Incommensurable magnitudes

The Greek discovery of incommensurable magnitudes changed the face of mathematics. At its most basic level it shed light on a glaring contradiction within the then current Greek conception of mathematical thought, which eventually resulted in a reformulation of both the methods and practice of mathematics in general. These reformulations brought about a new era in mathematics, and were the first stepping stones of some of our most important modern day conceptions, such as calculus.

The Greeks’ first dealings with incommensurables were borne by the Pythagoreans. Resulting from their intricate dealings with geometrical figures, they began to realize that “some ratios – for example, the ratio of the hypotenuse of an isosceles right triangle to an arm or the ratio of a diagonal to the side of a square – cannot be expressed by whole numbers.” [Kline, M. (1990). "Mathematical Thought from Ancient to Modern Times", Vol. 1. New York: Oxford University Press. (Original work published 1972). p. 32.] The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However Hippasus of Metapontum, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with an arm, then that unit of measure must be both odd and even, which is impossible. His reasoning is as follows:

- The ratio of the hypotenuse to an arm of an isosceles right triangle is a:b expressed in the smallest units possible.

- By the Pythagorean theorem: "a"² = 2"b"².

- Since "a"² is even, "a" must be even as the square of an odd number is odd.

- Since "a":"b" is in its lowest terms, "b" must be odd.

- Since "a" is even, let "a" = 2"y".

- Then "a"² = 4"y"² = 2"b"²

- "b"² = 2"y"² so "b"² must be even, therefore "b" is even.

- However we asserted "b" must be odd. *Here is the contradiction.* [Kline 1990, p.33.]

The Greeks termed this ratio of incommensurable magnitudes alogos, or inexpressible, but did not give Hippasus the respect he deserved. It is said that he made this discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios.” [Kline 1990, p. 32.] As can be noted by the actions of his fellow shipmates, Hippasus’ discovery posed a very serious problem to the Pythagorean notion of mathematics. Before this point, number and geometry were inseparable and formed the foundation of their theory. But with the discovery of numerically inexpressible geometric phenomena this foundation was shattered, bemusing mathematicians for many years to follow.

The discovery of incommensurable ratios was indicative of another problem facing the Greeks: the relation of the discrete to the continuous. Brought into light by Zeno of Elea, he questioned the conception that quantities are discrete, and composed of a finite number of units of a given size. Past Greek conceptions dictated that they necessarily must be, for “whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects.” [Kline 1990, p.34.] However Zeno found that in fact “ [quantities] in general are not discrete collections of units; this is why ratios of incommensurable [quantities] appear…. [Q] uantities are, in other words, continuous.” [Kline 1990, p.34.] What this means is that, contrary to the popular conception of the time, there cannot be an indivisible, smallest unit of measure for any quantity. That in fact, these divisions of quantity must necessarily be infinite. It seems like common knowledge to us nowadays, but consider a line segment: this segment can be split in half, that half spit in half, the half of the half in half, and so on. This process can continue infinitely, for there is always another half to be split. The more times the segment is halved, the closer the unit of measure will come to zero, but it will never reach exactly zero. This is exactly what Zeno sought to prove. He sought to prove this by formulating four paradoxes, which demonstrated the contradictions inherent in the mathematical thought of the time. While Zeno’s paradoxes magnificently demonstrated the deficiencies of current mathematical conceptions, they were not regarded as proof of the alternative. In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another, and therefore further investigation had to occur.

The next step was taken by Eudoxus of Cnidus, who formalized a new theory of proportion that took into account commensurable as well as incommensurable quantities. Central to his idea was the distinction between magnitude and number. A magnitude “was not a number but stood for entities such as line segments, angles, areas, volumes, and time which could vary, as we would say, continuously. Magnitudes were opposed to numbers, which jumped from one value to another, as from 4 to 5.” [Kline 1990, p.48.] Numbers are composed of some smallest, indivisible unit, whereas magnitudes are infinitely reducible. Because no quantitative values were assigned to magnitudes, Eudoxus was then able to account for both commensurable and incommensurable ratios by defining a ratio in terms of its magnitude, and proportion as an equality between two ratios. By taking quantitative values (numbers) out of the equation, he avoided the trap of having to express an irrational number as a number. “Eudoxus’ theory enabled the Greek mathematicians to make tremendous progress in geometry by supplying the necessary logical foundation for incommensurable ratios.” [Kline 1990, p.49.]

As a result of the distinction between number and magnitude, geometry became the only method that could take into account incommensurable ratios. Because previous numerical foundations were still incompatible with the concept of incommensurability, Greek focus shifted away from those numerical conceptions such as algebra and focused almost exclusively on geometry. In fact, in many cases algebraic conceptions were reformulated into geometrical terms. This may account for why we still conceive of x² or x³ as x squared and x cubed instead of x second power and x third power. Also crucial to Zeno’s work with incommensurable magnitudes was the fundamental focus on deductive reasoning which resulted from the foundational shattering of earlier Greek mathematics. The realization that some basic conception within the existing theory was at odds with reality necessitated a complete and thorough investigation of the axioms and assumptions that comprised that theory. Out of this necessity Eudoxus developed his method of exhaustion, and kind of reductio ad absurdum which “established the deductive organization on the basis of explicit axioms…” as well as “…reinforced the earlier decision to rely on deductive reasoning for proof.” [Kline 1990, p.50.] This method of exhaustion is said to be the first step in the creation of calculus.

Notes

References

*Zeno's Paradoxes and Incommensurability. (n.d.). Retrieved April 1, 2008, from http://www.dm.uniba.it/~psiche/bas2/node5.html