> I don't think you adddressed the question whether (Leibniz's)
>Euclid engaged in the practice of representing a mathematical entity by a
>raw equivalence class.
> It was Euclid's construction, not Leibniz's dissatisfaction with
>it, I had in mind as an example of taking a raw equivalence class to be
>the "object".
I apologize for misunderstanding your point. I mistakenly thought you were
attributing the use of equivalence classes as objects to Leibnitz.
But I think attributing this use to Euclid is also wrong, for the reason
stated by Mayberry. As far as I know, Euclid defines equality-of-ratio as a
quaternary relation between magnitudes or, perhaps more precisely, a binary
relation between pairs of magnitudes of the same kind. He doesn't have
ratios as objects.
> I think you missed a point, too, about set theoretic reduction.
>It really is logical sloppiness to set theoretically represent a non-set
>entity by a set. Call it "set theoretic reductionism" if you like, but
>that doesn't excuse it.
> I see your question as being about when
>mathematicians lost their fastidiousness and decided that it was okay to be
>sloppy because they knew how not to make any silly "mistakes".
This is a value judgment to which I am not entirely unsympathetic. I am
certainly aware that reductionism is problematic in all sorts of ways.
However, my question was concerned not with value judgments but with the
factual question as to who was the first to employ a certain reductionist
procedure.
Best wishes, Moshe'
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