Creating new mathematics ( ... )

Suppose your assignment was to teach a group of friendly aliens, just arrived from the Pleiadians, the rules of the game of chess. A student has completed the course with success if he is able to play a game according to the rules. That's not too difficult you think considering the number of six year olds able to play a descent game of chess. Your study materials are: lots of chalk and a blackboard. No chess pieces, no boards are available in class. Your teaching assistant will type your lecture as you speak, therefore it is not allowed to use drawings or symbols that can't be typed instantly.

This may seem difficult ( it is ), but compare it with the creation of new mathematics (...)

Mathematics is a parallel universe which we can enter with our mind only. Although bodiless, exterior, we are free to travel in this spectacular universe. When we come back however we lack the words to describe our observations, to communicate what we have seen with our mental ( mathematical ) eyes. Each and every observation must be recorded and analyzed before we can attempt to describe it. Definition by definition we try to create a consistent picture of what we have 'seen'. In this notion of mathematics, for example the number e always existed, it just took an Euler to describe it properly. Obviously the mathematical world is not some parallel library where we can go to and lookup the answers to the current unsolved problems. - In a BBC documentary about Andrew Wiles and Fermat's last theorem, Wiles describes his research as entering some space, then by touching things by hand in the dark he had to form a mental picture of what's in that space, and so on.

Describe what you observe (...)