A Is A

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Also known as the law of identity, is a worthless piece of philosophical drivel that plainly doesn't apply to the real world and seems to be unnecessary in mathematics. Objectivists use it as a shibboleth. Contrast "A is A" with "A is possibly A" or "A might not be A".

"A is A" cannot apply to the real world because the real world has time in it, and A at time 1 is not necessarily the same as A at time 2. It's never the same water in the river, and even protons can spontaneously decay. For "A is A" to apply to the real world, it has to be limited to an instant at time T. However, that makes it inappropriate for real-world issues, since they generally involve time spans. So at best, for real world issues, "A is A" is an approximation, that may vary wildly in accuracy over time for different subjects.

The modern formulation of identity is that of Gottfried Leibniz, who held that x is the same as y if and only if every predicate true of x is true of y as well. [Cribbed from wikipedia.] This makes it pretty obvious that A's at different times are not identical if only because there is a "time at" predicate.

Ludwig Wittgenstein writes (Tractatus 5.5301): "That identity is not a relation between objects is obvious." At 5.5303 he elaborates: "Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing." [Cribbed from wikipedia.]

The Ayn Rand Lexicon describes identity with a bunch of hand-waving, pseudophilosophical gobbledegook.

Links

Existential Comics 1134: The Philosophy Friends [More...]
Points out the ways in which "the river" is not "the river", but really just a convention used for inexact communication, and is recognized in flood or freeze or whatever else. Could also be extended to include the groundwater of the drainage basin.
NEW 7/08/2017: Hegel’s Science of Logic: Laws of Identity and Diversity [More...]
Hegel had a lot to say about this. Yet another example of Ayn Rand not giving credit.
The ABC of Materialist Dialectics [More...]
"The axiom ‘A’ is equal to ‘A’ appears on one hand to be the point of departure for all our knowledge, on the other hand the point of departure for all the errors in our knowledge."

Quotations

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