Proof in Mathematics

A guide to understanding proofs in Mathematics

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Mathematician Carl Gauss called Mathematics the "Queen of the Sciences". Why? Mathematics is at the heart of all of the Sciences, knowledge and understanding of this discipline is necessary to understand Physics, Chemistry, Biology and more.

So how is Mathematics different from the other Science disciplines? At the core of the answer is PROOF.

The Sciences rely on observation to formulate their theories. After observing something occur 99 times the scientist is well placed to hypothesise that it will happen again the 100th time. Not so in Mathematics, where the foundations are built on logical arguments, each step constructed on the last to arrive at not just evidence, not just a theory, not just a conjecture, but something that we know is true with absolute certainty. Once proved, theories become "theorems" which can be used to build other proofs.

The beauty of mathematics lies in its inescapable logic and marvellous application in helping us explain the world and universe around us. Truly, in the words of Galileo Galilei

 Mathematics is the language in which God has written the universe


A technical definition of proof misses the essential point - like a novel, a proof must tell an interesting story. The story must be convincing, and also describe the overall format to be used, but a good story line is the most important feature of all.

Very few textbooks say that.

When you watch a movie with lots of plot holes, we all have limits to what we are prepared to accept as credible. If in an otherwise realistic film a child saved the day by picking up a house and carrying it away, most of us would lose interest. Similarly, a mathematical proof is a story about mathematics that works. It does not have to dot every i and cross every t; readers are expected to fill in routine steps for themselves just as movie characters may suddenly appear in new surroundings without it being necessary to show how they got there. But the story must not have gaps, and it certainly must not have an unbelievable plot line. The rules are stringent: in mathematics, a single flaw is fatal. Moreover, a subtle flaw can be just as fatal as an obvious one.

Let's take a look at an example. I have chosen a simple one, to avoid technical background; in consequence, the proof tells a simple and not very significant story. I stole it from a colleague, who calls it the SHIP/DOCK Theorem. You probably know the type of puzzle in which you are given one word (SHIP) and asked to turn it into another word (DOCK) by changing one letter at a time and getting a valid word at every stage. You might like to try to solve this one before reading on: if you do, you will probably understand the theorem, and its proof, more easily.

Here's one solution:

There are plenty of alternatives, and some involve fewer words. But if you play around with this problem, you will eventually notice that all solutions have one thing in common: at least one of the intermediate words must contain two vowels.

O.K., so prove it.

I'm not willing to accept experimental evidence. I don't care if you have a hundred solutions and every single one of them includes a word with two vowels. You won't be happy with such evidence, either, because you will have a sneaky feeling that you may just have missed some really clever sequence that doesn't include such a word. On the other hand, you will probably also have a distinct feeling that somehow "it's obvious." I agree; but why is it obvious?

You have now entered a phase of existence in which most mathematicians spend most of their time: frustration. You know what you want to prove, you believe it, but you don't see a convincing story line for a proof. What this means is that you are lacking some key idea that will blow the whole problem wide open. In a moment I'll give you a hint. Think about it for a few minutes, and you will probably experience a much more satisfying phase of the mathematician's existence: illumination.

Here's the hint. Every valid word in English must contain a vowel.

It's a very simple hint. First, convince yourself that it's true. (A dictionary search is acceptable, provided it's a big dictionary.) Then consider its implications....

Can you prove it?