Maths - discovered or invented?

A question I often ask my students is whether they think that maths is discovered or invented. This is usually a simple question to answer. For example, finding out that the Earth is not flat - despite the evidence of our senses - but round - less intuitive despite gazing at the horizon of an ocean - is clearly a discovery and not an invention. Most scientific theories rely upon experimentation with the natural world and discovering how it behaves under this experimentation. On the other hand writing 'Hamlet' is an act of invention - we do not 'discover' our hero and his life story on the stage. Wheels do not exist in the natural world and have to be invented, as do dishwashers, TVs, money, drugs, ...

Most mathematicians seem to think they are discovering new mathematical theorems and then attempt to prove them. The famous French mathematician Pierre Fermat thought he had proved his famous 'last theorem' in the margins of his notebook, but a correct proof only emerged hundreds of years later. Goldbach conjectured that every even whole number over 2 can be expressed as the sum of two primes (so 28 = 11 + 17, for example, and 102 = 31 + 71 - I challenge you to find two primes which add up to 24,568, but no doubt a computer could). And no-one has found a counter-example yet, so we must think this is a true discovery. But no proof exists (and someone called Godel actually proved the existence of conjectures which cannot be proved or disproved.

In A level mathematics, you learn about calculus - indeed this is probably the most important 'discovery' you will make in your studies. Isaac Newton didn't 'invent' calculus - how could this be if Gottfried Leibnitz 'discovered' it several years after Newton, who hadn't bothered to tell anyone about his discovery/invention.

So maths is discovered, end of story... well, if and when the human race dies out, would mathematics still exist? Don't we need human brains to 'discover' it? Where would it exist without us?

Language is a human 'invention' - we certainly don't discover it and its rules are the invention of human society. But mathematics is also often described as a 'language' (sometimes as the'language of science'), and its development certainly relies heavily on inventing appropriate notation.

The ratio of the circumference of a circle to its diameter we call pi. The base of natural logarithms is called e. The square root of minus one is called i (although some call it j). We have to give these things names before we can 'discover' things about them, such as that e raised to power i x pi is equal to -1. You probably find this 'discovery' rather surprising if you don't believe that i can exist!

From 'The Road to Reality

One of the finest mathematicians of my generation, Roger Penrose, in his book 'The Road to Reality' suggests that mathematical objects and theorems inhabit its own Platonic world , linking to and overlapping with the 'Physical World' and the 'Mental World' . So when you are learning mathematics you are discovering things about this Platonic world.