Midlands Logic Seminars

Tuesday 24th Feb 2015 5pm
52 Pritchatts Road (Gisbert Kapp Building) Lecture Room 1
Preceded at 4pm by a talk by Richard Kaye on Combinatorial Games.

How can a physical universe produce mathematicians?

Metaphysical, Biological, Evolutionary Foundations for Mathematics
(As opposed to logical or set-theoretic foundations.)

(DRAFT SUMMARY: Liable to change)

Aaron Sloman
School of Computer Science, University of Birmingham

Background to talk: Evolution of construction kits
(Draft book chapter)

This is part of the Turing-inspired Meta-Morphogenesis project:

Shorter version of abstract for Midlands Logic Seminar talk.

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

No! Existence of humans, like existence of any other biological species,
depends on mathematical phenomena preceding humans by aeons!

I ask Kantian questions:
How is mathematics possible? (types of truth, necessity, etc.)
How can a physical universe produce mathematicians?
How can they come into existence in a universe with no life?
How can they acquire non-empirical non-trivial mathematical knowledge?

Conjectured Partial Answer:
Hopefully part of a progressive rather than a degenerating research programme in the sense of Lakatos.

The universe provides a fundamental construction kit (FCK) including space/time, physics and chemistry. Under certain conditions certain chemical structures begin to produce exact and nearly exact copies of themselves --morphogenesis. (Discreteness and reliability depend crucially on quantum mechanism, e.g. preventing degradation of information by thermal buffetting (as noted by several physicists).)
Under certain conditions this (eventually) leads (via the power of the FCK and the power of natural selection) to multiple branching layers of derived construction kits (DCKs), changing/enriching the processes of morphogenesis --meta-morphogenesis.

We need a richer theory of types of construction kit:
Some construction kits are concrete (physical) others abstract (e.g. grammars), others mixed, others using virtual machinery that's physically implemented but not adequately describable in the language of physics.

Possibilities and invariants (i.e. mathematical properties) in the FCK, then later in the DCKs. are "discovered" (blindly) and used first by natural selection, then later by its products: organisms, e.g. when they use invariant features of parametrised designs, such as negative feedback control loops, conditional switches, and increasingly complex information records, some used immediately, some later, some used online some offline. (Initially using only loop-closing semantics, then later more sophisticated semantics.)

Later, new or expanded construction-kits allow a subset of organisms to develop cognitive abilities to detect, reason about, and apply these mathematical properties, for instance in detecting and using affordances of various kinds. (More than J.J. Gibson noticed!).

More complex control functions require new DCKs supporting virtual machinery (re-discovered by 20th century computing engineers!), leading to an even greater variety of control mechanisms with new mathematical properties -- e.g. planning, reasoning and design capabilities, morphing later into story-telling, games, music, dancing. rituals???

Later, via various evolutionary and developmental steps some DCKs were able to produce meta-cognitive machinery, initially self-directed (e.g. for fixing planning processes), then later other-directed (e.g. for reasoning about information-processing in predators, prey, mates, offspring, competitors, etc.) and for cooperative reasoning/teaching/challenging/arguing.

Then meta-meta-meta-cognitive (didactic?) insights led to the assembly of previous discoveries into structured, more easily learnt and taught, knowledge repositories storing shared organised knowledge e.g. Euclid's Elements. (The most important book ever written???)

Continuation of the process of construction of new DCKs led to the invention/discovery of various fragments of logic and increasing formalisation of assembled bodies of mathematical knowledge.

Still later, a subset of thinkers attempt to produce a new fully formal organised encoding and call that 'foundations', without realising that they have changed the subject: they are producing and discussing new branches of mathematics that have interesting structural relations with the old branches. Frege pointed this out for geometry, but a similar claim can be made about his and others' attempts to logicize arithmetic, instead of basing it on topological(?) properties of one-one correspondences.

Later, some of the people involved in the formalisation processes either forgot about the earlier forms of mathematics or began to disparage them, leading to the education of an inferior brand of "formal" mathematicians... (lambasted by Mandelbrot).

Note on consciousness:
An implication of the ideas here is that theories/models of perception (e.g. vision) that cannot accommodate perception of mathematical (e.g. topological and geometrical) structures and relationships used in understanding proofs are inadequate.

Likewise, theories of consciousness that cannot accommodate consciousness of mathematical problems and solutions (e.g. in shirt mathematics also mentioned below ) are inadequate.

Note on modality:
This approach rejects 'possible world' semantics for modal concepts (e.g. must, may, can, cause, could have, couldn't have, if...then, necessary, possible, constrained, invariant, etc. etc.) and bases them on properties of portions of the world (or portions of construction-kits) and their properties and relationships. (This may be closely related to Alastair Wilson's ideas about causation and grounding. I am not sure.)

Does the mixture of discrete and continuous operation in chemistry-based construction-kits provide a richer space of information processing than Turing-equivalent machines?

Challenge: can you use your favourite form of mathematical foundation to do shirt-mathematics?


It is widely believed that children learn the language used in their environment. However, there is much evidence, including the episode of the Nicaraguan deaf children reported here: https://www.youtube.com/watch?v=pjtioIFuNf8 suggesting that humans don't learn languages, rather they create languages cooperatively, though usually the youngest collaborators are in a weak minority. This, and the debugging of overgeneralisations in "U-shaped" learning require powerful mathematical and software engineering competences. Some conjectures about evolution of (generalised) languages are here:

Compare the (messy, incomplete) survey of 'toddler theorems'

For a useful critical survey of some of the inadequate answers to questions about evolution of minds see:
A two part survey paper by Evelyn Fox Keller (MIT) Published in
Historical Studies in the Natural Sciences,
Vol. 38, No. 1 (Winter 2008), pp. 45-75 and Vol. 39, No. 1 (Winter 2009), pp. 1-31
Organisms, Machines, and Thunderstorms: A History of Self-Organization, Part One
Organisms, Machines, and Thunderstorms: A History of Self-Organization,
--Part Two: Complexity, Emergence, and Stable Attractors

This is closely related to a famous paper on levels of reality by a theoretical physicist:
Philip W. Anderson, (1972), More is different, Science, New Series, 177, 4047, pp. 393--396,

There are at least two distinct varieties of research on foundations of mathematics.
For more on complex virtual machines and a critique of simple versions of virtual machine functionalism (VMF) see:
Virtual Machine Functionalism
(The only form of functionalism worth taking seriously in Philosophy of Mind)

An extended discussion of construction kits and evolution can be found here:
[Significantly revised during February 2015]

An introduction to the scientific importance of explanations of possibilities is here:


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