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Nothing Really Matters

Nothing Really Matters
Sierpinski gasket

Originally Published 04 March 2002

by Dr David Wood, Department of Mathematics

Mathematics is full of the weird and wonderful, so where could 'nothing' possibly have a place? According to the Collins New English Dictionary:

nothing (pron) 1. not anything. 2. a matter of no importance. 3. absence of meaning, value, or worth. 4. the figure 0

So it may come as a surprise that, in fact, mathematicians actually spend a lot of the time doing nothing, but does nothing exist in mathematics?

The most obvious manifestation of nothing is zero, although what is zero?
0 degrees centigrade doesn't signify nothing, it's the same as 32 degrees Fahrenheit or 273.15 degrees Kelvin. In fact, zero as a number has actually only recently (in the grand scheme of things) become a vital component of arithmetic. The Romans certainly had no use for it, and the Greeks were very suspicious of the whole idea. For us, the number zero, rather than being nothing becomes a place holder for something that's missing, and without it 207 would be written as 27 which could be interpreted as 27, 207, 2007, 20007. Only the context would make the true size apparent. The masses of information shooting across the internet every second would be reduced from the binary 111000110100001... to a string of meaningless 'ones', 11111111111...!

So let's move away from zero, and try a new tack by trying to create nothing from something. Imagine a line of length 1 unit. It doesn't matter what unit we choose, be it 1cm or 1 mile. Now remove the middle third of the line so that you are left with two lines, each of length 1/3. Now we can do the same to each of these lines, and the same again to the four lines we now have and so on. This is a process mathematicians call 'iteration', and the idea is that if we could keep doing this forever, then eventually we would have removed the equivalent of a line of length one unit. So we would be left with nothing, since we have removed all of the line? But no, we are left with an infinite number of points. Moreover, if we take any of these points then there will be another point near it, as close as we want. This is the celebrated 'Cantor Set', named after the German mathematician Georg Cantor at the turn of last century, and has been a headache for undergraduates ever since.

Similar algorithms can be applied to squares and triangles to produce Sierpinski Carpets and Gaskets, wonderfully intricate patterns that have zero area, but infinite length (to confuse matters further, since it 'doesn't exist' in two dimensions, and is infinite in 1 dimension, mathematicians find a theoretical dimension between 1 and 2 where it has a finite length, but that's another story).

Next, what about looking at the mathematics of nothing happening? Does this give us nothing? Alas no. Take the simplest of equations governing, say, the motion of a pendulum driving a clock, or the chaos laden equations of the weather. The first thing a mathematician would do is not pronounce how the pendulum will swing, or how a cold front will come in from the east, no, instead the very first thing they would do (after getting a cup of coffee) is find out, under what conditions, and where, the equations will correspond to absolutely nothing happening! This then tells us an amazing amount of information about how such equations will behave.

In fact it turns out that there is only really one notion of true nothingness in mathematics, but be warned, even that has a sting in its tail!

When you have a collection of some objects, mathematicians call this a set and place brackets about the objects to signify this. For example, {oranges, apples, pears} is one set, containing the labels "oranges", "apples" and "pears". Now let us take another set, another collection of labels, {lemons, tomatoes, apples}.

We can now combine these sets in various ways, one of which is to take the intersection", which is just the set containing those labels that are in both sets, in our example it will be just {apples}. But what if the two sets don't have any elements in common? For example, the two sets {apples, pears} and {oranges, lemons}? The intersection is now what we call the ?empty set?, it's the set which contains nothing, signified 'Ø'

So is this the true mathematical nothing? Yes, and no. Now we have the empty set, we can have the set containing the empty set {Ø} (since Ø is now a label) with one object in it. We can also have the set containing: the empty set and the set containing the empty set ie { Ø, {Ø} } with two objects in. Here we find iteration again, and we have a set with three objects { Ø, {Ø}, {{Ø}} }, four, five, six... and before we know it we have constructed all the whole numbers from nothing, and once we have numbers the rest of mathematics follows in its wake.

So what does this all tell us? Nothing really matters!