Getting down to zero – John Barrow, Charles Seife

It’s fun to try and write about things you really don’t know about, but best not to do it too much of the time. That’s why biology comes up here a lot. Here’s a thing about physics and maths, for a change, though. I’m always interested in the areas – often featured in popular books – where theoretical physics (and maths) turn into metaphysics, which seem fair game for the lay commentator. In this case, the idea of nothing is the focus of attention. Turns out it’s quite a difficult notion to grasp. There’s also a chunk of comparison of two books on the same topic, which makes more work for the reviewer but can be illuminating.

Nothing will come of nothing, King Lear reproves his tongue-tied daughter. Nowadays, physicists routinely extract entire universes out of nothing – on paper, anyway. Once, multiplying or dividing by zero had terrifying results, reducing orderly arithmetic to instant nonsense. Today, it is a simple trick to recover all the numbers from nothing again. Zero, once a symbol of the void, of absence, of non-being, is now just a location on a number line, an innocuous entry in a string of binary code.

The history of how humans learned to love zero involves startling shifts in metaphysics, mathematics and science. These are hard for us to recover. Childhood rhymes are the nearest we moderns get to Platonic knowledge. When the last green bottle accidentally falls, there are no green bottles hanging on the wall. It feels as if we have always known this. So surely everyone else has, too? But the struggle to develop the idea, and the fact that entire civilisations got by without it, attests that zero is a deeply non-obvious notion.

The first cultures of counting had no need for zero because numbers were used for recording presence rather than absence. Five thousand years ago, if a victorious Pharaoh captured 100,000 goats, that was worth carving into rock. The fact that no slaves were seized this time did not detain the stonemasons. But the number symbols in Egyptian hieroglyphics, like other ancient numerals, were cumbersome. Ever more complex combinations were needed to denote larger and larger numbers. And each symbol meant the same, wherever it was put. They lacked the elegant economy of place value.

That idea, now familiar as the numbers, tens and units on which Mr Blunkett* will probably soon be testing three-year-olds, was an extraordinary discovery. What was needed, as the Babylonians first realised, was an extra sign. If a numeral has a different value depending on where it appears in a list, there must be a symbol for an entry in the list which is empty. Enter zero. Not to stand on its own, but to write, say, 1,001, keeping the ones in their place.

Eventually, through the Arab numerals we still use, zero became part of the global language of arithmetic. But not without a fight. While eastern cultures had long been comfortable with the idea of nothing as a something, the Greeks, for all their learning, could not countenance it. Their cosmos was both filled out and finite. Embrace zero, and the twin threats of the void and infinity might undermine the foundations of an Aristotelian universe. Not until the European renaissance and the scientific revolution did zero and the infinite come into their own in the west: in perspective drawing, in the infinitesimal divisions of Newton’s and Leibniz’s calculus, in the recognition that space might extend for ever but need have no content.

In some senses, we now know the Greeks’ fears of the void to have been unfounded. Since the advent of set theory, a mathematician starting with a representation of nothing can rebuild all the other numbers. The trick is to begin with the empty set, which has no members. Then you can define the number one as the set which contains the empty set as a member – like thinking about a thought. And so on, to infinity. Generated this way, zero nestles at the heart of things inside the endlessly embedded sets, a bootstrap that can pull out an endless ribbon of numbers even though it is tied to nothing. If you buy that, you may also be ready to contemplate modern physicists’ notion of the vacuum. They did away with the ether 100 years ago or so, and settled for empty space. They have now repopulated the void with a sea of “virtual particles”. Quantum mechanics depicts space as a seething foam of uncertainty, with unimaginably short-lived elementary particles continually appearing and disappearing. Contemporary cosmology even suggests that the whole universe might have appeared out of the quantum vacuum: the ultimate free lunch.

As these few highlights suggest, there is a lot more to nothing than meets the eye. So perhaps it is not surprising that, in one of those coincidences that amuse readers more than publishers, John Barrow and Charles Seife choose to tell the whole story in almost identical terms. Both are trained mathematicians and gifted popular writers, and both run through pretty much the sequence I have just sketched. Few people will have time for both, but choosing which to read is largely a matter of taste. Seife, in his first book, is breezier, more journalistic, occasionally preoccupied with the personal quirks of great men – as though he is not quite convinced the ideas alone will keep our attention.

Barrow, whose previous books include one on theories of everything, is very much at home with his new topic, from his splendid title onward. He is also a professional cosmologist and delves further into the physics, while Seife is mainly preoccupied with mathematics, especially the links between zero and infinity. Both tell a good story, from the Babylonians to Big Bang theorists. Barrow’s book is, to my mind, more satisfying, not just because he covers more ground. He has a firmer grasp of the deep metaphysics of being and non-being – though he balks at Sartre, and Heidegger hardly gets a look in. But readers of popular science are well served by Barrow’s apparently effortless range of reference. When it comes to explaining the trickiest ideas he often goes the extra mile, which helps you feel you really understand what is going on (while you have the book in your hand, anyway).

A telling comparison comes when he and Seife discuss the consequences of the zero-point energy of the vacuum, a property linked with the birth of all those virtual particles. Both are convinced that this is an amazing idea. The zero-point energy is, in theory, infinite. As Seife puts it: “According to the equations of quantum mechanics, more power than is stored in all the coal mines, oil fields, and nuclear weapons in the world is sitting in the space inside your toaster”.

Very striking, but why should we be convinced? Well, although the zero-point energy normally disappears in real life – particles and their twinned anti-particles constantly annihilate each other – it does, in some special circumstances, leave a faint residue that can be measured on an atomic scale. Siefe and Barrow both explain how this registers as a force between microscopically separated metal plates, in an experiment first suggested by the Dutch physicists B G Casimir and Dik Polder. Both explain the origin of the Casimir force in terms of the wavelength of particles. Move the plates close enough together and only some wavelengths will fit between them, creating a kind of pressure from an excess of waves outside that tends to push them closer together still.

Barrow, though, fishes a splendid trophy from the physics journals, and points out that a problem in handling sailing ships is just like the Casimir effect. Old nautical manuals warn that two sailing ships positioned side by side in a heavy swell will move toward each other and may collide. The reason turns out to be the same as the cancelling out of wave motions that impels atoms in a vacuum to move closer. This kind of link between the everyday world and the microworld is just what popular-science writers need to help one visualise the strange phenomena they report. Seife is pretty good at spotting them, but Barrow looks further and finds more. Some of the cosmological speculation is still hard to follow, but at least you do not feel that he is holding back from dealing with the most demanding stuff.

The same can hardly be said of David Bodanis’s E=MC2. Like Seife, Bodanis presents his book as a “biography”, in this case of Einstein’s famous equation. For both this means a sprinkling of personal anecdotes about people who worked with the ideas they write about. But in Bodanis’s book the anecdotes all but drive out the ideas. The result is a popular-science book in which the science is present in near homeopathic dilution. More seriously, the celebrated equation makes little sense by itself. As Bodanis says, it is simply a small part of the special theory of relativity, which in turn gave way to the general theory some years later. There is a good deal of history in his book, but little sense of the rich interconnection of a web of ideas that pervades Barrow and Seife’s narratives. In fact, they both explain more about relativity than Bodanis. For it turns out that in order to write convincingly about nothing, you really do have to explain just about everything. (2000)

The Book of Nothing, John D Barrow (Cape)

Zero: The Biography of a Dangerous Idea, Charles Seife (Souvenir Press)

E=MC2, David Bodanis (Macmillan)

*Mr Blunkett was once a minister in the government.

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