At around the same time, generative grammar came to the forefront in theoretical linguistics (Chomsky 1957, 1965). From the outset, it espoused a basic mathematical mindset—that is, it saw the study of language as a search for the formal axioms and rules that undergirded the formation of all grammars. As his early writings reveal, Chomsky was inspired initially by Markov’s (1906) idea that a mathematical system that has “n” possible states at any given time, will be in one and only one of its states. The generativist premise was (and continues to be) that the study of these “states” in separate languages will lead to the discovery of a universal set of rule-making principles that produce them (or reflect them). These are said to be part of a Universal Grammar (UG), an innate faculty of the human brain that allows language to develop effortlessly in human infants through exposure, in the same way that flight develops in birds no matter where they are in the world and to what species they belong. The concept of “rule” in generative grammar was thus drafted to be analogous to that in propositional logic, proof theory, set theory, and computer algorithms. The connection between rules, mathematical logic, and computation was actually studied insightfully by Alan Turing (1936), who claimed that a machine could be built to process equations and other mathematical forms without human direction. The machine he described resembled an “automatictypewriter” that used symbols instead of letters and could be programmed to duplicate the function of any other existing machine. His “Turing machine” could in theory carry out any recursive function—the repeated application of a rule or procedure to successive results or executions. Recursion became, and still is, a guiding assumption underlying the search for the base rules of the UG. Needless to say, recursion is also the primary concept in various domains of mathematics (as will be discussed in the next chapter).

The quest to understand the universal structures of mind that produce language and mathematics, considered to be analogous systems, goes actually back to ancient philosophers and, during the Renaissance, to rationalist philosophers such as René Descartes (1641) and Thomas Hobbes (1656), both of whom saw arithmetical operations and geometrical proofs as revealing essentially how the mind worked. By extension, the implication was that the same operations—for example, commutation and combination—were operative in the production of language. As the late science commentator Jacob Bronowski (1977: 42) observed, Hobbes believed in a world that could be as rational as Euclidean geometry; so, he explored “in its progression some analogue to logical entailment.” Hobbes found his analogue in the idea that causes entailed effects as rigorously as Euclid’s propositions entailed one another. Descartes, Hobbes, and other rationalist philosophers and mathematicians saw logic as the central faculty of the mind, assigning all other faculties, such as those involved in poetry and art, to subsidiary or even pleonastic status. They have left somewhat of a legacy, since some mathematicians see mathematics and logic as one and the same; and of course so too do generative linguists.

Since the early 1960s, mathematical notions such as recursion have influenced the evolution of various research paradigms in theoretical linguistics, both intrinsically and contrastively (since the paradigm has also brought about significant opposing responses by linguists such as George Lakoff). Mathematicians, too, have started in recent years to look at questions explored within linguistics, such as the nature of syntactic rules and, more recently, the nature of metaphorical thinking in the production of mathematical concepts and constructs. Research in neuroscience has, in fact, been shedding direct light on the relation between the two systems (math and language), showing that how we understand numbers and learn them might be isomorphic to how we comprehend and learn words. As rigid disciplinary territories started breaking down in the 1980s and 1990s, and with interdisciplinarity emerging as a powerful investigative mindset, the boundaries between research paradigms in linguistics and mathematics have been steadily crumbling ever since. Today, many linguists and mathematicians see a common research ground in cognitive science, a fledgling discipline in the mid-1980s, which sought to bring together psychologists, linguists, philosophers, neuroscientists, and computer scientists to study cognition, learning, and mental organization. So, in a sense this book is about the cognitive science of language and mathematics, but it does not necessarily imply that cognitive science has found the light at the end of the tunnel, so to speak. As mentioned in the preface, the basically empirical and theory-based focus of cognitive science will shed light on the math-language interface only from a certain angle. The hermeneutic approach espoused here is intended to insert other perspectives of a more critical nature into the disciplinary mix that might provide a clearer picture of how the interface unfolds intellectually and practically.

The purpose of this chapter is to provide an overview of the main areas that fall onto a common ground of interest and research in linguistics and mathematics. Then, in subsequent chapters, the objective will be to zero in on each of these areas in order to glean from them general principles that might apply to both systems. This is, in fact, a common goal today behind institutional initiatives such as the Cognitive Science Network at the Fields Institute for Research in the Mathematical Sciences at the University of Toronto (mentioned in the preface).

Perhaps the first detailed comparison of mathematics and language was Charles Hockett’s 1967 book, Language, mathematics and linguistics. Although a part of the book was devoted to a critique of Chomskyan grammar, a larger part dealt with describing properties that language and mathematics seemed to share and with what this implied for the study of both. Hockett was a structuralist, and his interest in mathematics was really an outgrowth of early musings on the links between language and mathematics within structuralism, such as those by Roman Jakobson, who claimed that notions such as the Saussurean ones of value and opposition, could be profitably applied to the study of mathematical structure (see Andrews 1990). Hockett’s book was an offshoot of Jakobson’s implicit entreaty to study mathematics from the structuralist perspective. Since then, much has been written about the relation between mathematics and language (for example, Harris 1968, Marcus 1975, 1980, 2003, 2010, Thom 1975, 2010, Rotman 1988, Varelas 1989, Reed 1994, MacNamara 1996, Radford and Grenier 1996, English 1997, Otte 1997, Anderson, Sáenz-Ludlow, and Cifarelli 2000, 2003, Bockarova, Danesi, and Núñez 2012). There now exists intriguing evidence from the fields of education, neuroscience, and psychology that linguistic notions might actually explain various aspects of how mathematics is learned (for example, Cho and Procter 2007, Van der Schoot, Bakker Arkema, Horsley, and van Lieshout 2009).

In a lecture given by Lakoff at the founding workshop of the Network mentioned above in 2011, titled “The cognitive and neural foundation of mathematics: The case of Gödel’s metaphors,” it was saliently obvious to those present—mainly mathematicians—that in order to study mathematical cognition at a deeper level than simply formalizing logical structures used to carry out mathematical activities (such as proof), it is necessary to understand the neural source of mathematics, which he claimed was the same source that produced figurative language. Lakoff discussed his fascinating, albeit controversial, view of how mathematicians formed their proofs and generally carried out their theoretical activities through metaphorical thinking, whichmeans essentially mapping ideas from one domain into another because the two domains are felt to be connected. The details of his argument are beyond the present purposes, although some of these will be discussed subsequently. Suffice it to say here that Lakoff looked at how Gödel proved his famous indeterminacy theorem (Gödel 1931), suggesting that it stemmed from a form of conceptualization that finds its counterpart in metaphorical cognition—an hypothesis that he had put forward previously in Where mathematics comes from (preface).

As argued in that book, while this hypothesis might seem to be an extravagant one, it really is not, especially if one assumes that language and mathematics are implanted in a form of cognition that involves associative connections between experience and abstraction. In fact, as Lakoff pointed out, ongoing neuroscientific research has been suggesting that mathematics and language result from the process of blending, which will be discussed in due course. It is sufficient to say at this point that Lakoff’s argument is highly plausible and thus needs to be investigated by mathematicians and linguists working collaboratively. The gist of his argument is that mathematics makes sense when it encodes meanings that fit our experiences of the world—experiences of quantity, space, motion, force, change, mass, shape, probability, self-regulating processes, and so on. The inspiration for new mathematics comes from these experiences as it does for new language.

The basic model put forth by Lakoff is actually a simple one, to which we shall return in more detail subsequently. Essentially, it shows that new understanding comes not from such processes as logical deduction, but rather from metaphor, which projects what is familiar through an interconnection of the vehicle and the topic onto an intended new domain of understanding. In this model, metaphor is not just a figure of speech, but also a cognitive mechanism that blends domains together and then maps them onto new domains in order to understand them. The two domains are the familiar vehicle and topic terms which, when blended together produce through metaphor new understanding, which is the intended meaning of the blend (see Figure 1.1).

Lakoff presents a very plausible argument for his hypothesis. But in the process he tends to be exclusive, throwing out other approaches, such as the generative one, as mere games played by linguists. While I tend to agree with the substance of Lakoff’s argument, as will become evident in this book, I also strongly believe that the other approaches cannot be so easily dismissed and, when looked at in a non-partisan way, do give insights into language and its mathematical basis, from a specific angle. Moreover, formalist models have had very fertile applications in areas such as Natural Language Programming in computer science and in Machine Translation, which have both become critical tools of the Internet (Danesi 2013).

While mathematicians are starting to look towards linguistics, and especially cognitive linguistics (which is what Lakoff’s approach is generally called), as a source of potential insights into questions such as what is number sense, one can also argue that linguistics, as a science, has always had an implicit interest in both mathematics as a system of understanding and in using mathematical techniques (such as statistics) to carry out specific kinds of research. For example, already in the nineteenth-century, the neogrammarians developed their theory of sound change on the basis of lists of frequently-used cognates. From their databases they extracted principles—or laws as they called them—of phonological change. Although they did not explicitly use statistical analysis (which was in its infancy anyhow in their era), it was implied in their modus operandi—that is, they developed their theories not through speculation, but by examining data in order to conduct analyses and develop theories from them.

The common ground for interdisciplinary research in linguistics and mathematics can be subdivided into several main areas, implied by work that has been conducted (and continues to be conducted) in both disciplines: