[FoM] What is a proof?

[A.P. Hazen]

    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.

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* 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