How To Prove Anything With Mathematics
4 - A Tutorial of Sorts

Written by Dr Yes

Dr Yes has been a little busy as of late. First there was the Korean incident, then the monkeys got loose. As a result there have not been nearly enough proofs coming the CEO's way and as a result, our esteemed correspondent has been copping some flak. In an attempt to divert some of this unwanted attention, Dr Yes is formally inviting proofs from the people (Nice try - Ed.). But, however hard as it may be to believe, we have standards here at Karate Party and so we give a few ground rules in a typically convulted manner;

What it is to prove something using mathematics?

A cornerstone of mathematics is the proof. A proof being a demonstration that, given certain assumptions or, if you like big words, axioms, some statement is necessarily true. The only other thing you have to know about proofs is that the must be based on logical argument rather than empirical (measured) evidence - sorry but there is little room for analytical science in the proofs corner. You could argue that because the sun has come up every day in living memory that it will come up tomorrow. This is not a proof but a hypothesis, and as mentioned earlier, we have no room for science in these parts, only cold hard mathematics. Incidentally, if you thought that math was warm and fluffy then you were wrong and I will prove this a little latter on.

So basically, to prove something using mathematics we must carefully define what it is that we are trying to prove and then logically prove that for a certain set of circumstances, or axioms if you are paying attention, that this is the case. One may make the mistake in assuming that logic is rather fluffy and can be crushed with semantics. This is not the route of choice here and if you are of persuasion you can slink off back to Philosophy 101. Instead, a good first step is to take a loosely defined statement that you are trying to prove e.g. Mathematics is cold and hard, and define it in a way that is most useful to your argument. This is where ones grasp of semantics is allowed to shine. However, a proof really tells the men from the boys in the choice selection of axioms used - the more perverse or convoluted the better.

We will now give a worked example: Prove that mathematics is cold and hard. Here at Karate Party we believe that everything on the internet is true so, as this an axiom, we could define mathematics, hard and cold in just about anyway we see fit. We could use the definition of mathematics from Wikipedia but it is quite long so, according to the more succinct answers.com, mathematics is "The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols". From the same fine source, we can define hard. Here we get a break as there are 24 definitions, many with sub-definitions. We will chose one that best suits our argument, number 3c: "Difficult to resolve, accomplish, or finish". Finally we define cold, again from answers.com. Here we only get 11 definitions, but number 3 will do: "lacking emotion". Thus, we must prove that math is both difficult to resolve, accomplish or finish and lacks emotion.

We will start with emotion. Wikipedia is more brief in its definition of emotion as"...a neural impulse that moves an organism to action". As we have defined mathematics as the study of something, it is clearly an exploration rather than an explorer and as such has no neurons, let alone a neural impulse. Thus mathematics is devoid of emotion and is thus cold.

Moving on to hard. Is mathematics difficult to resolve accomplish or finish? I could argue that this is true and cite the Millennium Problems - 7 unsolved mathematical proofs with a prize of US $1 million for each proof - as fiscal evidence that mathematics is difficult to accomplish - feel free to enter your proof there too by the way. However, this may be construed as using an empirical argument and this was disallowed in the definition of a mathematical proof. All we ask is that you play by the rules so its back to logic then. We defined mathematics as the "the study of the measurement, properties, and relationships of quantities and sets...". One can arbitrarily define an infinite number of quantities and/or sets, which will additionally change with time - the number of goldfish in Texas and the price of fish. Studying something that is in constant flux surely must make mathematics difficult to totally resolve or finish. Perhaps not impossible but certainly difficult. This is good enough though, as our definition of hard helpfully gives us this leeway. With this we can prove that mathematics is difficult to finish and thus, by our definition, hard. We have thus proven, with one choice axiom, that mathematics is both cold and hard.

Enjoy.

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