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Why current AI and neuroscience fail to replicate or explain ancient forms of spatial reasoning and mathematical consciousness

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If you have a question about this talk, please contact Hector Basevi.

It is widely, but erroneously, believed that Immanuel Kant’s philosophy of mathematics in his Critique of Pure Reason (1781) was disproved by Einstein’s theory of general relativity (confirmed by Eddington’s observations of the solar eclipse in 1919, establishing that physical space is non-Euclidean). My 1962 DPhil thesis (now online) defended a slightly modified version of Kant’s claim that many important mathematical discoveries are non-empirical, non-contingent, and non-analytic (i.e. not just logical consequences of axioms and definitions), but did not explain how brains or machines could make such discoveries. I later encountered AI, learnt to program, and hoped to show how to build a baby robot that could grow up to be a mathematician making discoveries like those of Archimedes, Euclid, Zeno, etc., and many other deep discoveries made long before the development of modern logic and formal proof procedures. I think those mathematical abilities are closely related to the spatial intelligence of pre-verbal human toddlers, and other intelligent animals, e.g. squirrels, elephants, crows, apes, and perhaps octopuses[†]—whose abilities are not yet replicated in AI/Robotics systems nor explained by current theories in neuroscience or psychology. Insofar as such mathematical discoveries involve necessity or impossibility they cannot be substantiated by mechanisms that collect statistical information and derive probabilities.

This version of Kant’s theory rules out natural and artificial neural nets and related forms of deep learning, E.g. they cannot learn that something is impossible, such as a largest prime number, or a finite volume bounded by three plane surfaces. I have a large, and steadily growing, collection of examples to be explained by any adequate theory of mathematical consciousness. I’ll present a small sample during the talk.[‡]

Alan Turing’s comments in his PhD thesis on the difference between mathematical intuition and mathematical ingenuity seem to me to echo Kant’s insights, and I suspect (though the evidence is flimsy) that Turing’s 1952 paper on chemistry-based morphogenesis (nowadays his most cited paper) was at least partly motivated by a search for a new model of computation, combining continuous and discrete components. The most likely location for such a mechanism is sub-neural chemistry, for reasons related to Schrodinger’s analysis in What is life? (1944) of the role of chemistry in reproduction. A few neuroscientists are exploring related ideas (e.g. Seth Grant in Edinburgh).

I’ll present examples of spatial/mathematical reasoning illustrating Kant’s claims. E.g. what sorts of brain mechanisms enable a child to understand that it’s impossible to separate linked rings made of impermeable material? Why are you sure that no planar triangle can have one side whose length exceeds the combined lengths of the other two sides?) Current neurally inspired AI mechanisms cannot discover, or even represent, necessity or impossibility, or understand paragraphs like this. Logic-based mechanisms don’t explain what was going on in mathematical brains before the development of logic in the last few centuries, or squirrel brains, or human toddler brains, e.g. this one: (Skip the introduction.)

The implications for the current wave of enthusiasm for deep learning are potentially devastating—but invisible to people who have never studied Kant, or philosophy of mathematics. Which is not to deny that deep learning can be very useful, if used properly.


[‡] A disorganised collection of additional examples can be found here, with links to many more: (also pdf)

This talk is part of the Artificial Intelligence and Natural Computation seminars series.

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