There’s a lot of buzz about inter-dimensional and extra-dimensional spaces these days. Making sense of these ideas is one of the goals of both mathematics and physics. But it immediately brings up a big question: is mathematics physical? The short answer seems to be “no.” Take the Banach-Tarski Paradox, for example. It’s a wild idea in mathematics that you can take a solid 3D ball, break it into a finite number of pieces, and then reassemble those pieces into two balls of the same size as the original. Clearly, this doesn’t happen in our physical universe due to the conservation of matter. So, the question becomes: what mathematics is physical?

This is one of the most subtle questions humanity has ever pondered. The mathematics of possible physics in the fourth dimension is particularly mysterious and intriguing to me.

In my last post, I talked about how transitioning from 3D space to 4D space could drastically change your sense of volume. Your local ball of influence would grow significantly, making space itself seem to expand. Now that you’re in 4D space, you might want to do some physics experiments to see how this new space compares to what you’re used to. Physics is all about motion, and the math that handles motion is calculus.

Since Einstein’s formulation of General Relativity, “doing calculus” has been tied to the idea of “smooth manifolds” or “differentiable manifolds.” A smooth manifold is like a shape with no sharp edges, such as a sphere or a donut surface. It’s a set of points where you can move smoothly from one to another without any hiccups, allowing for consistent definitions of speed, velocity, acceleration, and other physical quantities.

Applications of smooth manifolds in physics have been a triumph of modern science, helping us understand the universe. But when it comes to doing physics in 4D space, we hit a strange fact in mathematics.

Consider one-dimensional flat space, a line. It’s infinite in two directions, and you can move back and forth on it freely. There’s only one way to set up a smooth manifold structure on that line to do calculus. The same goes for 2D flat space (a plane) and 3D flat space. But 4D flat space is different.

In 4D flat space, there are infinitely many ways to do calculus, and these methods are mutually incompatible. This means there are potentially infinitely many ways to describe physics in 4D space. Imagine a creature living in our 3D space with one time dimension, becoming “timeless” and truly 4D. This creature would face an infinite choice of basic mathematical frameworks for their universe. Are all these choices physical? Do they all lead to the same kinds of universes?

In upcoming posts, I’ll explore some of the mathematics of flat 4D space and discuss how different approaches to calculus might lead to different physical experiences.