[FoM] What is a proof?
A.P. Hazen
a.hazen at philosophy.unimelb.edu.au
Tue Oct 28 03:09:16 EST 2003
Baldwin and Simpson have been talking about the notion of an
*explanatory* proof. Baldwin's proposed account of this notion is
that it is a proof in which every statement deduced contributes to
the final result: no obiter dicta. (I hope my paraphrase of his
suggestion is recognizable!) I'm afraid this sounds to me more like
a condition (of efficiency?) on the <i>presentation</i> of proofs
than a characterization of a quality -- explanatoriness -- of the
proofs themselves.
Simpson suggests something that is definitely a condition on the
proofs: a proof is explanatory if the (conjunction of the) axiom(s)
used in proving the theorem is actually equivalent to the theorem.
This is an interesting notion, having to do with what might
informally be called efficiency or economy: no more is used in the
way of axioms than what is strictly needed.
Mark Steiner, a long time ago ("Philosophical Studies" v. 34
(1978), pp. 135-151), published an article on mathematical
explanation, giving examples of proofs that did and proofs that did
not seem explanatory, and characterizing what he took to be the
essential properties of the explanatory ones.* Steiner's proposal
wasn't as precisely stated as Simpson's, but it may be precise enough
to lend itself to fom-ish investigation. AS I REMEMBER IT, he
thought an explanatory proof of a theorem about a kind of
mathematical object, X's, was one which appealed to properties of
X's that generalize to a broader category of objects. Thus,
typically (at least in the examples he gave) a proof of a theorem
about natural numbers is felt to be more explanatory if it appeals to
general semi-ring properties than is a proof of the same theorem
which goes by mathematical induction (and so depends on features
special to the natural numbers). Another example: the proof (whch
any good 10th grader comes up with (grin!)) that there are only 5
Platonic Solids because using hexagons as faces, or more than three
pentagons or squares meeting at a vertex, or more than five
equilateral triangles meeting at a vertex, fails to produce a convex
polyhedron, is NOT explanatory; the topological (graph theoretic?)
proof, which extends to polyhedra with irregular faces or faces of
different sizes, or non-euclidean polyhedra, IS.
Steiner's proposal, unlike Simpson's, seems to presuppose a
notion of one category of objects being a natural generalization
of a narrower one. Since premisses that generalize to a broader
range of cases are going to be logically weaker than premisses that
don't, however, I suspect that Steiner's and Simpson's proposals will
lead to the same judgments as to which proof is explanatory in many
particular examples.
---
* When I checked the "Philosopher's Index" for the reference, I found
three other papers with "Mathematical Explanation" in the titles,
which I have not read: by Mic Detlefsen in "International Studies
in the Philosophy of Science," 1988, by Sandburg in "British Journal
for the Philosophy of Science," 1998, and by Paolo Mancuso in
"Topoi," 2001. Going by the abstracts, the Sandburg article seems
the most likely to be relevant to this discussion: it considers (in
the light of some general, philosophy-of-science, proposals about
explanation) the distinction between explanatory and non-explanatory
proofs.
---
Allen Hazen
Philosophy Department
University of Melbourne
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