How does new math get invented/discovered?
In: Mathematics
A big part of the process of developing new maths is generalisation. Often someone will notice that multiple mathematical systems have some features in common and they will try to develop a framework that includes all of these systems. For example, suppose you notice that two simple models of unrelated real-world systems – a model of radioactive decay and a model of population growth under limited resources – both have something in common: over time they both get closer and closer to a fixed point in which nothing changes any more (maybe the amount of the radioactive substance approaches zero, while the population approaches the maximum that can survive on the available resources). You might try and develop a theory of all systems which approach a fixed point like this. This tends to be very fruitful as it helps you understand what exactly it is about these models that results in this behaviour, what would need to change for them to do something different, and what details of the models are irrelevant to this behaviour and can be safely ignored. And once you’ve developed this generalised theory of systems that converge to fixed points, it’s very straightforward to apply it to some new model that comes along: you don’t have to do everything from scratch, you just have to work out where this new model fits in your theory.
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