What We Cannot Know. Marcus du Sautoy

calculating all the intervening digits. Not something that I’m desperate to know. But this formula will only ever give me finite knowledge of a number that necessitates the infinite to fully embrace it.

      The discovery of these numbers seemed to imply an infinite divisibility of space. Only by dividing space infinitely would I be able to measure the precise dimensions of my simple cube. The discovery meant that in the West Aristotle’s continuous view of matter would remain dominant until the Renaissance.

       THE HARMONY OF TINY SPHERES

      With the insights of the science of Newton’s generation and beyond, the tide would turn in favour of the view that the universe is indeed made from basic building blocks. Newton’s contemporary, Robert Boyle, was perhaps the first to question the Aristotelian view of matter, which had dominated for nearly 2000 years. In his book The Sceptical Chymist, Boyle challenged the idea that matter was made up of the four elements of fire, earth, air and water. These might be good descriptions of the states of matter but not of the constituents.

      Instead, he argued for a new list of chemical elements. Not only that, he made what at the time was rather a heretical statement. He believed that these elements would be minute bodies or atoms differing only in ‘bulk, figure, texture and motion’. This was regarded as theologically dangerous: a materialistic godless view of the world in the eyes of the Church, which had always favoured the Aristotelian view. Some have declared Boyle the Galileo of the chemical revolution.

      Although Newton would concur with Boyle’s proposal for a material world made from indivisible units, the mathematical tools that Newton developed at the same time as Boyle’s work relied heavily on time and space being infinitely divisible. The calculus that allowed one to take a snapshot of a universe in constant flux made sense only as a process in which space is divided up into ever smaller pieces and then interpreting what happens in the limit as the pieces become infinitesimally small.

      The question of the infinite divisibility of time and space had fuelled philosophical argument since the ancient Greek thinker Zeno of Elea came up with paradoxes that seemed to arise from this cutting up of space. For example, Zeno proposed that an arrow can never hit its target because it must first cover half the distance to the target, then half the distance again, and then half again, requiring infinitely many moves if it is ever going to make it to the target. The success of Newton’s calculus reignited the debate. There were still those who considered such infinite divisibility as almost heretical.

      Bishop Berkeley dedicated a whole treatise called The Analyst to arguing that trying to make sense of dividing by zero was absurd, which the subtitle made clear: it was ‘Addressed to an Infidel MATHEMATICIAN’.

      While most infidel mathematicians quickly latched on to the power of the calculus, Newton’s other breakthroughs supported a view that, although space and time could be infinitely divided, matter could not. His idea of a world made up of indivisible matter would in time become the prevalent theory of the universe. But at this point it was still very much a theory without a great deal of evidence.

      His theory of forces acting on large objects like planets and apples had been so successful that Newton believed that if the ideas worked on the very large and the medium sized then why not on the very small. Why should there be a change in the way the laws of motion dictate the behaviour of the universe as we zoom in on my dice? The success of his calculus when applied to the movement of the planets depended on regarding them as single points with the mass concentrated at a point corresponding to the centre of gravity of the body. Maybe all matter consisted of particles like tiny planets whose behaviour was determined by his laws of motion. In his Principia he stated his belief that by applying his ideas to these individual particles one could predict the behaviour of all material things.

      Newton’s theory of light also contributed to the growing feeling that an atomistic perspective was the best way to understand the world. Regarding light as a particle seemed the easiest way to describe the phenomena he documented in his book Opticks. The way that light reflected seemed to mimic the behaviour of a billiard ball as it bounced off the sides of the table. But from a scientific point of view there was no empirical evidence for this model of a universe made from indivisible particles.

      Even with the microscopes that were emerging in the seventeenth century, you couldn’t see anything to justify this atomistic model. Even if you could see discrete objects, it wouldn’t prove anything about their indivisibility. But we can judge that the tide was turning by the fact that the atomistic view of matter made it into the popular culture of the time. Nicholas Brady’s ‘Ode to Saint Cecilia’, set to music by Purcell in 1691, talks of seeds of matter:

      Soul of the World! Inspir’d by thee,

      The jarring Seeds of Matter did agree,

      Thou didst the scatter’d Atoms bind,

      Which, by thy Laws of true proportion join’d,

      Made up of various Parts one perfect Harmony.

      The best evidence for an atomistic view of matter came a hundred years later from experiments that showed how matter combined to make new materials. And it was full of perfect harmony, as Brady had intimated.

       ATOMIC ALGEBRA

      It was the work of the English chemist John Dalton at the beginning of the nineteenth century that provided the first real experimental justification for thinking of matter as made of indivisible atoms. His discovery that compounds seemed to be made up of substances that were combined in fixed whole-number ratios was the breakthrough, and it led to the scientific consensus that these substances really did come in discrete packages.

      For example: ‘The elements of oxygen may combine with a certain portion of nitrous gas or with twice that portion, but with no intermediate quantity.’ It certainly didn’t constitute a proof that matter was discrete, and it was not strong enough to knock the belief of those who favoured a continuous model of matter. But it was highly suggestive. There had to be some explanation for the way these substances were combining.

      The notation developed to express these reactions added to the atomistic view. The combination of nitrogen and oxygen could be expressed algebraically as N + O or N + 2O. There was nothing in between. It seemed that all compounds came in proportions that were whole-number ratios. For example, aluminium sulphide was given algebraically as 2Al + 3S = Al2S3, elements combining in a 2-to-3 ratio. Elements never combined in a non-whole-number relationship. It was like musical harmony at the heart of the chemical world. The music of tiny spheres.

      The Russian scientist Dmitri Mendeleev is remembered for laying out this growing list of molecular ingredients in such a way that a pattern began to emerge, a pattern based on whole numbers and counting. It seemed that the Pythagorean belief in the power of number was making a comeback. Like several scientists before him, Mendeleev arranged them in increasing relative weight, but he realized that to get the patterns he could see emerging he needed to be flexible.

      He’d written the known elements down on cards and was continually placing them on his desk in a game of chemical patience, trying to get them to yield their secrets. But nothing worked. It was driving him crazy. Eventually he collapsed in exhaustion and the secret emerged in a dream that, when he woke, gave him the pattern for laying out the cards. One of the important points that led to his successful arrangement was the realization that he needed to leave some gaps – that some of the cards from the pack were missing.

      The key to his arrangement was something called the atomic number, which depended on the number of protons inside the nucleus rather than the combination of protons and neutrons that gave rise to the overall weight. But since no one had any clue yet about these smaller ingredients, Mendeleev was guessing somewhat at the underlying reason for his arrangement.

      It was a bit like recognizing that a conventional pack of cards can be laid out in suits, but also that across those suits there are cards that are of equal value. A periodicity of eight seemed to underlie the elements, so that elements eight