Numbers (NVGR)

This is a blog about math. If that scares you, go ahead and return to reading about video games. You won't hurt my feelings.

What's your favorite number? You might call it a lucky number; your go-to number when someone asks you for a numerical input. Do you genuinely feel that this number is better than others, or do you just feel a seemingly irrational attachment to it?

I like numbers. One of my favorites is 892. It's a nice, friendly number. Less than 1000, it's of manageable size, but has plenty of syllables, so it sounds impressive, but not too unwieldy. Beyond that, there's no particular reason I like it better than any others. Whenever I see it in some unexpected context, I smile and think, "There it is again." Other numbers that capture my interest are 232 and multiples of 3, especially those whose digits are all multiples of 3. I find them more elegant than their peers.



I've never felt that any number has brought me luck, since luck doesn't make sense to me. Truly random processes may or may not exist, but for all intents and purposes we can call the throw of a die a random event. Humans are good at seeing patterns, even patterns that don't exist. Attachment to items, even abstract items like numbers, can develop from exposure to apparently random experiences. We get a winning or losing streak and we associate it with some correlating event, like the presence of an item, the performance of a ritual, or the choice of a number. Your lucky number is no more likely to win at roulette than any other, but you have to choose something, right?

Math captured my interest around middle school, when I entered a mathematics program run by the University of Buffalo. I was introduced to number theory in seventh grade, when we were tasked with defining numbers in terms of set theory.

As it turns out, this can be done. We can then define a successor function that builds the next number by taking the previous number and appending the empty set to it. All we have to do is define 0 as the empty set:
0 = {}
1 = Succ({}) = { {} }
2 = Succ({ {} }) = { {}, { {} }}
...and so on.

Interestingly enough, this construction shows up in computer science. In 1936, Alonzo Church published a paper in which he definied lambda calculus. It was essentially a formal programming language with exactly three contructions:
names - x
function declarations - λx.y
function applications - (x y)

Names are handles by which functions are referred to and applications result in functions, meaning everything in the language is a function. The most basic function in lambda calculus is the identity function:
λx.x - Takes an argument and gives it back.

So doing something like this:
(λx.x 1)
would apply the identity function to "1", resulting in 1. Many functional programming languages, especially the Lisp family, are based on lambda calculus.

But hold on. This is a language without numbers. Where did I get that 1 from? And if everything is a function, isn't inherently weaker than a language that has more types (including numbers)?

As it turns out, we can build numbers out of these abstract functions. One way to do it is called the Church encoding, developed by the same Church I mentioned earlier. It looks like the following:
0 = λf.λx.x
1 = λf.λx.(f x)
2 = λf.λx.(f (f x))
3 = λf.λx.(f (f (f x)))
...and so on. Thus, every number is a function whose structure is both unique and usefully predictable.

Looks familiar, doesn't it? On the surface, at least, the Church encoding of natural numbers closely resembles the above set theory representation. Church and his students even constructed perfectly reasonable ways to manipulate these numbers, including your basic arithmetic operations.

My point is that numbers are interesting, even though we typically take them for granted. When you count something, you don't think about how you're counting, you just start at 1 (or, if you're a computer scientist, 0) and generate the successors automatically.

In fact, the concept of 0 as a number is an interesting topic on its own, and it might even be called the most interesting natural number. In some cultures, it started as a placeholder between symbols, so something like 102 could be distinguished from 12 or 10020. It only relatively recently that it was accepted as a number of its own right, for several philosophical and practical reasons.



On the other end of the spectrum, we have infinity. Most people don't consider infinite numbers. If you have two inifinite sets, they're both equally big, right?

As it turns out, that's not the case. Some sets infinite sets are countable and others are not. A countable set is one that is comparable to the natural numbers. That is, if we can assign one natural number to each element in some set, we have a countable set.

But if the natural numbers are inifinite, how can you have an uncountable set? After all, we can keep assigning numbers forever and never run out.

However, there are uncountably more real numbers than there are natural numbers. Intuitively, you might get an idea of why this is. Take any two natural numbers and count how many numbers are between them. This will obviously be some natural number. Now take any two real numbers and count how many numbers are between them. It can't be done. Regardless of how close together those numbers are, there is always an uncountably large amount of numbers between them. This is no proof, but it plants the idea in your head.

In fact, we have a hierarchy of infinite sets: the (aleph) numbers. (aleph-null) is defined as the size of the set of natural numbers, and higher subscripts describe the size of other, larger infinite sets. The implication of this structure is daunting: One set can be more infinite than another!

And let's not even get started on the transcendentals. Look up Euler's identity and try not to be impressed.

I'm a student of computer science, and I honestly chose the field because I like to program and mess with computers. But computer science reintroduced me to mathematics when I started to study complexity theory, and I found the parallels between counting and computation quite profound.

Back in high school, people would ask, "When are we ever going to use this stuff?" Today, I don't care if I never use number theory. I'm just glad I learned about it.

So, what spurred this appreciative rant of all that is numerical? I was re-playing Metal Gear Solid 4 the other day and saw the side of Drebin's APC:



"893," I thought. "That's only one less than 892, and 892 is a damned good number."