According to "The Words of Mathematics" by Steven Schwartzman, similarity in mathematics means "two similar things [that] look as if they're "one and the same" in respect to a certain property".
Examples thereof that come to my mind are:
- similar matrices, representing the same linear map up to change of base;
- similar triangles (or geometric shapes more generally), whose angles are the same, but the absolute size may differ.
According to the definition of Schwartzman one could apply the notion "similarity / similar" to a variety of concepts in mathematics that are given other names, e.g.:
- homeomorphic topological spaces;
- equivalence classes (e.g., in $\mathbb{Q}$);
- isomorphic groups/rings/fiels;
- equivalence of categories.
I am wondering how the term "similarity" and the mathematical concept of it emerged historically (e.g., Leibnitz introduced the symbol $\sim$ as far as I know, but what where the first concepts of similarity in mathematics?). And could one find a general (precise mathematical) definition of equivalence, that subsumes every of those mentioned cases?
More examples of similarities in mathematics are also appreciated.
In addition to the question of capturing similarity purely within mathematics, I would be interested to know what possibilities there are to classify/examine things/objects from everyday life mathematically for similarity.
Historically similarity is a "weak" form of the logic of identity.
Leibnitz's principle of the "equality of the indiscernibles" states that $x=y$ if every property of $x$ is also of $y$
$x=y\iff (F)(Fx\iff Fy)$
where $F$ is any predicate (property).
From this we can derive the reflexivity, symmetry and transitivity of the identity.
Now, similarity is a weak form since the condition of "all properties" is weakened to "some property".
Yet similarity is an equivalence relation. It is reflexive, symmetric and transitive.
In linear algebra for example two square matrices are similar if they have the property: $A\sim B\iff(\exists T)(det(T)\neq 0 \land B=T^{-1}AT)$
however they are not equal since they satisfy only a certain property.
The same is true for isomorphism of vector spaces and homomorphism of topological spaces