Are mathematical proofs incomparable with proofs in other disciplines?

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Vaughan,

I agree with your definition:

"A proof is sufficient evidence for the truth of a proposition,"

The article

http://en.wikipedia.org/wiki/Proof

does not discuss the idea (of Paul Lorenzen) that a mathematical proof 
is essentially a winning strategy in a formal game.
I first learned the idea from Andreas Blass
who introduced the game semantic of linear logic,

http://arxiv.org/abs/math/9310211

A proof can be viewed as an argumentation to convince others of the validity of a statement.
In mathematics, the argumentation must be solid enough to resist
any conter-argumentation by an ideal opponent.
It can be compared to a winning strategy in a game with two players, 
one defending a statement and the other attacking it.
Lorenzen associates to every mathematical statement S a formal game with two players G(S),
the defender and the attacker. The defender has a winning strategy iff the statement has a formal proof.
The rules of the games for a proof in intuitinistic logic differ
from the rules for a proof in classical logic.
In other words, the rules of the games are determining the logic and vice versa.

I believe that game semantic is putting some light on the origin of logic.
I guess that logic was discovered by peoples debating in a democratic manner.
All communities need to choose between different courses of actions. 
There are many answers to the question: how should this choice made? 
One was given by Plato who favored a government 
by the "philosopher king" who "loves the sight of truth":

http://en.wikipedia.org/wiki/Plato#The_State

Plato does not like Athenian democracy because it is imperfect. 
He observes that its political debates are manipulated by sophists. 

I agree with Plato that democracy is imperfect.
But it should be improved, not condemned.

Logic is anti-authoritarian since it wishes to convince, not to coerce.


Best,
André



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