[FOM] "Mere" correctness of a proof
Richard Kimberly Heck
richard_heck at brown.edu
Tue Sep 4 21:56:22 EDT 2018
On 09/04/2018 03:22 AM, José Manuel Rodriguez Caballero wrote:
> David Fernandez Breton wrote:
> > Although correctness in mathematics is not negotiable, and it
> > constitutes a "sine qua non" of mathematical practice, I believe (and
> > I'm sure others will agree) that correctness, by itself, is worth very
> > little, if it's not accompanied by some aesthetic appeal and sense of
> > understanding.
I don't think this is merely aesthetic. There's a difficult to
understand but
nonetheless very real sense in which a good proof doesn't just show that its
conclusion is true but does something like *explain why* its conclusion is
true. That may bring a sense of "understanding", but then again many
theories
of scientific explanation tie explanation to understanding. That is
part, too, of
why we routinely give and study multiple proofs of the very same theorem,
e.g., as Jamie Tappenden mentions somewhere, the three different proofs of
the Stone Representation Theorem in Herstein's _Abstract Algebra_.
One can also get a sense for the importance of explanatoriness by looking at
proofs that are wholly unexplanatory. My own favorite example (shared with
me many years ago by George Boolos) is the following, originally due to
Mieczysław Warmus. Say that a sequence of reals is "good" if the first
two reals are in different halves of [0,1]; the first three (including
the first
two) are in different thirds of [0,1]. the first four in different
fourths; etc.
Question: Is there an infinite "good" sequence? Or is there only a
sequence of
length n for any n? Or is there some bound on how long the sequence can
be? I won't spoil the fun by saying the answer. See
https://tinyurl.com/ycy24j23
But the proof is brute force calculation and does nothing to answer the
question why things should be as they are.
George also told me that Hilary Putnam expressed a simlar view about the
proof that cardinals do not collapse in Cohen's original model for ZF + ~CH.
As George related the story, Hilary is said to have remarked that the proof
was wholly unexplanatory and that the only reason the conclusion was true
was that God wanted CH to be independent.
Riki Heck
--
----------------------------
Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University
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