Evolution  the blind mathematician producing
increasingly sophisticated users of mathematical discoveries
Paper ID 10279
For 1st Mathematical Cognition and Learning Society Conference, (MCLS)
April 89 2018
Examination Schools building, Oxford, UK.
https://express.converia.de/frontend/index.php?folder_id=884
(DRAFT: Liable to change)
This is part of the Turinginspired MetaMorphogenesis project
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/metamorphogenesis.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/slomanmathcog18.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 nonempirical,
noncontingent, and nonanalytic (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 protomathematical 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. statisticsbased/probabilistic
learning
mechanisms *cannot* establish necessary truths and impossibilities, and logical
theorem provers starting from Euclid's axioms *cannot* replicate the original
nonlogical 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 MetaMorphogenesis
(MM) project: trying to discover or guess at relevant varieties of evolved
information processing mechanism between the very simplest organisms (or
prebiotic forms) and the most sophisticated, hoping to identify previously
unnoticed layers of mechanism that might be needed to enable eventual evolution
of (e.g.) Archimedeslike 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
nonnumerical) 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 nonmetrical 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 TMtape, or logical axioms, or arrays of
bits, it may be necessary to have multiple movable and deformable surfaces
(subneural membranes?) on which structures can be projected then moved and
deformed relative to one another, and possibilities and impossibilities
discovered. Whether this SuperTuring machine can be implemented as a virtual
machine on digital computers would be a secondary question.
REFERENCES AND LINKS
(To be expanded)

Euclid and John Casey,
The First Six Books of the Elements of Euclid,
Project Gutenberg,
Salt Lake City, Apr, 2007,
http://www.gutenberg.org/ebooks/21076
Also see "The geometry applet"
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
(HTML and PDF)
and this short (17 minute) introduction to Euclidean reasoning:
by Zsuzsanna Dancso at MSRI.
https://www.youtube.com/watch?v=6Lm9EHhbJAY

The MetaConfigured Genome (2017)
Based on work with Jackie Chappell
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/metaconfiguredgenome.html

Immanuel Kant, 1781,
Critique of Pure Reason

Aaron Sloman, (1962  digitised 2016),
Knowing and Understanding: Relations between meaning and truth, meaning and
necessary truth, meaning and synthetic necessary truth
DPhil Thesis, University of Oxford, Bodleian Library
http://www.cs.bham.ac.uk/research/projects/cogaff/sloman1962

Toddler Theorems
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddlertheorems.html

Some (Possibly) New Considerations Regarding Impossible Objects,
their significance for mathematical cognition,
current serious limitations of AI vision systems,
and philosophy of mind (contents of consciousness).
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html

Aaron Sloman (2018draft),
Can Digital Computers Support Ancient Mathematical Consciousness?
http://www.cs.bham.ac.uk/research/projects/cogaff/slomanmathcons.pdf

Aaron Sloman, (Work in Progress) 2018,
A SuperTuring Membrane Machine for Geometers, toddlers, and other intelligent
animals.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/superturinggeom.html

Aaron Sloman, (Work in Progress) 2018,
Construction kits for evolving life
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/constructionkits.html

Trettenbrein, Patrick C., 2016,
The Demise of the Synapse As the Locus of Memory: A Looming Paradigm Shift?,
Frontiers in Systems Neuroscience,
Vol 88,
http://doi.org/10.3389/fnsys.2016.00088

A. M. Turing, (1952)
"The Chemical Basis Of Morphogenesis",
Phil. Trans. Royal Soc. London B
237, 237, 3772.
Note: A presentation of Turing's main ideas for nonmathematicians 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