I recently read an entertaining thought experiment claiming to show that energy isn’t conserved in Newtonian mechanics. It involves a sneaky trick using an infinite sequence of balls of decreasing size. In this post I’ll discuss this and some other superficially similar paradoxes and consider how they might be resolved, starting with Zeno’s famous paradox.

The simplest version of Zeno’s paradox involves a woman walking along a chalked line. She starts at one end of the line and walks until she reaches the other end. For the sake of easy numbers we will imagine that the line is one metre long, that the woman is travelling at a constant speed of one metre per second, and that the woman is called Alice. After one second Alice will have walked the length of the line. The paradox comes from dividing the distance to be covered into infinitely many intervals of decreasing length. Label each point on the line by its distance in metres from Alice’s starting point (in particular, she starts at point 0 and must reach point 1). To walk the line, Alice must first get halfway across, i.e. travel from 0 to 1/2. We will call this the first task. In the course of getting the rest of the way along the line, she must travel from 1/2 to 3/4. We call this the second task. After performing the second task, she must travel from 3/4 to 7/8 (the third task), and so on. Continuing in this way we see that Alice must perform infinitely many tasks in order to walk the line: for each natural number $n$, she must walk from $(2^{n}-1)/2^n$ to $(2^{n+1}-1)/2^{n+1}$. Zeno claims that it is impossible to complete infinitely many tasks in finite time and so, contrary to our claim above, Alice never does make it to the far end of the line. In fact, we can scale down the intervals being considered to show that Alice cannot cover any positive distance in finite time and so in fact all movement is impossible. There are some obvious objections to this paradox and nowadays it’s generally considered to be resolved. Before discussing the standard resolution in detail, I’ll introduce the other paradoxes.

The previous paradox claimed to demonstrate something about reality. The remaining paradoxes discuss situations which appear to be clearly non-physical’ but which none the less manage to produce surprising outcomes. In the Ross-Littlewood paradox, we imagine a man (let’s call him Bob) who possesses an infinite collection of labelled balls: for each natural number $n$, he possesses exactly one ball with the label $n$. He also possesses a bag capable of containing all of these balls. We consider Bob to be capable of putting balls into the bag and removing balls from the bag instantaneously (it suffices for him to be able to perform these tasks arbitrarily quickly, but we’ll assume he can do them instantaneously when stating the paradox). Now imagine Bob performing the following task: at 11pm, Bob puts the ball with label 1 in the bag. After half an hour, he removes the ball labelled 1 and adds the balls labelled 2 to 10. Fifteen minutes later, he removes the ball labelled 2 and adds the balls labelled 11 to 20. After a further seven and a half minutes, he removes the ball labelled 3 and adds balls 21 to 30, and so on. i.e., we divide the time between 11pm and midnight into infinitely many intervals so that the first interval is of length 30 minutes and the $n+1$st interval is always half the length of the $n$th interval. At the end of each interval Bob removes the ball with smallest label from the bag and from among the balls which have not yet been in the bag adds those with the next ten smallest labels. As the time approaches midnight, the bag gets increasingly full of balls (every time Bob removes a ball he adds ten more). However, when we reach midnight the bag is empty. To see this, consider the ball with label $n$. At the end of the $n$th interval, this ball is removed from the bag and never returned. Thus none of the balls remain in the bag at midnight.

Thompson’s lamp

For this paradox we assume that a man (Thompson) possesses a lamp that he is capable of switching on or off instantaneously (as with the Ross-Littlewood paradox, arbitrarily quickly’ will suffice). At 11pm the lamp is off. At 11.30pm, he turns the lamp on. At 11.45pm he turns the lamp back off and he continues turning the lamp alternately on and off using the same time intervals as Bob did in the previous paradox. When we reach midnight, is the lamp on or off? There appears to be no definite answer to the question. Any time before midnight at which the lamp is turned on is followed by another time before midnight at which the lamp was switch off, and vice versa. Hence our completely deterministic process appears to have produced an indeterminate outcome.

Zeno’s balls

The final paradox involves a woman (Zeno) in possession of an infinite set of balls. For each $n$ she possesses a ball of radius $1/2^{n+1}$ of mass 1kg (the precise diameter doesn’t matter; we just need the balls to get smaller sufficiently quickly). We suppose Zeno possesses a frictionless table placed in a room with frictionless atmosphere and that when the balls are rolled at each other along the surface of this table they collide elastically. Zeno picks a point on the table (the origin’) and labels it 0. In some fixed direction, she then puts labels $1/n$ at distance $1/n$ metres from 0 for $n=1,2,3,...$. For each $n$ she places the $n$th ball on the point labelled $1/n$. Having done all this, she sets her timer to zero and pushes the first ball ball towards the origin, giving it a speed of 1m/s. The first ball then collides with the second ball, coming to a halt and transferring all of its energy. The second ball then rolls until it hits the third ball, which rolls until it hits the fourth ball, and so on. After one second, none of the balls are moving: for any $n$, the $n$th ball collided with the $n+1$st ball at time strictly less than one second and came to a halt. Thus energy is not conserved in Newtonian mechanics: at time zero the first ball has positive kinetic energy, while at time one all of the balls are stationary. Running this process in reverse, we have a row of balls that spontaneously start moving; now we’ve not only broken energy conservation, we also have an effect without a cause!

Some resolutions’

All of the paradoxes centre on the issue of performing infinitely many tasks in finite time. Zeno’s paradox asserts that this is not possible and claims to deduce that movement is impossible, while the others ask us to consider what happens after performing some specified infinite collection of tasks.

Since Weierstrass’s $(\varepsilon, \delta)$ definition of limit, and subsequent basic results on infinite series, Zeno’s paradox has largely been considered to be resolved (although this opinion is not universal). The generally accepted solution is that infinitely many tasks can indeed be completed in finite time if the time taken for each task decreases sufficiently quickly. Completing the first task takes Alice half a second, completing the second takes a quarter of a second and so on: in general, completing the $n$th task takes $1/2^n$ seconds. Thus Alice will complete the first $n$ tasks in $\Sigma_{j=1}^n 1/2^j = (2^n-1)/2^n$ seconds. As $n$ tends to infinity this sum tends to 1. Alice thus really can complete infinitely many tasks in unit time and she defeats Zeno. However, it’s worth pausing for a moment to consider this argument more carefully. What role does the technical machinery of series and limits play in it? On second glance, not an awful lot. The sum given above doesn’t tell us anything we didn’t already know; you don’t need a lot of maths to work out that when travelling at 1 metre per second it takes $(2^n-1)/2^n$ seconds to walk $(2^n-1)/2^n$ metres! And you certainly don’t need to use any technical machinery to calculate that it should take Alice one second to walk the entire line. So why do most people treat the machinary of limits and series as having resolved’ the paradox? It appears that the crucial role played by Weierstrass’s mathematical innovation is demystifying the notion of infinity. If you’re used to the idea of taking limits then it seems natural to state that Alice’s position at time 1 is the limit of her positions at times $(2^n-1)/2^n$ and the division of her task into infinitely many diminishing subtasks no longer appears problematic. The resolution to Zeno’s paradox is then just to realise that there doesn’t seem to be any good reason to deny that infinitely many tasks can be performed in finite time. The paradox is merely a linguistic trick. The remaining paradoxes however, appear to be less clear-cut.

Now consider the Ross-Littlewood paradox. The main issue with this paradox is that what happens in the limit (i.e. at time midnight) is not well-defined. We are told the state of Bob’s bag after any finite number of intervals (and so at any time before midnight), but the set-up doesn’t actually tell us what happens after an infinite number of intervals (i.e. at midnight). Let $\chi_n$ be the characteristic function of Bob’s bag at stage $n$. i.e., $\chi_n(x)$ equals 1 if the ball labelled $x$ is in Bob’s bag at the end of interval $n$ and 0 otherwise. Let $\chi$ be the characteristic function of Bob’s bag at midnight. The argument that Bob’s bag eventually contains no balls amounts to saying that $\chi$ should be chosen as the pointwise limit of the $\chi_n$, i.e. that for each $x$, $\chi(x) = \lim_n \chi_n(x)$. However, there doesn’t seem to be any particular reason to choose to take the limit of the $\chi_n$ in the norm of pointwise convergence. We could instead decide that the number of balls in the bag at midnight should be the limit of the number of balls in the bag at the end of each interval. If we adopt this approach then we will decide that the bag should contain infinitely many balls at midnight. One or the other of these arguments might appeal to you more. However, in this paradox I think the set-up leaves us with a genuine choice as to which limiting process we want to use to define the state of the bag at midnight. The paradox then amounts to the observation that when describing infinitary processes we should make sure that we carefully specify what happens when we take limits.

The case of Thompson’s lamp is a little different again. In this scenario, there is no sensible way to take a limit of the state of the lamp. Unlike the Ross-Littlewood paradox we don’t have a choice of which limit to take. Instead, we can only reasonably conclude that the information given tells us nothing about the state of the lamp at midnight. I don’t find this particularly troubling, as the ability to toggle a current on or off infinitely often in a finite period of time is clearly nonphysical (contrast this with Alice’s walk; her tasks’ become increasingly trivial while Thompson’s switching is just as tricky every time he does it). Our intuition that the lamp should have a definite state at midnight and that its state at all earlier stages should tell us this is based on our experiences with normal, finitary processes. In this case though, the talk of a man and his lamp is merely a way of tricking our intuition into treating the scenario as being somehow physical. The claim that not all sequences have limits is neither surprising nor mysterious. As with Zeno’s paradox then, it appears that this paradox dissolves when considered more carefully.

Finally, let’s consider the paradox of Zeno’s balls. An obvious first observation here is that we shouldn’t be unduly disturbed if it turns out that a model designed to predict the motion of finite numbers of sensibly behaved physical bodies gives screwy results when applied to a question involving an infinite number of objects of arbitrarily small diameter and arbitrarily high density. After reading my discussion of the previous paradoxes, you shouldn’t be too surprised to hear that my `resolution’ to this paradox is again that there’s nothing to resolve. Unlike the Ross-Littlewood paradox, we don’t appear to have a choice here as to which metric to take limits in. The velocities (and hence kinetic energies) of the balls tend pointwise to zero, while the kinetic energy of the whole system is constant for all times less than one, so I suppose we could decide that energy should change continously and conclude that the kinetic energy of the system is indeed conserved at time one. However, this invites the obvious question of where that kinetic energy lies! It seems ludicruous to try and claim that a collection of stationary balls somehow possesses non-zero kinetic energy. I think a better solution is just to say that this system is non-physical and so we shouldn’t be surprised that it doesn’t behave according to our physical intuitions. We’re perfectly free to decide that the state of the system at time one is indeed the pointwise limit of the states at earlier times, but we shouldn’t read much of anything into this.