Infinity in the Seventeenth Century
Hilbert saw the problem of the infinite as central to resolving foundational issues. Perhaps succumbing a bit to hyperbole, he said:
The infinite has always stirred the emotions of mankind more deeply than any other question; the infinite has stimulated and fertilized reason as few other ideas have; but also the infinite, more than any other notion is in need of clarification.[1]
People have pronounced and speculated about what is and isn’t true about infinity since they began thinking abstractly. Aristotle’s views on the subject in particular had a great influence. A discovery made by the Italian mathematician Torricelli in 1641 provides a very revealing example.[2] He found that the volume of a certain solid of infinite extent is finite. The solid in question is obtained by rotating about an axis a certain plane figure with infinite area. Specifically, in modern terminology, it is the figure bounded by the hyperbola whose equation is y = 1 /x, the line x = 1 and the horizontal asymptote of the hyperbola, namely the X-axis. Torricelli’s solid is formed by rotating this figure about the X-axis. Although showing that this solid of revolution has a finite volume is a routine “homework” problem in a beginning calculus course,
at the time it created a sensation because it contradicted prevalent views about the infinite. Torricelli himself remarked “.. .if one proposes to consider a solid, or a plane figure, infinitely extended, everybody immediately thinks that such a figure must be of infinite size.” In 1649, Petri Gassendi wrote,
Mathematicians .weave those famous demonstrations, some so extraordinary that they even exceed credibility, like what .Torricelli showed of a certain.solid infinitely long which nevertheless is equal to a finite cylinder.
Writing in 1666, Isaac Barrow found Torricelli’s result contradicting what Aristotle had taught. He referred to Aristotle’s dictum, “there is no proportion between the finite and the infinite”:
The truth of which statement, a very usual and well known axiom, has been in part broken by .modern geometricians [who] demonstrate .equality of .solids protracted to infinity with other finite .solids which prodigy .Torricelli exhibited first.
Much can be learned from this example about the way in which mathematicians expand the applicability of existing methods to new problems and with how they deal with the philosophical problems that may arise. Torricelli used a technique called the method of indivisibles, a method pioneered by Cavalieri that provided a short-cut for solving area and volume problems. Torricelli used this technique to prove that his infinite body had the same volume as a certain finite cylinder. The method conceived of each of the two bodies being compared as constituted of a continuum of plane figures. Although there was no rigorous foundation for this, Cavalieri and later Torricelli showed how effective it could be in easily obtaining interesting results. They were well aware of the Eudoxes-Archimedes method of exhaustion (which they called “the method of the ancients”), and used it to confirm their results and/or to convince skeptics.[3] But, Torricelli insisted on the validity of the new method.
What can we say about Torricelli’s methodology? He was certainly not seeking to obtain results by “cogent proofs from the definitions” or “in ontological terms, from the essences of things”. He was experimenting with a mathematical technique that he had learned, and was attempting to see whether it would work in an uncharted realm. In the process, something new about the infinite was discovered. I insist that this was induction from a body of mathematical experience.
