# [FOM] Euthyphro and proof

Vaughan Pratt pratt at cs.stanford.edu
Wed Jan 21 00:37:30 EST 2009

Timothy Y. Chow wrote:
> Keith Devlin uses the terms "right-wing" and "left-wing" to describe the
> two sides of this debate.
>
>   http://www.maa.org/devlin/devlin_06_03.html
>
> I personally don't care for the terms "right-wing" and "left-wing."
>
> My purpose here is not to argue for one side or the other, but to suggest
> that the debate be called a "Euthyphro dilemma."

What terms do *you* use then?  (You didn't say, you just named the dilemma.)

Here are two questions that might help in deciding for oneself which

1.  For any equational class or variety V, is an equation an identity
because it is provable, or provable because it is an identity?
(Identity in the sense of true for all values.)

2.  Does entropy seek a maximum or a minimum?

The answer to the first initially seems to depend on some combination of
nature and nurture.  By nature one may lean towards proof (the first) or
refutation (the second) as the criterion.  Nurture could make you switch
depending on the prevailing ideology of your culture.  Eventually you
learn Birkhoff's theorem for finitary algebras (every HSP-closed class
consists of the models of a set of equations) and its Galois-connection
dual (the completeness of congruence closure as a deductive system) and
conclude that the question is silly, forgetting that you may have left
others behind still leaning one way or the other.  (First-order logic
has its counterpart of course, as does every logic based on 2-valued
satisfiability.)

The second is a much more interesting question, being a statistical
impossibility---impossible to answer, that is.  Physicists are taught
that entropy seeks a maximum at equilibrium, the best statement of the
second law of thermodynamics.  Economists price derivatives under the
assumption that entropy seeks a minimum.  Both laws are good
approximations (though neither can be said to be exact, however
insistent your physics instructor may have been on that point in
dismissing the possibility of a Maxwell's Demon).  Which is correct
depends on whether nature or humankind is in the driver's seat.  It is
interesting to speculate on the possibility of a man-nature balance in
which entropy merely hovers, and whether this is in fact what has been
going on on Earth during say 1000-1700 AD, before which nature was
pushing entropy up even around these parts and after which things may
have started drifting the other way, certainly on Wall Street, but with
the Internet now so close at hand, also Main Street.

With this perspective in mind, one can apply it to Devlin's question
(which incidentally Keith claims to answer, strangely with a conclusion
opposite to the one I thought he had been adducing evidence towards all
along, but perhaps he is just testing us to see in which camp his own
proofs put his readership - do we put more faith in his arguments or his
personal conclusions?).

When the arrow of time is factored into the truth-or-proof question
(defining "truth" in this instance to be what the mathematical community
informs us mere mortals in the manner of the Oracle at Delphi as
correct, and taking proof to be the ultimate criterion for correctness
of arithmetic propositions), the question becomes that of whether the
status of conjectured propositions drifts towards truth or proof with
the passage of time.

I thought everything Keith wrote prior to his surprising (for me)
decision for truth pointed to proof (in the sense above).  But perhaps I
was reading what he wrote through the rose-colored glasses of one who
cherishes truth (in that absolute sense that Goedel has taught us is
unknowable without an Oracle) but reluctantly accepts that in the end
only proof can inform us at least for arithmetic propositions, in that
the fire of arithmetic can only be fought with the fire of proof, itself
an arithmetic endeavor.

(Give me a day or two to get over Obama's address and I should be able
to say all this with shorter words.  Hard not to like that CIC.  For
those who skipped to the end for my own conclusion, the mathematicians
sit on the right.  The left is occupied by the spectators.)

Vaughan Pratt

Mathematics is not a spectator sport.  --Julia Robinson, pre 1970