In the work I do as a physicist, I work with maths to attempt to explain and predict how physical systems behave.  However, in spite of my best efforts, the predictions I make are not exact, and never can be.  One major reason for this is that real-world physical systems are usually too complex and too poorly understood to permit an exact analysis.  For example, in my work on radioactive waste disposal, I can make numerical predictions that about disposal system behaviour at various times in the future.  However, the results I obtain are open to various objections, for example that I have used inappropriate simplifying assumptions or have conceptualised the system in an inappropriate manner. 

In the study of maths, however, it is possible to make predictions and assertions that can be proved to be correct and true beyond argument.  The objective of these articles is to take a look at some aspects of mathematical proof, and how mathematical proofs can be used to establish beyond doubt that statements and assertions are true.  Such statements and assertions are not subject to objections about assumptions made or other “approximations to the truth”.

In the first article, we will look at the basic ideas of proof by contradiction and proof by induction.  The remaining articles look at some simple and well-known applications of these techniques.  All the proofs in these articles are very well-known, I have tried to explain them as simply as possible.  This necessitates adding some extra steps to the proofs in some cases, making some of them slightly longer than versions to be found elsewhere.

Proof By Induction and By Contradiction

Sum of Cubes

Infinite Number of Primes

Induction in Differential Calculus

Irrationality of the Square Roots of Two and Three

Irrationality of e

Irrationality of π