Marcel Danesi’s *Language and Mathematics: An Interdisciplinary Guide* is the first in a series “Language Intersections” that will provide a guide to current research, discussing commonalities or differences between linguistics and a variety of disciplines. The bibliography of this book indicates that Danesi has been a significant contributor to research in the intersection of the disciplines, coming from a strong background in linguistics. “[His] goal is to show how this collaborative paradigm (often an unwitting one) has largely informed linguistic theory historically and, in a less substantive way, how it is starting to show the nature of mathematical cognition as interconnected with linguistic cognition.” [65]

The table of contents reveals the organization of the book: a first chapter that discusses the common ground between the two disciplines and divides it into the four topics that will be covered, chapter by chapter, in the rest of the book: Logic, Computation, Quantification, and Neuroscience. In each case, Danesi provides brief summaries of the current research in linguistics and the topics in mathematics that are related to these. “The balance tilts much more to the ‘linguistics-using-mathematics’ side than the ‘math-using-linguistics’ side.” [65]

The parallels set up by Danesi between linguistics and mathematics are often suggestive. Yet these parallels can be difficult to follow — there is a slippage throughout the book between language versus mathematics and linguistics versus mathematics. But linguistics is the study of language, so it would seem that if language and mathematics share commonalities, the comparison with linguistics would be with a discipline that studies mathematics, such as metamathematics or the philosophy of mathematics of perhaps mathematics education. All of these disciplines are identified by Danesi with mathematics itself. For example, Danesi writes that “Linguistics studies the final causes that constitute the phenomenon of language and mathematics the final causes that constitute math cognition.” [65]

I am not an expert in linguistics, so I am not able to evaluate that part of the book. The explanation of the mathematics, however, would have benefited from some interaction with mathematicians. There are some odd statements; here are just a few examples:

- “In the 1800s, mathematicians finally proved that the parallel postulate or axiom is essentially not an axiom.” [30] Instead, the mathematicians showed that it had to be assumed as a postulate or axiom in Euclidean geometry.
- “The Pythagorean theorem was not just a recipe for how to construct right angles.” [31] It’s not at all a recipe to construct right angles — the converse of the Pythagorean Theorem does that.
- After a discussion of Gödel’s incompleteness theorem, Danesi states that “Euclid’s fifth postulate is an example of an undecidable statement — it is obvious, but it cannot be decided whether it is an axiom or a theorem to be proved.” [36] I don’t even know what to say.
- Danesi sets \(\{\lambda\} =\) empty set, then notes that \(\lambda\) is the only element in \(\{\lambda\}\). [109] What?
- “Without notation, there would be no abstractions, theories, propositions, theorems, and so on in mathematics. There would be only counting and measuring practices.” [218] Surely this would have been a great surprise to pre-modern mathematicians.

In summary, while I believe this book could provoke useful dialogue between linguists and mathematicians, I can’t recommend that anyone read it for its mathematical descriptions. Danesi’s conclusion rings true: “Together with traditional forms of fieldwork and ethnographic analysis, the use of mathematics can help the linguist gain insights into language and discourse that would be otherwise unavailable (as we have seen throughout the book). That, in my view, is the most important lesson to be learned from considering the math-language nexus. The more we probe similarities (or differences) in mathematics and language with all kinds of tools, the more we will know about the mind that creates both.” [295]

Joel Haack is Professor of Mathematics at the University of Northern Iowa.