Series: Penn State Logic Seminar

Date: Tuesday, November 19, 2002

Time: 2:30 - 3:45 PM

Place: 312 Boucke Building

Speaker: Dale Jacquette, Philosophy, Penn State

Title: Are Irrational Lengths Artifactual?


A paradox concerning irrational lengths is explained. We are
accustomed to think of standard units of measurement as rational, or
as representing rational whole number increments of extension. Among
other units, these are usually specified as 1 (2, 3, etc.) micron(s),
milimeter(s), centimeter(s), meter(s), and the like. We recognize that
there can be practical difficulties in achieving precise measurements,
and that efforts to measure a given extension at best fall within the
margins of a range of accuracy values. The units of measurement
themselves, on the other hand, so to speak, in the abstract, are
supposed to be definite in dimension and in particular to correspond
by definition to rational numbers, stipulatively representing
rationally determinable lengths. If there are rational lengths, there
are also irrational lengths, already known to ancient Greek
mathematics. The Pythagorean theorem entails that the length of the
hypoteneuse of an equilateral right-angled triangle is an irrational
number; where the triangle's side length is 1 meter, the hypoteneuse
is the square root of 2 meters. If, however, the length of the
triangle's sides is the square root of 2 meters, then the Pythagorean
theorem implies that the triangle's hypoteneuse is not irrational but
rational. If we abbreviate the square root of 2 meters by means of an
invented name, such as 1 hoolaboola, then the hypoteneuse will be the
irrational square root of 2 hoolaboolas, but it will also be the
rational length 2 meters or rational square root of 4 meters. There is
no deep para-dox here, to the effect that the triangle's hypoteneuse
is both rational and irrational. We need only explicitly disambiguate
the distinct units of measurement, meters or hoolaboolas, relative to
which the hypoteneuse is respectively rational or irrational in
length. We see this clearly if we take the diagonal of a meter square,
with irrational length the square root of 2 meters, as in doubling a
square's area, making this the side of a new square, whose diagonal as
a result has the rational length of 2 meters. There are nevertheless
interesting conclusions to be drawn from the interrelation between
rational and irrational lengths according to the Pythagorean theorem
in the example. (1) Irrationality is irreducible, in the sense that
either the sides or the hypoteneuse of an equilateral right-angled
triangle must be irrational, regardless of whether the unit of
measurement is meters, hoolaboolas, or any other chosen standard of
length. (2) Whether in particular it is the sides or hypoteneuse of
the triangle that is irrational is an artifact of the unit of measure
stipulated as the length of the triangle's sides or hypoteneuse. There
is thus after all a paradox of philosophical interest to be discerned
in the impications of the Pythagorean theorem and the discovery of
irrational lengths. The paradox is that irrational lengths are not to
be found in nature but are artifactual, determined conventionally by
the stipulations of those who devise unit measures and decide
subjectively whether to apply a standard by which the length of the
sides or the hypoteneuse of an equilateral right-angled triangle are
to be measured by a rational whole number of some chosen value, and
yet irrational lengths are not eliminable by convention, stipulation
or mathematical artifactual policy.  Irrational lengths are
ineliminable despite the fact that they do not occur in nature, but
are essential artifactual. We cannot correctly and without
qualification say, as is otherwise maintained, that the diagonal of
every square or every equilateral right-angled triangle is
irrational. The paradox is explored against the background of an
Aristotelian inherence ontology of mathematical entities, according to
which mathematical objects exist only if and to the extent that they
are exemplified unequivocally in existent spatiotemporal entities.