X-Message-Number: 26734 Date: Fri, 29 Jul 2005 21:32:02 -0700 From: Mike Perry <> Subject: Further researches on "identity" References: <> Thomas Donaldson says: >However it's still important to remember that there is NO sense >of identity at all. We may apply the symbol "=" as an abbreviation >for the sense of equivalence we want to talk about. In a general context, this may be true. At least "identity" is used in different ways (not all of them referring to "equivalence" even, I would say). As for mathematics itself, I checked on the Web to see if there might be a special convention in use (as a general rule, maybe not always), such as "identity means equality." The situation seems somewhat confused; the equality symbol, for one thing, does not always mean "equality" in the sense that two things are one and the same. (T(n) = O(n^2) describes the growth of function T(n) as n-> infinity, as being like the second power of n, rather than saying that "T(n) is the same thing as O(n^2)".) However, my basic position about "identity in mathematics" is more or less captured in the following (found at http://encyclopedia.laborlawtalk.com/identity): In logic, the identity relation is normally, (by definition), the transitive, symmetric, and reflexive relation that holds only between a thing and itself. That is, identity is the two-place predicate, _=_, such that for all x, y, "x=y" is true iff x is y. except that maybe I should have said "in logic" rather than "in mathematics." (Albeit the two are closely connected; set theory was developed from logic and in turn served as a foundation of mathematics.) Here it is saying that identity is the equivalence ("transitive, symmetric, and reflexive") relation that holds only between a thing and itself (or between a thing and something that "is" that thing). So not just any old equivalence relation, unless you want to argue over the meaning of "is". (Note that here the equality symbol does have the meaning of "one and the same".) Anyway, what started this whole exchange was my statement in #26667: "In mathematics and some other settings, 'identity' holds between two objects if and only if they are the same in all respects." Again, maybe I should have said "in logic" instead of "in mathematics"--would anyone find that objectionable? Speaking of persons, though, I'm inclined to think one should not overuse the term "identity," in view of the confusions and disagreements it often causes. Be careful to explain what you mean if you do use it--and expect opposition! Mike Perry Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=26734