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? Abstract: 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.