Mathematics, Evolution and Development

Mathematical phenomena, their
evolution and development
(Examples and discussions on this web site.)
Aaron Sloman
School of Computer Science
University of Birmingham

"For mathematics is after all an anthropological phenomenon."
(Wittgenstein, Remarks on the Foundations of Mathematics)

No, though it is partly a biological phenomenon.

Installed:
29 Aug 2013
Updated: 13 Jul 2018; 24 Mar 2019; 4 Apr 2019
22 Sep 2013; 21 Nov 2013; 9 Sep 2014; 7 Dec 2014; 29 Mar 2015; 4 Sep 2016;

N.B. The order of items listed here is not significant.
The order may be revised later.
Many of the documents are available in both html and pdf formats. If you prefer pdf, try changing the url to end in 'pdf'. If that fails restore the 'html' suffix.

http://www.cs.bham.ac.uk/research/projects/cogaff/misc/kant-maths.html
Key Aspects of Immanuel Kant's Philosophy of Mathematics
Ignored by most psychologists and neuroscientists studying mathematical competences
The Triangle Sum Theorem
(Old and new proofs, including Mary Pardoe's Proof, using a rotating arrow, or pencil.)
Triangle Qualia/Triangle Theorems
Hidden Depths of Triangle Qualia (Especially their areas.)
Theorems About Triangles, and Implications for Biological Evolution and AI The Median Stretch, Side Stretch, and Triangle Area Theorems: old and new proofs.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
Toddler theorems (proto-mathematical discoveries) -- what they are and some examples
E.g. toddler exploring 3-D topology with pencil.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/impossible.html
On seeing that X is impossible, and seeing why X is impossible, with a large collection of examples.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/ p-geometry.html
Pardoe Geometry (P-Geometry: adding constrained motion to Euclid)
Compare the "neusis" construction in the next document:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/trisect.html
How to trisect an angle in P-geometry. Including a criticism of Poincaré's philosophy of mathematics.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-intuition.html
Alan Turing's 1938 thoughts on intuition vs ingenuity in mathematical reasoning.
Did he unwittingly re-discover key ideas first presented in Immanuel Kant's philosophy of mathematics?
Abstract of talk on this to Zurich philosophy students, March 2019.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/zurich-talk.html
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/deform-triangle.html
Non-monotonic angle change as a vertex moves on a line:
Discovery of a necessary connection between two different continua: variation in location of a point and variation in size of an angle, easily discovered by non-mathematicians. (And some surprisingly complex variants of a simple problem.)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/multicirc.html
Surprisingly Rational Circle Segments
http://www.cs.bham.ac.uk/research/projects/cogaff/crp/#chap8
CHAPTER 8 On Learning About Numbers: Problems and Speculations (1978)
Chapter 8 of The Computer Revolution in Philosophy: Philosophy science and models of mind
http://www.cs.bham.ac.uk/research/cogaff/62-80.html#crp
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/edsem-15.html
Biology, geometry, philosophy of mathematics and Kantian robotics?: The case of p-geometry and angle trisection
Edinburgh Informatics Forum, Mathematical Reasoning Group Seminar 25th Aug 2015,
http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#talk108
Talk 108: Why is it so hard to make human-like AI (robot) mathematicians?
Especially Euclidean geometers.
Talk at PT-AI 2013 Oxford, 22 Sep 2013
Closely related draft paper:
http://www.cs.bham.ac.uk/research/projects/cogaff/13.html#1304
Extending Turing's Pattern: From Morphogenesis to Meta-morphogenesis
Invited contribution to Cybertalk Magazine, September 2013
Biology, Mathematics, Philosophy, and Evolution of Information Processing
I try to show that biological evolution (blindly) made use of mathematical domains long before humans existed on this planet -- e.g. the domain of designs for homeostatic control systems, including negative feedback loops of various sorts, and sub-domains of the domain of chemically controlled morphogenesis discussed in Turing (1952), among many others.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/rings.html
Reasoning About Rings and Chains
(Impossible linking and unlinking)
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/torus.html
Reasoning about continuous deformations of curves, e.g. on a Torus.
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/shirt.html
Shirt Mathematics
Illustrating topological and semi-metrical reasoning in everyday life.
Knots: discovering theorems about them
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/mathsem.html
From Molecules to Mathematicians:
How could evolution produce mathematicians from a cloud of cosmic dust?
What might Turing have done if he had lived longer?
How might it have helped us understand biological evolution,
and how evolution produced mathematicians?
Notes for a presentation to the Midlands Logic Seminar: 1st November 2013
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/midlog-talk.html
How can a physical universe produce mathematicians?
Metaphysical, Biological, Evolutionary Foundations for Mathematics
(As opposed to logical or set-theoretic foundations.)
Midlands Logic Seminars, Tuesday 24th Feb 2015
http://midlandslogic.org.uk
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/schrodinger-life.html
What is Life?
Erwin Schrödinger on the Chemical Basis of Life
This is a collection of extracts from Schrödinger's 1944 book What is life? with some comments added by me, mainly trying to clarify what I think he is saying and why I think it is so important for anyone trying to understand how life as we know it is possible in this physical universe. This includes understanding how evolution could have produced mathematicians like Euclid.
The Meta-Configured Genome
Based on collaboration with Jackie Chappell
Different relationships between genome and competences
(Preconfigured vs meta-configured competences: Chappell and Sloman)
Incomplete note on problems of rotation in a discrete space
What could have led to euclidean geometry?
Why is it so hard to get machines to reason like our ancestors
who produced Euclidean Geometry?
Ideas about "Representational Redescription"
in Annette Karmiloff-Smith's Beyond Modularity (1992)
Abstract for talk to mathematics school leavers (17 April 2013)
Evolutionary transitions in information-processing
Far more and more varied than most people seem to have noticed?
Did life on earth start on dust particles?
The theory of Nasif Nahle, Monterrey, Mexico. If true this alters some of the domains of competence required by earliest organisms.