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)
This is part of the Turing-inspired Meta-Morphogenesis project:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
Shorter version of abstract for Midlands Logic Seminar talk.
No! Existence of humans, like existence of any other biological species,
depends on mathematical phenomena preceding humans by aeons!
Conjectured Partial Answer:
Hopefully part of a progressive rather than a degenerating research programme in
the sense of Lakatos.
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.)
Question:
Does the mixture of discrete and continuous operation in chemistry-based
construction-kits provide a richer space of information processing than
Turing-equivalent machines?
NOTE:
Compare the (messy, incomplete) survey of 'toddler theorems'
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html
This is closely related to a famous paper on levels of reality by a theoretical
physicist:
http://robotics.cs.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf
Philip W. Anderson, (1972), More is different,
Science, New Series, 177, 4047, pp. 393--396,
What is common to all varieties of the "modern" approach is the notion that doing mathematics requires abilities to reason from agreed starting points to conclusions. Foundations in this sense involves the study of different possible starting points (e.g. different formalisms or axioms), and different modes of reasoning (e.g. using only logic, or logic plus X's set theory, or ....).
Questions about foundations then focus on possible starting points, and possible modes of reasoning, usually modes that can be formalised, e.g. expressed in terms of truth-preserving operations on symbols with a precisely defined (discrete) syntax.
A problem about this is that working on foundations in this sense can be thought of as simply doing mathematics of a different kind, which raises the question whether further foundations are needed, and also questions about why work on foundations should be of greater interest (e.g. to philosophers) than work on any other mathematical domain.
A partial index of discussion notes is in
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/AREADME.html
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham
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