Mathematics

Euclid in "Elements" wrote that "things which equal the same thing also equal one another." Is this true in all cases? I've read that it is only true for "absolute entities," but not to "relations," although I do not understand this exemption. Are there any examples of things that are equal to the same thing but not to one another? Are relations really exempt from Euclid's axiom, and if so, why?

Comments

If by the adjective "equal" Euclid means "identical in magnitude" (which I gather is what he does mean), then his principle follows from the combination of the symmetry of identity and the transitivity of identity. The symmetry of identity says that, for any x and y, x is identical to y if and only if y is identical to x. The transitivity of identity says that, for any x, y, and z, if x is identical to y and y is identical to z, then x is identical to z. Therefore, Euclid's principle has exceptions only if the symmetry of identity sometimes fails or the transitivity of identity sometimes fails. But I don't think either of them ever fails.

Now, some relations that are similar to the identity relation aren't transitive. I might be (1) unable to tell the difference between color swatches A and B, (2) unable to tell the difference between swatches B and C, yet (3) able to tell the difference between swatches A and C. But color-indiscriminability-by-me isn't identity of color.

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