FOM: geometrical reasoning
Jerry Seligman
jseligma at null.net
Mon Feb 22 19:47:23 EST 1999
It might be useful to distinguish two theses:
(1) There is a complete formal characterisation of geometrical reasoning.
(2) The known axiomatisations of geometry in predicate logic provide good
models of geometrical reasoning.
Consideration of the importance of diagrams and the role of intuition in
geometric proofs are prima facie
evidence against (1), but have little weight in the light of known
axiomatisations. Future appeals to geometric intuition may persuade us to
change or add to the list of axioms, but would not count against (1).
(2) is a different matter. There is room for a case to be made against (2)
that does NOT appeal to a faculty of geometric intuition, and so bypasses
the debate about its existence. The case can be made on two grounds:
(i) diagrammatic proofs use a different set of basic concepts than those
employed in the know axiomatisations; and
(ii) the structure of diagrammatic proofs is very different from that of
the usual proof systems predicate logic.
The line of attack from (i) is weak. Any proposal for a new set of
primitive concepts is in danger of being met by an axiomatisation using the
new set, or an argument showing that they are all definable from the old
set.
(ii) presents the strongest threat to (2), especially if the attack is made
in a way that is compatible with (1).
The case could be made if there were detailed formal accounts of the syntax
and semantics of diagrammatic
systems of representation, together with provably complete systems of rules
for manipulating them. Such accounts exist - see, for example, work by
members of Jon Barwise's Visual Inference Laboratory
http://www-vil.cs.indiana.edu/Projects/diagram_logics.html. Typically,
diagrammatic systems of representation are expressively weaker than
predicate logic, and so it is not always possible to establish completeness
by translating into predicate logic.
It is important to see in what way all this is relevant to fom. The issues
raised by (ii) are part of the foundational study of mathematical proofs,
and so are relevant to fom insofar as "mathematical proof" is a basic
concept of mathematics. They are not directly relevant to the systematic
study of the basic concepts of geometry.
Jeremy M. Seligman
Department of Philosophy, The University of Auckland, Private Bag 92019,
Auckland, New Zealand
Tel: +64-9-373-7599 xtn. 7992, Fax: +64-9-373-7408, Time Zone: GMT +13 hours
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