FOM: geometrical reasoning

[Jerry Seligman]

This is in response to Steve Simpson's challenge (23 Feb 1999 21:13:25).

It is common to compare a sentential proof to a chain: the conclusion
is linked to the premises by sound inferences, which together support
the validity of the argument.  A diagrammatic proof is more like dam:
parts are combined to block the counterexamples, until the whole
structure is seen to be watertight.

A simple example is the method of Venn diagrams or Euler circles.
Given a premise, "all As are Bs", one constructs a diagram that
depicts precisely this information.  The content of further premises,
"No C is a B" and "c is a C", is incorporated by adding to and
modifying the diagram according to definite rules, until it depicts
the content of the conclusion, "c is not an A".  An important property
of the proof is that there is no loss of information. At every stage,
the diagram represents all the information in the premises, and the
procedure for drawing a conclusion is simply to "read off" the
diagram.

There are a number of widely-recognised differences between the
informational content of diagrams and sentences.  One is that the
information depicted by diagrams tends to be conjunctive: adding lines
and curves usually involves conjoining new information.  I assume that
this observation is related to Harvey Friedman's claim that his
axiomatisation is "purely geometric" whereas Tarski's is not.  It is
difficult (but not impossible) to represent disjunctive and negative
information using diagrams.  Another difference is that icons, the
diagrammatic equivalent of terms, are usually interpreted so that
distinct occurrences refer to distinct objects. In this way, the
identity relation - or, more accurately, the non-identity relation -
is depicted automatically.  A third difference is that diagrammatic
representations tend to be more compact: a picture is said to be worth
a thousand words.

(Digression: It would be more correct to say that the pictures *we
use* are worth a thousand words, because there are clear examples of
things that are easy to say with words, even the "words" of predicate
logic, but very difficult or impossible to convey with pictures.  Of
course, there is little reason to use pictures when using words would
be more efficient, and so the diagrammatic systems *in use* tend to be
the ones in which there is a clear advantage of expressive economy.)

The economy of diagrammatic representation is responsible for the
basic structure of diagrammatic proof.  The strategy of building up
representations that contain all the information in the premises only
works if the representations remain simple enough to allow that
information to be "read off" easily.  It breaks down when the
representations become too complicated. Just try to draw a Venn
diagram with more than five curves.

Representational economy can also be the downfall of diagrammatic
reasoning, if taken to excess.  In geometric constructions, one has to
be very careful not to read too much into the diagram. The successful
use of a particular triangle to stand for an arbitrary triangle
depends on keeping track of what information can be extracted without
error.  Shimojima has a nice discussion of this point in his article
in the Barwise/Allwein volume.  Geometric fallacies typically result
from extracting information illegitimately.  Nonetheless, the rules of
many diagrammatic systems *can* be made precise, so showing that
geometrical fallacies are no more mysterious than their syllogistic
cousins. (Shin has provided explicit rules for Venn diagrams; Luengo
did the same for geometry; and Hammer addressed a number of Pierce's
systems, some of which come much closer to predicate logic in
expressive power.  See also Hammer's book "Logic and Visual
Information" CSLI, C. U. P., 1996; ISBN: 1881526992.)

There is no general theory of diagram syntax and semantics, so it is
difficult to raise any of the above observations to the status of
theorems, or to see exactly what would have to be done to show that
diagrammatic proofs are all reducible to sentential proofs.  Perhaps
more seriously, there is no general notion of *mathematical proof* to
help us understand what is to be preserved when it is claimed that one
proof is or is not reduced to another.

My own work on diagrammatic reasoning concerns the Venn and Euler
methods.  The agglomerative nature of diagrammatic proofs struck me as
essentially algebraic, and I so identified the structure-preserving
maps between diagrams that underly the various operations used in
proofs.  In this way, I found a structural principle which accounts
for the correctness of Venn/Euler proofs. A Venn/Euler proof is valid
if and only if it yields a diagram that is related to the premises in
certain well-defined ways.  This sort of result is not possible with
systems of sentential reasoning.  There is no purely structural
(syntactic) relationship between premises and conclusion of a proof in
the predicate calculus.  Although the result is confined to a
particular system, I conjecture a fundamental difference between
diagrammatic and sentential proofs: diagrammatic proofs are
characterised in terms of a structural relationship between premises
and conclusion, whereas sentential proofs may only be defined
inductively.


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