THE FINAL MYSTERY

Figure 1. Professor John D. Barrow

If the thought of infinities gives you the jitters, just jump to the next page.

Real Infinities

Most people think that infinities are some sort of trickery with numbers, but they are in fact real, as real as the rest of maths. And because they are real, they are the most daunting objects known to humankind. John Barrow, makes the point that:

“Mathematicians have also had to face up to the reality of infinity. The issue was a big one, one of the biggest that mathematicians have ever faced .” 1

Figure 2. Mandelbrot computer image

Why can’t anything travel faster than light? Because to move anything to the actual speed of light would take an infinite amount of energy. At the Large Hadron Collider they accelerate tiny bits of atoms to 99.9999% of the speed of light, but astonishingly, to reach 100% would require an infinite amount of energy - more energy than there is in the entire universe! Perhaps the most terrifying infinity lies at the heart of black holes. A singularity where gravity becomes infinite and squeezes time and space out of existence. A massive one lies at the centre of our galaxy.

But infinities don’t just exist in exotic parts of cosmology, they exist in your laptop and mobile. Marcus Chown, a well known science writer and broadcaster, says:

“computers don’t just perform finite computations, doing one or a few things, and then halt. They can also carry out infinite computations, producing an infinite series of results.” 2

Veronica Becher, Associate Professor of Theoretical Computing at Buenos Aires University explains that:

“many computer applications are designed to produce an infinite amount of output.” 3

The examples given are web browsers such as Netscape and operating systems like Windows. So, although most of us don’t realise it, infinities are part of our everyday lives.

The infinity that most of us are used to is that of the natural numbers: 1,2,3,4,5,6,7,8,9,10,11…a hundred billion…and so on. There is no ‘biggest’ number. That’s because you can always just add another number to the one you just thought was the biggest. So the first one wasn’t the biggest! A bit strange isn’t it? Eli Maor a well known Israeli mathematician says:

“To the mathematician, infinity is a reality [his emphasis]. In fact, mathematics could hardly exist without it, for it is inherent already in the counting numbers, which form the basis of practically all of mathematics” 4

This is how deeply entwined infinity is with maths. Yet we have already seen all the evidence for mathematics being the blueprint of our reality. It’s a bit unnerving when you think about it, but it also presents us with a tremendous adventure.

The Great Internet Prime Search

One of the best pieces of evidence that numbers are independent of the human mind and also infinite comes from prime numbers. We all encounter them in school, but they never explain the ‘pure magic’ involved. Here it is. Prime numbers are numbers that can only be divided by 1 and themselves. They are all odd except for 2 which can only be divided by itself and 1.

Here are the first dozen: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37. Take the number 3 for example. You can divide it by either 1 or 3; nothing else goes into 3 without there being a remainder left over. This is also the case for the next prime number which is 5. But if you go to 6, then you can have 1x6 = 6, but you can also have 2x3 = 6, so 6 isn’t a prime. If you think about a number like 28. It can be: 1x28 = 28, 4x7 = 28, 14x2 = 28. So 28 isn’t a prime either. But the next number, 29, can only be 1x29 = 29. It seems strangely out of our control.

They start out being quite frequent, but as the numbers get bigger the number of primes reduces, but not in a way that anyone can predict. Literally, when it gets to bigger numbers there is no way to predict where the next one will pop up. They occur randomly. That’s why the military and government use them in secret codes. Now, if maths was created by the human mind we would surely be able to predict the next prime number, after all it’s us that is creating the numbers, right?

The truth is that we haven’t a blind clue about the next prime number. They have tried them out using supercomputers. The biggest prime discovered to date is over 17 million digits long, and there is still no pattern to predict where the next one will happen. Surely, if the human mind is creating mathematics, it would know when the next prime number would come up, but we have no idea. So numbers certainly seems to be independent of the human mind.

Figure 3. Georg Cantor

There are now thousands of enthusiasts all over the world using their PC’s to take part in an internet project to find the next prime number. Interestingly, it is the only game in town that can never end. Why? Because way back in 300 BC, Euclid proved that there must be an infinite number of primes!

Georg Cantor

It was a famous (Russian born) German mathematician named Georg Cantor in the middle of the 19th century who was the first to discover that there are different kinds of infinity. Not just a few either. He actually proved that there are an infinite number of infinities! According to John Barrow, Cantor was able to show that it is possible to generate:

“bigger and bigger infinite sets from ones that we already have. There is no limit to this escalation...By this means we can create an ever-ascending staircase of infinities...There is no end to this inconceivable infinity of infinities.” 5

This was a truly mind blowing revelation. Because it was so revolutionary, there was a lot of opposition to Cantor’s work at the time. It caused a civil war in maths. People didn’t want to include it in the main body of maths because it seemed so bizarre. It actually frightened them because ever since the ancient Greeks everyone had thought there could only be one infinity. Scholars and theologians used it as a proof for the existence of God.

Cantor showed that from any infinite set of things, it was always possible to make an infinitely larger one. But he also demonstrated that there had to be what he called an ‘Actual Infinite’, an infinite that was beyond human ability to understand. And the reason we could never understand it was precisely because it was beyond the power of maths to describe... Beyond the power of maths?

Numbers

I want to show you just very briefly, using the simplest example, why it scared them. It won’t take long. On one level it’s quite comical. We all know that the ordinary whole numbers are made up of ‘odd’ numbers and ‘even’ numbers. Here are the first 10.

Odd numbers: 1 3 5 7 9
Even numbers: 2 4 6 8 10

Together they make up the ordinary number line: 1,2,3,4,5,6,7,8,9,10… So it looks like the odd numbers are exactly half of all the numbers, and the even numbers make up the other half. Which is what most people think. But what happens when you double each number on the ordinary number line?

1……….(1+1)……....=2
2……….(2+2)………=4
3……….(3+3)………=6
4……….(4+4)………=8
5……….(5+5)………=10
6……….(6+6)………=12
7……….(7+7)……….=14
8……….(8+8)……….=16
9……….(9+9)……….=18
10……….(10+10)…….=20
...and so on forever.

You suddenly realise that although it seemed as though all the even numbers were only half of all the numbers, there are in fact as many even numbers as there are actual numbers! Magic?

This may seem trivial, but remember we are talking about mathematics…the blueprint from which ‘everything’ is constructed. Simple though it looks, it is telling us something deep about the nature of our reality. Something that is completely unique. Something quite extraordinary to our ordinary world of sense impressions. Infinite collections of things are ‘fundamentally’ different from finite ones because they can contain ‘themselves’ within smaller parts of themselves – something that finite things can’t do!

Figure 4. Galileo

Here’s another one which shows it even better. It comes from Galileo actually, but he thought it was merely a curiosity. He didn’t build it into a system as Cantor did. What happens if you multiply each number by itself instead of just adding them together? What’s called ‘squaring’ each number.

1x1=1
2x2=4
3x3=9
4x4=16
5x5=25
6x6=36
7x7=49
8x8=64
9x9=81
10x10=100

From this we can see that there are only 10 squares before you reach 100, so they make up only 1/10th of the numbers up to 100, so they must be pretty rare things right? But no, hang on. If you look at the first column of numbers it is just the ordinary number line (here 1 to 10) which can go on and on to infinity. Therefore, every number in the ordinary number line has a square (just x it by itself), so contrary to what we thought, there are just as many squares as there are ordinary numbers. More magic! Hope you are not getting bored. This is ‘real’ magic!

David Hilbert the most famous mathematician of his day, championed Cantor’s work referring to it as, ‘Cantor’s Paradise, from which no one will expel us’, which helped to bring it into the mainstream of maths where it has proved to be hugely influential. So important, that Hermann Weyl, one of the greatest of German mathematicians has described the whole of maths as the ‘science of the infinite’.

Magic Objects

Figure 5. Diagram depicting pi

Then there is the remarkable story of ‘Pi’, which is just the simple ratio of a circle’s circumference to its diameter, please don’t switch off now, everything that follows is very obvious. In other words, how much longer is the line around the circle, than the line that divides it in half. It’s an actual physical measurement, but one which can never ever be an exact amount no matter how big the circle. It always comes out as an infinite decimal expansion. Pi = 3.14159265…and so on, forever. You never come to a definite ‘whole number’ answer. Why is that? After all, this world is meant to be a physical place. Surely it has to have an exact answer?

On the other hand, perhaps this is a kind of hidden ‘proof’ that infinities are part of the ‘real’ world. Supercomputer calculations have determined pi to over one trillion digits, and it hasn’t ended. And like the prime numbers, no pattern in the digits has ever been found. Yet pi has proved to be enormously useful in physics, and it appears on a routine basis in equations describing the most fundamental laws of the universe.

Figure 6. Diagram of square root of 2

Similarly, all engineers rely on approximations to infinity in their calculations every day of their working lives because it gives them the greatest possible accuracy in designing things, like aircraft and bridges and shopping malls, and they are quite happy with it because it works!

Another piece of magic which contains an infinite expansion is the √2 which is equal to: 1.414213562…and so on forever. In ‘physical’ terms it could hardly be more simple. It is just the length of the diagonal of an ordinary square with sides that are equal to 1. But no matter how big or tiny or gigantic you make the square, the diagonal will never be equal to a ‘whole’ number! It seems very mysterious that something so ordinary and simple can hold within it an infinity that appears to be bigger than the universe. Why is that?

Fabulous Magic

More magic lies within a straight line. It’s about the simplest and least complicated thing we can think of, right? I mean, just an ordinary plain line. Yet it is actually shrouded in fabulous mystery. I’m sure you will not want to believe it, but consider this. We know that any and every line is made up of an infinite number of ‘points’ – that’s what ‘makes’ the line. This means that a single line a million miles long is made up of an infinite number of points, but so is a line just one inch long. See what I mean? I do hope you are enjoying this – it gets even better!

Infinities seem to define everything that ‘exists’. They appear to point to something that is not limited by ‘space’, like the length of a line. And also something that is not limited by ‘time’. Paul Davies (Director of the Centre for Fundamental Concepts in Science, Arizona State University) reminds us that:

“there are no more moments in all of eternity than there are in, say, one minute. In both cases there is an infinite number” 6

Those who think that these infinities are just something created by us should consider ‘now’. Is the ‘present moment’ perhaps one second, or is it half a second? Perhaps it’s a tenth of a second, or maybe a hundredth? Or is it a jiffy? The Planck time of: 1/000,000,000,000,000,000,000,000,000,000,000,000,000,000th of a second. A jiffy is just as real as a second. It seems to be telling us something very fundamental about all of reality. All these infinities, all the ones we’ve been looking at, seem to suggest that every possible option actually exists. It’s as though our reality, our physical world if you like, is just a sort of ‘shimmering arrangement’ of logical mathematical consistency:

“A quivering slice of mathematical stability [lost] in an infinite ocean of all possible possibility” 7

This takes us to the world of the very small and the world of the very big. From what happens inside atoms to the enormity of the multiverse.

The Book Chapter 23
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