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.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited