How does one “invent new maths”? Like Isaac Newton inventing Calculus, or John Napier logs. How does one answer a mathematical question that’s never been answered?
In: Mathematics
New mathematics comes about when someone has a new insight into a problem. A good example is topology. In the 18th century someone posed a question about the Prussian city of Königsberg. The city is on the river Pregel and it had seven bridges joining the two banks via a couple of islands. The question was, “Is there a path which goes over each bridge once and only once?”.
Nobody could devise such a route, but nobody could show that it was completely impossible. A mathematician named Leonhard Euler decided to try the problem. Euler’s great breakthrough was to realise that none of the usual measures made any difference to the problem – not the length of the bridges, or their distance apart, or their angles to each other. He realised that the key thing is how many paths into and out of each bridge there are: If you want to cross each bridge only once there must be as many paths out as there are in. Another way of saying this is that the total of paths in and out must be an even number (divisible by 2). Using this concept he showed that the Königsberg question had no solution – it was impossible. But he was able to go further and come up with a general rule for such problems – based on how many bridges have even numbers of entrances and exits.
Euler had distilled a certain problem into new abstract concepts: edges (the bridges), nodes (land masses) and the degrees of the nodes (paths in + paths out). This new view of the world turned out to be a very powerful tool for many different problems and developed into whole new branches of maths such as graph theory and topology.
See [Wikipedia](https://en.wikipedia.org/wiki/Seven_Bridges_of_Königsberg) for a better explanation of the problem and solution.
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