What is a proof?

Leichter, Jerry leichter_jerrold at emc.com
Mon Sep 10 10:52:33 EDT 2007

| If a proof is a record of a mental journey in which one person has
| discovered an important truth, and then made a record of that journey
| adequate so that a second person can walk the same path and see the
| same truth, then cryptography could do with more and better proofs.
Few results are discovered using the same journey as their accepted
proofs - much less than the elegant, beautified proofs that will appear
in the texts a number of years down the road.

| If, on the other hand, a proof is an argument impressively decorated
| with mathematical sounding jargon, cryptography could do with a good
| deal fewer of them.
Many, many years ago, a mathematician I knew asked a bunch of us
then-youngsters:  What's a proof?  Having been brought up in the
formalist tradition, we came back with definitions along the lines
of "a series of statements each of which is either an axiom or can
be derived from earlier lines by such-and-such a process".

His answer was:  No, a proof is something that convinces you of the
truth of something.  Interpreting this a bit more broadly:  A
*mathematical* proof is something that convinces *the relevant
set of mathematicians* of the truth of something.  I don't want to
sound like some post-modernist with statements about the relativity
of truth, but the fact is that this really is what proofs are about.
The standards for acceptable proofs are defined by the specialists
in the field.  If your proof has been vetted by those specialists,
it's accepted.  Until then, it's a *proposed* proof.

The standards of what constitutes an accepted proof have changed
over the years ago.  Consider the statement:  Any polytope has one
side on which it will rest stably.  (You can formalize this as:  If,
for each face, you draw a line from the centroid of the polytope
perpendicular to that face, than for at least one face the line
passes through the face.)  We can prove this by observing that if
this were not true, then if we put the polytope on the ground, it
will roll forever.

In the 19th century, this was accepted as a proof.  More recently,
early in the 20th century, a group of mathematicians in Italy created
the field of algebraic geometry.  No one outside that group really
understood what they were up to or accepted their techniques.  In the
1960's, the area became very "hot", and a number of PhD dissertations
were produced by finding some of the old results, recasting them in
modern terms, and producing proofs that were acceptable in modern terms.

Back in the 70's, Lipton, DeMillo and Perlis argued that program
proving was headed in a fundamentally untenable direction:  What
made mathematical proofs convincing was exactly that they were
reviewed by people who were legitimately convincing.  But it
was highly unlikely that anyone would ever review a proof of a
non-trivial program.  So we would have little reason to believe
in it.

There are a series of papers by Koblitz and Menezes, starting with
"Another Look at 'Provable Security'", that (I think convincingly)
argue that formal proofs in cryptography have been oversold:  The
availability of a formal proof should almost always add little to
your belief in the soundness of a system, and the *absence* of
such a proof should do little to decrease your belief.  They
demonstrate a number of cases where systems have been modified in
order to make formal proofs possible, but one can plausibly argue
that the modifications actually weakened the system.  (The lack of
adequate review of many of those proofs is one reason.  Bellare
and Rogaway's proof of the security of OAEP-RSA was accepted for
seven years - and OAEP-RSA got written into multiple standards -
before Shoup showed the argument was incorrect.  Proofs by
Bellare and Rogaway are well written and understandable.  Even
so, no one apparently looked closely for all that time.  All too
many proofs in the "provable security" match your description of
impressively decorated jargon.  Those will likely never get
reviewed by anyone, perhaps not even the authors!)

							-- Jerry

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