[FOM] request to moderator: Was Turing's 1938 intuition/ingenuity distinction unwittingly related to Kant's philosophy of mathematics? [REPOSTING]
Aaron Sloman
A.Sloman at cs.bham.ac.uk
Sat Nov 10 12:59:07 EST 2018
[I accidentally posted this from my home address (blueyonder) not my
academic address a.sloman at cs.bham.ac.uk -- I would be grateful if the
original post could be blocked and replaced by this one.
Sincere apologies.
Aaron Sloman]
I only recently learnt that in his thesis (Systems of Logic Based on Ordinals,
1938, Section 11) Alan Turing had briefly discussed a distinction between
mathematical intuition and mathematical ingenuity, suggesting that computers are
capable only of the latter. So he disagreed with the 'generalised Church-Turing
thesis'.
Trying to work out exactly what he was saying and what the implications are, led
to the conjecture that Turing had unwittingly arrived at a position closely
related to Immanuel Kant's philosophy of mathematics, including Kant's claims
that many important mathematical discoveries are non-empirical, necessary
(non-contingent) and synthetic (not derivable from definitions of the terms
using only logic).
Using mainly examples from ancient/elementary geometry and topology, I've tried
to spell out and defend this interpretation of Turing's distinction, and its
implications, not only for comprehensive foundations of mathematics, but also
regarding limitations of current AI, and inadequacy of research by most
psychologists and neuroscientists concerning mathematical cognition.
(E.g. neither artificial neural nets, nor known brain mechanisms can represent
or establish impossibility or necessity, a fact ignored by most researchers in
the field. Jean Piaget was an exception.)
I've assumed that for the ancient mathematicians, discoveries concerning the
natural numbers required what Turing called "mathematical intuition" because the
natural numbers were not *defined* by any formal system, but understood in terms
of implications and applications of one-to-one correspondences between
sets/collections of items of many types (objects, events, processes, parts of
structures, measures, plans, contents of thought, etc.). These aspects
are ignored by most researchers on human and animal number cognition and the
neural mechanisms used.
A draft discussion is here:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/turing-quotes.html
(or pdf)
Comments, criticisms and relevant references welcome.
Apologies for length -- connections kept emerging.
Note:
I don't think current AI systems are capable of replicating the spatial
reasoning of ancient mathematicians, e.g. Archimedes, Zeno, etc., including
discoveries of extensions to Euclid, e.g. the 'neusis' construction, or the
everyday spatial cognition of human toddlers, squirrels, weaver birds,
elephants, ... and many more.
I won't use a driverless car in cluttered urban environments (e.g. old European
towns and villages) in the foreseeable future.
All comments leading to changes will be acknowledged.
At present I have no publication plans.
Thanks.
Aaron Sloman
http://www.cs.bham.ac.uk/~axs
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/meta-morphogenesis.html
My 1962 defence of Kant (Oxford DPhil):
http://www.cs.bham.ac.uk/research/projects/cogaff/62-80.html#1962
School of Computer Science
University of Birmingham
UK
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