[FOM] Euthyphro and proof

[Vaughan Pratt]

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 
wing to sign up for.

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