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

$a * x = b,$

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

$a^{-1} * a * x = a^{-1} * b,$

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!

CC BY-SA 4.0 Samuel Belko. Last modified: May 29, 2024.
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