There are two kinds of scientific progress: the methodical experimentation and categorization which gradually extend the boundaries of knowledge, and the revolutionary leap of genius which redefines and transcends those boundaries. Acknowledging our debt to the former, we yearn nonetheless for the latter.

– Academician Prokhor Zakharov, “Address to the Faculty”

For those who learn English as a second or third language, there are certain words that are bound to cause confusion. One of them is “math”, or, as it is also known, “maths”. Depending on who you ask, you are likely to get different responses as to which one of these variants is correct. The key to understanding the issue is not to see it as a matter of correct or incorrect, but rather as regional variations, where one is applicable in some places and the other in others. As an outsider, taking a stance one way or another is not as important as understanding where each variant applies, and being flexible when encountering them in the wild. If one author talks about math and another about maths, they are most likely talking about the same thing – unless either of them is very technical, and talks about different kinds of math.

Nonlinear mathematics is a very specific kind of math. It is not an algebraic assertion along the lines that 2+2=5 due to a sudden nonlinear state of numbers. Rather, it is a specialized kind of math that has been developed for specific uses in specific circumstances. This type of math makes assumptions that do not apply in other kinds of math, but which nevertheless brings forth useful results. In this case, a particle impactor, or (as it referred to in non-technical terms) a big honking laser gun.

The nonlinearity refers to the fact that what is being calculated is chaotic, and thus behaves in ways that are difficult to predict. Not impossible to predict, mind, just difficult enough that simply relating one variable to another is insufficient to do the trick. Predicting the weather is an example of such nonlinearity: there are many variables which are relevant to the prediction, but there is no single equation (e.g. this plus that times this over that) which, once solved, will give you tomorrow’s weather forecast. This does not mean that math is useless in the predictive effort, but it does mean it will take more work than mere algebraic number-crunching to get it done.

The exact nature of this additional work differs from problem to problem, as you might imagine, and the details are bound to be plentiful and complicated. One of the implicit assertions of researching something in a video game is that all the things necessary for completing it have been smoothed out, mastered and put to use. It does not have to provide information about the steps involved, just proclaim it to be done – whatever is necessary is also what you just did, by virtue of completing the research project. The fact that Nonlinear Mathematics is part of a linear tech tree is ever so slightly ironic, and Zakharov’s quote is a very non-subtle nod towards that. Non-linear mathematics is all about many variables acting chaotically together towards complex outcomes; researching Nonlinear Mathematics is a binary proposition, where you have either done it or not. There are, indeed, two kinds of scientific progress.