Bruce J. Petrie – “The Roots of Transcendental Numbers: A Historical Perspective on the Development of Early Modern Transcendental Number Theory”
November 21 @ 12:00 pm - 1:00 pm
Please join us for our November colloquialism where Bruce J. Petrie will be giving a talk entitled,”The Roots of Transcendental Numbers: A Historical Perspective on the Development of Early Modern Transcendental Number Theory”
Abstract: The term transcendental was coined by Leibniz in 1673 and was applied to mathematical objects such as curves but he did not apply that label to individual numbers. The notion of a transcendental number is precisely defined today and various historians and mathematicians have speculated about the origin of that definition with some proposing that Euler was responsible. That definition is actually found in Lambert’s 1671 Memoire on squaring the circle, a famous problem from ancient Greek geometry. Euler did apply the transcendental label to individual values but his use and understanding of mathematical transcendence were not compatible with today’s definition.
The connection between the transcendental and classical mathematics is prima facie anachronistic. Certainly Lambert used his concept of a transcendental number to resolve a problem from antiquity but it must be understood that the transcendental did not exist within the ancient Greek paradigm of synthetic geometry. That said, mathematicians such as Pappus were cognizant of the limits of their practice of mathematics and some objects that they considered to fall outside of geometry were labelled mechanical and, strictly speaking, were unknowable. Some of these objects that were then unknowable are today called transcendental.
It is through the works of Euler that the transcendental became knowable. The transition from geometric-mechanical to algebraic-transcendental classification reflected a change in how Euler understood the natures and origins of mathematical objects. In Eulerian mathematics, the origins of the transcendental are revealed through the paradigm of algebraic analysis. I also propose a theory which I call inheritance, the means by which the explicitly transcendental nature of operations were passed onto other mathematical objects.