Abstract
Automated geometry theorem provers start with logic-based formulations of Euclid’s axioms and postulates, and often assume the Cartesian coordinate representation of geometry. That is not how the ancient mathematicians started: for them the axioms and postulates were deep discoveries, not arbitrary postulates. What sorts of reasoning machinery could the ancient mathematicians, and other intelligent species (e.g. crows and squirrels), have used for spatial reasoning? “Diagrams in minds” perhaps? How did natural selection produce such machinery? Which components are shared with other intelligent species? Does the machinery exist at or before birth in humans, and if not how and when does it develop? How are such machines implemented in brains? Could they be implemented as virtual machines on digital computers, and if not what human engineered “Super Turing” mechanisms could replicate what brains do? How are they specified in a genome? Turing’s work on chemical morphogenesis, published shortly before he died suggested to me that he might have been considering such questions. Could deep new answers vindicate Kant’s claim in 1781 that at least some mathematical knowledge is non-empirical, synthetic and necessary? Discussions of mechanisms of consciousness should include ancient mathematical diagrammatic reasoning, and related aspects of everyday intelligence, usually ignored in AI, neuroscience and most discussions of consciousness.
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Notes
- 1.
A 16 page paper introducing aspects of the Turing-inspired Meta-Morphogenesis project http://goo.gl/9eN8Ks submitted to the 2018 Diagrams conference, was accepted as a short paper. The original version is at http://goo.gl/39DRCT.
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- 3.
- 4.
However, modal operators, e.g. “necessary”, “impossible” should be analysed using “possible configuration” not “possible world” semantics.
- 5.
Some speculations about evolved construction kits are online here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/construction-kits.html.
- 6.
goo.gl/3N1yQV gives more detail (still expanding).
- 7.
Illustrated by the BBC here https://www.youtube.com/watch?v=6svAIgEnFvw.
- 8.
The case where A moves along a line that intersects BC outside the triangle is discussed in another document. See http://www.cs.bham.ac.uk/research/projects/cogaff/misc/apollonius.html. Surprising additional complexities are discussed in that and http://www.cs.bham.ac.uk/research/projects/cogaff/misc/deform-triangle.html.
- 9.
Later developments of the idea of a Super-Turing machine will be added here: http://www.cs.bham.ac.uk/research/projects/cogaff/misc/super-turing-geom.html.
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Sloman, A. (2018). What Sort of Information-Processing Machinery Could Ancient Geometers Have Used?. In: Chapman, P., Stapleton, G., Moktefi, A., Perez-Kriz, S., Bellucci, F. (eds) Diagrammatic Representation and Inference. Diagrams 2018. Lecture Notes in Computer Science(), vol 10871. Springer, Cham. https://doi.org/10.1007/978-3-319-91376-6_17
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