Having done two years of mechanical engineering, I’ve seen first hand how traditional mathematics has found ways to walk around the beauty and weirdness of non linear phenomena. Linearization was a common tool to transform non linear equations into linear ones, followed by algorithmic resolution techniques. Now, with a better understanding of nonlinearity and mathematics, it amazes me to realize that, unlike with linear problems, one does not get linear equations as a result of non linear mathematics. But rather geometric patterns. Fractals are famous examples of that!
However, I still do not know what that means for mathematics at large. How does our capacity to mathematically look “under the hood” of nonlinearity help us to deal with nonlinear phenomena? That is, how do the results we get from nonlinear problems, like fractal geometric patterns, help us to make sense of these problems and translate that into the “real world”?