Thursday, November 24, 2011

A Collection Of Nothings Means Everything To Mathematics

THE mathematicians' version of nothing is the empty set. This is a collection that doesn't actually contain anything, such as my own collection of vintage Rolls-Royces. The empty set may seem a bit feeble, but appearances deceive; it provides a vital building block for the whole of mathematics. It all started in the late 1800s.
While most mathematicians were busy adding a nice piece of furniture, a new room, even an entire storey to the growing mathematical edifice, a group of worrywarts started to fret about the cellar. Innovations like non-Euclidean geometry and Fourier analysis were all very well - but were the underpinnings sound? To prove they were, a basic idea needed sorting out that no one really understood. Numbers. Sure, everyone knew how to do sums.
Using numbers wasn't the problem. The big question was what they were. You can show someone two sheep, two coins, two albatrosses, two galaxies. But can you show them two? The symbol "2"? That's a notation, not the number itself. Many cultures use a different symbol. The word "two"? No, for the same reason: in other languages it might be deux or zwei oribtatsu. For thousands of years humans had been using numbers to great effect; suddenly a few deep thinkers realised no one had a clue what they were. An answer emerged from two different lines of thought: mathematical logic, and Fourier analysis, in which a complex waveform describing a function is represented as a combination of simple sine waves.
These two areas converged on one idea. Sets. A set is a collection of mathematical objects - numbers, shapes, functions, networks, whatever. It is defined by listing or characterising its members. "The set with members 2, 4, 6, 8" and "the set of even integers between i and 9" both define the same set, which can be written as {2, 4, 6, 8}.
Around 1880 the mathematician Georg Cantor developed an extensive theory of sets. He had been trying to sort out some technical issues in Fourier analysis related to discontinuities — places where the waveform makes sudden jumps. His answer involved the structure of the set of discontinuities. It wasn't the individual discontinuities that mattered, it was the whole class of discontinuities.
How many dwarfs?
One thing led to another. Cantor devised a way to count how many members a set has, by matching it in a one-to-one fashion with a standard set. Suppose, for example, the set is {Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, Dopey}.
To count them we chant "1, 2, 3..." while working along the list: Doc (i), Grumpy (2), Happy (3), Sleepy (4), Bashful (5), Sneezy (6) Dopey (7). Right: seven dwarfs. We can do the same with the days of the week: Monday (0, Tuesday (2), Wednesday (3), Thursday (4), Friday (5), Saturday (6), Sunday (7). Another mathematician of the time, Gottlob Frege, picked up on Cantor's ideas and thought they could solve the big philosophical problem of numbers.
The way to define them, he believed, was through the process of deceptively simple process of counting. What do we count? A collection of things — a set. How do we count it? By matching the things in the set with a standard set of known size. The next step was simple but devastating: throw away the numbers.
You could use the dwarfs to count the days of the week. Just set up the correspondence: Monday (Doc), Tuesday (Grumpy)... Sunday (Dopey). There are Dopey days in the week. It's a perfectly reasonable alternative number system. It doesn't (yet) tell us what a number is, but it gives a way to define "same number". The number of days equals the number of dwarfs, not because both are seven, but because you can match days to dwarfs. What, then, is a number? Mathematical logicians realised that to define the number 2, you need to construct a standard set which intuitively has two members. To define 3, use a standard set with three numbers, and so on.
But which standard sets to use? They have to be unique, and their structure should correspond to the process of counting. This was where the empty set came in and solved the whole thing by itself. Zero is a number, the basis of our entire number system (see "From zero to hero", page 41). So it ought to count the members of a set. Which set? Well, it has to be a set with no members. These aren't hard to think of: "the set of all honest bankers", perhaps, or "the set of all mice weighing 20 tonnes". There is also a mathematical set with no members: the empty set.
It is unique, because all empty sets have exactly the same members: none. Its symbol, introduced in 1939 by a group of mathematicians that went by the pseudonym Nicolas Bourbaki, is 0. Set theory needs 0 for the same reason that arithmetic needs o: things are a lot simpler if you include it. In fact, we can define the number o as the empty set. What about the number 1? Intuitively, we need a set with exactly one member. Something unique. Well, the empty set is unique. So we define ito be the set whose only member is the empty set: in symbols, {0}. This is not the same as the empty set, because it has one member, whereas the empty set has none.
Agreed, that member happens to be the empty set, but there is one of it. Think of a set as a paper bag containing its members. The empty set is an empty paper bag. The set whose only member is the empty set is a paper bag containing an empty paper bag. Which is different: it's got a bag in it (see diagram). The key step is to define the number 2. We need a uniquely defined set with two members. So why not use the only two sets we've mentioned so far: 0 and {0}? We therefore define 2 to be the set {0, {0}1. Which, thanks to our definitions, is the same as fo, Now a pattern emerges. Define 3 as {o,1, 2}, a set with three members, all of them already defined. Then 4 is 10,1, 2, 31, 5 is 1, 2, 3, 41, and so on. Everything traces back to the empty set: for instance, 3 is {0, {0}, {0, {0}1} and 4 is {0, {0}, {0, {0}1, {0, {0}, {0, {0}111.
You don't want to see what the number of dwarfs looks like. The building materials here are abstractions: the empty set and the act of forming a set by listing its members. But the way these sets relate to each other leads to a well-defined construction for the number system, in which each number is a specific set that intuitively has that number of members. The story doesn't stop there. Once you've defined the positive whole numbers, similar set-theoretic trickery defines negative numbers, fractions, real numbers (infinite decimals), complex numbers... all the way to the latest fancy mathematical concept in quantum theory or whatever. So now you know the dreadful secret of mathematics: it's all based on nothing.


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