Stranger Than We Can Imagine (14 page)

O
n the last day of the nineteenth century, six hours before midnight, the British academic Bertrand Russell wrote to a friend and told her that ‘I invented a new subject, which turned out to be all mathematics for the first time treated in its essence.’ He would later regard this claim as embarrassingly ‘boastful’.

Russell was a thin, bird-like aristocrat who compensated for the frailness of his body through the strength of his mind. He became something of a national treasure during his long life, due to his frequent television broadcasts and his clearly argued pacifism. His academic reputation came from his attempt to fuse logic and mathematics. Over the course of groundbreaking books such as
The Principles of Mathematics
(1903) and
Principia Mathematica
(1910, co-written with Alfred North Whitehead), Russell dedicated himself and his considerable brain to becoming the first person to prove that 1 + 1 = 2.

Russell had a solitary childhood. He was brought up in a large, lonely house by his stern Presbyterian grandmother following the death of his parents, and he lacked children of his own age to play with. At the age of eleven, after his older brother introduced him to Euclidian geometry, he developed a deep love for mathematics. In the absence of other children he became absorbed in playing with numbers.

Yet something about the subject troubled him. Many of the rules of mathematics rested on assumptions which seemed reasonable, but which had to be accepted on faith. These assumptions, called axioms, included laws such as ‘For any two points of space there exists a straight line that connects those two points’ or ‘For any natural number x, x + 1 is also a natural number.’ If those axioms
were accepted, then the rest of mathematics followed logically. Most mathematicians were happy with this situation but, to the mind of a gifted, lonely boy like Russell, it was clear that something was amiss. He was like the child in
The Emperor’s New Clothes
, wondering why everybody was ignoring the obvious. Mathematics, surely, needed stronger foundations than common-sense truisms. After leaving home and entering academia, he embarked on a great project to establish those foundations through the strict use of logic. If anything could be a system of absolute clarity and certainty, then surely that would be logically backed mathematics.

In the real world, it is not too problematic to say that one apple plus another apple equals two apples. Nor is there much argument that if you had five scotch eggs, and ate two of them, then you would have three scotch eggs. You can test these statements for yourself the next time you are in a supermarket. Mathematics, however, abstracts quantities away from real-world things into a symbolic, logical language. Instead of talking about two apples, it talks about something called ‘2’. Instead of three scotch eggs, it has the number ‘3’. It is not possible to find an actual ‘2’ or a ‘3’ out in the real world, even in a well-stocked supermarket. You’ll find a squiggle of ink on a price tag that symbolically represents those numbers, but numbers themselves are immaterial concepts and hence have no physical existence. The statement 1 + 1 = 2, then, says that one immaterial concept together with another immaterial concept is the same as a different immaterial concept. When you remember that immaterial concepts are essentially things that we’ve made up, the statement 1 + 1 = 2 can be accused of being arbitrary. Russell’s project was to use logic to prove beyond argument that 1 + 1 = 2 was not an arbitrary assertion, but a fundamental truth.

He nearly succeeded.

His approach was to establish clear definitions of mathematical terms using what logicians then called
classes
, but which are now better known as
sets
. A set is a collection of things. Imagine that Russell wanted a logical definition of a number, such as 5, and that he also had a vehicle with a near-infinite storage capacity, such as
Doctor Who’s TARDIS. He could then busy himself travelling around the world in his TARDIS looking for examples of five objects, such as five cows, five pencils or five red books. Every time he found such an example, he would store it in his TARDIS and continue with his quest. If he successfully found every example of a five in the real world then he would finally be in a position to define the immaterial concept of ‘5’. He could say ‘5’ is the symbol that represents the set of all the stuff he had collected in his magic blue box.

Producing a similar definition for the number 0 was more complicated. He could hardly travel the world filling his TARDIS with every example of no apples or no pencils. Instead, he defined the number 0 as the set of things that were not identical to themselves. Russell would then fill his TARDIS with every example of a thing that was not the same as itself and, as there aren’t any such things in the world, he would eventually return from this fruitless quest with an empty TARDIS. Under the rules of logic there is nothing that is not identical to itself, so this was a valid representation of ‘nothing’. In mathematical terms, he defined the number 0 as the set of all
null sets
.

If Russell could use similar, set-based thinking to produce a clear definition of ‘the number 1’ and the process ‘plus 1’, then his goal of being able to prove beyond doubt that 1 + 1 = 2 would finally be achievable. But there was a problem.

The problem, now known as Russell’s paradox, involved the set of all sets that did not contain themselves. Did that set contain itself? According to the rules of logic, if it did then it didn’t, but if it didn’t, then it did. It was a similar situation to a famous Greek contradiction, in which Epimenides the Cretan said that all Cretans were liars.

At first glance, this may not appear to be an important paradox. But that wasn’t the point. The problem was that a paradox existed, and the goal of rebuilding mathematics on the bedrock of logic was that it would not contain any paradoxes at all.

Russell went back to first principles, and a number of new definitions, arguments and fudges were proposed to avoid this problem.
Yet every time he built up his tower of mathematical logic, another problem revealed itself. It felt like paradoxes were unavoidable aspects of whatever self-contained system mathematicians produced. Unfortunately, that turned out to be the case.

In 1931 the Austrian mathematician Kurt Gödel published what is now known as Gödel’s Incompleteness Theorem. This proved that any mathematical system based on axioms, complex enough to be of any use, would be either incomplete or not provable on its own terms. He did this by coming up with a formula which logically and consistently declared itself unprovable, within a given system. If the system was complete and consistent then that formula would immediately become a paradox, and any complete and consistent system could not contain any paradoxes. Gödel’s theorem was extremely elegant and utterly infuriating. You can imagine how mathematicians must have wanted to punch him.

This did not mean that mathematics had to be abandoned, but it did mean that mathematical systems always had to make an appeal to something outside of themselves. Einstein had avoided the contradictions in the physical world by going beyond normal three-dimensional space and calling on the higher perspective of space-time. In a similar way, mathematicians would now similarly have to appeal to a higher, external system.

If any branch of thought was going to provide an omphalos which could act as an unarguable anchor for certainty, then common sense said that it would have been mathematics. That idea lasted no longer than the early 1930s. Common sense and certainty were not faring well in the twentieth century.

For people with a psychological need for certainty, the twentieth century was about to become a nightmare.

The central monster in that nightmare was a branch of physics known as quantum mechanics. This developed from seemingly innocuous research into light and heat radiation by scientists at the turn of the century, most notably Einstein and the German physicist Max Planck. This spiralled off into an intellectual rabbit hole so
strange and inexplicable that Einstein himself feared it marked the end of physics as a science. ‘It was as if the ground had been pulled out from under one,’ he said, ‘with no firm foundation to be seen anywhere, upon which one could have built.’

Einstein was not alone in being troubled by the implications of this new science. The problem was, in the words commonly attributed to the Danish physicist Niels Bohr, ‘Everything we call real is made of things that cannot be regarded as real.’ As Richard Feynman, arguably the greatest postwar physicist, later admitted, ‘I think I can safely say that nobody understands quantum mechanics.’ The Austrian physicist Erwin Schrödinger probably summed up the situation best when he said, ‘I do not like [quantum mechanics], and I am sorry I ever had anything to do with it.’ But people not liking quantum physics does not change the fact that it works. The computer technology we use every day is testimony to how reliable and useful it is, as the designs of computer chips rely on quantum mechanics.

Quantum mechanics was entirely unexpected. Scientists had been happily probing the nature of matter on smaller and smaller scales, oblivious to what horrors awaited. Their work had proceeded smoothly down to the level of atoms. If you had a pure lump of one of the ninety-two naturally occurring elements, such as gold, and you cut that lump into two pieces and discarded one, what remained would still be a lump of gold. If you continually repeated the process you would still find yourself with a piece of gold, albeit an increasingly small one. Eventually you would be left with a piece that was only a single atom in size, but that atom would still be gold.

It is at this point that the process breaks down. If you were to split that atom into two halves, neither of those halves would be gold. You would have a pile of the bits that once made a gold atom, the discrete
quanta
which give the science its name, but you would not have any gold. It would be like smashing a piñata and ending up with a pile of sweets and broken papier mâché, but no piñata.

At first, things looked neat enough. An atom consisted of a centre or nucleus, which was made up of things we called protons
and neutrons. These were orbited by a number of electrons, which were much smaller and lighter than the protons and neutrons. In time, it became clear that some of these bits could be broken down even further. A proton, for example, turned out to be made up of a number of smaller things called quarks. An atom was constructed from a family of a few dozen different building blocks, which soon gained exotic names like leptons, bosons or neutrinos. All these were given the generic name of subatomic particles.

The problem here is the word ‘particle’. It seemed like a reasonable word at first. A particle was a tiny object, a discrete thing with mass and volume. Scientists liked to imagine particles as being like snooker balls, only smaller. They were actual things that you could, in theory, put in a cupboard or throw across the room. Physicists measured an aspect of these particles which they called ‘spin’, which they said could be either clockwise or anticlockwise, as if the tiny snooker ball was rapidly rotating. The classic illustration of the atom was a cluster of snooker balls in the centre, with a few more circling in clearly marked orbits. The study of the subatomic world, it was assumed, was like studying how snooker balls collide and behave, except smaller. Or at least, that’s how people instinctively assumed it should be.

But it wasn’t.

As research progressed, scientists found themselves in the strange position of knowing a lot about how subatomic particles behaved, but knowing nothing about what they actually were. One suggestion, which has been studied in great detail from the mid-1980s onwards, was that subatomic particles are all we can see of multidimensional vibrating strings. We are still unable to say whether or not this idea is actually true. The only thing we know with any certainty is that we don’t know what these things are.

We know that these building blocks of atoms are not tiny snooker balls because they also behave like waves. Waves, such as sound waves or waves on the sea, are not discrete lumps of stuff but repeating wobbles in a medium, such as air or water. Experiments intended to show how subatomic particles behaved like waves
conclusively proved that, yes, they did indeed behave like waves. But experiments that were intended to show that these things were discrete particles also conclusively showed that they behaved like individual particles. How was it possible that the same thing could behave like tiny snooker balls and also behave like waves? It was like finding an object that was simultaneously both a brick and a song.

Studying objects which were two contradictory things at the same time was something of a challenge. It was like Zen Buddhism with extra maths. It emerged that these subatomic things could be in more than one place at once, that they could ‘spin’ in different directions at the same time, move instantaneously from one place to another without passing through the distance in between, and in some way communicate instantaneously over great distances in contradiction of all known laws. All this was bad enough without having to assign such behaviour to objects viewed as both waves and particles at the same time. Yet incredibly, following some serious intellectual arguments in the first half of the twentieth century and some really expensive experiments in the second, much of the behaviour of subatomic thingamabobs is now predictable.

One result of the simultaneous acceptance of both the ‘wave’ and ‘particle’ models was that these objects were considered to be extremely strange. This could not be more wrong. Their behaviour is the most commonplace and unremarkable thing in the universe. It is occurring, constantly and routinely, everywhere around you, and so is surely the opposite of ‘strange’. The reason why we think subatomic particles are strange is because they are so different to how things appear at a human-scale perspective. It is once again down to the observer as much as the observed. It is our problem, not the universe’s.