ABSTRACT
When we do mathematics, we study the structures that underlie the various phenomena encountered in the world. For this to work, mathematics has to be able to filter out all the detail that is not relevant to the particular structure being studied. Equivalence is the filter. The most basic example is the concept of number. What does the number
3 stand for? A set X has 3 elements if and only if there is a set isomorphism
f : {1, 2, 3} → X (3.1) counting off the elements of X as f(1), f(2), and f(3). The function f has to be injective, so that no element of X gets counted twice. The function f has to be surjective, to make sure that each element of X gets counted. The only problem here is the circularity. To characterize the number 3, we
have used that number in the domain of the function (3.1). To escape the circularity, we can decide to consider two sets as equivalent for the purposes of counting whenever they are isomorphic. The number 3 then emerges as the property which is common to each of the sets that are isomorphic to some given 3-element set (for instance {1, 2, 3} or {∅, {∅}, {{∅}}}). The particular details of the elements in the sets are not relevant to the problem of counting. They are filtered out by the equivalence. Equivalence relations play a key role in the analysis of general functions.
Each function determines an equivalence relation on its domain, identifying two elements whenever they have the same function value. Conversely, it transpires that every equivalence relation is of this type.
