Metaphor and Mathematics

224 Pages·2014·1.28 MB·English

Metaphor and Mathematics
A Thesis Submitted to the
College of Graduate Studies and Research
in Partial Fulfilment of the Requirements
for the degree of Doctor of Philosophy
in the Unit of Interdisciplinary Studies
University of Saskatchewan
Saskatoon
By
Derek Lawrence Lindsay Postnikoff
c Derek Lawrence Lindsay Postnikoff, April 2014. All rights
(cid:13)
reserved.
Permission to Use
Inpresenting thisthesis inpartialfulfilment oftherequirements foraPostgraduatedegree
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it freely available for inspection. I further agree that permission for copying of this thesis in
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or the Dean of the College in which my thesis work was done. It is understood that any
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or part should be addressed to:
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Abstract
Traditionally, mathematics and metaphor have been thought of as disparate: the for-
mer rigorous, objective, universal, eternal, and fundamental; the latter imprecise, deriva-
tive, nearly — if not patently — false, and therefore of merely aesthetic value, at best. A
growing amount of contemporary scholarship argues that both of these characterizations are
flawed. This dissertation shows that there are important connexions between mathematics
and metaphor that benefit our understanding of both. A historically structured overview of
traditional theories of metaphor reveals it to be a notion that is complicated, controversial,
and inadequately understood; this motivates a non-traditional approach. Paradigmatically
shifting thelocusofmetaphor fromthelinguistic totheconceptual —asGeorgeLakoff, Mark
Johnson, and many other contemporary metaphor scholars do — overcomes problems plagu-
ing traditional theories and promisingly advances our understanding of both metaphor and
of concepts. It is argued that conceptual metaphor plays a key role in explaining how mathe-
matics is grounded, and simultaneously provides a mechanism for reconciling and integrating
the strengths of traditional theories of mathematics usually understood as mutually incom-
patible. Conversely, it is shown that metaphor can be usefully and consistently understood
in terms of mathematics. However, instead of developing a rigorous mathematical model
of metaphor, the unorthodox approach of applying mathematical concepts metaphorically is
defended.
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Acknowledgements
The support I have received from my community of colleagues, family, and friends over
the course of this decade-long project has seemed non-compact — that is, without limit.
Many individuals deserve recognition for a variety of important contributions. My supervi-
sor, Dr. Sarah Hoffman, has been unwaveringly encouraging; this document owes much to
her scholarly guidance and her generosity. I also owe a debt of gratitude to my advisory com-
mittee: Dr. Eric Dayton, Dr. Karl Pfeifer, Dr. Murray Bremner, Dr. Florence Glanfield, and
Dr. Emer O’Hagan. They provided many insightful suggestions, willingly took on the many
responsibilities associated with advising an interdisciplinary student, and were patiently sup-
portive when personal crises slowed my progress. Thanks also to my external examiner, Dr.
Brent Davis, for his useful and flattering comments.
In addition to my committee, I wish to thank three individuals for contributing to the
content ofmy dissertation: Dr. JohnPorter (who provided invaluable assistance withAncient
Greek); Dr. Ulrich Teucher (who introduced me to the work of Cornelia Mu¨ller); and Ian
MacDonald (who suggested I read Jesse Prinz). I would also like to thank my peers and col-
leagues in the Philosophy Department for valuable discussions and camaraderie, particularly
Aaron, both Wills, Leslie, Sean, Scotia, Kevin, Mark, Diana, Jeff, Ian, and Dexter.
A very sincere thank you to the support staff of the university — custodians, secretaries,
heating-plant employees, librarians, and so on — whose efforts to maintain our institution go
unnoticed far too often. Your work is appreciated! In particular, thanks to Della Nykyforak,
Debbie Parker, Susan Mason, and Alison Kraft for their immense help in navigating the
administrative aspects of my program.
This research was funded in part by various scholarships awarded by the University of
Saskatchewan; byemployment opportunitiesgivenbythePhilosophyDepartment, St. Peter’s
College, the Mathematics and Statistics Help Centre (Holly Fraser), The Ring Lord (Jon and
BerniceDaniels), andDABWelding (DaveBodnarchuk); andby subsidies fromthetaxpayers
of Canada. I am supremely grateful for these financial contributions.
Finally, this document would not exist without the emotional, financial, and intellectual
support of my family: thank you all!
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To Thora, point of my compass, patiently steadfast and
wholeheartedly supportive while I ran in academic circles.
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Contents
Permission to Use i
Abstract ii
Acknowledgements iii
Contents v
List of Abbreviations vi
1 Introduction 1
2 A Philosophical History of Metaphor 7
2.1 Ancient Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Early Modern and Modern Philosophy . . . . . . . . . . . . . . . . . . . . . 23
2.3 Twentieth Century . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Conceptual Metaphor 56
3.1 Theories of Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Conceptual Metaphor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Criticisms and Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4 Mathematics is Metaphorical 120
4.1 Traditional Theories of Mathematics . . . . . . . . . . . . . . . . . . . . . . 122
4.2 Conceptual Metaphor Theory and Embodied Mathematics . . . . . . . . . . 133
4.3 Yablo’s Mathematical Figuralism . . . . . . . . . . . . . . . . . . . . . . . . 157
4.4 Conceptual Metaphor Theory and the Philosophy of Mathematics . . . . . . 164
5 Metaphor is Mathematical 183
5.1 Computational Linguistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
5.2 Mathematical Metametaphors . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6 Conclusion 204
References 208
v
List of Abbreviations
BCE Before the Common Era
CE Common Era
CMT Conceptual Metaphor Theory
OED Oxford English Dictionary
UP University Press
vi
Chapter 1
Introduction
Mathematics is the classification and study of all possible patterns.
Pattern is here used...in a very wide sense, to cover almost
any kind of regularity that can be recognized by the mind.1
— W.W. Sawyer
What cognitive capabilities underlie our fundamental human achievements?
Although a complete answer remains elusive, one basic component is a special
kind of symbolic activity — the ability to pick out patterns, to identify
recurrences of these patterns despite variation in the elements that compose
them, to form concepts that abstract and reify these patterns, and to express
these concepts in language. Analogy, in its most general sense, is this ability to
think about relational patterns. As Douglas Hofstadter argues...
analogy lies at the core of human cognition.2
— Holyoak, Gentner, and Kokinov
In the Western tradition, mathematics and metaphor have long been thought of as in-
habiting opposite ends of the intellectual spectrum. Mathematics is generally considered the
paradigm case of rigor and objectivity. Mathematical theorems supported by valid proofs
seem to express truths that are exact, eternal, and evident. The precision and certainty
afforded by mathematical techniques underpin the extraordinary success of the quantitative
sciences. Even though many people find the practice of mathematics difficult and unenjoy-
able, most nonetheless acknowledge the contribution mathematics makes to the flourishing
of our species. Metaphor is generally not regarded so highly. At best, tradition regards it as
a convenient linguistic device for communicating subjective experiences (as in poetry) and as
a temporary measure when precise, literal language does not yet exist. At worst, metaphor
1W.W. Sawyer, Prelude to Mathematics (Harmondsworth, Middlesex: Penguin, 1955), 12; emphasis his.
2KeithHolyoak,DedreGentner,andBoichoKokinov,“Introduction: ThePlaceofAnalogyinCognition,”
The Analogical Mind: Perspectives from Cognitive Science,Ed. Gentner,Holyoak,andKokinov(Cambridge:
MIT Press, 2001), 2.
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is considered an unnecessary and avoidable figurative impediment to clear communication,
a mere step away from outright prevarication. Thus, metaphor has often been conceived as
antithetical to but also disparate from mathematics.
There are a variety of reasons to question this traditional view. For one, mathematics
and metaphor do not seem as disparate as suggested above. In my decades of experience
as a mathematics student, educator, and researcher, I have observed the use of metaphor
and analogy at every level of mathematical practice, from young novices learning to add to
conference presentations by Fields Medallists. Three specific examples will help substanti-
ate this observation. First, when children are first learning addition their teachers draw a
comparison between the addition of numbers and the act of combining collections of physical
objects. Second, in order to help students understand the somewhat difficult notion of a
mathematical function, teachers and professors often describe functions metaphorically as
machines that take in numerical inputs and perform various manipulations and operations
upon them to yield numerical outputs. Third, some mathematicians (myself included) use
analogy in trying to understand multidimensional spaces by way of their experiences of the
three spatial dimensions they inhabit. For example, the vertices of the two-dimensional ge-
ometric object known as the Penrose tiling are sometimes understood as projections of a
five-dimensional cubic lattice in a similar way to how three-dimensional objects project two-
dimensional shadows.3 These examples are not isolated instances; many more mathematical
metaphors reveal themselves once one starts looking out for them.
Anotherquestionableaspectofthetraditionalviewisitssuspicionofandhostilitytowards
metaphor. Historically, several authors have spoken out against this traditional dismissive-
ness, claiming that metaphor hasbeen significantly undervalued andmischaracterized. These
scholars argue that metaphor should be embraced as a fundamental and pervasive part of hu-
man experience, not seen merely as an obfuscating derivative of literal language that should
be avoided whenever possible. The advances and increased interest in language scholarship
that occurred over the past century have correspondingly generated a substantial literature
on metaphor, much of which views metaphor in a more positive light than earlier works.
3N.G. de Bruijn, “Algebraic theory of Penrose’s non-periodic tilings of the plane I,” Indagationes Mathe-
maticae 84.1 (1981): 40.
2
Some contemporary authors even claim that metaphor is conceptual in nature and therefore
frequently precedes literal language rather than being derived from it. If metaphor is a basic
cognitive mechanism that helps structure our conceptual system then it is plausible that
metaphor could play a constitutive role in our understanding of mathematics. Even if one
rejects the idea that metaphor is conceptual, it seems that metaphor is a more complex,
legitimate, and widespread phenomenon than was previously suspected.
While scholarship of the last century generally improved metaphor’s reputation, it simul-
taneously brought mathematics down to earth a little. Mathematics was revered since the
time of the Ancient Greeks as the closest we flawed, mortal humans could come to knowing
objective Truth. The dramatic mathematical advances of the nineteenth century revealed
that mathematics is more varied, expansive, and complicated than was previously thought.
Inparticular, the discovery ofnon-Euclidean geometries calledinto question the long-heldbe-
lief that mathematical axioms are uniquely self-evident. These developments, among others,
brought about a foundational crisis in mathematics at the end of the 1800s that prompted
philosophers and mathematicians to search for a way to ground and unify an increasingly
abstract and voluminous discipline. While a variety of popular foundational theories have
been defended, all of them are controversial in some way and there is no consensus; thus, over
a century later, the foundational crisis still lacks definitive resolution. What seems clear is
that mathematical results such as the non-Euclidean geometries and G¨odel’s incompleteness
theorems have shown that mathematics is less certain and absolute than was once believed.
If mathematics and metaphor are not antipodal then the question remains of how they
are related to each other. This dissertation argues that important connexions exist between
mathematics and metaphor, and that exploring and developing these connexions improves
our understanding of both topics. On the one hand, metaphor and analogy seem to comprise
a fundamental mode of human reasoning. This is particularly evident when one considers
that we frequently learn by understanding the unknown in terms of the known. Insofar
as metaphorical reasoning is basic and ubiquitous, one expects it would play some role in
mathematics. Conversely, the modeling capabilities of mathematics are justifiably renowned;
it thus seems reasonable that one could, to at least some extent, mathematically model
metaphor. Both of these approaches are considered below, though more emphasis is placed
3