Geometry is Space, and Algebra is Time

 This is an exploration of some interesting parallels.

One might say that the categories of geometry and algebra cover most of mathematics. I think most mathematicians see them not as categories but as different ways to see the same object. Time and again, geometric intuition uncovers hidden dimensions in algebraic objects: the algebraic geometric of elliptic curves, as well as the standard practice of studying a large set of functions by considering each function as a point in a "function space," serve as ready examples. On the other hand, geometric spaces that we're interested in, such as the fabric of space-time, or crystal lattices, often have underlying symmetries that can be encoded as algebraic structures which afford us a compact, high-level language to state and prove far-reaching claims on the properties of the geometry.

Icosahedron, CC-BY-SA 3.0 by Kjell André

Representation of the group of symmetries of an icosahedron. Exactly one rotation or reflection brings each triangle to another.

Why are these two viewpoints so effective, so illuminating, so incisive? If we see geometric intuition as an extension of our human capability for spatial visualization and reasoning, then I'd say that's a pretty fundamental source for geometric intuition—an internalization of our everyday visual and tactile interaction with out environment. What about algebra then? We could tie it to our faculties of language, which associate meanings to symbols or strings of them (literally called a word in group theory), communicate that meaning, and process the information in the symbols by manipulating them. We really do have a "language of algebra." So the first parallel goes from geometry and algebra to spatial reasoning and linguistic ability.

But I like to go even further, rise higher in abstraction: is there any more fundamental difference, or similarity, between our faculties of spatial reasoning and language? How about space vs. time? Obviously geometry is space. If we think of multiplication in algebra as composing transformations one after the other, such as the linear transformations in linear algebra (Cayley's theorem guarantees that we can view any group as a transformation group), then algebra is a form of "discrete time travel."

What about other fields of math? If geometry is space and algebra is time, then perhaps analysis is change, and topology is connection.