming plans, the cognitive processes become increasingly disembodied, i.e. disconnected from current states, and current actions, and therefore disconnected from sensory input signals and motor output signals.

Modern engineering and architectural design illustrate this: typically modern engineers and architects don't have physical contact with the things they design until long after all the main design decisions have been taken. Minor changes can sometimes be made at a late stage.

Likewise explanatory theories in modern science (e.g. subatomic physics, astrophysics, and theories about chemical processes in biological reproduction and gene expression) are also disconnected from the physical interactions of scientists with their environments, and from their sensory and motor signals. Humanly perceivable and manipulable laboratory objects are only very indirectly related to the concepts used in the theories being tested or applied.

All this is ignored in fashionable theories of embodied cognition [Note: It's hard to give references without giving offence!]

needed]. Moreover, insofar as attempts are made to demonstrate the powers of embodied robots without complex computational control mechanisms, the abilities of the robots are usually pathetically limited, like the passive walker robots: the demonstrators fail to point out what will happen if you place a brick in front of the passive walker, or try to get it to walk up or down a staircase. A nice example:

Theories of embodied intelligence that I have encountered also ignore the commonalities between normal humans and those born blind or deaf, or lacking arms and legs, or inseparably conjoined with a twin, etc.

This tutorial is about some of the requirements for more intelligent systems.

Evolved construction-kits
Construction of those mechanisms in individuals requires increasingly complex construction kits, all ultimately based on the fundamental construction kit provided by physics/chemistry (about which there are still major gaps in our knowledge): As individuals develop new physical features and mechanisms they also build increasingly complex construction kits for building those new components.

Likewise, construction of new information processing mechanisms, requires construction of new construction kits for building those mechanisms, including meta-cognitive construction kits that build new mechanisms for meta-cognition.

This contradicts theories of learning and reasoning that assume uniform learning mechanisms throughout an individual's life.

Evolved mathematical (especially spatial) cognition
I'll give examples of requirements for construction kits and reasoning mechanisms that seem to have been used in ancient mathematical discoveries -- e.g. the amazing geometrical and topological discoveries reported in Euclid's Elements, extending earlier forms of spatial intelligence to use forms of spatial reasoning (e.g. "Diagrams in minds") that are still not replicated in current AI systems, using mechanisms as yet unknown to neuroscience, e.g. because current neural models explain only discoveries involving classification, correlation. and probabilities, but not discoveries of impossibility and necessity noticed by Immanuel Kant as essential features of mathematical knowledge.

I suspect that when Turing wrote (Turing 1939, Sec. 11)

"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct. . . . The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings."
he was beginning to move toward ideas something like those presented here. I suspect, but cannot demonstrate, that this was connected with his thinking in the paper on morphogenesis.

What sorts of mechanisms can explain mathematical/spatial cognition?
The tutorial will analyse some requirements for such mechanisms and ask whether all are implementable as virtual machines on digital computers.

There are aspects of familiar spatial/diagrammatic/geometric reasoning that seem to be very hard to implement on digital computers/Turing machines, unlike simple types of logical/algebraic reasoning -- where everything is discrete and finite. Spatial reasoning, and perception and use of spatial affordances essentially involves continuous structures and processes. Can digital mechanisms go beyond approximate simulation of spatial structures and processes to include discovery of impossibilities and necessary truths about them? Examples:

Could new kinds of chemical computers help? Could that be why Alan Turing's last great publication, in 1952 was about "The Chemical Basis of Morphogenesis"?

What sort of Super-Turing multi-membrane machine might suffice
I'll introduce the idea of a Super-Turing membrane computer that might be able to explain aspects of human spatial/diagrammatic/geometric/topological reasoning. This will introduce some of the ideas developed in this draft, incomplete, paper, posing some hard problems that I think nobody currently knows how to answer, though I suspect Turing was thinking about this in the last few years before his death in 1954.

An incomplete document (work-in-progress) attempting to make the ideas more precise is here:
This requires solutions to problems in physics, chemistry, neuroscience, psychology, philosophy, AI, robot-engineering, and perhaps new kinds of mathematics. How much of this was was Turing thinking about while writing his paper on chemistry-based morphogenesis? In his 1948 paper he showed that he was interested in intuitive thinking in mathematics as something distinct from "mechanistic" logic-based thinking. "Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity."

This tutorial will add much detail to my short paper accepted for the main Diagrams conference, available here:

Participants will be invited to comment on several types of example, listed in the documents linked below.

An illustrative but incomplete list
A newly-discovered way of proving the triangle sum theorem.
Reasoning about what happens if you deform a triangle by moving just one vertex indefinitely along a line. (There's an interesting discussion of what happens in the limit, in Beccuti(2018)).
Invited talk (video lecture) for a workshop on cognition at IJCAI 2017.
Many examples of reasoning about spatial impossibilities and necessity.
(Needs better organisation.)
"Toddler theorems" -- mathematical discoveries used (unwittingly) by pre-verbal human toddlers.
The theory of evolution by natural selection presupposes but does not describe increasingly complex and powerful construction kits produced and used by evolution, and by products of evolution, to enable new designs to be used. including increasingly complex construction-kits for increasingly complex information-processing systems (many still unknown).

Overview of the Turing-inspired Meta-Morphogenesis project


Francesco Beccuti, Stretching a triangle to infinity, May 20, 2018

K. S. Lashley, The Problem of Serial Order in Behavior, in Cerebral mechanisms in behavior,
Ed. L.A. Jeffress, Wiley, New York, 1951, pp. 112--131,

A. M. Turing, 1939, Systems of Logic Based on Ordinals, Proc. London Mathematical Society, pp. 161-228,
Also reprinted in Alan Turing: His work and impact Elsevier 2013.

Eds. S. Barry Cooper and J. van Leeuwen, Alan Turing: His Work and Impact, 2013, Elsevier, Amsterdam,

References to be expanded later.
See also the references in the conference paper:

This document is available as html and pdf
A partial index of discussion notes in this directory is in

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