I would like to get temporarily sidetracked. The straight road is often not always the best road, at least not when the journey matters at least as much as the destination. Basho often took sidetracks, and they often provided some of the highlights of his journeys. Wandering off track gives you the opportunity to learn a little more about the country that you are travelling through; that’s precisely the sort of sidetrack we’re about to take.
We’ve spent some time contemplating and discussing the intricacies of the infinite. We started off with a very natural abstraction, and quickly got lead into a mire of technicality and complexity. With a little work we came through unscathed and ended up with a new appreciation for the unfolding beauty and complexity that we were lead to. All of that, however, was a matter of “head in the clouds” contemplation as to what it truly meant to be infinite, and what the continuum actually was. Now that we have that understanding it is time to wander off the path and explore the countryside that such understanding begins to open up for us.
We are going to take a rather circuitous route through all of this, so please bear with me as we wander in odd directions. I want to start by picking up from Paradoxes of the Continuum, Part I and Zeno’s paradoxes — in particular it is finally time to have a look at the previously undiscussed third paradox of Zeno, knows as The Arrow. The paradox generally runs as follows: picture an arrow speeding toward its target. Now imagine freezing time to just a single moment as the arrow travels; the arrow will be stuck, stationary and not moving. And yet this will be true at each and every moment of the arrow’s flight — the arrow is always stationary! Thus, would conclude Zeno, the concept of movement and change is false; merely an illusion.
As with Zeno’s other paradoxes, also meant to demonstrate that movement is an illusion, most people aren’t convinced. It sounds all well and good, but I think we all tend to suspect there’s a trick somewhere in that reasoning. And indeed there is, but it is a subtle one, and teasing it out will require our new understanding of the continuum. To start, however, we’re going ask some questions about speed.
We tend to have intuitions about speed (or more correctly velocity, but I’m going to avoid vectors for now), and it is these inexact intuitions that Zeno preys upon. We think of speed as the rate of movement of an object. An object that has zero speed is stationary — not moving. But how do we know the speed of an object? We observe its change in position over time, and then normalize that value based on the span of time we were observing the object for. This works well enough, giving an average speed over reasonable spans of time, but it fools us when we have to deal with Zeno’s thought experiment: If we freeze time then surely the change in position will be zero, and doesn’t that mean that the speed will be zero? Well no, contrary to intuition the instantaneous speed isn’t zero; to see that, however, we need to think about time as a continuum.
This is the heart of the trick in Zeno’s third paradox: he effectively manages a straddle, viewing time as either discrete or continuous at different times to suit his needs. If we think carefully, and view time as truly continuous, the problem will evaporate.
Given a span of time it is easy enough to calculate a change in position by taking a difference (that is, subtracting the first position from the second). What are we to make, however, of a change in position when there is no span of time involved? What we need, is a continuous difference between positions — the difference as positions change smoothly and continuously. So let’s think about a moment in time: it is a point on the continuum. We have, however, learned about continuums, and found that a point on a continuum is really an infinite sequence of ever more accurate approximations — a Cauchy sequence. That is, a point on the continuum is an infinite sequence of rational points that “home in” around the desired point; for many points on the continuum (indeed, for most of them!) we can only specify them as an infinite sequence of approximations that get ever closer to, but never quite reach, the point. More importantly, however, we learned that there are many different Cauchy sequences that all refer to the same point (in much the same way as there are many fractions that refer to the same ratio): we simply pick a representative sequence. We could, of course, choose two different representative sequences for the same point. If we’re careful we can choose sequences that have no terms in common — perhaps one sequence is consistently increasing, each term larger than the next, but by a smaller and smaller amount, toward our desired “limit point”, while the other is consistently decreasing, each term slightly smaller than the next, down toward the same desired “limit point”.
To make this a little easier to talk about we’ll provide some labels. Let
t1, t2, t3, t4, …
be the moment in time expressed as the sequence tending from below, and let
T1, T2, T3, T4, …
be the same moment, expressed as the sequence tending from above. Now, for each term in each sequence, we’ll have an associated position of the arrow at that time (the term is, in a sense, shorthand for a constant sequence). Thus for the first sequence we’ll have a sequence of positions:
p1, p2, p3, p4, …
and for the second sequence we’ll have a different sequence of positions:
P1, P2, P3, P4, …
With all of that in mind we can set about calculating the speed of the arrow at the chosen moment in time. How do we do that? Well we can certainly find the difference between positions p1 and P1, and since t1 and T1 are different we can normalize against the span of time between those positions to get a speed — call it s1. That is, we can calculate
s1 = (P1 – p1)/(T1 – t1)
with solid assurance that neither the numerator nor denominator are zero (we specifically chose our sequences tn and Tn so this would be the case). Indeed, we can do the same calculation for each set of terms, and find
s2 = (P2 – p2)/(T2 – t2)
s3 = (P3 – p3)/(T3 – t3
and so on, giving us an infinite sequence s1, s2, s3, …, which shouldn’t be that daunting since, if we were being realistic, we were expecting speed to be a continuum as well, and thus we expect speeds to be, ultimately, Cauchy sequences too. So is the sequence Cauchy — is it an ever closer approximation of some single value? In the case of our arrow, the answer is yes. Does the sequence get closer and closer to zero? In the case of our arrow, the answer is no. And so we have a non-zero speed that we calculated by finding the change in position between a moment, and itself (normalized, of course, against span of time over which we observed the change). In other words, at any given moment the arrow isn’t actually stationary — it has some non-zero speed.
There is a technicality that it is worth being aware of: for all of this to make sense we want to know that the speed we get is independent of our choice of Cauchy sequence to describe the chosen moment of time. That is, we don’t want to find that, if we choose two new and different Cauchy sequences that express our chosen moment and proceed with the calculation as above, we end up with a different speed as the result. We can, of course, end up with a different sequence, as long as the sequence tends to the same limit (recall that the same point on the continuum can be expressed as many different Cauchy sequences). Clearing this particular hurdle will be easier later when we have better mathematical machinery and notation, so for now it will suffice to say that, presuming our arrow travels as any normal arrow, we won’t have an issue here.
The real question, however, is: what just happened? With a wave of my hand and a dance of symbols we’ve apparently made the problem disappear, but I can certainly forgive you if you don’t feel any more enlightened as to where Zeno’s paradox breaks down. Let’s go over it all again, but at a higher level, to try and get a feel for what’s really going on here.
Zeno wants us to conclude that the arrow is frozen in a moment in time, that it is stationary. We intuitively think this is the case because we presume that, in a moment of time, the change in position of the arrow is zero, and from this we think of it as having zero speed. The catch is that, for speed to make sense, we require the moment of time in which the arrow is frozen to be non-zero — that is, a moment should capture some span of time. We can think of the moment that way because we are used to thinking of things in terms of discrete units, and we intuitively associate a moment with a discrete unit of time. We think of it as the smallest possible span of time; the fundamental tick of the universe that takes us from one moment to the next. The problem is that time is a continuum, and continuums don’t work that way: there isn’t any clear “next moment”; more specifically, as noted in Paradoxes of the Continuum, part II, the continuum contains incommensurable points — there simply is no fundamental unit for continuums, it cannot exist.
Now, in Paradoxes of the Continuum, Part II we made sense of the continuum, and these baffling points, by working in terms of Cauchy sequences. The points of the continnum are, in some sense, evanescent; we cannot pint them down, but instead must specify them as an infinite sequence of progressively more accurate approximations. Or, to put it another way, a point on the continuum, a moment in time, is a completed infinite, and as we saw in A Transfinite Landscape we must be careful of our intuitions when dealing with completed infinities.
Now, just as a moment is expressible as an infinite increasingly accurate approximation, the speed of the arrow in that moment (which exists in the continuum of speeds — since we expect speeds to change smoothly just as time does) will also be expressible as an infinite increasingly accurate approximation. How do we find increasingly good approximations of the speed at a given moment? By building each approximation from approximations of the moment in time! This is, in essence, exactly what we have done above. The evanescent nature of points on the continuum — that they can only be apprehended as ever better approximations rather than clear discrete points — is the catch here. Zeno’s paradox relies on us failing to fully grasp this bizarre aspect of the continuum.
To defeat the paradox we have shifted from thinking in terms of discrete differences in position, to continuous (normalized) differences in position — differences that don’t require a discrete “next moment”. A natural question begins to arise: if we can take continuous normalized differences instead of the usual discrete differences, can we evaluate continuous normalized sums instead of the usual discrete sums? And what do we mean by that anyway? It is time to follow our nose, and attempt to resolve these questions, for in doing so we will eventually come to a better understanding of continuous differences as well!
Before trying to work out what we mean by a continuous normalized sum, let’s sort out what a discrete normalized sum looks like, and what would have to change for it to be a continuous sum. A discrete normalized sum is, in practice, an average. If we have a list of populations for each country in the world we can find the average population for a country. To do that we sum up all the populations, and divide by the number of countries.
That covers discrete normalized sums — so what does it mean to be continuous? All we need to do is shift to values that are spread over a continuous rather than discrete domain. Suppose we have a metal bar with a temperature that varies (continuously!) along it’s length. We can reasonably ask for the average temperature of the metal bar as a whole. Now, however, we don’t have discrete values as we did with the populations of countries; the temperatures are spread over a continuum. Our usual methods of summation won’t work — we need a continuous normalized sum!
As with continuous differences we should expect to find our answer takes the form of an infinite sequence of ever better approximations since we expect our answer to, itself, be a value on a continuum. How do we arrive at these approximations however?
If we look back at how the continuous differences case worked, we see that we arrived at the desired continuous difference via a series of ever better approximations using discrete differences. Following that line of thought, we should look to create discrete sums that approximate our desired continuous sum. To create a discrete sum we can simply pick a finite number of points along the bar, sum up the temperature values at those points, and normalize by dividing by the number of points on the bar we picked — that gives us a discrete normalized sum that we can calculate. Now, obviously in picking merely a finite number points to sample the temperature at we are getting only an approximation; if we add more points to our existing sample, however, the approximation will improve. Thus if we choose to sample the temperature at more and more points for each successive approximation we will have a sequence of approximations that will get more and more accurate. It’s not hard to see that such a sequence will be a Cauchy sequence, and thus we have specified a point on the continuum of temperatures that will be the average temperature of the bar.
As with continuous differences, we again have the technical issue: in this case ensuring that which finite set of points we choose for each approximation doesn’t effect the final result; that is, at each stage we have a choice of which (and how many!) new points to add to our sample, and we want to be sure that our choice of points doesn’t actually change the end result: the Cauchy sequences of approximations are allowed to differ, but their limit, the point of the continuum that they specify, must be the same. For now that’s a little to technical to get into, but suffice to say that, with sufficient care, we can surmount this issue.
Another issue is that of why we are using normalized sums; why not just use ordinary sums instead? This comes down to the nature of continuous sums, and can be elucidated by our examples of populations of countries and the temperature of a metal bar. In the discrete example, summing populations of countries, there is a clear way of finding the ordinary sum, the total population of all countries. We can do that because we have a clear sense of units that we are summing over: a country is a unit, each country has a population value, and all of those population values are of equal weight with regard to one another. Indeed any discrete sum, by default, defines the units that are being summed over, since we can regard each discrete value as a discrete unit. In contrast, when we have a continuous domain to sum over, such as the metal bar, there are no basic units since such a thing simply does not exist for a continuum! Given that we have no units to sum over, and thus cannot ensure that the values we are summing are of comparable weight to one another, a standard sum no longer quite makes sense. Instead we simply need to pick an arbitrary unit and convert (i.e. normalize) our values into that unit system.
In the case of the calculation we made for the metal bar our unit was the bar itself, and thus the continuous normalized sum was the average temperature for the length of bar (the average, normalized per our unit of choice). We could equally well have worked in meters and normalized according to that unit; what we would get is the sum of the temperatures along the bar averaged over one meter of bar. What exactly does that mean? Well if the bar was, for example, three meters long, then the resulting sum over the length of the bar normalized to meters would be three times that of the average for the bar — we’re summing over three meters of bar, but normalizing (averaging) that total over only one meter of bar. Of course if we wanted the average temperature of just the first meter of the bar, and summed only over that first meter things would again make sense as a standard average. The key, however, is having a standardized unit to normalize each individual contribution to the sum against.
This same sort thinking applies to the discrete differences example: we choose an arbitrary unit of speed (be it meters per second, feet per hour, or what have you) and, for each discrete speed calculation we do, normalize to those units so each successive approximation is comparable to the last. Thus we see that our continuous differences and sums are necessarily normalized differences and sums because we simply don’t have a default set of units in the continuous case, and thus have to pick and arbitrary unit and consistently normalize to those units.
So, we now have a method for finding continuous sums, as well as continuous differences. The natural question is: how do they relate to one another? If they are to behave at all similar to discrete sums and differences, we would expect them to be inverses of one another. Does this work? Let’s go back to our example of the arrow, and recall that we can calculate, for each point in time, the speed of the arrow at that moment — the continuous (normalized) difference in position. It should be possible, using continuous sums, to sum up all those speeds. And indeed we can, but if we look a little closer at exactly what is going on, we’ll find something very interesting is happening.
Both the continuous differences and continuous sums need to be normalized; the continuous differences in position are normalized with respect to time, giving us speed, while a continuous sum of speeds is also normalized with respect to time. Intriguingly, because of the way these normalizations work, they will cancel out. Recall that, if we normalized our metal bar against meters, we got a average temperature scaled by the length of the bar that we summed over in meters. If we were to sum up the continuous differences, the speeds, then we would get a average speed scaled by the amount of time we were summing over — but what is an average speed multiplied by the amount of time over which the average was maintained? Why it is the total change is position over that time! Thus if we take a continuous sum of the continuous differences in position, we arrive back at the total change in position. Likewise if we were to sum up instantaneous acceleration values (continuous differences in speed) we expect to get the total change in speed, and so on. This relationship between continuous sums and continuous differences (as relatively clear as it is when phrased in terms of sums and differences) is The Fundamental Theorem of Calculus!
Indeed, what we have been doing here is the heart of calculus. Ultimately this is what calculus is: the arithmetic of the continuous. And this also begins to explain why it is that calculus is so important: the continuous is all around us; time and space are continuous, and so are many things that inhabit it such as electromagnetic fields. And even things that aren’t strictly continuous, such as the flow of water, can be modelled or approximated as being continuous (calculating the movement of every molecule of water is intractable, but if we approximate water as a continuous medium the problem becomes quite manageable). When so much is best handled as continuous, it is clear that an arithmetic of the continuous, rather than the basic discrete arithmetic we are used to, is vital: it opens up vast new areas of our universe for mathematical exploration that were inaccessible using only discrete arithmetic. And so it was that, with the development of an organised calculus, an arithmetic of the continuous, in the 17th century, physics and our understanding of the universe underwent a stunning revolution that is still on-going today. And at the heart of this remarkable revolution was a deeper understanding of of the age old abstractions of the infinite and the continuous.