Identity

Identity is a primary, axiomatic concept. It refers to the "this"-ness of an object, its quality of being this and not that.

All identification is conceptual identification. To identify something

[TODO] "this (c) is a cow (C)" is in my notation $c:C$. "cows (C) are animals (A)" is in my notation $C:A$.

Mathematics conceptualizes identity using the equals ("=") symbol.

Equivalence relations

Things can be identical in some respects, but not others. Standard mathematics formalizes this idea as follows.

Let $S$ be a set. An equivalence relation is a relation $\sim$ on $S$ satisfying three axioms. For any $a,b,c\in S$,

1. (Reflexivity) $a\sim a$,
2. (Symmetry) If $a\sim b$ then $b\sim a$,
3. (Transitivity) If $a\sim b$ and $b\sim c$, then $a\sim c$.

If $a\in S$, we define the equivalence class of $a$ to be the set $[a]:=\{b\in S:b\sim a\}$. Note that if $a\sim b$, then $[a]=[b]$. Define the quotient of $S$ by the equivalence relation $\sim$, or $S/\sim$, as the set of all (distinct) equivalence classes of elements in $S$, i.e. $S/\sim \ :=\{[a]:a\in S\}$.

We can think of $a\sim b$ as being the judgment that "$a$ and $b$ are similar (in the respect represented by the equivalence relation)." Passing from $S$ to $S/\sim$, we have $[a]=[b]$, which is the judgment that "$[a]$ and $[b]$ are the same (in the respect represented by the equivalence relation)."

Although the mathematical expressions in the former and latter judgments are both perfectly precise, the English words used in the latter judgement ("the same") are more precise than those used in the former judgment ("similar"). Indeed, many real-life "similarity" relations are not equivalence relations, because do not satisfy the transitivity axiom. For example, light of wavelength 400nm (purple) is similar to light of 401nm (also purple), and 401nm light is similar to 402nm light, and so on; but light of wavelength 600nm (orange) is not similar to light of 400nm (purple), even though we can connect it by a chain of "similarity" to 400nm light.

It is preferable to be in a situation where one can think of things as being the same rather than similar, because the former judgement is binary and the latter judgement is analog. In the context of computing, it is well-known that analog processes accumulate errors to a much greater extent than digital ones, and a similar principle holds in epistemology. This is another perspective on what I said in the previous paragraph, regarding the wavelength of light.

Real-life examples of equivalence relations

Let $S$ be a set of potential lengths. Given a ruler $R$, we could say of two lengths $\ell$ and $\ell '$ that $\ell \sim _{R}\ell '$, if their measurements would fall between the same two notches in the ruler. So for example, two shoes have whatever lengths they have, $\ell$ and $\ell '$, but if we use a ruler $R$ with notches every 0.5cm, and we measure both their lengths to be between 30.0 and 30.5 cm, then we can say that $\ell \sim _{R}\ell '$. One easily verifies that $\sim _{R}$ is truly an equivalence relation (at least if you have a consistent way of dealing with boundary cases). Note that it is impossible to ever know that $\ell =\ell '$, unless the "two" shoes in question are quite literally the same shoe (not just the same make of shoe). However, if we pass to the quotient, then we can say $[\ell ]_{R}=[\ell ']_{R}$, or in English "these shoes have the same length (with respect to the tolerance set by the ruler $R$)."

Let $S$ be a set of possible three-dimensional rotations of a sphere about its center. By a possible three-dimensional rotation, I mean something like a movie; a rotation begins happening at some time, does some stuff, and ceases at a later time. We may define an equivalence relation on $S$ by saying that two rotations $r,r'$ are equivalent if they lead the sphere to be rotated in the same way (the "same" way, with respect to the tolerance set by some given standard of measurement). We might define a coarser equivalence relation: If the top half of the sphere is painted black, and the bottom half of the sphere is painted white, then we might say that two rotations are equivalent if they lead the sphere to be rotated in a way that fixes the circle bounding the black and white regions (since the results of two such rotations would be visually indistinguishable from one another). We might also define a stronger equivalence relation, using a concept from algebraic topology: Two rotations are equivalent if their "movies" are homotopic.

Let $S$ be a set of possible initial conditions for coin flips. Position of the hand, angle of the hand, temperature, wind speed, height from the ground, impulse exerted by the finger doing the flipping, fingernail length, etc. etc. There is a huge amount of things determining the outcome of a coin flip; one might not even know all the relevant variables [TODO but we actually do. Link experiment], let alone the values of those variables. But an equivalence relation can be defined nonetheless, by saying that two initial conditions are equivalent if they led to the same outcome (heads or tails). This is a very coarse equivalence relation.

Summary

Equivalence relations among some things always come from other things that actually are the same.