Eudoxus and Dedekind Irrational Numbers and Mathematical Development - Essay Example

The theory, as stated, was very oblique and difficult. It was pondered by mathematicians until it was superseded in the nineteenth century. His definition of proportions in Euclid's work exemplifies the struggle taking place in the Greek mind to get a handle on this problem. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples taken in corresponding order.

What could such an inscrutable statement possibly mean It seems that Eudoxus (through Euclid) must have sat up nights trying to write something that no one could comprehend. To understand this statement we must remember two things about Greek mathematics. First, Eudoxus was not talking about numbers, but magnitudes. The two were not the same and could not be related to each other. Second, the Greeks did not have fractions, so they spoke of the ratios of numbers and ratios of magnitudes. Hence, our fraction 2/3 was for them the ratio 2:3.

For their geometry, they also needed to talk about ratios, not of numbers, but of geometric magnitudes. For example, they knew that the ratio of the areas of two circles is equal to the ratio of the squares of the diameters of the circles. We can show this as (Flegg, 1983)(area of circle A):(area of circle B) (radius of circle A)2:(radius of circle B)2 The Greeks had to be sure that when these ratios of magnitudes involved incommensurable lengths, the order relationships held. In other words, would their geometric proofs be valid when such proofs involved ratios of incommensurable lengths The definition developed by Eudoxus was an attempt to guarantee that they would.

The magnitudes in the ratios have the following labels: first: second = third: fourth. Eudoxus said that the first and second magnitudes have the same ratio as the third and fourth if, when we multiply the first and third by the same magnitude, and multiply the second and fourth both by another magnitude, then whatever order we get between first and second will be preserved between the third and fourth. This explanation, simple as it is, can be rather confusing. An example will clarify the matter.

We will assign the following lengths to the four magnitudes: 3:6 = 7:14. From this we get the following inequalities: 3 < 6 and 7 < 14. Eudoxus says that if we multiply both 3 and 7 by any magnitude A and we multiply both 6 and 14 by any magnitude B, then the order relationship between the altered 3 and 6 will be the same as between the altered 7 and 14. Let A = 5 and B = 2. Multiplying we get A3:B6 = A7:B14 or 15:12 = 35:28. Now clearly 15 > 12 and 35 > 28. Hence, multiplying by 5 and 2 preserved the order of the two ratios.

Eudoxus' definition says that for two ratios to be equal, all values of A and B will preserve the order between the corresponding magnitudes. This gave Greek geometry the definition of magnitudes of ratios it needed to carry out the various proofs relying on proportion. However, magnitudes are not numbers, and the requirement that all values of A and B satisfy the definition introduced, through the back door, the notion of infinity. While Eudoxus' work satisfied the needs of geometers, it was