The Language of Mathematics

Ganesalingam’s book is based on his dissertation work, which won the Association for Logic, Language, and Information’s 2011 Beth prize. This is an honor indeed, and is an indication of both the quality of thought in the volume and the sheer magnitude of the project he has taken on. At the outset, Ganesalingam tells us that his aim is “to give a formal, objective, and above all precise analysis of the language used by mathematicians in textbooks and papers.” This seems a worthy (if grand) goal, particularly as it becomes clear that by “the language of mathematics” he means not formal logic but the actual text and notation used in publications.

Be warned that the author assumes a background knowledge of linguistics that most readers of this review probably do not possess. This is not a criticism of the book; it is written for philosophers, not for mathematicians. With chapters on topics such as Disambiguation and Typed Parsing, the mathematician-reader can be forgiven for finding the table of contents somewhat daunting. Be reassured, however, that the ideas contained in the book are fascinating for anyone who takes an interest in the foundations of mathematical thought and understanding, and that, with a bit of bracing for the sticky bits, such a reader can grasp those ideas quite readily.

The book begins with an analysis of the difference between mathematical text and mathematical symbols. An example of the former is the statement “every natural number has a successor,” while the latter is something like “\( \forall n\exists m (m = S(n)) \).” More often than not these two modes are mixed, as in the statement “every bounded sequence in \( \mathbb{R}^n \) contains a convergent subsequence,” and it is this interdependence that necessitates the new theory of language the author seeks to create. Ganesalingam is at pains to explain that the development of this new theory faces significant challenges unique to the language of mathematics, and much of the first part of the book is taken up with outlining these challenges and the strategies that will be used to surmount them.

To this reader, the most interesting such challenge is the so-called ambiguity of mathematical language. As mathematicians we pride ourselves on the lack of ambiguity in what we do, but from the perspective of linguistics our statements are hopelessly ambiguous because the same string of symbols can have many meanings. Consider the expression given above, \( \forall n\exists m (m = S(n)) \). This could mean any number of things depending on, for example, the set from which \(n\) and \(m\) are being chosen, the definition of the predicate \(S\), and the model of set theory in which one is working. Beyond that, mathematical language is highly type-dependent, which simply means that the way we write down a piece of mathematics depends on which mathematical objects occur in it. This is not the way “natural” languages (the ones we use to talk to each other) work, or anyway not nearly to the same extent. The author’s explication of these difficulties in Chapters 1 and 2 and subsequent resolution of them in Chapters 4, 5, and 6 are, if not precisely page-turners, certainly very compelling.

Chapter 7 is, alas, where the author and the reviewer part ways. Full disclosure: this reviewer is a set theorist, and as such is suspicious of any effort to “reconstruct the foundations of mathematics so that natural numbers [are] not sets.” Why, one might reasonably ask, would anyone want to do such a thing? The answer seems to be that defining numbers as sets creates what the author deems a “mismatch between formal predictions and belief.” This is territory first explored by Paul Benacerraf in his groundbreaking 1965 essay What Numbers Could Not Be, a provocative and compelling examination of the problems with mathematical Platonism. In this context, however, the issue seems to be that using sets to define numbers presents certain linguistic difficulties having to do with the aforementioned types. Of particular concern is the fact that the relationships between \( \mathbb{N}\), \(\mathbb{Q},\) and \( \mathbb{R} \) with which we are all familiar (to wit: \( \mathbb{N} \subset \mathbb{Q} \subset \mathbb{R} \)) are not true in the constructed set-theoretic universe. That is, \( \mathbb{Q} \) is a set of equivalence classes of pairs of integers, while \( \mathbb{N} \) is… well, is not. Therefore, set-theoretically, \( \mathbb{N} \not\subset \mathbb{Q} \). Why this is so upsetting, I could not say, but the author resolves it by introducing the notion of ontogeny in mathematics, which the reviewer confesses not to have understood.

On the whole, Ganesalingam’s work is thoughtful and thought-provoking. He gives an analysis of mathematical language at a depth not previously attempted, and the results are impressive. Beyond that, in the process he achieves the difficult task of bringing new insight to some very old questions in the philosophy of mathematics. I recommend the book to anyone with an interest in those questions.

Kira Hamman teaches mathematics at Pennsylvania State University, Mont Alto. She justifies her not-very-socially-aware interest in logic and the philosophy of mathematics by also working on issues in the intersection of mathematics with social justice and civic engagement.