Abstract
This paper poses, discusses, but does not definitively answer, the following questions: What sorts of reasoning machinery could the ancient mathematicians, and other intelligent animals, be using for spatial reasoning, before the discovery of modern logical mechanisms? “Diagrams in minds” perhaps? How and why did natural selection produce such machinery? Is there a single package of biological abilities for spatial reasoning, or did different sorts of mathematical competence evolve at different times, forming a “layered” system? Do the layers develop in individuals at different stages? Which components are shared with other intelligent species? Does some or all of the machinery exist at or before birth in humans and if not how and when does it develop, and what is the role of experience in its development? How do brains implement such machinery? Could similar machines be implemented as virtual machines on digital computers, and if not what sorts of non-digital “Super Turing” mechanisms could replicate the required functionality, including discovery of impossibility and necessity? How are impossibility and necessity represented in brains? Are chemical mechanisms required? How could such mechanisms be specified in a genome? Are some not specified in the genome but products of interaction between genome and environment? Does Turing’s work on chemical morphogenesis published shortly before he died indicate that he was interested in this problem? Will the answers to these questions vindicate Immanuel Kant’s claims about the nature of mathematical knowledge, including his claim that mathematical truths are non-empirical, synthetic and necessary? Perhaps it’s time for discussions of consciousness to return to the nature of ancient mathematical consciousness, and related aspects of everyday human and non-human intelligence, usually ignored by consciousness theorists.
Highlights
Spectacular examples were produced by biological evolution and its products long before there were any human control engineers. These examples illustrate the fact that the ability to discover and use mathematical facts does not presuppose the ability to recognize what has been achieved as a mathematical discovery that is independent of its practical applications
In at least some humans, evolution seems to have produced abilities to absorb hard-won mathematical discoveries that have been made by previous generations, and use those as a platform on which to build yet more kinds of mathematics, either as a kind of playful activity that is enjoyed for its own sake, or as a goal directed activity seeking a kind of mathematics that allows new solutions to be found for old or recently encountered practical problems
What will happen to the angle at A if it continually moves further from the opposite side, BC, along a line that intersects BC and passes through A, as illustrated in Figure 4? I have informally given the problem to at least 40 people, most of them non-mathematicians, including many who have never studied Euclidean geometry, and they all seem to have been able to discover the same effect of moving the vertex further from BC along a line passing between B and C
Summary
There have been theories of consciousness that make use of mathematics, e.g. mathematical models of patterns of activity in neural nets, but no theory of brain function or automated reasoning that I have encountered explains how brains enable great mathematical discoveries to be made, e.g. the discoveries in geometry and topology, reported many centuries ago in Euclid’s Elements [1], describing not just observed regularities but necessary connections or impossibilities, some of which, e.g. Pythagoras’. Possibilities for creating 3-D structures by repeated folds of a flat sheet of paper (Origami) can produce combinations of lines that cannot be achieved in Euclidean geometry, including trisection of an arbitrary angle (for more details see https://en.wikipedia.org/wiki/Origami) Another example not derivable within Euclidean geometry is the Neusis construction that was known to ancient mathematicians, but not included in Euclid’s Elements. I do not believe there is any sense in which ordinary individuals can, or need to, refer to complete alternative universes when making discoveries about geometrical or topological possibility, impossibility or necessity It is not clear what enables humans to understand concepts like necessity and impossibility: neural nets that merely record categories encountered so far and their relative frequencies cannot express these concepts, which, as Kant pointed out, characterise mathematical knowledge. (The discovery was crucial to Newton’s mechanics and the invention/discovery of differential and integral calculus, on which a great deal of modern science and engineering depends.)
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