FOM: Equivalence classes as objects

Moshe' Machover moshe.machover at
Sat Aug 1 07:16:02 EDT 1998

I hope that--being a layman as far as history of maths is concerned--I will
be forgiven for raising a simple question which, I am sure, the HOM
professionals can answer without difficulty.

In present-day maths, a routine step has the following general form.

(*) 	Given an equivalence relation R on a class A, we wish to assign to each
 	x in A an object x* such that, for all x and y in A,  x* = y* iff xRy.

The routine reductionist step is to define x* as the equivalence class of x
mod R.

The earliest example of this I have come across is Frege's definition of
number. (I am aware that I am being a little imprecise here, but I think it
is near enough.)

Formerly, when faced with a problem of the form (*), the standard practice
was to _invent_ primitive entities of a new sort and assign one such new
entity to each equivalent class. I believe that this is what Cantor and
Dedekind used to do. And if I am not mistaken v. Staudt's definition of
_direction_ also follows this pattern (where R is the relation of

My question is: who was the first to use the reductionist ploy of using the
equivalence classes _themselves_ as the required objects x*.

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