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The main purpose of theoretical or pure mathematics is to “prove” a certain calculation or theorem. Just as in English where a sentence or word may be misspelled, or a turn of phrase not make sense, or sound right, so a mathematical calculation can be said to be “proved” if it works, gives the necessary answer and can be repeated with the same result.

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## Proof - You Can’t Handle The Proof …

Proofs - mathematical proofs, geometry proofs, algebraic proofs and so on, go to the core of mathematics.

The whole definition of this is neatly encapsulated in this quote by Bertrand Russell, “Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

As you can see, this is all a bit nebulous and esoteric, but proofs are the means by which these ideas are strung together and made to work. Most proofs are arrived at by following a “formula,” or “equation,” which is a kind of framework or template on which the numbers (or “variables,” are hung. For example, 2+2=4 is a framework; if I change the variables to 3 rather than 2, I get 3+3=6; the basic equation x+y=z is the same, and each use of the equation with different variables becomes “proof” that the equation works.

The same holds true of, G(3)(1) = 3 = 21 + 20 andP(3)(1) = f(G(3)(1)) = ω1 + ω0 = ω + 1, for instance, but it is far more complicated and difficult to understand, and unless you were well versed in higher mathematics and logic you could be forgiven for being a little overwhelmed by having such an equation to prove.

You really need to be honest and face up to whether or not your abilities stretch to such abstractness and reasoning powers. We find that students sometimes completely underestimate the vast gap between college level mathematics, and the knowledge base necessary for advanced level mathematical proofs at degree level and above.

### Saying It Is One Thing

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