The Infinite Divisibility of Space

A good philosopher who concerns himself with science brings new light to the subject and adds conceptual and historical depth to the big scientific questions. This was my first thought after reading The Paradoxes of Zeno, a sophisticated short book by Vincenzo Fano, an Italian philosopher who teaches at Urbino and who is equipped to deal with science.

Zeno was also an Italian philosopher who was interested in science. He taught in Elea, in what is now Cilento, in the province of Salerno, 2,400 years ago. Fano’s book places his work in a contemporary light, close to questions of theoretical physics that are at the heart of my research.

Zeno has passed into history on account of several arguments known today as ‘Zeno’s Paradoxes’ – arguments which, in the words of Bertrand Russell, are ‘immeasurably subtle and profound’. They came at the beginning of the great era of Greek thought. In the preceding decades, the philosophers of the naturalistic school of Miletus, in what is now Turkey, had begun to appreciate that things are not necessarily as they seem, but that their nature can be investigated and understood by reason. In Italy, the philosophers of the school of Elea, following in the footsteps of Parmenides, had taken this intuition to a radical extreme, maintaining that true reality is only what can be reconstructed by reason, therefore denying reality to mere appearances. Zeno presents his ‘paradoxes’ as arguments to show that movement is inconceivable: hence movement is just one of the false ‘appearances’, and not a true reality. A daring idea.

The most famous of these paradoxes is that of Achilles and the tortoise. Achilles and the tortoise challenge each other to a race, with Achilles reckoning that he runs ten times faster than the tortoise and conceding to his opponent a hundred-metre advantage. How long will it take for Achilles to catch the tortoise?

Before reaching the tortoise, Achilles will have to cover the one hundred metres he has given as a head start, and this will take a certain amount of time. In the meantime, the tortoise will have covered a certain amount of ground: ten metres, say. Achilles will have reached the tortoise’s point of departure, but the tortoise will be ten metres ahead. Before reaching the tortoise, Achilles will also have to cover these ten metres, which will take more time. But, meanwhile, the tortoise will have covered a bit more ground, and so on to infinity. Hence before reaching the tortoise, Achilles will have to cover an infinite number of portions of space, and each one will take time. Hence Achilles will require an infinite number of times – that is, according to Zeno, an infinite amount of time.

In other words, he will never catch up with the tortoise. If we do actually see Achilles catching up with and overtaking the tortoise, after we have demonstrated that it is not possible, it follows that what we are seeing is an illusion. This is Zeno’s argument.

But the reasoning is wrong. The error is to think that a sum of infinite intervals of time must result in an infinite time. This is not true. To see this, we need only think of taking a piece of thread one metre long and cutting it at fifty centimetres. Then cutting it at twenty-five centimetres, then at twelve and a half, and so on, always adding to the cut pieces a length half of the one before it. If we continue this exercise to infinity, will the cut pieces add up to a thread of infinite length? Obviously not, because taken all together the segments cannot possibly be longer than the metre that we had at the outset. Hence the sum of infinite lengths can easily be a finite length, when the lengths added together are gradually shorter. In the same way the sum of infinite times can very well become a finite time. Zeno’s reasoning is mistaken at the point when he deduces that the sum of infinite times covered by Achilles must add up to an infinite time.

This consideration seems to settle the matter. It is a solution that twenty-four centuries ago perhaps escaped the best minds, because familiarity with infinite sums only developed later. They began to be understood a couple of centuries later, by Archimedes in Italy, but clarity was achieved regarding them only in the modern era: mathematicians today call them ‘convergent series’. Complete clarification was achieved gradually, but there can be no doubt now that Zeno’s argument about Achilles and the tortoise is misconceived. That’s why, before reading Fano’s book, I had always considered the paradoxes of Zeno to be of little interest.

But then why did Bertrand Russell, who certainly knew something about mathematics and was far from naïve, consider those paradoxes to be ‘immeasurably subtle and profound’? And why has Vincenzo Fano, an acute philosopher, dedicated a book to Zeno at this time? The point is this: are we really certain that the solution I have just given is the right ‘physical’ response to the question posed by Zeno? Is this really what happens when Achilles races after the tortoise? Does he really cover an infinite number of segments of ever-decreasing length?

Let’s put the question another way, still following in the steps of Zeno. At school we learned that space is a set of points. But a point has no extension. Not even two points have extension, or three. As a matter of fact, however many points I add together, I will always have something without extension. So how can I obtain an extended space by piling points together? Fano’s book, which is didactic and intelligent, erudite and exhaustive, lays out before us the many difficulties entailed by the concept of continuous space, the theoretical acrobatics required to get round them, the reflections and the objections that these difficulties have generated over the centuries. And he brings us to the threshold of today’s version of the problem.

In order to understand the profundity of that problem, let’s follow another historical thread. Zeno had a friend called Leucippus. Leucippus was the first philosopher to propose an idea that was to have a brilliant future: the atomic hypothesis. Motivated by the difficulties of the idea of divisibility ad infinitum, on which his friend Zeno was so insistent, Leucippus proposes the idea that matter is made up of indivisible units. The tremendously important development of this idea will fall to his disciple Democritus, one of the greatest philosophers of all time.

Those who study Democritus consider the loss of his texts, probably censored during the ‘devout centuries’, to be one of the greatest tragedies to have befallen world culture. Perhaps the world would have been a better place if, instead of getting rid of the works of Democritus and preserving those of Aristotle, our forefathers had lost all copies of Aristotle’s books and managed to preserve those of Democritus instead.

Well, Leucippus and Democritus explore the idea that matter may not be divisible ad infinitum. It is not possible to divide a drop of water into an arbitrarily large number of increasingly smaller drops. There is a minimal unit of water: a water molecule. According to Democritus, the universe in its variety and complexity may be understood by being thought of as made up of indivisible units of matter, which he called ‘atoms’, and by their dance in space. The vision offered by Democritus, that we know above all in the version in verse given to us by Lucretius, inspired the birth of modern science and has been splendidly confirmed in recent times – and it is the one that every child learns at elementary school today: matter is made up of atoms: it cannot be infinitely divided.

And what about space? The original problem posed by Zeno, on the other hand, that of the infinite divisibility of space, is still very much open. What’s more, it is at the centre of theoretical research in fundamental physics. The physics of Newton assumed that space was real and infinitely divisible. During the course of the nineteenth century, mathematicians developed refined theorems to account for the peculiarities of the continuum. These provide, as I’ve already explained, a possible solution to Zeno’s paradoxes, but it is an abstruse one. In the ancient and profound arguments of the philosopher of Elea there remains something disquieting that is properly highlighted by Fano. What remains is the physical question posed by him: is space and is time in reality infinitely divisible? And what if this proved not to be the case?

Is it reasonable to think of physical space preserving its continuous structure, all the way down to infinity? To infinity, plunging into an abyss of the ever smaller, is an awful long way down …

Today, indeed, there are indications that the correct solution to the paradoxes of Zeno is not to be found in the continuum. The insight of Leucippus and Democritus may prove to be right not just for matter but for space itself.

Twentieth-century physics has demonstrated the relevance to the structure of the universe of three physical constants: the speed of light, the constant of universal gravity and the Planck constant that establishes the scale of so-called quantum phenomena. By combining these constants we obtain a length, called the ‘Planck scale’. It is a very small scale indeed (a decimillionth of a billionth of a billionth of the nucleus of an atom), but one that is finite. At this scale we expect ‘quantum’ phenomena, and the most typical ‘quantum’ phenomenon is granularity. Electromagnetic waves, for example, behave like a swarm of particles: the famous photons or ‘quanta of light’. In the same way it is reasonable to expect that space will also show aspects of granularity at the Planck scale. Space might well be made up of elementary ‘atoms of space’, providing a limit to divisibility.

This granularity of space is a key element in various theories that are currently being developed. The theory in which it is studied most explicitly is ‘loop’ quantum gravity, in the context of which my own research unfolds. In loop theory, a centimetre of space is not continuous: it is a collection of a very large but finite number of ‘atoms of space’. Hence Fano’s essay on Zeno inserts itself not just into historical-philosophical debate, but also offers arguments and points of reflection for theoretical physics. The conceptual complexities entailed in the notion of continuous space, well described by Fano, may not be the best way to think about the real world.

If loop theory is correct, Achilles will not have to take an infinite number of steps in order to overtake the tortoise. The bounds required of our hero may be many, but their number will be finite.