28.3544, Review: Applied Linguistics; History of Linguistics: Danesi (2016)

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LINGUIST List: Vol-28-3544. Mon Aug 28 2017. ISSN: 1069 - 4875.

Subject: 28.3544, Review: Applied Linguistics; History of Linguistics: Danesi (2016)

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Date: Mon, 28 Aug 2017 12:43:53
From: Bev Thurber [bat at alum.mit.edu]
Subject: Language and Mathematics

 
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Book announced at http://linguistlist.org/issues/27/27-3558.html

AUTHOR: Marcel  Danesi
TITLE: Language and Mathematics
SUBTITLE: An Interdisciplinary Guide
SERIES TITLE: Language Intersections 1
PUBLISHER: De Gruyter Mouton
YEAR: 2016

REVIEWER: Bev Thurber

REVIEWS EDITOR: Helen Aristar-Dry

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 language-mathematics
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 in-depth 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 math-language 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,
quantitative-probabilistic, 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 single-page 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 GW-BASIC. 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, tool-making, 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 Indo-European homeland in
the discussion of recent applications of glottochronology and related methods.
Danesi first describes the broadly accepted idea that the Indo-Europeans
“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 Indo-European
languages (2003), “support their theory by taking into account the fact that
Indo-European originated in Anatolia and that Indo-European 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. Language-tree divergence times
support the Anatolian theory of Indo-European 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/.


ABOUT THE REVIEWER

Bev Thurber is an independent researcher who is interested in historical and
computational linguistics and the history of ice skating.





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