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new opportunities, new kinds of truths. Too bad almost
everybody steered clear of these issues for more than
two millennia. His paradox demonstrates how impor-
tant it is to keep separate the language of maths (the ut-
terances and symbols we use to express mathematical
statements) from the metalanguage, the language we
use to talk about mathematical statements, to evaluate
and prove them. Both paradoxes stem from the use
of the metalinguistic element false (or its derivative
liar in the case of Epimenides) in a statement whose
truth we wish to evaluate. To see the key role played by
this element in the paradoxes, replace liars by adul-
terers in Epimenides paradox, replace false with
short in Eubulides . The paradoxes go away. Meta-
language and language make for an explosive mix.
Another key ingredient in Eubulides paradox is the use
of the phrase this statement, a magical spell that cre-
ates self-reference, introspection. An utterance that
speaks about itself dangerous stuff. Fortunately, the
language of mathematics is free of such introspective
capabilities, isn t it? I mean, you don t expect a geomet-
ric statement, a sketch of lines and circles, to start talk-
ing to you about its own correctness, do you? Or a
theorem about the whole numbers to proclaim its own
unprovability? Well, so we thought for the longest
time . . .
30 Christos H. Papadimitriou
Anyway, on with our story. I believe that you know well
the historical epoch that we have reached. 200 B.C. Dif-
ficult times. Greece has expanded from Italy to India
and is now shrinking. Fast. Its socioeconomic system is
crumbling what with the dwindling population of
slaves, the limitations of its agriculture, the difficult co-
existence of the archaic with the asiatic mode of pro-
duction. Mass urbanisation, the rising prices of grain,
lawlessness and piracy, competition and military pres-
sure from both east and west. Not a good environment
for seeking truth, is it? What a powerful metaphor, the
illiterate Roman soldier passing a sword through Archi-
medes, as the prince of ancient mathematicians was ab-
sorbed by the sketch he had drawn in the sand. His last
words were You fool, you ruined my circles. How I
wish I knew what he was calculating at the time or
was he designing something?
But how about the Neoplatonic philosophers?
Alexandros interjects.
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Yes, they did try hard, didn t they? But, as Plato himself
would have observed, what a disappointing version of
the original ideal. Now a new kind of truth is conquering
the souls, focusing almost exclusively on questions of
theology and ethics, its methodology based on faith and
mysticism. The world kept going, of course, a great em-
pire rose and fell, the victorious German tribes built a
new, more efficient economic system on its ruins. But,
all said, very little progress on the truth front, as you and
I understand it.
And all for the better, if you ask me. By 200 B.C., Greek
philosophy had run its course, had reached its limits.
You see, all ancient Greek philosophers, for all their
dazzling diversity of ideas, subscribed to certain work-
ing hypotheses about the truth that they were seeking:
Turing (A Novel about Computation) 31
First, the truth would be simple and elegant, easy once
you think of it. Four elements; unsplittable atoms; ide-
als and their images; either all change or no change
at all you get the idea. Second, all Greek philoso-
phers seemed to expect that pure thought and pas-
sive contemplation of the world is all that is needed
to discover truth. Both of these assumptions were
tremendously influential and productive, and quite
appropriate for that age. But, ultimately, they were
terribly limiting. Modern science succeeded in ex-
panding our knowledge of the truth only through ac-
tive interaction with the world, by experiments that
either support or topple theories. And we are con-
stantly discovering that the world is more compli-
cated than we had thought and wished; truth-seekers
today must be prepared for truths of mind-boggling
complexity often complexity is the only truth there
is. For real progress to be made, ancient wisdom had
to be forgotten, buried deep in darkness. Like a seed.
And, as you know, that is exactly what happened.
32 Christos H. Papadimitriou
HUBRIS
There is a long pause in the stream of text flowing from
the screen. Alexandros lights a cigarette. This last part
of the lesson has struck close to home. His life s work is
about a strange artifact from exactly that period. The
Kythera gearbox seems to be a voice from the end of the
third century B.C. crying out against the decline of the
Hellenic world, an ill-fated force struggling in vain.
But Turing s lecture has now resumed:
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Who had expected the winter to be so long, who would
think that the spring would bud so forcefully. We are in
Italy, in the fourteenth century A.D. Suddenly the feudal
system is collapsing, the cities are thriving, the bur-
ghers are rich and powerful. The rulers long for the
company of accomplished artists and wise thinkers.
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