How can a physical universe produce mathematicians? Metaphysical, Biological, Evolutionary Foundations for Mathematic (as opposed to logical or set-theoretic foundations)
Abstract:
I ask Kantian questions:
Conjectured Partial Answer:
The physical 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 mechanisms.) 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 -- hence: the Turing-inspired meta-morphogenesis project described herehttp://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
Some construction kits are concrete (physical) others abstract (e.g. grammars), others mixed, others using virtual machinery that's fully 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. Later new 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 (J.J.Gibson).
Still later meta-cognitive mechanisms evolve allowing such knowledge to be made more explicit internally, then later communicated, asked about, and eventually recorded externally. Still later meta-meta-cognitive mechanisms allow challenges and rebuttals, leading to attempts to produce shared systematically organised knowledge with transitions that can be defended against criticism (e.g. Euclid's Elements ).
Still later (about two thousand years 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 attempts to logicise arithmetic, instead of basing it on topological (?) properties of one-one correspondences.
An expanded version of this abstract (subject to change) is here:http://www.cs.bham.ac.uk/research/projects/cogaff/misc/midlog-talk.html
Draft book chapter expanding the ideas.http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.html
Background surveys by Evelyn Fox Keller:
Dr Walter Dean
Department of Philosophy
University of Warwick
http://go.warwick.ac.uk/whdean