What We Cannot Know: Explorations at the Edge of Knowledge. Marcus Sautoy du

of the fun things about the cello is the possibility of sliding your finger up the string to create a continuous glissando of notes. Not something I can do on my trumpet, which is an instrument of discrete notes corresponding to the different combinations of fingers I put down. It turns out that this tension between the continuous glissando of the cello and the discrete notes of the trumpet is relevant to my attempts to predict the behaviour of my dice.

       ZOOMING IN

      To predict how the dice might land I need to know what my cube is made from. Denser acetate in one corner of the shape will lead to one side of the dice being favoured over the others. So if I am going to attempt to apply Newton’s laws to my dice as it tumbles through the air, I need to know how my dice is put together. Is it a continuous structure, or, if I look closely, is it made up of discrete pieces?

      If I accept the limits of my own vision, as Schopenhauer’s quote at the beginning of this Edge suggests I tend to, then I can’t see anything other than the clear red acetate that makes up the dice. But with an optical microscope I can magnify the dice by a factor of 1500, which would scale my dice up to the size of a large building. Peering inside this huge dice still won’t reveal much about the secrets of how it is built. Everything still looks pretty smooth and continuous.

      In the twentieth century microscopes exploiting different bits of the electromagnetic spectrum have allowed scientists to create images which magnify things a further 1000 times. Now my dice will span from one side of London to the other. At this magnification the dice is looking grainier. The sense of the continuous structure is giving way to something more discrete. Current electron microscopes allow me to zoom in another 10 times closer, at which point I might start to see the carbon and oxygen atoms that I know are some of the ingredients of the acetate from which my dice is made.

      The intriguing thing is that scientists had already formulated an atomic view of matter long before I could actually see these atoms under the modern microscopes in the labs across the road from the mathematics department. And it is a combination of a mathematical and theoretical perspective with a physical vision that is the best tool for knowing what my dice is made from.

      But atoms like oxygen and carbon turned out not to be as atomic as the name suggests. Beyond the atomic structure revealed by current electron microscopes, I know that there is more internal structure. Atoms give way to electrons, protons and neutrons. Protons and neutrons in turn give way to quarks. In 2013 quantum microscopes even captured pictures of electrons orbiting the nucleus of a hydrogen atom. But is there a theoretical limit to how far I can dig down inside my dice?

      What happens, for example, if I take my dice and keep dividing it in half? Just how far can I go? The mathematical side of me says: no problem. If I have a number I can keep dividing it by two:

      There is no point mathematically where I have to stop. Yet if I start trying to do the same thing with the physical dice sitting on my desk and cut it in half, then in half again, just how far can I keep going?

      The tension between the continuous versus the discrete nature of matter, between what is possible mathematically versus the limits placed by physical reality, has been raging for millennia. Is the universe dancing to the sound of my trumpet or shimmying to the glissando of my cello?

       THE MUSIC OF THE SPHERES

      How did I personally come to know about these electrons and quarks that are believed to be the last layer of my dice? I’ve never seen them. If I actually ask myself how I know about them, the answer is that I’ve been told and read about them so many times that I’ve actually forgotten why or how I know. Or, come to think of it, was I ever told how we know? Is it a bit like the way I know Everest is the tallest mountain? I know that only because I’ve been told it enough times. So before I ask whether there is anything beyond this layer, I need to know how we got to these building blocks.

      Reading through the history, I am surprised that it is only just over a hundred years ago that convincing evidence was provided for the fact that things like my dice are made of discrete building blocks called atoms and are not just continuous structures. Despite being such a relatively recent discovery, the hunch that this was the case goes back thousands of years. In India it was believed that matter was made from basic atoms corresponding to taste, smell, colour and touch. They divided atoms into ones that were infinitesimally small and took up no space, and others that were ‘gross’ and took up finite space – an extremely prescient theory, as you will see once I explain our current model of matter.

      In the West it was the ancient Greeks who first proposed an atomistic philosophy of nature, advocating the reductionist view that physical reality could be reduced to fundamental units that made up all matter. These atoms could not be broken down into anything smaller, and their properties should not depend on some further complex inner structure. One of the seeds for this belief in a universe made from indivisible building blocks was the Pythagorean philosophy that number is at the heart of explaining the secrets of the universe.

      The conviction in the power of whole numbers had its origins in a rather remarkable discovery attributed to Pythagoras: namely, that number is the basis of the musical harmony that both my cello and my trumpet exploit. The story has it that inspiration struck when he passed a blacksmith and heard the hammers banging out a combination of harmonious notes. (We can’t be sure whether this and similar stories told about Pythagoras are true, or even whether he really existed and wasn’t an invention of later generations used to promote new ideas.)

      This story goes that he went home and experimented with the notes made by a stringed instrument. If I take the vibrating string on my cello then I can produce a continuous sequence of notes by gradually pushing my finger up towards the bridge of the cello, making a sound called a glissando (although the question of whether this is truly producing a continuous sequence of notes will be challenged in the next Edge). If I stop at the positions that produce notes that sound harmonious when combined with the open vibrating string, it turns out that the lengths of the strings are in a perfect whole-number ratio with each other.

      For example, if I place my finger at the halfway point along the vibrating string I get a note which sounds almost like the note I started with. The interval is called the octave, and to the human ear the note sounds so similar to the note on the open string that in the musical notation that emerged we give these notes the same names. If I place my finger a third of the distance from the head of the cello, I get a note which sounds particularly harmonious when combined with the note of the open string. Known as the perfect fifth, what our brains are responding to is a subliminal recognition of this whole-number relationship between the wavelengths of the two notes.

      Having found that whole numbers were at the heart of harmony, the Pythagoreans began to build a model of the universe that had these whole numbers as the fundamental building blocks of everything they saw or heard around them. Greek cosmology was dominated by the idea of a mathematical harmony in the skies. The orbits of the planets were believed to be in a perfect mathematical relationship to each other, giving rise to the idea of the music of the spheres.

      More importantly for understanding the make-up of my dice, it was also believed that discrete numbers rather than a continuous glissando were the key to understanding what constituted matter. The Pythagoreans proposed the idea of fundamental atoms that, like numbers, could be added together to get new matter. The Greek philosopher and mathematician Plato developed the Pythagorean philosophy and makes these atoms into discrete pieces of geometry.

      Plato believed the atoms were actually bits of mathematics: triangles and squares. These were the building blocks for the shapes that he believed were the key to the ingredients of Greek chemistry: the elements of fire, earth, air and water. Each element, Plato believed, had its own three-dimensional mathematical shape.

      Fire was the shape of a triangular-based pyramid, or tetrahedron, made from 4 equilateral triangles. Earth was cube-shaped like my Vegas dice. Air was made from a shape called an