Mathematical Accuracy Is “Shallow Pedantry”?

Mathematical Accuracy Is “Shallow Pedantry”?


So we see that even when Fortuna spins us downward, the wheel sometimes halts for a moment and we find ourselves in a good, small cycle within the larger bad cycle. The universe, of course, is based upon the principle of the circle within the circle. At the moment, I am in the inner circle. Of course, smaller circles within this circle are also possible. – John Kennedy Toole


Briefly: a mathematician’s casual correction of a philosopher’s incorrect statement led to a comically explosive diatribe about “definite descriptions”, “modern semantics”, and who-the-fuck-knows.[1]

Longly: in the October 8 issue of the New York Review of Books (NYRoB), novelist and philosopher Rebecca Newberger Goldstein (married to famed cognitive scientist Steven Pinker) wrote a piece titled “What Philosophers Really Know” which reviews Colin McGinn’s Philosophy of Language: The Classics Explained. Early on, Goldstein establishes that “philosophy of language has been, from the beginning, close to the center of analytic philosophy,” and that the origins of analytic philosophy are in the mathematical logic works of Gottlob Frege and Bertrand Russell. Great. So we would expect Goldstein to be acutely sensitive to both the symbolic forms of mathematical statements as well as any plain language used to present them. However, late in the piece she writes:

A rigid designator, in contrast, inflexibly connects to the same referent in all possible worlds. “The square root of 4” is a rigid designator, referring to 2 in all possible worlds. The distinction has implications that reach well beyond language.

This marks the beginning of several problems. First is the obvious: the statement “the square root of 4” does not “refer to 2 in all possible worlds”. I’ve seen enough red pen throughout my mathematical life to know that the square root of four can refer to both positive 2 and negative 2. When a mathematician pointed that fact out to Goldstein, she went sort of batshit defensive. The other problem is more fundamental and more savage: the incorrect “referring to 2 in all possible worlds” statement was made when Goldstein tried to interpret a classic example of a “rigid designator” in her own words. When coupled with her ridiculous response, this casts doubt on the overall value of the philosophy of language.

The Domain of Discourse


In an upcoming issue of the NYRoB, financial mathematician George Szpiro responds:

Rebecca Newberger Goldstein writes in her review of Colin McGinn’s Philosophy of Language: The Classics Explained [NYR, October 8]: “‘The square root of 4’ is a rigid designator, referring to 2 in all possible worlds.”

Oh no, this is so only in the world of positive numbers. In the world of real numbers, it also refers to –2. Loose use of language can lead even philosophers astray. Rigorous use of mathematics won’t allow it, however.

When Szpiro refers to “the world of real numbers” he invokes a common mathematical concept called the Domain of Discourse. Most simply, the domain of discourse (a.k.a. “the universe”) is the set of all of the things which we might be talking about. For example: if I asked you for the best animated Disney character, you wouldn’t be able to answer with Bugs Bunny, because Bugs is not in the universe of Disney characters. In Goldstein’s example, if the domain of discourse were limited to the “natural numbers” (positive integers) then “the square root of 4” could only be positive 2. But as Szpiro points out, in the world of real numbers (a quantity along the continuous line which also contains all integers, the natural numbers, and rational numbers), we could also be talking about negative 2. Therefore, what “the square root of 4” refers to depends on our world.[2]

Now here’s where shit gets weird. Instead of saying, “yes I should have specified the domain of discourse”, Goldstein embarks on an incoherent 400-word rampage, starting with:

The first error in George Szpiro’s letter is his assertion that the singular definite description “the square root” refers to two entities, a number’s positive and negative square roots. Grammar and semantics preclude using a definite description to refer to two things in a given context. Given the semantics of definite descriptions, “the square root of 4 are even numbers” is ungrammatical. So perhaps a definite description should not have been used by me at all? Not so, as the Wikipedia entry for “square root” makes clear: “Although the principal square root of a positive number is only one of its two square roots, the designation ‘the square root’ is often used to refer to the principal square root” (that is, the positive one; italics in the original).

Bertrand Russell would be ashamed. Of course a “definite description” (or definition) can refer to two things, because the “thing” it can refer to can be a set which contains only two things. Therefore, in English we would say that the definition refers to either A or B (or both or neither, if you’re picky). It’s actually that simple. There are no great mysteries of philosophy or language to be found here.[3] Finally, the last part using Wikipedia to define the “principal square root” works against her original point — if you need to specify it as the “principal square root” then there’s no way “the square root” is unambiguous. All of us can agree, even Goldstein herself, that she did not specify her universe appropriately.

Bad Plagiarism

My underwhelming, but most likely true, hypothesis about this whole situation is that Rebecca Goldstein simply didn’t understand the example she was borrowing from other philosophers. Remember, the statement in question is:

“The square root of 4” is a rigid designator, referring to 2 in all possible worlds.

A little Google-Fu turns up that Goldstein probably borrowed this example from one of a dozen potential sources. However, most of those sources use “the square root of 25”, and almost none of them say what it “refers to” — i.e. Goldstein’s nonsense rant about modern semantics isn’t often supported by the people she lifted her example from.


Sybil Wolfram Philosophical Logic: An Introduction

Kripke subdivided rigid designators into:

a) strongly rigid designators: these not only cannot refer to anything but what they do but also must refer to what they do refer to. Examples are ‘5’ or ‘9’, ‘the square root of 25 or of 81′(Kripke (2971) 1977: 78)

Peter A. French Contemporary Perspectives in the Philosophy of Language, Volume 1

Now Kripke clearly points out that this distinction is not co-referential with the distinction between proper names and descriptions; after all, “the square root of 25” is certainly a rigid designator

While neither of these examples try to claim this “rigid designator” evaluates to 5, I must admit that there are other texts which do claim that it “refers to 5” and “cannot refer to any number other than 5.” But I think this is wrong. “The square root of 25” can be a rigid designator and still refer to a single object which contains two more objects. No amount of highfalutin semantic hocus pocus can attain the specificity of a well-defined mathematical statement.

Identify Then Decide

Here’s the closing of Goldstein’s rant:

Szpiro’s additional error is to confuse the vernacular term “world” (as in “the world of positive numbers”) with the technical term “possible world,” which, as McGinn also discusses, has become a central construct in modern semantics, helping to explain the concepts of logical necessity and possibility and the meanings of names and natural kinds. A possible world describes the way the world might have been. There is no possible world in which positive integers exist but negative ones do not. Szpiro’s attempt to explain the semantics of this particular definite description by distinguishing between “the possible world” of positive numbers and “the possible world” of real numbers commits him to a category error.

Szpiro’s letter inadvertently underscores the value of the modern philosophy of language, as explicated in McGinn’s book. It gives our understanding of linguistic meaning a depth and rigor that are commensurate with the complexity of language itself, rather than leaving it as an opportunity for shallow pedantry.

lol. It doesn’t sound like Szpiro confused “the vernacular term ‘world'”. In fact, his usage of it is more specific as it identifies a domain of discourse. As far as the claim that there is “no possible world in which positive integers exist but negative ones do not”, one only needs to look at the world of the natural numbers, which are non-negative. Rather than Szpiro “inadvertently underscoring the value of the modern philosophy of language”, I think it is Goldstein who inadvertently puts any potential value into doubt.






Footnotes    (↵ returns to text)
  1. I’m clearly skipping an introduction about where I’ve been and what I’ve been doing over the past year — unmiraculous and immaterial — perhaps another day
  2. In my tiny little niche of quantitative trading and finance, the idea of “the universe” is very important. When you meet with team members or potential investors to propose an automated, logic-based trading system, you can’t just write your math on a piece of paper and say “it works in all possible worlds.” In fact, accurately specifying the universe is a crucial step on one’s way to claiming real knowledge of markets. I suspect the same is true elsewhere.
  3. I’m also giving Goldstein a lot of the benefit of the doubt with respect to how she handles “languages” and “grammars”, which are now rigorously formalized and studied. None of her references in the past half-century involve the people who have actually worked on mathematically formalizing these concepts.