NOTE added 2 Sep 2021
Notes for a talk to be presented at the MORCOM conference which is part of the IS4SI 2021
summit, can be found here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/sloman-morcom.html
(For presentation on Wed 15th September, to which Mike Levin, Tufts
university, will reply.)
For example ancient mathematicians found multiple proofs of what we call
Pythagoras' theorem long before pythagoras was born!
See
https://en.wikipedia.org/wiki/Pythagorean_theorem
Includes several different proofs of the theorem, e.g. one based on moving
triangles.
Many of these ancient discoveries used methods of reasoning with diagrams that
for many centuries were taught in school mathematics. Examples are presented in
this tutorial:
https://www.youtube.com/watch?v=6Lm9EHhbJAY
by Zsuzsanna Dancso
Unfortunately, since the mid 20thC (except in a few exceptional schools) they are no longer taught. This decision was taken for bad reasons that we can discuss later.
As a result most of our brightest young learners grow up totally ignorant of some of the most important forms of mathematical discovery and reasoning in the history of mathematics, science, engineering, architecture, and other fields of application..
This talk is mainly about the problem of trying to explain what the reasoning powers involved in these discoveries are, and what brain mechanisms make them possible.
I used to think it must be possible to replicate them using AI (though not neural nets) but I now suspect they depend essentially on sub-neural chemical computation mechanisms in brains, which are merely some of the latest developments of an extended chain of chemical mechanisms that have been used ever since life began, long before there were any brains.
For reasons that I'll come back to, it may be impossible to replicate them using processes running on computers, because the chemical mechanisms use a mixture of discrete and continuous processing, and computers are capable only of discrete processing.
But I am not sure of this.
Replicating ancient spatial reasoning using AI is hard
Those geometrical/topological discoveries (e.g. about properties of triangles,
circles, 3D shapes, etc.) cannot be replicated by current neural models using
statistical/probabilistic mechanisms.
They cannot discover, or even represent, impossibility or necessity: those are not points on a probability scale.
They also cannot be replicated by AI theorem provers, that have to start from logically formulated axioms or rules that were neither needed nor used by ancient mathematicians,
It's not just human intelligence that involves recognition of spatial necessity and impossibility.
Varieties of spatial intelligence in squirrels, crows, elephants, octopuses, orangutans, etc., are unlikely to be based on logic and definitions.
There is fragmentary evidence in Turing's 1950 paper suggesting that he thought the required mechanisms of mathematical reasoning, mathematical intuition, made use of sub-neural chemistry (which combines discrete and continuous processes).
In brains, chemistry is at least as important as electricity.
His 1952 paper on chemistry-based morphogenesis of patterns in a surface may have been intended as a precursor to a much deeper theory about brain mechanisms - a development cut short by his death two years later.
I'll give examples of spatial reasoning competences that seem to require such mechanisms and outline a Meta-Morphogenesis research project that aims to continue Turing's investigation, going far beyond current AI, neuroscience, and psychology, exploring possible sub-neural chemical mechanisms and their ancient evolutionary precursors, mechanisms also used in producing an alligator, or a chick, from a fertilised egg.
Neither logic-based AI, nor "deep learning" in neural nets can match these achievements.
(Which is why I would not trust a 'self-driving' car in cluttered narrow streets in busy parts of small towns, as opposed to driving on motorways.)
It is possible that finding detailed explanations of natural spatial reasoning
capabilities will reveal previously unnoticed features of the physical universe
that are essential for development and evolution of an enormous variety of forms
of life.
Physics may not be as simple as suggested by Neil Turok in a lecture at
the Perimeter institute.
Maintained by
Aaron Sloman
School of Computer Science
The University of Birmingham