What is Maths?

Is maths the language of reality?

"Mathematics is the abstract key which turns the lock of the physical universe." This quote by John Polkinghorne captures the most amazing characteristic of maths: that somehow it, better than anything else, can be used to describe and explain natural phenomena.

Much of physics, for example, is written in maths. Newton's F=ma or Einstein's E=mc2 are examples of equations that capture the heart of their theories. Is reality mathematical or is maths an invented 'language' useful for modeling natural phenomena? Is maths discovered or invented?
We may never know the answer and it may not even matter because maths works very well for our purposes. Abstract mathematical explanations make sense and empirical evidence seems to support them. Galileo said "the laws of nature are written in the language of mathematics"... we just do not know who is writing!

What is maths based on?

In early 20th century Bertnard Russell and Alfred Whitehead published The Principia Mathematica which tried to prove that maths was based on logic (maths is reducible to logic). The Principia was celebrated but as it turned out the thesis was unsustainable.
In 1931 Kurt Gödel proved that it was impossible create logical systems that are both consistent and complete.
Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.
This came to be known as Gödel’s Incompleteness Theorem and it meant that there are some mathematical truths that cannot be deduced in any formal system. Maths cannot be reduced to logic. We do not know what maths is based on. This makes maths fascinating and mysterious but it also means we cannot totally trust it. (Gödel's proof is not easy to follow, but in the end it hinges on something like the liar's paradox known since antiquity. Check his proof here).

How many sides?

During the introductory lessons we learned about many marvels of maths such as game theory, chaos, fractals and topology (an area of geometry). Topology studies continuity of space and spatial properties.

An example of topologically interesting object is the Möbius Strip. You can make one by taking a strip of paper, giving it half a twist and joining it into a loop. An object that had two sides now only has one.
In 1882, Felix Klein imagined sewing two Möbius Loops together to create a single sided bottle with no boundary. Its inside is its outside. It contains itself. It is not very useful as a bottle but it is quite beautiful mathematically (you can find one here, scroll down, it is the last picture on the page).

By the way, topologically speaking vests worn under the jacket are not really under the jacket ... if I ever dress like that I will demonstrate that by removing my vest without taking off my jacket first.