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

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      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 СКАЧАТЬ