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"If man is five then the devil is six. If the devil is six then God is seven" - Monkey Gone To Heaven, The Pixies.
BBC4's recent three (appropriately enough) part series Magic Numbers:Hannah Fry's Mysterious World of Maths spent a lot of time, possibly a little too much, trying to ascertain if mathematics was something we've discovered, or something we've invented. Are numbers, sums, logarithms, and trigonometry a gift from some kind of God, something already extant in the universe, or something our incredibly complex brains have dreamed up to help us make sense of the wonderful and frightening world that we all must navigate our way through?
That's not to say it was a bad series. It wasn't. It was great. A lot of that is down to Fry herself. She's enthusiastic, well informed, and she manages to relay often baffling concepts across to laymen such as I in a style that's neither patronising nor alienating. She's game too. Witness her riding rollercoasters, cycling around Bloomsbury, singing, having her brain scanned at UCL, and measuring nautilus shells. Actually, sounds like a great gig! Although I may have balked at descending (headfirst) the world's fastest zip wire (in north Wales) which Fry does to illustrate Newton's law of universal gravitation.
But some credit must go to her co-stars. Magic Numbers:Hannah Fry's Mysterious World of Maths was like watching every episode of Sesame Street ever. Not that it featured guest appearances from Big Bird, Bert & Ernie, and Mr Snuffleupagus but in that it was bought to us by the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and every single combination of them imaginable - plus a little more.
Some would be alienated by the idea of watching a three part documentary about mathematics but not me. I love maths. But loving maths doesn't make me particularly knowledgeable about it so this series had the potential to be both educational and entertaining and so it proved to be. At times it seemed to be more an existential quest to travel to the heart of maths than simply a history of famous mathematicians, their theorems, and how those ideas affect the world we live in but, fortunately, there proved to be plenty of that too.
Fry began by asking why does maths infiltrate everything? Many see the patterns created by mathematics as the underlying language of the universe. They're hidden in flowers, shells, and viruses and those of you who are familiar with the Fibonacci sequence will see that nature seems to have evolved into mathematical formulae and that humans, in their attempts to harness and ape the beauty of nature, have, in art and architecture, utilised the same numerical patterns.
In ancient Greece, Pythagoras and the Pythagoreans believed numbers were a gift from God and that that gift had arrived from the almighty in the form of music. Octaves, vibrating strings, and perfect fifths all corresponded to mathematical formulae. Whilst bones dating back to the Paleolithic era, almost three million years ago, with indentations marked into them are believed to have been used for tallying it was with the Greeks that the study of numbers, what they were and how they worked, really took hold.
Plato, like Pythagoras before him, also believed mathematics originated from God. He cited the example of the perfect circle. Something that doesn't, and can't, exist in our material world but does live in some form of mathematical heaven somewhere. If that sounds a little crazy, Plato also believed, or at least hypothesized in his 360BC page turner The Timaeus, that everything in the universe could be created by five solid objects.
These came to be known as the Platonic solids and they were the cube (representing earth), the tetrahedron (fire), the octahedron (air), and the icosahedron (water). "But, Dave, you've only passed four Platonic solids" I imagine you saying. That's because the fifth one, the dodecahedron, was anointed by Plato the great honour of representing nothing short of 'the entire universe'. Big daddy dodecahedron indeed.
These five solids are the only objects that exist, or seemingly can ever exist, where each side is exactly the same shape and size. Maths geeks love 'em. Fry presents them to a few and they unwrap them and remove them from bubblewrap with the glee of children at Christmas.
Plato didn't stop there, though. His ideas got even more adventurous. He asked his readers to imagine humans locked in a cave, shackled by their necks, and forced to stare for eternity at a blank wall completely unaware of the raging inferno that was burning above their heads. Above this fire was a road in which all other life passes through but all the prisoners can see are their shadows in the fire. This, Plato thought, was an allegory that spoke deep truths about our own concept of reality. We, the humans, are the prisoners staring at the reflections on the wall and mistaking it for all there is while the real world, or the world that is more real than real, exists in a spiritual plane beyond our comprehension.
Crikey! Euclid was another brainbox too. Around 300BC in Alexandria he wrote The Elements and it now appears as homespun as Plato's theories appear hi-falutin'. Euclid posited evidently simple truths such as the undeniable fact that a straight line can be drawn between any two points and that all right angles are the same and then he tested these, and other, fundamental building blocks of maths to see if they were assumptions or absolute truths. The application of logic to mathematics was a new thing and since the invention of the printing press it is estimated that Euclid's Elements is second only to the Bible in the number of editions published. Until the advent of the 20c it was considered to be something that all educated people had read.
Yet, one number didn't appear in Plato's Timaeus or Euclid's Elements and that number was our slippery friend, zero! That didn't wash up on European shores until the 7th century and it wasn't until about 400 years later, at the time of Crusades, it really gained any traction as a concept. Obviously, as with anything Arabic at the time (and, sadly, now) zero was treated, initially, with both suspicion and contempt.
It had lead a long and happy life in India (one minor gripe about this programme is that it does take a very occidental view of the history of mathematics) before it ventured across Arabia to the Mediterranean where it was finally given a warm welcome by the Pisan mathematician Leonardo Pisano Bigollo, better known as Fibonacci.
Roman numerals had had no need for a zero but Fibonacci had been educated in North Africa (Bugia which is now the city of Bejaia in modern Algeria) where he'd learnt the Hindu-Arabic numeral system. That's the one we all use now. Even Jacob Rees-Mogg. As with the immigration of people, the immigration of concepts proved beneficial. Roman numerals were long and unwieldy and this 'new' system was much easier and much more effective. In fact it was as easy as 1 2 3.
What wasn't so easy to get one's head around were negative numbers (-1, -2) and even more confusing was the idea that if you multiplied negative numbers by each other you always end up with a positive number. In fact, there is no multiplication that can be done that can result in a negative number as its solution.
It's at this point I start scratching my head and, here, Fry doesn't really help, conceding, disappointingly if understandably, that it would take too long to explain before moving on to the even more baffling concept of 'imaginary' numbers. Numbers like 'I' (!) were invented so that squaring, multiplying a number by itself, could, finally, result in a negative.
A number had been invented, rather than discovered, and, in a development that utterly discombobulates me still, this number, or these numbers, became vital in the development of radar, waves, and aeronautics. It is proposed that without the invention of the imaginary number we'd have never been able to fly in aeroplanes, so vital were these numbers in the science behind flight.
Jetting forward to the 17c we find Rene Descartes, like Euclid before him, daring to question all the prevailing philosophical and mathematical assumptions. Inspired by a night of bad dreams, he became the first person to invent a formula that could describe shapes and thus lay the foundation for modern science. Descartes', at the time, radical idea was that experience and reason were the soil in which ideas grew rather than tradition, authority, or most iconoclastically of all, divinity. You'll not be surprised to read that, in 1663, the Catholic Church prohibited his books.
But the ideas were out there anyway and horizons were expanding. Aristotle's seemingly obvious assertion that heavier objects fall faster than lighter ones had remained unquestioned for years until Galileo Galilei, in the 16c, came up with a different explanation.
It's true that if you drop a hammer and a feather from the same height the hammer will reach the ground first but that's because of air resistance. Dropped in a vacuum they reach the ground at the same time which was something that, more than three hundred years later, American astronaut David Scott was able to prove when he repeated the experiment on the moon.
Galileo saw the world as a book written in the language of mathematics (and it's highly debatable how far into it we are) and Isaac Newton was determined to write his own chapter. Unafraid too. Newton was not a big fan of theory and, as such, performed experiments on himself. Like poking himself in the eye with a blunt needle so he could understand better how optics work.
Of course, Newton's most famous work was in the field of gravity. Gravity, like zero, is a slippery beast. It's strong enough to hold us down but we can defeat it, slightly, simply by using our muscles to lift our arms. Newton's law of universal gravitation provided the maths behind the science that powered the Industrial Revolution and changed the world. The invention of the steam engine opened up the country to people and goods, but also to ideas. We had entered the age of machines and mathematics was making these machines faster and more effective.
Scientists started to look into the invisible links between electricity and magnetism and Michael Faraday was the first to connect the two. Using an electric wire and a magnetic needle that affected each other without making contact. James Clerk Maxwell did the drudge work of distilling these links into equations that proved electricity and magnetism were inextricably linked.
He coined the term 'electromagnetic field' and described how they created 'electromagnetic waves' that travel through time at 300,000 kilometres per second. The speed of light. That couldn't be a coincidence. Light had to be an electromagnetic wave and, with this, we had a new understanding of what light is. We had, quite literally, seen the light.
Equations were revealing new truths about the universe, and they were getting more and more complicated and confusing too. Not least when scientists started questioning the long held assertions of Euclidean geometry. What if geometry only worked like physical space on a local scale, but on an astronomical scale, one almost too great to imagine, worked very differently?
Rules were being changed. The shape of the world, the shape of the universe, was now under question and was put back under the metaphorical microscope. This new found freedom for exploring more abstract scientific ideas gave birth to a difficult child. A child they called infinity.
Infinity is hard to think about it, and it's hard to describe. One professor claims to be "tormented by infinity" while another says it is as real (or as unreal) as 1 or 0. German mathematician David Hilbert imagined a large hotel with an infinite number of rooms and wondered what would happen if all the rooms were full and a weary traveller looking for a bed for the night was to show up.
In the Hotel Infinity you can't simply move in to the last room because there is no last room, infinity never ends, so instead you have to get the person in room one to move to room two, the person in room two to move to room three, and so on. Because infinity + 1 = infinity and infinity + 2 also equals infinity.
Another German, Georg Cantor, set out to "tame the infinite beast" by asking "how big is infinity?", a simple sounding question that caused a revolution in mathematics, had Cantor denounced as a "scientific charlatan" and a "corruptor of the youth" by people who tried to sabotage publication of his works, and also, possibly, led to a nervous breakdown and a spell in a psychiatric hospital with what we'd now recognise as a bipolar disorder.
The concept of infinity had been known about since the times of the ancient Greeks but Cantor felt we couldn't just accept it, we had to understand it. But it's almost impossible to understand except in an abstracted way. There can be infinities between 0 &1, between 1 & 2, 2 & 3, and, in fact, between any numbers you wish to think of. 0.01 is smaller than 0.1, 0.001 is smaller than 0.01, and so on ad infinitum! Cantor's conclusion was that some infinities are bigger than others although he neglected to add that some infinity's mothers are bigger than other infinity's mothers.
You don't tend to find infinity (or multiple infinities) in the physical world - because it has a beginning and end - so we are now deep into the realm of abstract mathematics, a journey that will only get more and more complicated. A visit to the Culham Centre for Fusion Energy in rural Oxfordshire is where the show really starts to lose me.
It's still fascinating but both the maths and science are proving very elusive to someone who long ago accepted that one and one is two. At Culham they're trying to harness the power of a star, by bending maths, to improve our future. They're attempting to recreate the conditions on the 'face' of the sun to make plasma and then hold it in place at 200,000,000 degrees celsius.
Subatomic reactions we know little of take place making the movements of this plasma difficult to predict but it needs to be controlled because if it hits the wall of the doughnut shaped tube it's encased it in it will immediately cool. Which at least means the good people of Abingdon should, hopefully, be safe from 200,000,000 degree celsius fireballs should something go wrong.
The logic of mathematics starts breaking down as maths redefines the nature of space and time. Algorithms so powerful appear that they can, mostly, recognise human faces and set theory, a language that talks about numbers as groups (odd numbers, even numbers) creates a rather confusing paradox they call 'who shaves the barber?'.
Hannah Fry visits an actual barber to try to explain it to us. It's a contradiction where the barber both shaves himself and does not shave himself, where he is both a barber and, simultaneously. not a barber. Bertrand Russell realised that, with this paradox, mathematics was more fluid than we had previously thought. Perhaps mathematics was not even grounded in logic at all?
Albert Einstein crops up, surprisingly for the first time, with his big hit E=MC2. The 'world's most famous equation' posited that matter and energy are equivalent and, because of that, nothing can travel faster than the speed of light. It means we weigh a tiny bit more when we travel in a a car or on a plane.
Objects travelling close to the speed of light become considerably heavier and this explains how the stars convert mass into energy as they burn brightly in the night sky. When we look at the sun we see what it looked like eight and a half minutes ago. Some of the stars we see are not as they are now, they may not even exist anymore, but as they were thousands or even billions of years ago.
It's a lot to get your head round and it works in a more local way too. Even if you're stood right next to somebody it takes time, admittedly a negligibly minute amount of time, for you to see them. The universe expands not just in space but in time and Einstein understood that the concept of time is relative dependent on the position of the observer. Einstein overturned the ideas of not just the ancient Greeks but also of Isaac Newton, when he moved on to gravity.
Einstein thought that gravity worked as a result of distortion, a warping of space/time in the vicinity of massive objects. He thought we should be able to observe light from distant stars being bent by the power of our sun and studies of solar eclipses proved Einstein to be correct. Space was no longer the stage on which things happened, Space was a key player.
His findings are still used in GPS software and in satellite technology. In the Prussian city of Konigsberg in 1930 at a maths conference there was a clash of two other giants of mathematics. David Hilbert and the younger Kurt Godel argued the toss in the ol' discovery/invention debate to which Fry keeps returning us to.
Hilbert felt humanity would soon know all the maths there is to know but Godel disagreed entirely. He felt we could never know all the rules and even, head-scratchingly, that he could prove that some mathematics is unprovable. Godel's incompleteness theorem.
Godel proposed that some maths was actually faith based. If so could maths one day 'prove' the existence of a God? Or, possibly, something far stranger? Alice in Wonderland, written in 1865, was believed to contain passages that acted as a thinly veiled satire on that era's development in mathematics but, surely, Lewis Carroll would have had a field day with quantum physics which was brewing in his lifetime but became a much bigger concern in the 20c.
Quantum physics, or quantum mechanics, is a very uncertain science which, Fry tells us, developed around Erwin Schrodinger's observations of the seemingly unpredictable behaviour of subatomic particles. Particles that can be both up and down at the same time. Particles, even, that can both be and not be at the same time.
Imagine a cup that is both full of water and completely empty at the same time. It's not easy. Because this doesn't happen in the physical world. But we don't live in the quantum world so we can't understand the rules. We are the humans locked in Plato's cave looking at shadows on the wall and imagining this is all there is when, in fact, there is so much more. Things we can barely comprehend.
It's a world where lights can be on and off at the same time. It's a world where, quite literally, you CAN have your cake and eat it. It's also a world which can create a phenomenon as bizarre as 'entanglement' where electrons as far as 1200kms apart can affect each other instantly. There appears to be no direct or indirect causal link.
The word that describes their nature is synchronicity and it's something we still don't understand or know how it works. Even if every leaf on every tree in every forest in the world has been following the rules of synchronicity and entanglement for millions of years. Without quantum physics there would be no trees, no oxygen, and no life on Earth whatsoever.
But how does this weirdness solidify into our day to day, comparatively normal, existence? Our reality? Us humans are never in a 'superposition' where we both exist and do not exist at exactly the same time. Or perhaps we are and we just don't know.
The possibilities of entanglement and synchronicity taken to the next level suggest the idea of multiple universes and if there are multiple universes there would have to be an infinite number of multiple universes and if this, this multiverse theory, was true that would mean that everything that could ever happen has happened and, is in fact, happening now, happened in the past, and will happen in the future all at the same time.
That is what you call a happening, man and at happenings people went on trips. A trip was, in a very different way, exactly what Hannah Fry took us on during her often baffling, occasionally weird, but always fascinating journey into the world of mathematics. We were joined along the way by an assortment of clever professors and boffins but the main credit has to go to Fry and her team. There was a fair bit of repetition but a mathematics dunce like me needed that to help me stay on track.
Now, at the end of this trip, I'm both less confused and more confused about the possibilities of mathematics, almost as if I'm in a superposition, but those possibilities now look endless. Like, David Hilbert, I thought we knew most of what there is to know about numbers and how they work but Hannah Fry disabused me of that notion and pushed me over to the side of Kurt Godel.
I feel as if mathematics is a huge forest that, like the universe (or multiverses) it both forms and informs, expands much much faster than I could ever hope to travel through it. Certainly before my number is finally up. Guess I need to just enjoy the trip.
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