Paper ID 10279
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
(DRAFT: Liable to change)
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.
Note: A presentation of Turing's main ideas for non-mathematicians can be found in
Philip Ball, 2015, "Forging patterns and making waves from biology to geology: a commentary on Turing (1952) `The chemical basis of morphogenesis'",
http://dx.doi.org/10.1098/rstb.2014.0218
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham