REVIEWS EDITOR: Helen AristarDry
SUMMARY
Language and mathematics: An interdisciplinary guide is the first volume in a new series, Language Intersections, published by de Gruyter. It provides an extremely broad survey of what Danesi calls “the languagemathematics interface” (64) in a short preface, five substantial chapters, a bibliography, and an index. The goals of the book, as stated in the first chapter, are “to give a generic overview, not an indepth description and assessment of all the many applications and connections between the two disciplines” and “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). This book is clearly intended for linguists rather than mathematicians; Danesi concludes that “the most important lesson to be learned from considering the mathlanguage nexus” is that “the use of mathematics can help the linguist gain insights into language and discourse that would be otherwise unavailable” (295).
Chapter 1, “Common Ground,” is a whirlwind tour through the history of mathematical logic, linguistics, computation, and neuroscience. The four sections following its “Introductory remarks” are the same as the other four chapters of the book, which focus on “formalist, computationist, quantitativeprobabilistic, and neuroscientific” approaches to studying language and parallels in mathematics (64). Formalist approaches include those of authors such as Aristotle, Euclid, Lobachevsky, and Chomsky. Computation begins with the invention of computers and flows from the formalism of the previous section into cognitive science. The section on quantification describes mathematical methods and their applications in linguistics. Neuroscience is introduced by means of Gödel’s incompleteness theorem as an example of blending, “the most promising [model] for getting at the core of the neural continuity between mathematics and language” (62). The final section, “Common ground,” is a singlepage summary of the chapter.
The goal of Chapter 2, “Logic,” is to “look more closely at the main techniques and premises that underlie both formal mathematics and formal linguistics, as well as at the main critiques that can be (and have been) leveled at them” (66). The story begins with ancient Greek philosophers and the notion of proof with examples from geometry. It then moves into symbolic logic and the role of computers. Next, Danesi summarizes the development of set theory before segueing into a section on formal linguistics that mostly focuses on generative grammar. This is followed by a brief section on cognitive linguistics as a response to formalism. The chapter ends with a section entitled “Formalism, logic, and meaning” that draws logic, mathematics, and language together.
Chapter 3, “Computation,” begins with a discussion of the power and limitations of computers. Danesi argues that “because language and mathematics can be modeled computationally in similar ways, this can provide insights into their structure and, perhaps, even their ‘common nature’” (134). To connect with the previous chapter, computation is described in terms of formalism. The chapter introduces computer science using Euclid’s algorithm for finding the prime factorization of a number. It continues with a brief history of computing that focuses on artificial intelligence and a discussion of programming, which includes simple example programs in GWBASIC. The next section, “Computability theory” discusses the types of problem that can be solved using an algorithm. The chapter then proceeds to tighten its focus in sections on computational linguistics and natural language processing, which feature machine translation. The chapter ends with a section entitled “Computation and psychological realism” that includes a discussion of the Turing Test and a summary of the chapter.
Chapter 4, “Quantification,” provides a summary of concepts from statistics and probability and applications to linguistics, particularly corpus linguistics. Among the specific topics covered are Zipf’s law and stylometry. The section on probability is dominated by a discussion of Bayesian inference and its applications with the goal of showing that “the world seems to have probabilistic structure and its two main descriptors – mathematics and language – are themselves shaped by this structure” (230). The penultimate section, “Quantifying change in language,” brings together the mathematical ideas discussed in the previous sections in a description of techniques that have been used in historical linguistics: lexicostatistics, glottochronolgy, and the economy of change. Danesi introduces Swadesh’s wordlists and the initial work with them, summarizes positions for and against the results taken by scholars, and concludes on the hopeful note that “good glottochronological analyses are becoming more and more a reality, thus validating Swadesh’s pioneering work” (244), before discussing recent advances in the field. The chapter ends with a brief review of the ideas discussed that connects them using the principles of economy and least effort. The final section summarizes the development of quantum mechanics as an example of the interplay between mathematics and physics.
The last and shortest chapter, “Neuroscience,” begins with a summary of ways in which neuroscience connects with mathematics. Computational neuroscience and responses to it (connectionism and modularity) are discussed in the first section. The next topic, mathematical cognition, creates a bridge to the discussion of mathematics and language in the third section. The chapter includes summaries of many experiments that have been conducted to determine how the brain works, particularly with regard to mathematical or linguistic ability. The philosophers discussed include Immanuel Kant and Charles Peirce, who explained the importance of visualization to mathematical cognition. Danesi goes on to summarize studies arguing that “several key evolutionary factors” connect mathematics and language (280). These factors include bipedalism, brain size, toolmaking, and social structures. The book concludes with the idea that mathematics is an important tool for linguists.
The book ends with a long list of references and a short index.
EVALUATION
The book’s goals are quite broad, and the book achieves them, especially the first, which is supported by the historical approach taken. Danesi covers the foundational research in each area and shows how ideas developed over time. One thread that runs through the book to support these goals is the concept of blending. This idea is introduced abstractly as the process that results in both mathematics and language (4) and is developed throughout the book. The discussion of this concept peaks during the sections on neuroscience, when blending is defined as combining information from two sources to create something new (56–57). Danesi links this concept to metaphors in the section on cognitive linguistics and notes that metaphors are what connect language and mathematics (285).
The mathematical examples provided throughout the book are simple ones that mathematicians know well, such as the proof that the square root of two is irrational (77–78) and the Cantor set (92). The explanations are clear and should make sense to readers with little mathematical background; complex details are avoided. However, readers must be wary of oversimplifications and imprecise uses of mathematical terms. Most of these may be forgiven as part of explaining complex concepts to readers who are presumed to have little mathematical knowledge, such as the absence of negative values from the set of integers (97). Others are more awkward, such as the assertion that “A valid proof has what mathematicians call consistency, completeness, and decidability” (81); these terms are relevant to the notion of proof in mathematics, but are not properties of proofs. Still others may confuse readers, such as the seeming change of stance on the IndoEuropean homeland in the discussion of recent applications of glottochronology and related methods. Danesi first describes the broadly accepted idea that the IndoEuropeans “lived around ten to five thousand years ago in southeastern Europe, north of the Black Sea” (241). A few pages later, he states that Gray and Atkinson, who completed a mathematical analysis of the spread of the IndoEuropean languages (2003), “support their theory by taking into account the fact that IndoEuropean originated in Anatolia and that IndoEuropean languages were transported to Europe with the spread of agriculture” (245). In fact, Gray and Atkinson analyzed the steppe and Anatolian hypotheses, and their results supported the latter.
The book also contains some errors. The source code provided in the description of a simple example of natural language processing does not provide the output listed (174–175) or even run as written (line 100 should read “IF LEN(A$)...” instead of “IF LEN(A)$...”). This source code is very similar to the code provided for the earlier cookies program (147); there are minor changes in the text strings (cookies become children) and the for loop prints the wrong number of stars, but both listings have the same error. There are a few other typos, including “The” for “This” (175), “aculculia” for “acalculia” (282), and extraneous equals signs in logarithms (203, 204).
Despite these issues, the general ideas come across clearly, and the book achieves its stated goal of being a guide to the connections between mathematics and language. Its covers much of the same material as Gödel, Escher, Bach (Hofstadter 1999), but is broader in scope. It does not include discussions of specific mathematical tools used in linguistics today, such as the Natural Language Toolkit (2015) for Python, which is in keeping with the book’s goal of providing a general overview. Overall, the book is a concise summary of an extremely broad range of material. It provides many starting points for people who want to understand how mathematics can make them better linguists and who are interested in the connections between mathematics and language.
REFERENCES
Gray, Russell D. and Quentin D. Atkinson. 2003. Languagetree divergence times support the Anatolian theory of IndoEuropean origin. Nature 425. 435–439.
Hofstadter, Douglas. 1999. Gödel, Escher, Bach: An eternal golden braid. 20th anniversary edn. New York: Basic Books.
Natural Language Toolkit. 2015. http://www.nltk.org/.
