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 $(G, *)$ from algebra. Consider an equation

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

hence $x = a^{-1} * b$. 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!