Can current AI reasoning mechanisms be used to model ancient geometric reasoning, illustrating Immanuel Kant’s ideas about mathematical knowledge?
Prof A Jung
Prof A Sloman
No more applications being accepted
Funded PhD Project (European/UK Students Only)
This project is supervised by a theoretical computer scientist (Jung) and a philosopher of mathematics who has worked in AI and Cognitive Science (Sloman).
Immanuel Kant’s Critique of Pure Reason (1781) claimed that mathematical knowledge about necessary truths and impossibilities differed from empirical knowledge (which requires extensive testing in different contexts) and also differed from truths derived logically from definitions (analytic knowledge). Many 20th Century thinkers believed that Kant was proved wrong, e.g. because Einstein’s general theory of relativity, supported by Eddington’s observations of a 1919 eclipse of the sun showing apparent displacement of distant stars seen through the sun’s gravitational field, showed that Euclidean geometry could be refuted empirically, while use of logic and David Hilbert’s axiomatization of Euclidean geometry, showed that truths of Euclidean geometry were analytic.
Sloman’s 1962 DPhil thesis, recently digitized, http://www.cs.bham.ac.uk/research/projects/cogaff/sloman-1962/, was an attempt to defend Kant. A much stronger defence of Kant could come from an AI model of ancient mathematical thinking (e.g. Archimedes, Euclid, Pythagoras, Zeno, etc.) But so far all the geometrical theorem provers, starting from Gelernter’s 1964 theorem prover
https://pdfs.semanticscholar.org/2edc/8083073837564306943aab77d6dcc19d0cdc.pdf are based on logic and the Cartesian coordinate representation of geometry. Is it possible to use AI techniques to replicate the ancient modes of geometric discovery? Or is that beyond the scope of digital computer programs?
This project will investigate evidence collected so far, including the kinds of examples of mathematical discovery discussed in this invited lecture at an IJCAI workshop in 2017 http://www.cs.bham.ac.uk/research/projects/cogaff/misc/ijcai-2017-cog.html. Can new ideas about sub-neural, molecular mechanisms, combining discrete and continuous processes, be replicated and shown to be better able to model human geometric discovery processes and the spatial intelligence of squirrels, crows, elephants, and pre-verbal human toddlers?
See Trettenbrein 2016, The Demise of the Synapse As the Locus of Memory: A Looming Paradigm Shift?, Frontiers in Systems Neuroscience, Vol 88, http://doi.org/10.3389/fnsys.2016.00088
The research could be purely theoretical or could include development of a working model, or discuss the adequacy of current forms of computation for modelling ancient discovery processes. Or it could survey and criticize publications on the nature of mathematical knowledge e.g. whether it is innate.
2:1 Honours undergraduate degree and/or postgraduate degree with Distinction (or an international equivalent). This project has many facets that might suit students with different backgrounds, including mathematics, computer science, AI/Robotics, philosophy, psychology or neuroscience, though previous experience of the use of diagrams in geometrical and topological reasoning and some AI programming experience will be particularly useful.
If your first language is not English and you have not studied in an English-speaking country, you will have to provide an English language qualification.
The position offered is for four years of part time (75%) study with 456 teaching hours per year. The value of the award is £18,552 pa.
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FTE Category A staff submitted: 40.60
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