Biological/Evolutionary Foundations of Mathematics (BEFM)
Biological/Evolutionary Foundations of Mathematics (BEFM)
How can evolution produce babies that grow up to be
Can we produce baby robots that grow up to be
This paper is partly superseded by a new paper (November, 2016):
Several Types of Foundation For Mathematics (e.g.):
Neo-Kantian (epistemic) foundations,
Do we need to understand all of these in order to build artificial mathematical
minds comparable to ancient mathematicians, e.g. able to work out how to extend
Euclidean geometry to make triangle trisection not only possible, but easy? See
Many philosophers and mathematicians have sought to discover or create
foundations for mathematics. I argue that mathematics takes different
forms with different sorts of foundations, and present examples.
This is work in progress and liable to be drastically revised.
a.sloman [at] cs.bham.ac.uk
"For mathematics is after all an anthropological phenomenon"
(Wittgenstein, Remarks on the Foundations of mathematics).
No! The existence of humans, like the existence of any other biological species,
is a combined instance of a collection of mathematical phenomena.
19 Nov 2014
24 Nov 2014; 15 Dec 2014 (See the 2016 update above)
This paper is
A PDF version may be added later.
A partial index of discussion notes is in
Claim: there are different foundations for different mathematical phenomena.
The BEFM approach asks questions partly inspired by Kant's ideas about how
mathematical knowledge of non-analytic necessary truths is possible and partly
inspired by Turing's 1952 paper on morphogenesis, hinting at what he would have
done if he had lived more than two years longer.
Above all it is inspired by a collection of biological issues: what happened
when, as schoolboy I was introduced to Euclidean geometry and learnt that I
could actually prove
things about lines, circles, triangles, etc.? How was
whereas in all my other subjects I had to take everything on trust
or do experiments in a laboratory, where inaccurate measurements, or equipment
problems could lead to incorrect results? What made it possible for brains to do
such things? Moreover, whereas I was guided by a teacher and textbooks, there
must originally have been humans, or perhaps ancestors of humans, who had no
mathematics teachers, yet made the original discoveries that were later
organised and presented in a (fairly) systematic way in Euclid's
Elements, over 2,500 years ago.
Thinking about these issues, and about aspects of the design of many highly
functional organisms, led me to the conclusion that evolutionary mechanisms were
implicitly discovering and using mathematical theorems long before organisms
evolved that could also make such discoveries, notice that they were making
them, and then turn the making, discussing and teaching, of such discoveries
into a socially organised activity (though sometimes the relevant society had
one individual, e.g. in the case of
for a while).
And how is all that related to the intelligence of other animals, e.g. birds who
make extraordinarily complex nests apparently requiring insight into spatial
structures and constraints, even if they can't talk about that (e.g.
knot-tying weaver birds),
defeat "squirrel-proof" bird-feeders, and many more, including human toddlers
who, if observed carefully,
seem to be capable of mathematical insights that they
can use before they can talk about them -- and possibly before they can think
about what they are doing? (Illustrated in a growing collection of
here. I think Jean Piaget also noticed such phenomena.)
Yet, some of those discoveries have so far not been matched by even the most
sophisticated automated theorem provers, and although robots can be trained to
do many very impressive things there is so far (as far as I know) not one that
can explain why it succeeded and what it would have had to change if the
situation had been slightly different, and why the task would become impossible
if changed in other ways: forms of reasoning about what J.J.Gibson called
"affordances" (which we can generalise to proto-affordances, vicarious
affordances, epistemic affordances, deliberative affordances, as explained
Perhaps, if we can continue lines of thought that apparently preoccupied Alan
Turing and discover what biological reasoning systems are doing that we have
not learnt to replicate in machines, we'll have a new deeper understanding of
foundations of mathematics than ever before. I suspect this is deeply connected
with the fact, without which biological evolution as we know it would have been
impossible, that the physical/chemical structure of the universe provides
collections of "construction kits" with deep mathematical properties (e.g. in
some cases including unbounded extendability, and types of "emergence" of
qualitatively new kinds of mathematics extending old kinds). See
What mathematical properties of the physical universe make possible
known features of biological evolution? E.g.
- Evolution of organisms (implicitly) using mathematical structures in their
environment (e.g. homeostatic control mechanisms based on negative feedback;
avian flight control mechanisms managing instability; syntax-based descriptions
of environmental structures and processes, e.g. "A is between B and C"),
- Evolution of mathematical abilities to notice and reason about those
structures, including inferring properties and relations, and constructing
explanations ("Why do I see more of the cave as I move closer to the entrance?",
"If I move three paces then five paces forward where will I be?")
- Evolution of mathematical abilities to explore possible extensions of those
structures and alternatives to those structures? (E.g. adding length and other
metrics to systems of comparison: from "longer" to "three paces long", or
thinking of lengths of curves, areas
of irregular shapes, volume of an egg...)
- Evolution of meta-cognitive meta-mathematical abilities to reflect on
those reasoning abilities? (Would the answer be different in
another village? On a higher mountain? When the wind is blowing? Why not? What
makes P necessarily true? Where do constraints come from? (Kant))
- Evolution of meta-meta- abilities to reflect on possible alternatives to
those reasoning and exploration abilities? (Attempting to organise results into
an ordered derivational system with explicit axioms. Attempting to find
syntactic criteria for valid inference. Attempting to find indisputable starting
points. Engaging in foundational debates: Can arithmetic be reduced to logic?
Can geometry be reduced to arithmetic? Did Hilbert's axiomatisation
of geometry change the subject(Frege 1950)? Did Frege's logicisation of
arithmetic change the subject?)
After reading this document David Mumford pointed out that "the big thing that
Hilbert contributed to Euclid was his observation that betweenness was an
essential component in making Euclid fully rigorous -- a topological concept
that the Greeks ignored as being too obvious" -- a view of Euclid that I had
previously learnt as a student. However, Guggenheimer
(1977) suggests that the axioms of betweenness are not needed since the
concept B is between A and C is equivalent to length(AB)+length(BC)=Length(AC).
- Products of evolution (e.g. humans) discovering how to produce working
replicas (proof checkers, automated reasoners, automated mathematical
discoverers...) But why have we not yet been able to mechanise some
competences, e.g. topological reasoning about continuous deformation of closed
curves? -- as illustrated
Some universes (e.g. one made of Newtonian point masses) could not support
reasoning mechanisms [Why not? Could a useful computer be constructed out of
point masses with only mass/inertia, elasticity, location, direction,
velocity, acceleration, etc. of motion, and gravitational attraction?].
A key feature of our universe is chemistry
, making possible an enormous
variety of stable structures of varying size and complexity able to store usable
energy, create motors, create sensors, create enclosing and supporting
structures, create self-moving machines, and create stable
information structures (using catalysis, thanks to quantum
mechanics). (Ganti 2003). Turing machines are more restricted, unable to support
mixed discrete and continuous operations.
That leads to a host of new questions: if X is possible what
mathematical properties of the processes of biological evolution
possible ... e.g. make possible mechanisms able to discover Euclidean geometry?
Was Kant on the right track? Was it really pure logic somehow disguised?
Something else? (J.S.Mill? Wittgenstein? ...?) If it was logic, what made
organisms able to do logical reasoning?
This leads to further questions: e.g. what new possibilities are enabled by
mathematical properties of the products of biological evolution
Products include information-based control mechanisms of many kinds. What
mathematical properties of various information structures and information-using
mechanisms make possible various forms of perception, learning, reasoning,
deliberating, acting, communicating, ... and mathematical discovery?
Example: what biological mechanisms made it possible for our ancestors to
discover bits of geometry that led to the production of Euclid's
many discoveries in arithmetic-long before the development of formal proof
methods or attempts to base mathematics on some well defined foundations. I
think those early non-empirical mathematical modes of reasoning have never been
fully understood. Euclid's Elements
clearly reported and to some extent
organised mathematical discoveries, not empirical discoveries (even if they
started as empirical). Could the abilities of ancient mathematicians be
connected with the abilities to perceive and reason about affordances in the
environment - what is and is not possible, and why? (J.J.Gibson)
Those modes of perception and reasoning must in part be products of biological
evolution that served needs of some intelligent animals, e.g. the need to
acquire and manipulate information about what is and is not possible in an
individual's environment (e.g. in a spatial configuration where some things move
and others do not), and to discover consequences of realising some of the
possibilities (e.g. how they would alter subsequent possibilities and
Example: what biological mechanisms made it possible for Descartes to notice the
structural correspondences between parts of Euclidean geometry and sets of
numbers, sets of pairs of numbers, sets of triples of numbers, and equations
relating numbers - a mathematical discovery linking two domains? (No current
robot could do that. What needs to be added?)
What mechanisms made it possible for Newton to use Descartes' results to
discover and prove things about previously unnoticed aspects of the physical
world, expressible in a new mathematical formalism? Abilities to invent, see
and use syntactic structures are also mathematical competences.
And later on: what features of the forms of meta-cognitive information
processing in humans make it possible for some of them to notice and begin to
investigate systematically possible alternatives
to the mathematical
structures discovered and thought about up to any particular time - including
non-Euclidean geometries, non-standard logics, non-standard arithmetics,
different transfinite structures,...
[All this presupposes a non-standard theory of the semantics of modality,
Some aspects of abilities that underlie adult mathematical competences seem to
play a role in various kinds of animal intelligence and pre-verbal human
intelligence (We need examples of toddler discoveries, as well as examples from
other animals, e.g. squirrels, weaver birds, elephants...).
Example (toddler?) theorem: If a shoe-lace goes through a hole in the shoe you
can extricate it by pulling one end or the other end, but not both ends
simultaneously - even if the lace is stretchable. Why not? Why is it a mistake
to try to put your shirt on by pushing a hand into a cuff, and pulling the
sleeve up the arm? Your answers need not depend on statistical evidence from
failed attempts, because you can reason mathematically about everyday things,
even unwittingly. It's very hard to axiomatise such knowledge, and to justify
required premises/axioms/inference rules?
Examples of spatial intelligence in young humans and other animals suggest
that biological evolution (blindly) "discovered" and made effective use of
mathematical structures in both the environments in which organisms evolved, and
also in the space of possible biological information-processing mechanisms that
we (human mathematicians, philosophers, computer scientists) have not yet
understood, and which may provide new answers to old questions about the nature
and scope of mathematical knowledge and how it differs from other kinds.
Those (object)-mathematical abilities can be present in some animals and
young humans without the meta-cognitive abilities required to think about the
discoveries and the reasoning processes, or to communicate
discoveries to others, or give reasons. What has to change?
The meta-cognitive abilities are clearly not available at birth in humans, and
seem to depend on later development of brain mechanisms that for important
biological reasons are delayed in some intelligent species. Kant:
"...faculty of knowledge ... awakened into action...", i.e. there is
unreflective and (later) reflective mathematical learning.
Are foundations of mathematics and foundations of meta-mathematics necessarily
This may eventually provide new support for a modern variant of Kant's ideas:
mathematical knowledge is synthetic and a priori
, non-empirical, but
neither innate nor infallibly derived (Lakatos 1976), and mathematical truths
are necessary but not a subspecies of logical truths.
There are many unanswered questions about mathematical discoveries (e.g.
discoveries about equivalence classes of curves on a planar, spherical or
toroidal surface) that are close to human common-sense reasoning about spatial
structures and processes but beyond the scope of current automated theorem
provers. Can that be fixed without radical innovation? Brain science has shed no
light on any of this. New advances in AI are needed to model these processes.
What sort? Then perhaps neuroscientists will know what to look for.
It may also turn out that there are forms of computation (e.g. perhaps, as
Turing hinted in 1950, chemistry-based computation, with its mixture of discrete
and continuous processes) that provide a different space of mathematical
mechanisms from Turing-equivalent mechanisms. Turing was thinking of chemical
processes before he died (1952). He had a deep interest in how living things
I think there's a rich space of mathematical structures waiting to be explored
concerned with such evolutionary and developmental processes, which ultimately
depend on the mathematical structures and processes supported by physics and
chemistry - work in progress.
There are connections with Jean Piaget's developmental psychology and the work
of neurodevelopmental psychologist Annette Karmiloff-Smith (e.g. her ideas about
representational redescription), and probably others I've not yet identified.
We need to attract clever, young, researchers from several disciplines to
contribute to this type of research on foundations of mathematics. We may need
new forms of education.
Incomplete (illustrative) Bibliography
(More to be added later)
Frege G (1950) The Foundations of Arithmetic: a logico-mathematical enquiry
into the concept of number
. B.H. Blackwell, Oxford, (Tr. J.L. Austin.
Ganti T (2003) The Principles of Life
. OUP, New York, Eds. Eörs
Szathmáry & James Griesemer, Translation of the 1971 Hungarian edition
Gibson JJ (1979) The Ecological Approach to Visual Perception
Mifflin, Boston, MA
Guggenheimer, Heinrich, (1977) The Axioms of Betweenness in Euclid
, Vol. 31, No. 1/2, pp. 187-192, Wiley
Hilbert D (2005) The Foundations of Geometry
. Project Gutenberg,
, translated 1902 by
E.J. Townsend, from 1899 German edition
Ida T, Fleuriot J (eds) (2012) Proc.
9th Int. Workshop on Automated Deduction
in Geometry (ADG 2012)
, Informatics Research Report, University of
Kant I (1781) Critique
of Pure Reason
. Macmillan, London, translated (1929) by
Norman Kemp Smith
Karmiloff-Smith A (1992) Beyond Modularity: A Developmental Perspective on
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McCarthy J, Hayes P (1969) Some philosophical problems from the standpoint of
AI. In: Meltzer B, Michie D (eds) Machine Intelligence 4
University Press, Edinburgh, Scotland, pp 463-502,
Lakatos I (1976) Proofs and Refutations
. Cambridge University Press,
Lenat DB, Brown JS (1984) Why AM and EURISKO appear to work. Artificial
Mueller I (1969) Euclid's Elements and the Axiomatic Method. British
J for Phil of Science
Sauvy J, Sauvy S (1974) The Child's Discovery of Space: From hopscotch to
mazes - an introduction to intuitive topology
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Harmondsworth, translated from the French by Pam Wells
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of Knowledge Representation and Reasoning:
Proc. 5th Int. Conf. (KR `96),
Morgan Kaufmann Publishers, Boston, MA, pp 627-638,
A. M. Turing, 1952,
The Chemical Basis Of Morphogenesis,
Phil. Trans. R. Soc. London
Winterstein D (2005) Using Diagrammatic Reasoning for Theorem Proving in a
. PhD thesis, University of Edinburgh, School of
Wittgenstein L (1978) Remarks on the Foundations of Mathematics, 3rd Edition.
Blackwell, Oxford, Eds. G. H. von Wright, R. Rhees & G. E. M. Anscombe
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On 19 Nov 2014, 01:15.