Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Many early Greeks, particularly the followers of Pythagoras, were fascinated by the idea that whole numbers and their properties provided the first principles upon which all else could be built. Numbers, to the Pythagoreans, were discrete building blocks, like atoms. One of their basic assumptions was that there was always some indivisible unit that could be used to compare any two quantities in nature.
The Eleatic philosopher Zeno proposed a series of philosophical challenges to the notions of multiplicity and motion that demolished the idea of fundamental units of both space and time. That these arguments are paradoxical is due, in large part, to the role of infinity.
Let's first look at Zeno's arguments against multiplicity. In the Pythagorean view, all things in nature could be measured as multiples of a standard unit. For instance, they viewed a line as a collection of discrete points. An infinite line would be an infinite collection of points, but only in the sense of potential infinity, because it was impossible for any person to create a real infinity. A bounded line, a line segment, would, therefore, be constructed of a finite number of points. Zeno, representing the views of the Eleatic school, argued against this view by pointing out that a line segment of any given length can always be bisected, or cut in half. Such a division creates two line segments, each of which can be bisected again, and again, ad infinitum.
To a Pythagorean, it was perfectly acceptable to think of an indivisible unit, an "atom," with which magnitude could be "built." Hipassus' argument against commensurability complicated this view somewhat. Hipassus showed that there could be no fundamental common unit between the side and diagonal of a square. As we saw in Theodorus' proof using triangles, this idea of incommensurability implies that a magnitude can be divided as many times as one wishes. Accepting the notion that a magnitude, such as a line segment, can be infinitely divided, or bisected, leads ultimately to the conclusion that any fundamental, atom-like unit must have zero length. This creates a paradox: how can one construct a line segment out of pieces that have no length? One can add zero to zero as many times as one likes and the result will always be zero.
The Eleatic view that a line segment can be infinitely bisected requires that the segment be a continuum with no firm boundaries between one location and the next. The Pythagorean view is based on the concept of discrete parts. Hipassus and Theodorus argued against the Pythagorean view, but Zeno presented a series of four situations that undercut both views. Zeno's paradoxes, although primarily constructed to refute the idea that motion is real, simultaneously manage to argue for and against continuous space (and time), invoking infinity and the absurdities that so often accompany it.
Zeno's most famous arguments have to do with both time and space. He showed that viewing space as a multitude of points and time as a multitude of discrete "moments" forces us to believe that motion is an illusion. Common sense argues against this view, but common sense is informed by our senses, which could, in the view of a philosopher in love with rational deduction, be deceiving us. Zeno presented four arguments against motion: The Dichotomy, The Arrow, Achilles and the Tortoise, and the Stade. Let's look at two of these, the first an argument against continuous space, the second an argument against discrete space and time.
The Dichotomy is very similar to the bisecting line argument we saw in the prior section. In Zeno's example, a horse is trying to traverse the distance from point A to point B. Before it can reach point B, it obviously must first cover half the distance. Before it can cover half the distance, it just as surely must cover a quarter of the distance, and so on. If space is composed of a multitude of points, it must cover an infinite number of these points in a finite time, which is contradictory. Hence, by this line of reasoning, the horse can never make it from point A to point B.
This last of Zeno's arguments is more easily understood in a modern example. Suppose that there are three trains, each composed of cars of equal size. Train A is at rest; train B is moving to the left relative to train A; and train C is moving to the right relative to both of the other trains and is traveling at the same speed as train B.
Let's say it takes a time, T, for one car of train B to pass completely by one car of train A.
Because train C is moving at the same absolute speed as train B, it also takes time T for one car of train C to pass one car of train A.
How far do trains B and C move relative to each other in the given time, T? Because train B moves one car to the left and train C moves one car to the right, they move two whole cars relative to each other. On the basis of this reasoning, we would be perfectly justified in defining a new smallest unit of time, , as the time it takes for train C to move one car relative to train B. This effectively treats train B as being at rest, and we could imagine a new train, train D, and repeat the argument ad infinitum.
The point here is that it is contradictory to imagine time as a series of discrete moments, because those moments can be infinitely subdivided.
Philosophers and scientists throughout the centuries attempted to resolve Zeno's paradoxes by a variety of arguments. Some denied that space and time exist in any meaningful sense. Some asserted that space and time are not, in fact, infinitely divisible, and moved on. Others used the paradoxes as evidence that our ability to reason is itself contradictory. Still others regarded the distinction between the many and the one to be false, a concept reminiscent of the Eleatic world view that helped spawn the paradoxes in the first place.
Whatever the putative resolutions, it would be a stretch to call any of them mathematical. Mathematicians after Zeno had to accept the existence of actual infinity, even though it does not make intuitive sense. For example, to resolve the paradox of the The Dichotomy, we can look to the convergence of a geometric series, 1 + x + x^{2} + x^{3}…. It is not hard to show that a general geometric series converges to as long as |x| is less than 1. To do this requires that we examine the behavior of the series as it approaches infinity. Note that a general geometric series begins with 1, so if x = , the sum of the series is then 2.
The Dichotomy paradox essentially presents an infinite sum of terms of decreasing size , which we can recognize to be . However, unlike a general geometric series, the series implied by The Dichotomy does not start with 1. Consequently, the sum of The Dichotomy series is actually − 1, which, with x = , equals 1. In other words, the horse makes it from point A to point B.
So, infinity became a tool that could be used, as long as one didn't look too closely at exactly how it worked. Mathematicians came to accept that one could indeed have a finite limit to an infinite sum. This concept made it possible to arrive at a finite magnitude by summing an infinite number of infinitely small pieces. Such pieces, which became known as infinitesimals, have, in some disturbingly vague sense, arbitrarily small but non-zero magnitudes. The great Newton, one of the fathers of the calculus, the revolutionary new theory of the 1600s that described motion both in the heavens and on Earth, at first based his ideas on these troublesome infinitesimals. It wasn't until the 1800s that Augustine Cauchy turned matters around and developed a sound base for the subject by speaking of limits. This had a profound effect on the concept of "number," for Cauchy also found a consistent way to give meaning to irrational quantities, essentially defining them as limits of sequences of rational quantities. For example, let's return to the mysterious quantity 0.101001000100001000…. Cauchy would define this as the limit of the sequence of rationals, 0.1, 0.101, 0.101001, 0.1010010001, …. This shift of perspective represented a marrying, of sorts, of the potential and the actual infinite, and it brought some logic to the concepts of the infinity of irrationals and the infinite sums that arise in calculus.
As calculus began to assume a larger and larger role in both math and science, the need to understand infinity became greater. This quest for understanding ultimately required a shift in thinking, away from looking at whole numbers and magnitudes, toward thinking about sets. In the next section, we will see some of the fundamental ideas in this new way of thinking.
Next: 3.5 Re-Learning to Count
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