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.
Even in his earliest, professional publication (A Treatise on Universal Algebra, 1898) Whitehead was aware that ideas about “equals” and “equivalence” required a measure of care in treatment that they often did not receive. When his Principle of Relativity was published in 1922, Whitehead devoted an entire chapter (III) to the topic of “equality.” This chapter bears some closer scrutiny, because the ideas Whitehead presents here are of considerable importance, yet that importance has generally been overlooked because of the awkward symbolism Whitehead employed in presenting those ideas. As this is not a scholarly essay, I’ll forego page numbers and citations: anyone who wishes can click on the link above, jump to chapter III, and see for themselves, all without any particular difficulty.
When ever a claim that fits the abstract pattern “A = B” is made, before any attempt to evaluate if the claim is true or false, first it must be meaningful. And in order for meaningfulness to occur, A and B must first be comparable before that comparison (demanded by the “=”) can be evaluated. Thus, A and B might represent the “same” numerical value, but if one is referring to a linear measure, while the other to a cubic volume, then there is no basis of comparison of the two (in point of fact, different) numbers. Alternatively, “A” might represent the formula “1 + 1”, while be might stand for the number “2”. But unless one has taken prior steps to ensure the homogeneity of what is being discussed, the statement has no general claim to truth. For example, one quart of water plus one quart of alcohol does not equal two quarts of fluid.
So in order to make explicit this needed situational attention, Whitehead embedded the formula “A = B” with an “arrow gamma,” which then reads off something like, “A equals B in the context gamma.” His symbolism for this was: A = B → γ
This is not very satisfactory because, quite frankly, it leaves it ambiguous if the “→ γ” refers only to “B” or to the entire equality. One can disambiguate the formula with parentheses, thus giving us:
(A = B) → γ
But at this point, the arrow is redundant; why not simply write “(A = B)γ”? There is yet one more step I wish to take here: let us now move the gamma to a subscript, thus (A = B)γ.
Unless you’ve a significant background in philosophical logics, this last move might seem rather gratuitous. But for persons with such a background – and everyone else, once I’ve finished explaining it – the symbolism of the subscript gamma is very (and very deliberately) reminiscent of “possible worlds” or, more appropriately, “possible models” interpretations of modal logics. Modal logics go beyond standard proof-theoretic systems that examine truth or falsity in various engines of proof, or model and algebraic methodss that look at truth with more of an eye toward the formal structures of systems of propositions, and ask as well about the mode of that truth or falsity: Is it necessarily true, or only possibly true, etc? A statement that is true in one context or model might not be true – or even meaningful! – in another. The subscript letter (in our case, the gamma) is an “index” that is used to indicate in the abstract how these contexts &/or models are connected to one another, if they are connected at all.
The technical details are both very technical and very detailed, and I’ll not try to explore them here. (Some basic facts can be found HERE, introductions (only) to many aspects of the field can be found HERE, and a full text may be downloaded HERE.) It is sufficient for our purposes to notice that these modal characteristics introduce a number of complications to the nominally simple idea of “=”; some of these complications can literally be intractable (unsolvable by any computer. See the discussions on “decision problems” in the linked documents above.) Whitehead’s insight here is one (among so many of his) that has been all but entirely overlooked. I know of no logic text that treats “=” as anything other than a two-place, first-order predicate that is directly introduced as a quasi-logical operator of effectively trivial simplicity. Recognizing that equality (even in its formal, mathematical sense) is characterized by intrinsically modal factors, is a radical decentering of traditional views. (The previous qualifier, “ formal, mathematical sense,” is to prevent any confusion with the social, economic, and political issues of equality. Not because they are irrelevant, and definitely not because my comments here have no bearing upon them. But in keeping things limited to the simplest case possible, it keeps open their application to the more complex situations.)
In Whitehead’s case, the consideration of equality came in the context of a theory of measurement – or, more specifically, the lack of a coherent such theory within Einstein’s general theory of gravity and the gravitational cosmology that is based upon it. There is, in general, a rather too sanguine attitude toward the presuppositions that lie behind every act of measurement throughout the physical sciences. For Whitehead, the question was focused on how one determines that such-and-such a number “equals” the distance to some astronomical object. But his considerations carry over to any act of comparison (metrical in character or not) where it is asserted that one thing (one quart of fluid plus another?) is equal to a second thing (two quarts of fluid?). No such comparison can even be meaningful, much less true or false, unless the modal factors that are inherent in every act of comparison (and silently buried inside the loaded concept of “equals”) are organized so as to allow that comparison to stand; so as, in a word, to make the modality of that comparison possible. A few works in the literature touch on such issues, though they also fail to highlight the modal character of such contextual issues as make comparison possible. And all universally fail to note Whitehead’s position at the foundation of their studies.