For 1st Mathematical Cognition and Learning Society Conference, (MCLS)
April 8-9 2018
Examination Schools building, Oxford, UK.
https://express.converia.de/frontend/index.php?folder_id=884

Evolution -- the blind mathematician producing
increasingly sophisticated users of mathematical discoveries

Partly with the unwitting help of those users.

(DRAFT: Liable to change)

Aaron Sloman
http://www.cs.bham.ac.uk/~axs/
School of Computer Science, University of Birmingham

This is part of the Turing-inspired Meta-Morphogenesis project
investigating changes in forms of information and forms of informatin processing during evolution, including changes that lead to new products of evolution.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html

Conference organizers: Roi Cohen Kadosh and Francesco Sella,
Department of Experimental Psychology, University of Oxford.

Conference timetable:
https://express.converia.de/frontend/converia/media/MCLS2018/ProgrammeTable.pdf
Detailed programme:
https://express.converia.de/frontend/converia/media/MCLS2018/Programme_15March.pdf
Submitted Abstracts:
https://express.converia.de/frontend/converia/media/MCLS2018/BookAbstracts_26March.pdf
Installed: 6 Mar 2018
Last updated: 16 Mar 2018; 29 Mar 2018
This paper is
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/sloman-mathcog18.html
A partial index of related discussion notes in this directory is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html

ABSTRACT
Session: 16:00 - 17:00 Mon 9th April Room 7

After a degree in Maths and Physics I switched to a philosophy DPhil (Oxford, 1962) defending Kant's claim that mathematical discoveries are non-empirical, non-contingent, and non-analytic (despite the evidence that our space is not Euclidean, wrongly thought by many to demolish Kant).

Later, after being introduced to AI by an inspiring vision researcher (Max Clowes), and learning to program, I decided that by building a baby robot that could "grow up" to be a mathematician like the ancient mathematicians (e.g. Archimedes, Euclid, Zeno, etc.) I could produce a much stronger defence of Kant.

Nearly half a century later, with a varied collection of examples of the sorts of mathematical discoveries, and evidence of proto-mathematical competences in toddlers, weaver birds, squirrels, elephants and other intelligent animals, neither I nor anyone else (as far as I can tell) knows how to create such a machine, and I believe there are no accurate explanatory theories/models in psychology or neuroscience either. (E.g. statistics-based/probabilistic learning mechanisms *cannot* establish necessary truths and impossibilities, and logical theorem provers starting from Euclid's axioms *cannot* replicate the original non-logical discoveries leading to those axioms.)

In 2011 I was invited to comment on Turing's 1952 paper on chemical morphogenesis for a centenary volume, which led me to wonder what Turing would have done if he had not died two years later. Perhaps the Meta-Morphogenesis (M-M) project: trying to discover or guess at relevant varieties of evolved information processing mechanism between the very simplest organisms (or pre-biotic forms) and the most sophisticated, hoping to identify previously unnoticed layers of mechanism that might be needed to enable eventual evolution of (e.g.) Archimedes-like organisms (and before that squirrels, etc.).

This includes collecting examples of evolution's mathematical discoveries (e.g. uses of homeostatic control) and other relatively simple mathematical (mostly non-numerical) discoveries not necessarily previously documented, and informally exploring abilities of colleagues and students to make those discoveries (e.g. if ABC is a planar triangle what happens to the size of angle A if A moves away from the side BC along a line that passes between B and C?, and many others involving topology and geometry including non-metrical relations such as partial orderings).

The forms of reasoning used don't seem to map onto any known mechanism, so, using many such examples, I have begun to collect requirements for mechanisms that might provide a basis for implementing the required mechanisms. E.g. instead of a discrete TM-tape, or logical axioms, or arrays of bits, it may be necessary to have multiple movable and deformable surfaces (sub-neural membranes?) on which structures can be projected then moved and deformed relative to one another, and possibilities and impossibilities discovered. Whether this Super-Turing machine can be implemented as a virtual machine on digital computers would be a secondary question.

REFERENCES AND LINKS
(To be expanded)

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