A mathematician’s legacy: the exception to the rule
Maths is often about trying to show that a statement is definitely and irrefutably true for all cases. The area of the square of the hypotenuse is always equal to the areas of the squares on the other two sides, and the square of an odd number is always odd. These are the certainties which maths often deals in.
Proving that something is definitely true for all cases (e.g. numbers, right-angled triangles, etc) can be very tricky, and in general, you need to show that your hypothesis is true by only using properties shared by all members of the set.
For example, we can efficiently prove there are infinitely many prime numbers (whole numbers divisible only by themselves and one) by showing that for every prime number there is a bigger one. For example, take to be the largest prime (where there are a finite number of primes). Multiplying all the imagined primes together gives , which is divisible by all the primes. Add one, and is not divisible by either or any of the smaller primes. So, is either prime or divisible by another prime greater than, and we have shown that our original assumption was wrong and that no prime can be the largest prime.
This sort of argument is the only way to be certain. French mathematician, Pierre de Fermat, found that is prime for , and conjectured that all Fermat numbers (numbers of this form) were prime. But, in 1732, Leonhard Euler found that the 6th Fermat number, 4294967297, could be factorised, thereby destroying Fermat’s conjecture with a single counter-example.
Like proofs, counterexamples are a cornerstone of mathematical thinking.
The Pythagoreans, an ancient Greek community of mathematicians, believed that all quantities (numbers) could be expressed as a ratio of two whole numbers (a fraction). However, it was shown, allegedly by a Pythagorean called Hippasus, that there were numbers, such as , that could not be expressed as a fraction. One legend claims that the Pythagoreans were so outraged by the existence of this singular number that defied their claim (an ir-ratio-nal number), they drowned Hippasus for his discovery.
Other counterexamples challenge assumptions which seem blindingly obvious. Familiar shapes with finite area have perimeters with finite lengths; circles, squares and even dodecagons are all witness to this fact. However, in 1904, the Swedish mathematician, Helge van Koch, showed that it was possible, by adding incrementally smaller triangles to the sides of an equilateral triangle, to construct a shape with finite area but with infinite surface area!