Many mathematicians spend their time proving results. The (very old) joke is that they are machines for turning coffee into theorems. A theorem is a statement that has been shown, by a sequence of irrefutable steps, to follow logically from a set of fundamental assumptions known as axioms.
These axioms themselves may be self-evident, or may simply be assumed to be true. Given this, the statement contained in a theorem is known with certainty to be true.
The Bedrock of Mathematics
Proof is the bedrock of mathematics, distinguishing it from the natural sciences. But how do mathematicians formulate theorems? How do they know which statements are likely to be true and which to be false? Frequently, they look at particular cases to build up evidence.
Let’s take a simple example, the sum of consecutive odd numbers. Starting with 1, we have 1+3=4, then 1+3+5=9, then 1+3+5+7=16, and so on. The sequence 1, 4, 9, 16 reveals a pattern: they are the squares of the first four natural numbers! But will this pattern continue? We can compute further sums and they will always be squares; but computation alone will never prove the general result, because there is an infinitude of cases.
There are several ways to prove the result. The sum of odd numbers is an example of an arithmetic series, and there is a formula ― learned in school and soon forgotten by most of us ― that we can apply to get the result. Or we can prove it by a process called mathematical induction, or by a geometric argument. In any case, it is a simple matter to show that in every case the sum is a perfect square.
Conjectures that are shown by proof to be true become theorems. Rigorous logical proof is the gold standard of mathematics, but computation is playing an ever-increasing role. While some mathematicians are proving theorems, others are computing special cases to look for patterns that provide evidence for conjectures or, by producing counter-examples, to show them to be invalid.
The use of computers to look for patterns that indicate general results is called experimental mathematics. In a sense, mathematics has always been experimental, but computers have expanded our ability to crunch out vast numbers of special cases.
Open a mathematical journal and you will find theorems stated in Spartan style, with cold logical proofs devoid of any hint of the inspiration behind them. To appreciate the origin of mathematical ideas, you must either read expository review articles or study the unpublished notebooks of mathematicians.
These notebooks are often peppered with experimental calculations and speculative explorations that illustrate the circuitous process by which results are achieved, involving much trial and error and pursuit of blind alleys. What appears in the published journals is just the essential core: the heart is there but the soul is imperceptible.
In 1742, in a letter to the great Swiss mathematician Leonard Euler, Christian Goldbach stated a conjecture: Every even number is the sum of two primes. Recall that a prime number is one that cannot be evenly divided by another. There is cast-iron evidence that Goldbach was right.
All even numbers to beyond a quintillion (one million million million) have been shown to be sums of two primes. But the greatest mathematicians have been unable to prove that the assertion is true in all cases, so the matter remains open. Perhaps you can win enduring fame by proving it!
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