calculus book - Mathematics and Computer Science

1.3. LOGIC 15
over A. Wiles’ proof of “Fermat’s Last Theorem”. Wiles used techniques
from “hyperbolic geometry”, which the columnist thought was
self-contradictory because “It is possible to square the circle in hyperbolic
geometry”, while every school student of the columnist’s generation
learned that “It is impossible to square the circle”. The columnist
was, presumably, remembering the conclusion of a celebrated theorem
of 19th Century mathematics:
Theorem 1.2. Let the axioms of Euclidean geometry be assumed. If a
line segment of unit length is given, then it is impossible to construct a
line segment of length π in finitely many steps using only a straightedge
and compass. Consequently, it is impossible to construct a segment of
length √ π, that is, to “square the circle”.
As a general lesson, Theorem 1.2 says nothing about the possibility
of constructing such a line segment with tools other than a straightedge
and compass, nor about the possibility of obtaining better and better
approximations with a straightedge and compass, thereby (in a sense)
achieving the construction in infinitely many steps. The relevant shortcoming
in this story is that the theorem says nothing unless the axioms
of Euclidean geometry are assumed. 2
If there is a linguistic lesson to be gleaned from mathematics, it
is that words themselves are merely labels for concepts. While our
minds react strongly to words, 3 it is the underlying concepts that are
central to logic, mathematics, and reality. Good terminology is chosen
to reflect meaning, but it is a common, human, mistake to assume an
implication is obvious on the basis of terminology. Mathematicians
remind themselves of this with the red herring principle:
In mathematics, a ‘red herring’ may be neither red nor a
herring.
Theorem 1.2 is remarkable for another reason: It asserts the impossibility
of a procedure that is a priori conceivable (namely, that is not
2 The columnist’s error was not this glaring; they argued that because theorems
of hyperbolic geometry can be interpreted as statements in Euclidean geometry,
a “hyperbolic” proof is self-contradictory. The resolution to this objection is that
while “squaring the circle in hyperbolic geometry” can be interpreted as a statement
about Euclidean geometry, the interpretation is markedly different from “squaring
the circle in Euclidean geometry”, and does not contradict Theorem 1.2.
3 To the extent that nonsensical rhetoric can be persuasive, or that it is illegal in
the U.S. to broadcast certain words by radio or television.