How do Mathematicians Prove Things?

Establishing ‘proof’ in mathematics leaves no room for error or doubt. Here are some common ways mathematicians meet this high burden

The word ‘proof’ has different implications in different contexts.

In the field of jurisprudence, for example, the phrase ‘proof beyond reasonable doubt’ is a common phrase used for the burden of proof. This suggests there is room for some doubt in a conviction, as long as it is not considered ‘reasonable’. Statisticians will often reject a hypothesis on the basis of a very low likelihood of the observed data sample occurring if the hypothesis were true. Again, there remains room for doubt in the conclusion, albeit very small.

In the field of Pure Mathematics, however, the burden of proof is absolute. There can be no room for doubt whatsoever. Given a statement, the mathematician must be able to describe a logical path that leads to a conclusion that is 100% certain. 99.99999% is no good, because that remaining 0.00001% means that the statement is not completely proven. The logical path that the mathematician describes must depend only on definitions, things that are already known and proved true, or on a very small set of very basic axioms that are accepted to be self-evident by the mathematical community without need for proof. For example, one such basic axiom states that two sets with exactly the same elements are the same set.

So what techniques do mathematicians use to meet this incredibly high burden of proof? In this article I will outline the most common three methods and their logical foundations, and later I will mention some other aspects of mathematical proof which I find interesting, including weird and very rare situations where it is impossible to prove or disprove a statement using the axioms of mathematics.

Method 1: Direct proof

In direct proof, the mathematician starts with definitions, axioms or statements that have already been proven to be true and outlines a series of direct logical consequences which lead to the statement needing to be proved. Here is an example of a statement that we can prove directly: