In this post, I'll address the seems-to-be-everlasting objection:
Mathematics is bullshit - there's axioms that are just true and you don't prove them, and then you just pull out more bullshit out of them.
Seriously, that's a cleaned up version of an objection a few friends made me the other day. This is the biggest beef I have with them about maths.
It's a valid objection. If you don't have a clue about mathematics.
Disclaimer: The following is not an official opinion of all mathematicians. Just the true ones. ;)
So, let me go into what axioms, lemmas, theorems and proofs are. And yes, I hear you back row, there will be examples.
I already likened a theory to a picture before, and let me do that again (no, this is not the example, this is a parabol or simile). The entire picture is a theory - static, beautiful, vast. Ungraspable all at once sometimes (like standing too close to the screen).
The first few strokes of a brush are the axioms. Some artists will tell you that the picture cannot be changed once the first strokes are done, they just paint the rest like it's already there in their heads. Only the few first strokes are free. Then again, some artists don't say that. We will not concern ourselves with such amateurs (oww, please, no more flying brushes, I was kidding!).
And now it is time to begin our example. Our theory will revolve around a game called rock-paper-scissors. You've heard of it, right? It just so happens that I don't yet know any maths around it, so let's just see how it develops into a mathematical theory, shall we?
We first need a set of axioms that will axiomatize our theory. That means that everything in our theory will be derived from them. They lay the groundwork. If you are one of those who voice the objection, imagine an implicit Let us assume that in front of every statement, because it's actually supposed to be there, but nobody writes it because, well, every mathematician knows it's there.
How do we go about this axiomatizing business? Do we want to simulate playing the game? Well, that's a process, and we can define a function (map, um... I haven't defined what functions are here. Let's say they are black boxes that take some stuff in on one side and spit out appropriate stuff on the other. We'll deal with them later.)... So, yes, we can define a function that will take what the players played and produce a result of who won. Yeah, that can work. Who won? Well, the one whos symbol destroys (incapacitates, decapitates, pulverises or otherwise inconveniences) the opponent's, obviously. This function business seems superflous though - let's ditch it and deal with the interesting part - who trumps who?
So, the axioms, try one:
1) There are three objects called Rock, Paper, Scissors (R, P, and S for short).
2) For every pair of objects (a, b), one of them trumps the other, unless a = b. (Let's write a < b ifb trumps a, shall we?)
3) The number of objects that an object trumps is equal for every object.
4) The number of objects that trump an object is equal for every object.
Does this look like what you would want for a rock-paper-scissors game? Well, let's see.
Now, for all of you who still cling to that objection, here's the real catch that renders it moot - now, we will try to apply the theory to something that we actually do want to study. In order to do that, what we are studying must satisfy the axioms, and here, we will have to prove the axioms, in a way. (Actually, we have to prove that the game rock-paper-scissors as we know it satisfies the axioms).
This is trivial -
1) yes, we have those 3 (and only those 3) objects in the game
2) of course
3) each object trumps one other
4) each object is trumped by one other.
Seems it does satisfy the axioms. Great!
You might say that what we just did is stupid - we wrote the axioms to suit what we are studying, of course they're going to fit! You must consider this then: what if we wrote the axioms and built a theory on top of them (painted some of the picture) and then discovered that the thing we actually want to study doesn't satisfy them? Well, it's obvous, but a bit dissapointing - we would not be able to use the theory for our object of study. The cool thing about it, though, is that we would be able to use the theory for stuff that does satisfy the axioms - all of it. This is the real secret - mathematicians often do not care about the object of study - physicists, engineers and other, more real world people think about that. Mathematicians are painters - they see the lines on the canvas, not the table of fruit and assorted meatloves they're painting (The weird ones don't even get hungry). The real world people are then happy to look at some of the spots in the picture and use them to their advantage, but for a mathematician, the picture is the thing that they are painting, not the moot real-world idiocies their peers find so attractive. (It must be noted the table of fruit often does provide ideas for exploration though, and this is a very puritan view.)
Here, we should ask ourselves two questions
a) Is the above axiomatization complete? Meaning, is *our* definition of rock-paper-scissors the only game that satisfies those axioms?
b) Is it minimal? (Is one of the axioms actually a theorem, meaning that it follows from the ones that were set before it?)
Answers:
a) No, it is not complete. If you think about it, we haven't defined who trumps who. We could just as easily say that paper beats scissors, scissors beat rock and rock beats paper. (That this satisfies the axioms is an excercise for the reader).
b) No. The 4th axiom follows from the 1st three.
Here we embark on our first actual proof. Proofs are important: if you think of statements as coloured dots on our canvas, proofs are the lines of joined dots that lead from several dots to the statement we are trying to prove. Only statements we can join with the axioms through some set of lines belong in the picture, all others are just not there.
The proof is easy, in a way. I'll even prove it for not just games with 3 objects - I'll prove it for games with n objects - as long as n is finite, and larger than 2. (That means we replace the 1st axiom with the one saying "we have n objects, called 1,2,...,n". Do note that these are just names, 1 < 2 (2 trumps 1) may or may not be true, we don't know).
So, how do we prove this? Well, we have n objects, right? We also have axiom 3 that says that each object trumps the same number of objects. Let's call that number m. Then we have axiom 2. Because of it, we can write the number of objects that trump an object a down: it is n - m - 1. Why? Well, we have n objects all together, a trumps m of them, none of which trump it back, and a does not trump itself. All othersMUST trump a, because of axiom 2. Notice we didn't require anything of this a, just that it was an object. So, this argumentation is valid for any object. It is therefore valid for all of them. QED.
(QED means Quod Erat Demonstrandum, which is fancy latin for "what we wanted to show", or "which is what we wanted to get you to understand the whole way".)
Since the text is long, this is how mathematicians do the proof:
We can get rid of axiom 4 then. Good riddance. We now have less to check when we want to see whether something satisfies the axioms.
There is the question of problem b. We can either dismiss it as not-a-problem because it's irrelevant how we name things anyway. I'd like to get into a quick discussion here though: how many relationships do we have to specify in order for the game to be unambiguously defined? One? Well, let's try:
Let's assume S < R. This forces P < S because otherwise Rwould trump two objects and we can't have that (excercise). The same thing now forces R < P. So, I guess 1 is the answer for an n = 3 game.
Let us now replace axiom 1 with the one mentioned above the proof for real, because the standard rock-paper-scissors just got boring. Let's study games of n objects. (And so we widen our canvas)
The first interesting thing we can ask ourselves is what numbers of objects are ok. For instance, can we make a game with 4 objects? Well, if we try, it doesn't work, but what do the axioms say?
Whatever result, it would certainly help if we knew what m and p are - they being the number of objects a trumps and the number of objects that trump a. (every a). From seeing our real-world rock-paper-scissors game, we can get the idea that they should probably be equal. So here it is:
Lemma: p = m
(lemmas are "support for proof of theorems", but sometimes they are more important than the theorems themselves. Go figure.)
Proof: We are going to be counting relations here. A relation, in this case, is that a trumps b. For each pair (a, b) we have one such relation (and only one), because either a trumps b or the other way around, but not both. There are n(n-1)/2 such pairs in our little game. (If you don't believe me, try to prove that yourself!). We can also count the number of relations another way - we have objects, each of which has exactly m relations where it trumps another. If you think about it, these are all the relations there are, because the ones where the object is trumped by another are counted by that object's m. We have now counted the relations in two ways, and we can set an equation: n(n-1)/2 = n m.
We cross out the n on both sides, do a bit of jiggling and out falls m = (n-1)/2.
It should be obvious that n = m + p + 1. We plug m in and we get n = (n-1)/2 + p + 1. When we shake this around we get p = (n-1)/2. Great, now we know m = p. QED.
Here's how I did it:
So, can we make a rock-paper-scissors style game with 4 objects? Let's see:
n = m + p + 1, 4 = m + p + 1, which means 3 = m + p. There's a problem though - m and p are supposed to be equal, and here that means that m = p = 1.5. Hm... That doesn't make sense, since they MUST be whole numbers so we can make a game (try having scissors trump paper and half or a rock). This means that 4 is not a permissible number of objects.
Corrolary:
It follows simply from the above that the only permissible numbers of objects are odd ones.
So, what have we learned today?
We saw that axioms aren't pieces of bullshit you don't prove. They are statements you assume to be correct in order to build a theory. You still have to prove the axioms to *use* a theory.
We learned that systems of axioms are sometimes not optimal, and need to be pruned.
We have also learned that theorems are statements that follow from the axioms, or assumptions.
Lemmas are statements that are made to support a theorem and proven.
Corrolaries are statements that follow directly from a theorem and their proof is trivial.
Perhaps I should point out where the beauty lies here. We have taken a children's game and expanded it to as many objects as we liked. We now know that for whatever odd n, we can think of a rock-paper-scissors like game. We have reasoned about all possible such games at the same time! The power!
Casually reading about it isn't that illuminating though. You have to go and start deeply thinking about something simple. It will then show itself to your mind's eye in complex shapes and interconnects, the simplicity and, at the same time, complexity of the structure playing in a caleidoscope of logic. I cannot describe how that feels. You really have to experience it yourself.
Anyway, I had fun writing this (and a very restless hour trying to find a proof for the m = p theorem. Thanks to #math on freenode for a nudge in the right direction). I hope you had fun reading about it. I hope you saw some of the beauty of math already, and that I didn't get overly technical.
'Till next time,
Gašper
