Square Roots

In response to:

What Philosophers Really Know from the October 8, 2015 issue

To the Editors:

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.

George Szpiro
Neue Zürcher Zeitung
New York City

Rebecca Newberger Goldstein replies:

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).

So even if, defying grammar and common usage, the expression “the square root of 4” could refer either to one or two numbers, my particular use of it could only refer to one number, for reasons that the book I was reviewing makes clear. McGinn recounts how Paul Grice’s seminal essay “Meaning” explains how terms that are ambiguous in an isolated sentence may be unambiguous in the context of the discourse. According to Grice’s Maxim of Relevance, the only intelligible interpretation of “the square root” in the context is “the principal square root.”

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.