Simple Complexity

Analogies and Math

By Samuel Belko, published on May 4, 2024.

I have read somewhere that math is a study of analogies. This became even more pronounced while attending a math talk, where someone in the audience asked about an analogy to a different but in some abstract way related concept. Reflecting on the question, I realized that abstraction itself is defining an analogy.

For instance, let's take one possible motivation for a definition of a group (G,)(G, *) from algebra. Consider an equation

ax=b,a * x = b,

where a,x,ba,x,b are elements of GG. What does it mean to solve for xx? Well, we multiply both sides by the inverse element of aa and obtain

a1ax=a1b,a^{-1} * a * x = a^{-1} * b,

hence x=a1bx = a^{-1} * b. So really, the group structure is one possible abstraction for scenarios where we can solve equations, and induces an analogy between specific examples of groups that can carry additional structure.

That's all I wanted to share on the topic, thanks for reading!

CC BY-SA 4.0 Samuel Belko. Last modified: May 29, 2024. Website built with Franklin.jl and the Julia programming language.