In my opinion, the ways in which mathematicians use this symbol today can be roughly summarized as to express (i.e., to make explicit) the co-extensionality among referential expressions with different (discursive) descriptions --- whatever this means.(*) On this ground, I think that one of the most important mathematical merits of this metalinguistic resource is to facilitate the analysis of the co-intensionality (i.e., theory-laden equivalence of some sort) among predicative or functional expressions (of the same type but) with different (theoretical) descriptions. In other words, I think that while the main function of the symbol = is to make possible certain (peculiar) kinds of referential analysis, the upshot of such referential analysis is that it also makes possible certain kinds of inferential move which are otherwise unavailable to mathematicians. As such, modern mathematicians' use of the symbol = is, in my opinion, an integral part of their systematic and methodical use of various metalinguistic vocabularies and expressions (including the symbol =) with which they articulate their referential and inferential (speech) acts with precision. Such vocabularies/expressions include, among others, those for quantification, negation, propositional-connection and those for (what may be casually called) intension-definition (i.e., positing of an axiom-set), extension-definition (i.e., "naming ceremony," e.g., by a locution of a form like "Let a be any element of G such that ..."), and positing of a hypothesis (e.g., by a locution of a form like "Suppose that there is an element a in G such that ..."). In short, I think that we can discern something of "mathematese" in the totality of the sophisticated systematic ways in which mathematicians nowadays use these metalinguistic vocabularies/expressions.
(*) As anyone who has read Frege (1892) can see through this passage, I'm a partial Fregean when it comes to the analysis of the so-called "identity statement." However, my semi-Fregean position on this issue is quite different form his original in that my position is based on a kind of epistemological dualism (which I have been developing for about ten years now, and) on which my structuralist and pragmatist view of language (outlined in various places in this website, e.g., here) is also based. Therefore, my summary of the mathematical uses of the symbol = above (the italicized text) is very crude, to say the least, from my epistemological-dualistic structuralist/pragmatist point of view. To see how crude it is, please wait until I finish the revision of the entry on Mar. 5, 2015 (on which I've been working now). If anyone is interested in the epistemological dualism, there is a very early exposition of it available here. However, this paper ("Two ways of identification") is written relatively early in my development of this position that it contains some nontrivial claims which I no longer hold. I will revise this paper entirely, too, to update its content to my current view of the matter, as soon as I can. |
Now, there are people who may be casually called "native mathematese speakers" --- that is, those working mathematicians who have mastered "mathematese" so thoroughly that they feel as if it were nothing but a natural culmination of the rational aspect of their respective first languages, which just happens to be a cognitive prerequisite for doing exact mathematics. Typically, those people feel this way about "mathematese" either because they have picked it up through actual studies of mathematics (especially abstract mathematics), rather than by taking a formal course on modern logic, or else, because they have "internalized" it thoroughly through their (abstract) mathematical studies (even if they are at first initiated to it by a formal course on logic). For those people, it may be tempting to think that acquiring "mathematese" is a matter of acquiring certain logical concepts, which of course includes (according to their "native intuition" as "native mathematese speakers") the concept which is commonly denoted by the symbol = (or by equivalent locutions in forms like "A is B") --- that is, the concept of "equality" or "identity," or, (though subtly different in nuance) "uniqueness." It can probably naturally occur to many of those "native speakers" that they can use "mathematese" because they have acquired those "logical concepts," which dictate how to employ those "logical" vocabularies/expressions of "mathematese" accurately. I might have created an impression of taking some distance from those "native speakers" by calling them "them." This impression is not entirely wrong, but, I actually think I am a working mathematician who has internalized "mathematese" thoroughly enough through actually doing various mathematical studies (though at a very beginning level). So, even I am partly inclined to say, e.g., that if we ("native mathematese speakers") use the symbol = in the alleged epistemologically mixed and mixing way, that is because the concept of "equality" (or "identity" or "uniqueness") which we all share as "native mathematese speakers" is epistemologically mixed and mixing. (Indeed, even in this very essay, I have already revealed this part of myself, inadvertently, at various occasions, I think. For instance, when I identified the basic vocabularies/expressions of "mathematese" other than the symbol =, I listed them as "vocabularies/expressions for quantification, negation, ...".)
However, those "logical concepts" are something which we possess implicitly, and which, therefore, I think, we can ultimately attribute to ourselves only based on the ways in which we actually use certain vocabularies/expressions which are presumably metalinguistic (i.e., "logical"), including the ways in which we actually correct one another when we find others using them in some "ungrammatical" ways, etc. This is somewhat like a "radical translation" situation (to borrow a phrase of Quine 1960). Suppose, for instance, that some non-speakers of English encounter the language of English for the first time in the history of their native language A, directly or indirectly. (So, there has been no intermediating language B such that there has been someone who has some command of each of the three languages, A, B, and English.) In such a situation, if those first-contact makers (the speakers of the language A) can come to judge at all that certain vocabularies and expressions of the strange new language (English) express (what they recognize as) "logical concepts," that is only because of the ways in which English-speakers overtly use those vocabularies and expressions. (And, we must notice, it is also because of the "existence" in their language, A, of those "logical concepts," whatever such "existence of logical concepts in A" amounts to.) I think the essentially same "behavioristic" approach can be, and should be, adopted for the analysis of "mathematese." A crucial difference from a typical "radical translation" situation is that I'm proposing to adopt this approach as a "native mathematese speaker." That is, I'm making this proposal to the community of "native mathematese speakers," or, to a philosophically oriented sub-community of it, as a member of that community. That is, I am proposing this approach socially self-reflexively, for the analysis of our own concepts, not theirs, as I think that is the proper concerns of philosophy --- which is because I think philosophy is ultimately the search of what I am, where I think I am, and we are, essentially social-semiotically inter-related and co-realizing processes or phenomena. I'm proposing this "philosophical behavioristic" approach for the philosophy of mathematics, that is to say, I'm proposing to analyze what mathematics is from the point of view of the linguistic/semiotic pragmatician, primarily by analyzing what we are doing when we are doing mathematics. Behind this proposal, I have a hope that efforts to answer this question will also bear some sort of answer to questions like what it is to know a statement to be true of a certain mathematical situation and/or what it is to know a stated inference to be valid in a certain class of mathematical situations, etc., and, ultimately, what we are qua "native mathematese speakers," which I think is a kind of dual question to what mathematics is.
As far as I'm concerned, this "philosophical behavioristic" approach to the philosophy of mathematics is merely a special application of a certain "philosophical behavioristic" methodology for (general) philosophy. This is because for me philosophy is ultimately about what I am, and I adopt, in addition, the aforementioned social-semiotic view of what I am/we are. (Which, by the way, I take to be inseparably related to the structuralist-pragmatist view of language and the epistemological dualism, both mentioned in the above side note.) Now, this general philosophical methodology can be characterized as a method of making explicit what we, as speakers of some language, already know implicitly in virtue of being able to use that language. (Notice that our concepts are included among such "things we know implicitly when we know some language.") And, as such a method of explication, it is comparable to a method which has been widespread especially in the analytic philosophical community, which is based on what may be called (from my point of view) philosopher's "native's intuitions." I'm talking about the intuition-based method whose naive assumption of the universality of intuitions under investigation has been recently increasingly criticized by an "empirical" movement called "experimental philosophy." But, I believe that the "philosophical behaviorist" method I propose is better than that intuition-based method.
Actually, both of the widespread method and the method I'm proposing are based on one's own "native intuitions." The crucial difference between them consists in the kind of "inference" for which the "native intuitions" are used. So, the widespread method uses philosophers' own "native intuitions" as "evidence" for (what is usually and naively understood as) inductive reasoning. For example, when Gettier (1963) offered the famous counter-examples to the traditional "justified true belief definition" of the concept of "knowledge," he made an appeal to his own intuitions (that his examples did not count as cases of "knowledge," despite their satisfying the traditional definition), apparently tacitly assuming his intuition to be shared by other human beings universally. This use of "native intuitions" can be described as using one's own "native-intuitive" judgments as "evidence" for induction, i.e., as given singular truths against which some general hypotheses are tested. For that matter, even experimental philosophy seems to use "native intuitions" as "evidence" for induction exactly in the same sense, only taking into account "intuitions" of diverse populations to put the tacit universality assumption to "empirical" test. By contrast, the "philosophical behavioristic" method I'm proposing uses shared "native intuitions" of a homogenous population (that is, a population which is by assumption homogenous in all the relevant respects because it is identified by the "native intuitions" in question), as "evidence" to be explained for by a sort of abduction. In other words, this method uses those shared "native intuitions" as "evidence" in the sense of explanandum. What does play the role of explanans? The hypothetical system of concepts does, which is to be articulated structurally inferentially, and which constitutes a theory which is laden in the language of which the "intuitions" in question are native.
This sort of abduction is precisely what I elsewhere called rational abduction.
From this "philosophical behavioristic" point of view, it is more natural to say that, if we ("native mathematese speakers") share epistemologically mixed/mixing conceptions for some of the "logical concepts" (such as the concept of "identity"), that is because we share the same epistemologically mixed/mixing use of certain "logical" (or metalinguistic) vocabularies.
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PS: By the way, if anyone is interested, I have written a short essay (in Japanese) which is in effect another defense of the "behaviorist" approach to the philosophy of mathematics, titled (in English translation) "Is mathematics speech act?".