Anyone who is reading this post – indeed, anyone who can read at all – has some minimal exposure to mathematical ideas, even if that exposure goes no further than elementary arithmetic and not, as I am only half-jokingly known to say, actual mathematics. (Well, since I’ve mentioned it: the thing I’m known to say is, “that’s not mathematics, that’s arithmetic.” This is always in response to someone who has protested something along the lines, “I’m no good at mathematics; I can’t even balance my checkbook.” The humorous, yet legitimately educative nature of MY statement always strikes me as obvious, yet I am constantly amazed by the numbers of people who get lost in the elementary rhetoric of my statement.)

In any event, even such minimal exposure is typically enough to satisfy most people, even most mathematicians (I suspect), that they have a pretty good handle on what that equals sign (“=”) means as it is expressed in, say, the title of this little essay. Clearly I wouldn’t be writing about it if such an impression was even remotely true. For one thing, how do we read “A = B”? Does it say, “A equals B”, or does it say “A is B”, and is there a difference between those two? Spoiler: yes. Yes to both questions, depending on how crudely one is using one’s language, which makes the fact that “equals” does not equal “is” an especially problematic conflation of terms. “Is” tends to mean “identity” in such a context, which is tricky enough in its own right (I wrote an MA thesis on the subject). The reading of “=” as “equals” helps to emphasize a somewhat more functional approach to matters, though it is still more rigid and “substantive” than such formal notions as “equivalence” and “isomorphism.” topics I’ll likely blog about in the future because I can already hear the math-phobes screaming in horror. For now, I want to focus on the logical issues of “equals,” as a formal relation. Thus, the word “equality” may also find a use here, but that use should not be mistaken for the political, economic, cultural, &/or social senses of the term. (On the other hand, I do not preclude in advance that what I say here will have no bearing on those uses, either.) Obviously, the starting point for the primary discussion is with the work of Alfred North Whitehead. Continue reading →