Neither Discovered Nor Invented: Grant Sanderson on the Cycle Between Math and the World

Fridman asks whether math is discovered or invented, in the context of whether an alien species would arrive at the same mathematics. Sanderson says everyone thinks about it differently, but his view is that a cycle is at play. Discoveries about the universe tell you what kind of math will be useful, and that math is invented in a sense, but the choice among all possible maths is guided by discoveries about the world.

He uses the Pythagorean theorem as an example. Historically it was a discovery, probably because people were building intuition with physical objects. He adds that modern mathematics has moved far from that, and that the mathematics of 20th and 21st century physics goes beyond what our minds can comprehend. Even so, he argues, a five-dimensional manifold can be useful for describing a three-dimensional world through state spaces, so the physical world need not itself be five-dimensional.

Returning to the theorem, Sanderson notes that geometric proofs make it feel like a discovery, yet when we formalize the plane as `R^2` we define distance so that the theorem holds. What informed that choice, he says, was physical observation. Other metrics give consistent mathematics but would not describe the physical world we study, so a real discovery informed an invention, which then raises new questions that may yield more discoveries. In his words, it is not an either/or, because math plays a different role at different times.

Fridman asks about the difference between math and physics, raising the intuition physicists seem to have alongside the rigor of mathematics. Sanderson describes math as the study of abstractions over pure patterns and logic, and physics as grounded in the desire to understand the world we live in. He expects different mathematicians to answer differently because the field is diverse.

He sketches three kinds. Some are driven by pure puzzles, such as combinatorics lovers who build problem-solving tools for pure patterns. Others are physically motivated, such as in chaos theory, which is pure math heavily driven by applications largely in physics. A third group loves abstraction and generality, such as those who moved from topology to category theory, seeking systems that describe truths about as many things as possible.

Each camp gives a different answer. Vladimir Arnold, who wrote many books on differential equations, would say math is a branch of physics, while category theorists see it as the power of generality and treat usefulness for the real world as a happy coincidence. Sanderson says that raises the surprising question of why something about pure puzzles and abstraction also describes the fundamentals of quarks.

Fridman asks why fundamental physics is so compressible into clean, relatively simple equations, mentioning Stephen Wolfram's idea of simple rules producing arbitrary complexity. Sanderson suggests there could be a filter: physicists may find interesting only the systems simple enough to describe mathematically, while anything sufficiently complex becomes biology. He also says that when you wash away unimportant parts of what you study, something simple will always remain, and he has trouble conceiving what intrinsically complicated laws would even mean.

Fridman pushes the other way: laws could differ everywhere, like a deeply distributed system, or be too chaotic to model, and we may take for granted that equations for gravity and atoms exist. He wonders if our minds are biased toward the compressible parts of the universe, or whether a world with incompressible laws could even allow beings who ask such questions. Sanderson acknowledges an anthropic-principle flavor and admits he doesn't know what he is talking about.

Fridman raises a Chomsky-style argument that we are limited biological computers perceiving only certain things in a much more complicated world. Sanderson replies that we can do remarkable things, like flying spaceships, and so our simplified models must have some connection to reality. He takes it as a reality check that the laws we work with are working well.