the new Truth that Black was White and White was Black
Forwarded conversation
Subject: [FOM] A puzzle concerning truth
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From: Vaughan Pratt
Date: Thu, Dec 24, 2009 at 6:34 AM
To: Foundations of Mathematics
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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From: Andrej Bauer
Date: Fri, Dec 25, 2009 at 9:22 AM
To: pratt@cs.stanford.edu, Foundations of Mathematics
When Big Brother declared the new Truth that Black was White and White
was Black, classical mathematicians had no problem adapting, thanks to
the blackwhite principle (known as the Duality principle before the
times of newspeak). They flourished under the new circumstances just
as before, albeit standing on their heads (or maybe not, nobody
remembered how many times Black and White had been switched around).
But when the intuitionistic mathematicians heard of the new Truth,
they wondered about shades of grey. The wanted to know whether almost
black was still almost black, or perhaps it was now almost white. Such
crimethink was quickly disposed of, with several intuitionistic
mathematicians becoming unpersons. The rest kept silent, and always
applied the automorphism x -> 1 - x of the closed interval [0,1]
before they spoke to a classical mathematician about shades of grey,
lest they become unpersons too. But in their hearts they knew that the
true world was not made of just Black and White, or even shades of
grey. They sneaked out of the grey industrial cities into the nature,
where they enjoyed green trees, brown soil, and colorful flowers and
bugs. Little did they know that Big Brother was watching them all the
time. When they were eventually rounded up on charges of crimethink
and sexcrime, their classical brothers could do little to help them,
especially since their blackwhite thinking made it difficult for them
to understand such simple words as green, brown, red and blue. In the
end, all that remained of intuitionistic mathematics was a historical
note about a certain extreme faction of the Brotherhood which denied
not only Ingsoc but also such self-evident truths as the fact that
there were only two colors, Black and White.
Happy holidays!
Andrej Bauer