FOM: objectivity, reality, truth
rhersh at math.unm.edu
Fri Jan 16 19:15:45 EST 1998
Thanks for your interest in extricating me from my predicament.
Let me respond at first to your objection to my claim that mathematics
exists in the shared consciousness of people (especially mathematicians)
and that it differs from other sets of ideas and beliefs existing in
shared consciousness in its reproducibility (which Martin Davis wishes
I had referred to as "consistency.")
You object that this kind of truth or knowledge "depends on us."
First, I would point out that many facts of a social-cultural nature
are objective. For instance, who won the World Series, what is the address
of the New York Stock Exchange, how much you get in your
monthly pay check. Etc. Facts, all plain facts. Not facts
about physical matter, not facts about Transcendental Reality, facts
about the social-cultural reality where we live.
All these facts depend on us, in the sense that they could
be other than what they are. They are in a sense arbitrary. Not
completely arbitrary, for each of them came about by taking into account
various desiderata in a more or less rational way. It's not arbitrary
that the World Series has 7 games, yet it's not really absolutely necessary
Nevertheless, that fact that the World Series has 7 games is
completely certain, as much as whatever theorem you like. The fact
that there wasn't always and won't always be a world series is another
matter. The claim that your theorem always was and always will
tbe true is an assertion of philosophical belief, not
an established fact.
Now, let's talk about the Pythagorean theorem. Of course it's true.
But true in what sense? If you think Hilbert's axioms capture
2-dimensional Euclidean geometry, you can say it's true in the
sense that it follows logically from Hilbert's axioms. (Although
you'd have trouble finding writen down anywhere a complete, logical
depending at all on diagrams.) But maybe the artificial theorem-
proving people in Austin did it, or have good
reason to be sure they could. Then the Pythagorean theorem is
proved within Hilbert's geometry. Does that make it true? Only
as far as Hilbert's axioms are true. I don't think Hilbert or anyone
else claimed that the axioms were true. Only that they were
intuitively acceptable, and served as a "foundation" for
However, Euclid already had a proof of the Pythagorean theorem.
To Euclid, Newton, Gauss and many others, his proof seemed complete.
After Hilbert, we see that Euclid's was always incomplete. Strange that
Euclid, Newton and Gauss could have blundered so trivially!Iin their day
the Pythagorean theorem
was true but they couldn't know it was true. We today do know
it's true, but only as a consequence of Hilbert's axioms.
As an offensive piece of ethnomathematics, I mention that historians
are generally agreed that before the Greeks the Egyptions, Babylonians,
Indians, Chinese and Japaness knew "Pythagoras" theorem. In what
sense did they know it? Not, it seems, as a theorem in a deductive
system, even an incomplete, inconclusive system like Euclid's.
Very likely they knew it empirically, from measuring right triangles.
But of course measuring one right triangle proves nothing about the
next right triangle. From what we know of measuring, we can be sure
that their measurements could not have been 100% precise. But the
Pythagorean theorem is exact, 100% precise. So the Babylonians etc.
not only could't have known the theorem was true, they couldn't even
have made a corrrect statement of it. But they did know something,
Is the Pythagorean theorem about the material world or about
some ideal immaterial things? Euclid already knew it was about ideal
things. But the meaning of it, the interest of it, the use of it, all
have to do with material things. For instance, right triangles
carefully drawn on the ground. No matter how big. But if a triangle
is big enough, the curvature of the Earth will have a measurable
effect, and the Pythagorean theorem becomes false. Moreover, there
is no reasonable way of deciding when it becomes false. To a high
enough degree of precision, it is false even for the most carefully
drawn right triangle on the ground.
Pardon this long drawn out survey. The question now is, in
what sense is the Pythagorean theorem true? In physical application,
it's only apprximately true. In other words, false. In ideal
interpretation, it's true, but true about what? About any object
with 3 vertices satisfying Hilbert's axioms. Was it true about
Euclid's axioms? They are so incoherent that it's meaningless to
say anything about their truth.
Since Hilbert's axioms were concocted with
the inetntion of being a foundation for Pythagoras theorem and the rest
of Euclid's theorems, it seems more accurate to say that Hilbert's
axioms are true because the Pythagorean theormee is true, rather
than vice versa. An axiom set for Euclidean geometry that didn't
yield Pythagoras would be thereby unacceptable.
One way of answering is to say that the Pythagorean theorem
is true independently of Euclid, Hilbert, or anyone else, true
universally and eternally. Go out to Sirius and measure a right
triangle. If Pythagoras isn't satisfied, the triangle isn't flat, or
Now, I say the Pythagorean theorem is true, because I
find the proof convincing, and I know countless thousands of others
have found the proof convincing, and dozens of other proofs have been
found, all with the same conclusion. I conclude that the idea of a
right triangle is so clearly understood by all who venture
into geometry that this idea is held in comomon and shared, so that
different people, thinking along agreed upon lines, about right
trianges, come to the same conclusion--Pythagoras theorem. In other
words, this shared idea is real--in the sense that other shared ideas
are real--and it is held in common so well that different people come
to the same conclusions about it. I consider this an underlying
reality, which accounts for the consensus in people's thinking.
The force of the consensus is based on closely shared ideas.
And the existence of the closely shared ideas is demonstrated by the
Now you can say, that can't be right, because mathematical
truth doesn't depend on us. Well, it doesn't depend on us, in the
sense that we are not free to make it other than it is. In other
words, mathematical problems have solutions that are determined by
the problem itself, not by our wishes. And yet it does depend on us, in
the sense that mathematical ideas are ours. Their locus is in collective
human thinking, but their properties are determined by the nature of
the mathematical ideas. Once we conceive of a triangle, the Pythagorean
true, that's why we are able to discover it. But it is true about the
triangle in our heads, the ideal triangle, and it remains a truth,
remains anything at all, only so long as it is part of our collective
And what about a real physical right triangle made of wood or
aluminum, isn't Pythagoras
theorem true there? Only as a loose approximation and only with
reference to the possibility of measuring the three sides, which
possibility is a human artifact.
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