Any theory of consciousness that does not include and explain
ancient forms of mathematical consciousness is seriously deficient.
7 Jun 2018 Parts of a paper on deforming triangles have been moved into this paper.
A partial index of discussion notes in this directory is in
This is part of the Turing-inspired Meta-Morphogenesis project
Which is part of the Birmingham Cogaff (Cognition and Affect) project
Being a "Blind Watchmaker" in Richard Dawkins' sense is a side effect of this.
If AI researchers wish to produce intelligent organisms they will need to understand the deep, pervasive, and multi-faceted roles of mathematics in the production of all organisms, including reproduction and development, in addition to many and varied uses of mathematical mechanisms and competences in meeting the practical information processing challenges of individual organisms interacting with their physical environments and other organisms (including in some cases intelligent prey or predators).
A consequence is that many forms of (human and non-human) consciousness involve deep mathematical (e.g. topological) competences. The ideas of theorists like Immanuel Kant, von Helmholtz, Jean Piaget, Richard Gregory, Noam Chomsky, Max Clowes, Margaret Boden, Daniel Dennett, James Gibson, David Marr and many others (my sample of names should be regarded as quirky and random), all contribute fragments towards a deep theory of consciousness. And all have errors or omissions, either because they focus on a restricted set of phenomena or because their explanatory accounts are inadequate, or both.
For example, any theory of consciousness that says nothing about mathematical consciousness, e.g. the forms of consciousness involved in ancient mathematical discoveries by Archimedes, Euclid, Zeno and others (including pre-verbal human toddlers), must be an incomplete, and usually also incorrect, theory of consciousness. That rules out most of them!
However, "What is it like to be a mathematician?" or "What is it like to understand a mathematical discover?" are not helpful questions about mathematical consciousness. Compare: What is it like to be a rock? Some of my examples below can be construed as partial answers to "What is it like to be a mathematician?" or "What is it like to make a mathematical discovery?", but the work is still at a stage that's too early for a clear structure to determine the order of presentation.
Biological evolution (The blind mathematician) made many mathematical discoveries and put them to good use in control mechanisms, long before any individual organism was aware of using them.
But what that claim means is not obvious. In part, it involves development and use of powerful "construction kits" with meta-mathematical properties: e.g. the meta-grammatical competences allowing the human genome to produce thousands of languages using grammars with different mathematical properties.
Some of the mechanisms use abstractions that allow for changing parameters, e.g. a type of organism using an epigenetically modified control mechanism whose parameters change as the size, strength, and speed of the organism change.
There seems to be a huge variety of such mathematical discoveries, some used in control of physical/chemical growth and development, and others in particular forms of sensing and action control.
In later stages, evolution provides itself with mechanisms able to discover and use important mathematical structures in forms that can be parametrised so as to produce different examples, either in different species, or in different individuals, or in the same individual at different stages of development.
The discovery of those "re-usable" and "variable" features can be regarded as meta-mathematical discoveries. They involve generic mechanisms that allow individual organisms to make mathematical discoveries, e.g. about how to derive information about the environment from sensory-motor data, or how to control actions to maintain speed while avoiding obstacles, and how to manage tradeoffs between speed, accuracy and other features.
Many examples involve using structures and processes in the optic array (not just structures in retinal projections) to infer structures of perceived objects and changing relationships between them, including structures and relationships never previously encountered -- a point that is also often made about language understanding: e.g. people reading this sentence for the first time, and constructing an interpretation.
Even if depth measures are not available, inferred 3-D structures can be complex and useful, as shown by work on "scene analysis" in the 1960s and onwards, surveyed in Ballard and Brown (1982). Some of the mechanisms use mathematical aspects of the projection process to derive "reverse" projections that can be useful despite information loss. This generally requires constraint-propagation to remove ambiguities.
Many animals, including pre-verbal human toddlers, can do that sort of thing without knowing that they are doing it or how and why what they do works.
Some competences produced by evolution, or learning, or some combination, involve more complex and messy structures than those normally studied by mathematicians, but that does not make them non-mathematical, e.g. a carnivore purposefully changing topological relationships between parts of a prey animal while dismembering it after capture -- a process that is not normally considered mathematical. If decisions about what to do next are merely innate reflex responses the mathematical "reasoning" must have been done by evolution. But there is too much variation in structures and processes involved in eating (and sharing) prey for every response to have been acquired by evolution. An animal that can work out how to deal with a new configuration, like a crow deciding where to insert the next twig in a part-built nest may be using a mixture of topological and geometric reasoning. Compare Betty the hook-making crow Weir et al. (2002)
Only much later in our evolutionary history could individuals have begun making mathematical discoveries that they were aware of making and using, with the ability to describe them and motivation to try to understand how they worked, and in some cases (much later?) the ability to communicate them to others and debate the merits of alternative modes of reasoning.
Later still, evidence suggests that social/cultural applications, practices, and institutions allowed new forms of discovery and development of mathematical competences and knowledge -- including, in some cases, restricting the processes (keeping knowledge secret).
I suspect there is a vast amount of unrecorded pre-history of human and non-human mathematical competence, that can be inferred only from indirect clues, including varied kinds of proto-mathematical intelligence in non-human species coping with different environments, different physiological needs, and different body structures -- sensor and motor mechanisms. Not all relevant cases are human precursors, or even vertebrates: some of the sensory-motor control mechanisms, e.g. in winged insects catching, escaping, mating, feeding and laying eggs.
Even the amazingly reliable transformations between larval and flying stages via a chemical soup must depend on mathematical properties of the genome and its products, which in turn depend on mathematical features of molecular structures and processes discussed by Schrödinger(1944), and many others influenced by him.
We also need deep new theories about the mechanisms and capabilities of
biological evolution, e.g. the (growing) theory of evolved construction-kits of
Compare: Stewart Shapiro, 2009 We hold these truths to be self-evident: But what do we mean by that? The Review of Symbolic Logic, Vol. 2, No. 1
It may appear that I am using 'self-evidence' as a type of justification. I
don't! I am concerned with explanatory mechanisms, not
For a short discussion of 'self-evidence' and how it differs from the notion of
non-empirical discovery of necessary truths see:
Extreme (and wrong) answers refer to social conventions, aesthetic/moral decisions, pragmatic claims about usefulness, etc. ....
"Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct. . . . The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings."he was moving toward ideas something like the ideas presented here. But that assumes a connection between his thinking in the late 1930s and his thinking around 1950. For a more detailed summary and discussion of his views see:
A.A.S Weir, J. Chappell, and A. Kacelnik,(2002),
Shaping of hooks in New Caledonian crows,
Science, vol 297, page 981.
(The videos on the laboratory web site show more complex and varied solutions than the paper reports.)