[FOM] A puzzle concerning truth
Vaughan Pratt
pratt at cs.stanford.edu
Thu Dec 24 00:34:11 EST 2009
Some axioms of mathematics seem more inevitable than others. Thus
Foundation and Choice lack the apparent inevitability of Singleton (for
every x there exists {x}) and Union.
Yet more basic even than these is that theories form, at a minimum,
filters. If P is true and Q is true, then so is P&Q. And if P is true
and P entails Q then Q is true. Tampering with that principle is surely
like rewiring one's house at random and hoping it still works.
It is hard to imagine a more extreme example of tampering with the logic
of truth than that in Orwell's Newspeak in 1984, where black is white
and white is black.
Orwell imagined that this would change the face of society. Perhaps so,
but when the mathematicians adapted to Newspeak they found that they
were no more or less productive than before. All their theorems were
now false, but somehow this didn't cause the confusion they were expecting.
Eventually the explanation was found. These were classical
mathematicians, for whom Boolean logic enjoys the Duality Principle.
They then realized that their intuitionistic brethren must be in
trouble. Rushing to their rescue, they found them floundering in a
strange mathematics previously unknown to them. Only its classical part
continued to work normally for them.
What does this parable tell us about the inevitability of mathematics,
at least of the classical kind? Is it really so inevitable after all?
Pointers to earlier instances of this parable would of course be
appreciated.
Vaughan Pratt
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