> > If I am not mistaken, the ideas of equivalence relation and equivalence
> > classes first appeared in "Disquisitiones arithmeticae" by Gauss.
>
> Certainly Gauss used equivalence classes, both in modular arithmetic
> and binary quadratic forms, and often had modern-looking notation.
> Offhand I don't know what may have appeared before Gauss.
>
> Michael Josephy josephy@cariari.ucr.ac.cr
The original question was posed by Moshe' Machover. I quote him.
BEGIN QUOTE
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
parallelism).
My question is: who was the first to use the reductionist ploy of using the
equivalence classes _themselves_ as the required objects x*.
END OF QUOTE
Gauss did not introduce the integers n* mod p; rather he continued to speak
of the integers n under the relation of congruence mod p.
It would be ironic if Frege was first to use the device of equivalence
classes, since his use of it to define the cardinal numbers is inconsistent
(e.g. his number 1 is the `set' of all unit sets). However, he did give as a
preliminary example in *Grundlagen der Arithmetik* the equivallence relation
`parallel to', yielding the objects L* = the direction of the line L. I had
inferred from this that von Staudt had used this device; but Moshe' seems to
be implying that he did not. One more piece of a priori history shot down.
Bill Tait