You’ve may have heard of Kirby Ferguson’s documentary series “Everything is a Remix,” which argues that all creative work follows a simple pattern: copy, transform, combine. Ferguson shows how this applies to music, film, and technology, but I want to argue that mathematics provides perhaps the purest example of this principle in action.
Consider the development of calculus. Newton and Leibniz didn’t create something from nothing, they were master remixers. They copied existing techniques: Greek methods of exhaustion, Cartesian coordinate geometry, algebraic manipulation methods. They transformed these ideas, pushing them to handle infinitesimal quantities and continuous change. Finally, they combined these transformed elements into a unified system that could solve problems about motion, areas, and optimization that had puzzled mathematicians for centuries.
The result felt revolutionary, but it was really sophisticated recombination. Even more tellingly, two people independently arrived at essentially the same remix, suggesting that the mathematical “ingredients” were ready to be combined in this way.
This pattern appears everywhere in mathematics. Galois theory emerged from combining group theory with field theory to understand polynomial equations. Algebraic topology was born from applying algebraic methods to topological spaces. Analytic number theory brought the tools of analysis to bear on problems about integers.
Even our most celebrated “original” results follow this pattern. Gödel’s incompleteness theorems ingeniously combined formal logic, arithmetic, and self-reference in a way that seemed impossible before, but each ingredient existed independently. The breakthrough was in the combination and transformation, not in creating entirely new mathematical objects.
What makes this perspective powerful is that it demystifies mathematical creativity. We don’t need to wait for lightning strikes of pure inspiration. Instead, we can actively work to become better remixers: absorbing more mathematical “vocabulary,” developing skill at transformation and generalization, and most importantly, learning to see connections between seemingly disparate areas.
The collaborative nature of mathematics, building on others’ work, citing predecessors, improving and generalizing existing results, makes it perhaps the most natural example of Ferguson’s thesis. Every mathematical paper is essentially a remix of the mathematical literature that preceded it.
But this raises a crucial question: if everything is a remix, what distinguishes profound mathematical work from merely derivative applications? That’s what we’ll explore in the next post.