Was mathematics invented or discovered?
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Where does maths come from? Is it just in our minds or is it something intrinsic, universal? How do mathematical patterns occur in nature and do animals other than humans use maths? Would aliens use maths in the same way? Does maths really ‘exist’ or is it more or less a highly viable belief system? Is it just an advanced language? To what degree can we shape it as we choose and to what degree must we follow rules beyond our control?
Maths can be scary for those of us who haven’t touched the bleak hours of head scratching at school. That said, I think it’s possible to get through a good discussion about the philosophical background for maths without delving too deep into any mind-bending concepts or excessive arithmetic. If you’re one of these people – you are not alone and I urge you to come along; I’ve prepared a ‘survival guide’ below which should aid next week’s discussion. I intend to break the attendees up into groups based upon your experience of maths, so that those with relatively little knowledge of the area don’t get swamped by advanced maths from the more experienced.
To address the question at hand we need to ask ‘what is mathematics?’ and ‘how did it develop?’ http://en.wikipedia.org/wiki/Mathematics gives a decent overview of the subject though https://www.youtube.com/watch?v=Rte4Lslg9BM is more fun. ‘Maths’ has meant different things at different times in history and still means different things to different people today (see http://en.wikipedia.org/wiki/Definitions_of_mathematics ). For shopkeepers, maths might be limited to adding up costs and prices, keeping stock etc. For the ancient Greeks, the focus of their enquiries was geometry and number theory ( http://en.wikipedia.org/wiki/Number_theory ). They would not have recognised dimensions or powers beyond three or many other things from modern mathematics. ‘6 squared’ or ‘6^2’ would be seen as making a square from a line of length 6 – and an area of 36 – i.e. 6 x 6. Then ‘6 cubed’ or ‘6 ^ 3’ would have been similarly a cube with edges of length 6 and a volume of 6 x 6 x 6 = 216. ‘6 ^ 4’ would have meant nothing to them – it would have been to call on a non-existent fourth dimension.
Euclid captured much of the geometric knowledge of the day in his work ‘Elements’ using an axiomatic approach – i.e. to build theorems from axioms (undisputable basic facts) and other theorems previously established in this way. So it progresses like a house of cards, each theorem dependant on everything it stands on, rising to the pinnacle of the first book, the well-known Pythagoras theorem describing the relationship between the sides of a right-angled triangle. Though Euclid is not known to have developed the mathematics himself, the area is named Euclidean geometry after him and encompasses most of the geometry we still learn today at school.
To give the briefest whistle-stop tour of the next 2000 years of mathematics: Algebra was developed by Arabic and Indian mathematicians. Descartes unified algebra and geometry by developing ways to represent lines and graphs as numbers, symbols and equations ( http://www.youtube.com/watch?v=N4nrdf0yYfM ). By the 19th and 20th centuries, maths explores more abstract areas such as set theory, logic and non-Euclidean geometry. The later was developed by varying the set of axioms used in Euclidean geometry and exploring the impact of such changes ( http://www.youtube.com/watch?v=Jvs_gTrP3wg ); although not originally developed as models of the world, physicists were able to utilise non-Euclidean geometries to understand relativity.
It might also be worth considering in our discussion the idea of negative numbers (e.g. I can have -£5 in my bank account, but there can’t be -1 cow stood in the field) and imaginary numbers (also called complex numbers – see http://www.mathsisfun.com/numbers/imaginary-numbers.html ). The renaissance mathematician Cardano explored the possibility of imaginary numbers and dismissed them pretty much as nonsense, hundreds of years before they were developed more formally. Today, I understand, they have practical applications, though I’m not well-informed enough to tell exactly where!
Other concepts it might be worth familiarising yourself with before the discussion are prime numbers ( https://www.youtube.com/watch?v=yyZdnOt5TUo&list=PL1DB7660127D36D82&index=9 ), square numbers, triangular numbers and perfect numbers ( https://www.youtube.com/watch?v=ZfKTD5lvToE ). It might be worth also considering whether numbers actually exist:
Was mathematics invented or discovered?