Recently I have been wondering what different notions of equivalence are possible. Thinking about one of my favorite fourfolds — Structure-Function — with help from the fourfold of The One and the Many, I have (naturally) come up with four notions: Identities, Isomorphisms, Confluences, and Indiscernibles.
Identities: One as One. In mathematics, an identity is an equivalence between two (or more) things that are really just the same thing. One can say there is an equivalence relation between the things and they are part of the same equivalence class. In equations, each thing can be substituted for the other thing because they are really just the same thing! In my consideration, Actions have these kind of identities, and so it is a reasonable term to use for Actions in the Structure-Function fourfold.
Isomorphisms: One as Many. In mathematics, an isomorphism is an equivalence between two (or more) things that have the same (mathematical) structure. Another way to consider this is to say that there is a paradigm or model that is representative of all the things that have this same structure. Thus it is a good term to use for Structures in the Structure-Function fourfold.
Confluences: Many as One. In logic, computer science (rewriting theory), and mathematics, a confluence is an equivalence between two (or more) things that can each be transformed into the same, maybe different, thing. Confluences can be used for this notion of equivalence because if one says that two rivers are confluent, then that means that they both flow into another larger river. Thus I think it is a valuable term to use for Functions in the Structure-Function fourfold.
Indiscernibles: Many as Many. In philosophy and perhaps physics, an indiscernible is an equivalence between two (or more) things where one cannot tell the difference between them. Thus it is a useful term to use for Parts in the Structure-Function fourfold. Another term to consider using is Substitutivities. Thus I do not believe I agree with the principle of the Identity of Indiscernibles since I think that would collapse my two notions of Identities and Indiscernibles into one. For instance, atoms of gold may be indiscernible from one another but that doesn’t mean they are the same atom.
A recent foundational project for mathematics starts with the following:
Univalence Axiom: (A = B) ~ (A ~ B): Identity is equivalent to equivalence.
At this time I do not understand the implications of this axiom and how it might impact my four notions of equivalence.
Perhaps Extensionalities would be a better choice than Confluences.
The HoTT Book: Homotopy Type Theory: Univalent Foundations of Mathematics