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I have been trying to come up with an abstract definition of what an equation is. What I have so far is the following:

Let $X$ and $Y$ be two not necessarily unique sets such that $Y$ has an equivalence relation $\sim_Y$ defined on it.

An equation is an expression of the form

$$f(x) = g(x)$$

where $f,g: X \to Y$ are arbitrary functions. A solution to the equation is any $x^\ast \in X$ such that $f(x^\ast) \sim_Y g(x^\ast)$.

Have you ever stumbled upon someone who discusses this topic? If so, could you point me to their work?

Martin Brandenburg
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mdkovachev
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  • A very detached (may seem like abstract nonsense) way to see it using category theory would be the study of equalisers of two morphisms $f,g$ which have the same domain and codomains. In this case, X and Y are not necessarily sets. But it would be pretty general already. See https://en.wikipedia.org/wiki/Equaliser_(mathematics) – julio_es_sui_glace Nov 20 '24 at 21:51
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    This may help https://math.stackexchange.com/questions/2738360/what-exactly-is-an-equation/2738382#2738382 – Ethan Bolker Nov 20 '24 at 21:56
  • @EthanBolker indeed, this answer points out a distinction I had not considered. When talking about equations, we want $f(x^\ast) = g(x^\ast)$ to mean that the two things on the sides of the equal sign are actually the same object. What I defined as an equation in my question would be a bit more general. Perhaps "equivalation" would be a better name for it. – mdkovachev Nov 20 '24 at 22:09
  • See also this post as well as this one. – Mauro ALLEGRANZA Nov 21 '24 at 09:02
  • In general, to "find a solution" to an equation $Ax=Bx$, i.e. to $Ax-Bx=0$ means to check if the formula $\exists x (Ax-Bx=0)$ is *s – Mauro ALLEGRANZA Nov 21 '24 at 09:09

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