FOM: Geometric proofs

Neil Tennant neilt at
Tue Feb 23 16:43:00 EST 1999

The interesting discussion currently under way on fom about the relationship
between diagram-mediated informal proofs in geometry, and their formalizations
as proofs in axiomatic systems of elementary geometry, prompts one to consider
the following rough analogy.

Let me set this analogy up by first describing some stages in the evolution of
philosophical positions concerning the relationship between mind and body.

For a long time followers of Descartes took the mind to be a separate entity,
distinct from the body (hence also from the brain plus nervous system). The
mind and the body *interacted*, as discrete entities. Hence the label
`Interactive Dualism'.  The view recognized the causal efficacy of mental
states, but at the cost of offering no account of the mechanism of that

Then came philosophical attacks on this view, in the age of a more mature
physical science. Ryle accused Descartes of making a `category mistake', and
re-conceptualized mind as a set of dispositions to behavior, thereby creating
the position known as Logical Behaviorism. The Logical Behaviorist interprets
the attribution of mental states as a device for anticipating or explaining
bodily behavior in various circumstances.

This view was subsequently refined by Putnam into the position known as 
Functionalism (which Putnam has since abandoned). According to Functionalism,
a mental state is to be thought of as like a state of a Turing machine. 
`Mind' is the program, or state transition table, of the machine. The machine
(that is, the brain + CNS) receives its inputs from sensory transducers, 
and its outputs are signals to the motor system. Mind (the program) mediates 
between perception and action. In principle, any hardware could realize
the program. Thus robots, silicon-based life forms, etc. could have minds.

A roughly contemporaneous and competing account was Davidson's
doctrine of anomalous monism. The `monism' part adverted to the belief
that there really is only one Reality, and it is wholly physical. Moreover,
that physical reality is causally closed. The `anomalous' part is that
some parts of that physical reality---namely, human bodies---behave in such
a way as to be better understood in terms of mental states and events: 
perception, belief formation, desire, intentional action.  Same physical
reality, but under a description that avails itself of mental attributes.
Presumably, given the causal closure of the physical, the same events could
be explained in wholly physical terms; but that would be awfully awkward
and unwieldy. One makes better sense of people's behavior by employing the
mentalistic idiom.

More recently, there has been the extreme development that could have been
anticipated from all this: the Eliminative Materialism of Paul and Patricia
Churchland. According to EM, it's a waste of time to persist in cranking out
mentalistic descriptions or explanations of human behavior. If it's all
really just physical, why, then, we should expend our scientific efforts
on perfecting the underlying neuroscience that would divulge the Real Truth
about what makes people tick. We should talk in terms of nerve-firings rather
than in terms of headaches. So-called Folk Psychology, the everyday mental
idiom, will prove itself to be a false theory, based on a conceptual scheme
that will one day be obsolete, because of its inability to compete with more
comprehensive and accurate explanations and predictions of human behavior 
that will be based on some future neuroscience.

The analogy I want to suggest is this:


	Informal diagrammatic   <---->  Mental events/states

	Formalized first-order  <---->  Physical events/states

In the domain of mind/body, a phenomenon is real only if it is physical;
while it has experiential significance for the person concerned only if
it is mental. A healthy person is not aware of their pancreatic secretions,
but is certainly aware of those nerve-firings/secretions/neurochemical
fluctuations or what-have-you, that go to make up a migraine.  They are
aware of the latter, however, as splitting headaches, not as the physical
processes that, necessarily, correspond (?= are identical) to the having of 
such a headache. Knowing what it is like to have a migraine is knowledge 
that could never be attained simply by studying the neurophysiology of 
migraine sufferers.  The essential dimension of human suffering would 
simply be missing.

In the domain of geometry, a demonstration is sound only if it is formalizable
in the appropriate elementary axiomatic system; while it is intuitively 
comprehensible, convincing, and suggestive of further developments for the
thinker concerned only if it is presented as an informal piece of diagrammatic
reasoning. A good intuitive geometer does not have to be aware of the existence
of the abstract proof-types afforded by the axiomatic system, but is certainly
able to follow the diagrams, and constructions within those diagrams, that go 
to make up a satisfying demonstration in the manner of Euclid.  They are aware 
of these thought-processes, however, as geometric demonstrations, not as the 
formal axiomatic proofs that, necessarily, correspond to them. Knowing the 
grounds for the truth of a geometric claim is knowledge that could never be 
attained simply by checking the correctness of a completely formal proof in 
an axiomatic system of elementary geometry. The essential dimension of 
geometrical grasp or understanding would simply be missing.

Neil Tennant

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