Paradoxes of the Continuum, Part I

Infinity is a slippery concept. Most people tend to find their metaphorical gaze just slides off it, leaving it as something that can only ever be glimpsed, blurry and unfocused, out of the corner of their eye. The problem is that, for the most part, infinity is defined negatively; that is, rather than saying what infinity is, we say what it is not. This, in turn, is due to the nature of the abstraction that leads to the concept of infinity in the first place.

The ideas of succession and repetition are fairly fundamental, and are apparent in nature in myriad ways. For example, the cycle of day and night repeats, leading to a succession of different days. Every such series of successive events is, in our experience, bounded — it only extends so far; up to the present moment. Of course such a series of events can extend back to our earliest memories. Via the collective memory of a society, passed down through written or oral records, it can even extend back to well before we were born. Thus, looking back into the past, we come to be aware of series of successive events of vastly varying, though always bounded, length. We can then, at least by suitable juxtaposition of a negation, form the concept of a sequence of succession that does not have a bound. And thus arises the concept of infinity. Is the concept coherent? Does succession without bound make any sense? With this conception of infinity it is hard to say, for we have only really said it is a thing without a bound. We have said what property infinity does not have, but we have said little about what properties it does have.

Indeed, despite the basic concept of infinity extending back at least as far as ancient Greece, whether infinity is a coherent concept has been a point of bitter debate, with no significant progress made until as recently as the end of the 19th century. Even now, despite having a fairly well grounded definition and theory for transfinite numbers, there is room for contention and differing conceptions of infinity, and in particular of the continuum. Such modern debate divides over subtle issues which we will come to in due course. First, however, it will be educational to look at some of the more straightforward reasons that people have difficulty contemplating infinity: the apparent paradoxes and contradictions that arise.

Some of the earliest apparent paradoxes that involve the infinite are from ancient Greece. Among the more well known are the “paradoxes” proposed by Zeno of Elea. Interestingly Zeno’s paradoxes (of which there are three) were not originally intended to discredit the concept of infinity — on the contrary they assume the coherency of infinity as a concept to make their point. Zeno was a student of Parmenides, who held that the universe was actually a static unchanging unity. Zeno’s paradoxes were intended to demonstrate that motion, and change, are actually just illusions. The paradoxes have, however, come to be associated with the paradoxical nature of the infinite.

The first of Zeno’s paradoxes, the Dichotomy, essentially runs as follows: Before a moving body can reach a given point it must traverse half the distance to that point, and before it can reach that halfway point it must traverse half of that distance (or one quarter of the distance to the end point), and so on. Such division of distance can occur indefinitely, however, so to get from a starting point to anywhere else the body must traverse an infinite number of smaller distances — and surely an infinite number of tasks cannot be completed in a finite period of time?

The second paradox, the most well known of the three, is about a race between Achilles and a tortoise, in which the tortoise is granted a head start. Zeno points out that, by the time Achilles reaches the point where the tortoise started, the tortoise will have moved ahead a small distance. By the time Achilles catches up to that point, the tortoise will again have moved ahead. This process, with the tortoise moving ahead smaller and smaller distances, can obviously occur an infinite number of times. Again we are faced with the difficulty of completing an infinite number of tasks. Thus Achilles will never overtake the tortoise!

The third paradox, the Arrow, raises more subtle questions regarding the continuum, so I will delay discussion of it until later. Taken together the paradoxes were supposed to show that motion is paradoxical and impossible. Few people are actually convinced, however: everyday experience contradicts the results that the paradoxes claim. The common reaction is more along the lines of “Okay, sure. What’s the trick?”. The “trick” is actually relatively subtle, and while rough and ready explanations can be given by talking about convergent series, it is worth actually parsing out the fine details here (as we’ve seen in the past, the devil is often in the details), as it will go a long way toward informing our ideas about infinity and continuity.

Let us tackle the Dichotomy first. To ease the arithmetic, let us assume that the moving body in question is traversing an interval of unit length (which we can always do, since we are at liberty to choose what distance we consider to be our base unit), and that it is travelling at a constant speed. We can show that, contrary to Zeno’s claim, the object can traverse this distance in some unit length of time (again, a matter of simply choosing an appropriate base unit) despite having to traverse an infinite number of shorter distances along the way. To see this, consider that, since the body is travelling at a constant speed, it would have to cover a distance of 1/2 in a time of 1/2, and before that it would cover a distance of 1/4 in a time of only 1/4, and so on. The key to resolving this is that the infinite sum 1/2 + 1/4 + 1/8 + 1/16 + … is equal to 1, and thus the infinite tasks can, indeed, be completed in finite time. This tends to be the point where most explanations stop, possibly with a little hand-waving and vague geometric argument about progressively cutting up a unit length. It is at this point, however, that our discussion really begins. You can make intuitive arguments as to why the sum turns out to be 1, but, given that we weren’t even that clear about what 1 + 1 = 2 means, a little more caution may be in order — particularly given that infinity is something completely outside our practical experience, so our intuitions about it are hardly trustworthy.

Since we can’t trust our intuitions about infinite sums yet, it seems sensible that we should look at finite sums instead. Certainly we can calculate the sum 1/2 + 1/4 = 3/4, and 1/2 + 1/4 + 1/8 = 7/8, and so on. Each of these sums will, in turn, give a slightly better approximation of the infinite sum we wish to calculate; the more terms we add, the better the approximation. The obvious thing to do, then, is to consider this sequence of ever more accurate approximations and see if we can say anything sensible about it. To save myself some writing I will use S_n to denote the sum 1/2 + 1/4 + 1/8 + … + 1/2ⁿ (thus S₂ = 1/2 + 1/4 and S₄ = 1/2 + 1/4 + 1/8 + 1/16, and so on), and talk about the sequence of partial sums S₁, S₂, S₃, …

It may not seem that we’ve made much improvement, having shifted from summing up an infinite number of terms to considering an infinite sequence of sums, but surprisingly infinite sequences are easier to deal with than infinite sums — and we at least only have finite sums to deal with now. The trick from here is to deal with the n^th term of the sequence for values of n that are finite, but arbitrarily large. That means we get to work with finite sums (since for any finite n, S_n is a finite sum) which we can understand, but at the same time have no bound on how large n can be, which brings us into contact with the infinite. In a sense we are building a bridge from the finite to the infinite: any given case is finite, but which term the case deals with is without bound. Before we can get to the arbitrarily large, however, we must first deal with the arbitrarily small.

In some ways it was the arbitrarily small that lead to this problem — the paradox is founded on the presumption that the process of dividing in half can go on indefinitely, resulting in arbitrarily small distances to be traversed. It is precisely this property of infinite divisibility that is a necessary feature of the idea of a continuum: something without breaks or jumps. The opposite of the continuous is the discrete; a discrete set of objects can only be divided into the finest granularity provided by the discrete parts, since any further “division” would involve a reinterpretation of what constitutes an object. In presuming indefinite divisibility we have moved away from discrete collections of objects, and into the realm of continuous things. In the world of the continuous we may talk about the arbitrarily small (a result of arbitrarily many divisions — note the relationship between the infinite and the continuous). What we are really after is a concept of convergence; the idea that as we move further along the sequence we get closer and closer, and eventually converge to, some particular value. That is, we want to be able to say that, by looking far enough along the sequence we can end up an arbitrarily small distance away from some particular value that the sequence is converging to. This, in turn, leads us to the next concept: distance.

We need to be careful here because while the original problem was about a moving object covering a certain distance in the real world, we have abstracted away these details so as to have a problem solely about sequences of numbers. That means we are no longer dealing with practical physical distance, but an abstract concept of distance between numbers. So what does it mean for one number to be “close” to another? We need a concrete definition rather than vague intuition if we are to proceed. Since numbers are purely abstract objects we could, in theory, have “close” mean whatever we choose. There is a catch, however: when talking about numbers we generally assume that they are ordered in a particular way. For example, when arriving at rules for algebra we included rules for ordering numbers. This implicit ordering defines “closeness” in the sense that we would like to think that x < y < z means that y is “closer” to z than x is. Looking back at the rules regarding ordering we find that this means that the closer z − y is to 0, the closer y is to z. That’s really just saying that the smaller the difference between y and z, the smaller the distance between them, and so the definition of distance we need is the difference between y and z! The final catch is that we would like to be able to consider the distance from z to y to be the same as the distance from y to z, but z − y = −(y − z). The solution is simply to say that the direction of measurement, and hence the sign of the result, is irrelevant and take the absolute value to get:

The distance between y and z is |y − z|.

As a momentary aside, it is worth noting that we have defined a distance between numbers to be another number, but that the number that defines the distance is, in some sense, not the same type of number. The number defining the distance is a higher level of abstraction, since it is a number describing a property of abstract objects, while the numbers that we are measuring distance between are describing concrete reality. For the most part these differences don’t matter — numbers are numbers and all behave the same — but as we move deeper into the philosophy of mathematics teasing apart these subtleties will be important. Now, back to the problem at hand…

It is time to put the power of algebra — the ability to work with a number without having to specify exactly which number it is — to use. Let ε be some non-zero positive number, without specifying exactly what number (I’m using ε because it is the traditional choice among mathematicians to denote a number that we would like to presume is very small — that is, very close to zero). Then I can choose N to be a number large enough that 2^N is bigger than 1/ε, and hence 1/2^N is less than ε. Exactly how big N will have to be will depend on how small ε is, but since there is no bound on how big N can be, we can always find a big enough N no matter how small ε turns out to be. Now, if we note that, for any n, S_n = (2ⁿ−1)/2ⁿ (which you can verify for yourself fairly easily) then, if we assume that n is bigger than N, we find that the distance between 1 and S_n is:

|1 − S_n| = |2ⁿ/2ⁿ − (2ⁿ−1)/2ⁿ| = |1/2ⁿ| < |1/2^N| < ε.

That may not look that profound because it is buried in a certain amount of algebra, but we are actually saying a lot. The main point here is that ε was any non-zero positive number — it can be as small as we like; arbitrarily small even. Therefore, what we’ve just said is that we can always find a number (which we denoted N) large enough that every term after the N^th term is arbitrarily close to 1. That is, by going far enough down the sequence of partial sums (and there are infinitely many terms, so we can go as far as we like), we can reach a point where all the subsequent terms are as close to 1 as we like. This is what we mean when we say that a sequence converges. We have shown that the further along the sequence you go, the closer and closer you get to 1. It follows then, due to the way the sequence was constructed by progressively adding more terms to the sum, that the more terms of the sum we add together, the closer the sum gets to one. There is no limit on how close to 1 we can get, since there is no upper limit on the number of terms we can add. In this sense the infinite sum (which has no bound on the number of terms) is equal to 1 (since we are infinitesimally close to 1 by this point).

The key points here were the ideas of distance between numbers, and of convergence, which lets us show in concrete terms that we can end up an arbitrarily small distance away from our intended target, just by looking far enough (and we can look arbitrarily far) along a sequence. These ideas — of defining abstract distance, and of convergence as defined in terms of that distance — will continue to be increasingly important as we progress down this road.

Zeno’s second paradox, about Achilles and the tortoise, can be tackled in a similar manner. Once we abstract away the details of the problem and arrive at the question of whether we can sum together all the times for each ever smaller distance that Achilles must run to catch the tortoise, we find that the same basic tools, involving sequences of partial sums, and convergence, will yield the same kind or result — Achilles will overtake the tortoise in a finite period of time. I leave the proof, and the determination of how long it will take Achilles, as an exercise to the reader. So we have resolved two of Zeno’s paradoxes; in so doing, however, we have developed a much richer theory. I would like to pause and ask you to contemplate what we’ve actually done here. It is easy to get mired in the details, but the bigger picture is truly remarkable. Through the concept of convergence we have built a bridge between the finite and the infinite, between the discrete and the continuous. Convergence provides a tool that allows us to extend our concrete reasoning about the finite and the discrete, step by inexorable step, into the realm of the infinite and the continuous. It is a tool that allows us push out the boundaries of what we can reason about from restricted and mundane confines of everyday experience to the very limits of possibility and beyond: we can reason about a lack a bounds!

When we next deal with this stretch of road we will continue to develop our understanding of the continuum, and the infinite. Next, however, we will start down a different road, and consider other basic abstractions of a finite collection.