''Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being'' (hereinafter ''WMCF'') is a book by
George Lakoff
George Philip Lakoff (; born May 24, 1941) is an American cognitive linguistics, cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain comple ...
, a
cognitive linguist, and
Rafael E. Núñez Rafael E. Núñez is a professor of cognitive science at the University of California, San Diego and a proponent of embodied cognition. He co-authored '' Where Mathematics Comes From'' with George Lakoff.
External links
Academic home page* Rafa ...
, a
psychologist
A psychologist is a professional who practices psychology and studies mental states, perceptual, cognitive, emotional, and social processes and behavior. Their work often involves the experimentation, observation, and interpretation of how indi ...
. Published in 2000, ''WMCF'' seeks to found a
cognitive science of mathematics
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes ...
, a theory of
embodied mathematics based on
conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...
.
''WMCF'' definition of mathematics
Mathematics makes up that part of the human conceptual system that is special in the following way:
:It is precise, consistent, stable across time and human communities, symbolizable, calculable, generalizable, universally available, consistent within each of its subject matters, and effective as a general tool for description, explanation, and prediction in a vast number of everyday activities,
anging fromsports, to building, business, technology, and science. - ''WMCF'', pp. 50, 377
Nikolay Lobachevsky said "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." A common type of
conceptual blending process would seem to apply to the entire mathematical procession.
Human cognition and mathematics
Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
ically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path.
''WMCF'' builds on earlier books by Lakoff (1987) and Lakoff and Johnson (1980, 1999), which analyze such concepts of
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
and image schemata from second-generation
cognitive science. Some of the concepts in these earlier books, such as the interesting technical ideas in Lakoff (1987), are absent from ''WMCF''.
Lakoff and Núñez hold that mathematics results from the human cognitive apparatus and must therefore be understood in cognitive terms. ''WMCF'' advocates (and includes some examples of) a ''cognitive idea analysis'' of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to them. A standard mathematical education does not develop such idea analysis techniques because it does not pursue considerations of A) what structures of the mind allow it to do mathematics or B) the
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
.
Lakoff and Núñez start by reviewing the psychological literature, concluding that human beings appear to have an innate ability, called
subitizing
Subitizing is the rapid, accurate, and confident judgments of numbers performed for small numbers of items. The term was coined in 1949 by E. L. Kaufman et al., and is derived from the Latin adjective '' subitus'' (meaning "sudden") and captures ...
, to count, add, and subtract up to about 4 or 5. They document this conclusion by reviewing the literature, published in recent decades, describing experiments with infant subjects. For example, infants quickly become excited or curious when presented with "impossible" situations, such as having three toys appear when only two were initially present.
The authors argue that mathematics goes far beyond this very elementary level due to a large number of
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
ical constructions. For example, the
Pythagorean position that all is number, and the associated crisis of confidence that came about with the discovery of the irrationality of the
square root of two, arises solely from a metaphorical relation between the length of the diagonal of a square, and the possible numbers of objects.
Much of ''WMCF'' deals with the important concepts of
infinity
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
and of limit processes, seeking to explain how finite humans living in a finite world could ultimately conceive of the
actual infinite
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, ext ...
. Thus much of ''WMCF'' is, in effect, a study of the
epistemological
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
foundations of the
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. Lakoff and Núñez conclude that while the
potential infinite
In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities as given, actual and completed objects. These might include the set of natural numbers, ext ...
is not metaphorical, the actual infinite is. Moreover, they deem all manifestations of actual infinity to be instances of what they call the "Basic Metaphor of Infinity", as represented by the ever-increasing sequence 1, 2, 3, ...
''WMCF'' emphatically rejects the
Platonistic philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
. They emphasize that all we know and can ever know is ''human mathematics'', the mathematics arising from the human intellect. The question of whether there is a "transcendent" mathematics independent of human thought is a meaningless question, like asking if colors are transcendent of human thought—colors are only varying wavelengths of light, it is our interpretation of physical stimuli that make them colors.
''WMCF'' (p. 81) likewise criticizes the emphasis mathematicians place on the concept of
closure. Lakoff and Núñez argue that the expectation of closure is an artifact of the human mind's ability to relate fundamentally different concepts via metaphor.
''WMCF'' concerns itself mainly with proposing and establishing an alternative view of mathematics, one grounding the field in the realities of human biology and experience. It is not a work of technical mathematics or philosophy. Lakoff and Núñez are not the first to argue that conventional approaches to the philosophy of mathematics are flawed. For example, they do not seem all that familiar with the content of Davis and
Hersh Hersh ( yi, הערש) is a given name and surname. People with the name include:
Given name
* Hersh Leib Sigheter
* Hersh Wolch
Surname
* Arek Hersh, German Nazi Holocaust survivor and writer
* Kristin Hersh, American singer-songwriter
* Patricia ...
(1981), even though the book warmly acknowledges Hersh's support.
Lakoff and Núñez cite
Saunders Mac Lane (the inventor, with
Samuel Eilenberg, of
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
) in support of their position. ''
Mathematics, Form and Function
''Mathematics, Form and Function'', a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.
Mathematics and human activities
...
'' (1986), an overview of mathematics intended for philosophers, proposes that mathematical concepts are ultimately grounded in ordinary human activities, mostly interactions with the physical world.
Educators have taken some interest in what ''WMCF'' suggests about how mathematics is learned, and why students find some elementary concepts more difficult than others.
However, even from an educational perspective, WMCF is still problematic. From the conceptual metaphor theory's point of view, metaphors reside in a different realm, the abstract, from that of 'real world', the concrete. In other words, despite their claim of mathematics being human, established mathematical knowledge — which is what we learn in school — is assumed to be and treated as abstract, completely detached from its physical origin. It cannot account for the way learners could access to such knowledge.
WMCF is also criticized for its monist approach. First, it ignores the fact that the sensori-motor experience upon which our linguistic structure — thus, mathematics — is assumed to be based may vary across cultures and situations.
Second, the mathematics WMCF is concerned with is "almost entirely... standard utterances in textbooks and curricula",
which is the most-well established body of knowledge. It is negligent of the dynamic and diverse nature of the history of mathematics.
WMCF's logo-centric approach is another target for critics. While it is predominantly interested in the association between language and mathematics, it does not account for how non-linguistic factors contribute to the emergence of mathematical ideas (e.g. See Radford, 2009; Rotman, 2008).
Examples of mathematical metaphors
Conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...
s described in ''WMCF'', in addition to the Basic Metaphor of Infinity, include:
*
Arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
is motion along a path, object collection/construction;
*Change is motion;
*
Sets are containers, objects;
*
Continuity is gapless;
*Mathematical systems have an "essence," namely their
axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
atic
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
;
*
Functions are sets of
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s, curves in the
Cartesian plane;
*Geometric figures are objects in space;
*
Logical independence In mathematical logic, independence is the unprovability of a sentence from other sentences.
A sentence σ is independent of a given first-order theory ''T'' if ''T'' neither proves nor refutes σ; that is, it is impossible to prove σ from ''T' ...
is geometric
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
ity;
*
Number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s are sets, object collections, physical segments, points on a line;
*Recurrence is circular.
Mathematical reasoning requires
variables ranging over some
universe of discourse, so that we can reason about generalities rather than merely about particulars. ''WMCF'' argues that reasoning with such variables implicitly relies on what it terms the Fundamental
Metonymy
Metonymy () is a figure of speech in which a concept is referred to by the name of something closely associated with that thing or concept.
Etymology
The words ''metonymy'' and ''metonym'' come from grc, μετωνυμία, 'a change of name' ...
of Algebra.
Example of metaphorical ambiguity
''WMCF'' (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set
Then recall two bits of standard terminology from
elementary set theory:
#The
recursive construction of the
ordinal natural numbers, whereby 0 is
, and
is
#The
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
(''a,b''), defined as
By (1), ''A'' is the set . But (1) and (2) together say that ''A'' is also the ordered pair (0,1). Both statements cannot be correct; the
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
(0,1) and the unordered pair are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any
Platonistic foundations for mathematics.
While (1) and (2) above are admittedly canonical, especially within the consensus
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
known as the
Zermelo–Fraenkel axiomatization, ''WMCF'' does not let on that they are but one of several definitions that have been proposed since the dawning of set theory. For example,
Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philo ...
, ''
Principia Mathematica'', and
New Foundations
In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of ''Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundations ...
(a body of
axiomatic set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
begun by
Quine in 1937) define
cardinals and
ordinals as
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es under the
relations
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
of
equinumerosity
In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', ther ...
and
similarity, so that this conundrum does not arise. In Quinian set theory, ''A'' is simply an instance of the number 2. For technical reasons, defining the ordered pair as in (2) above is awkward in Quinian set theory. Two solutions have been proposed:
*A variant set-theoretic definition of the ordered pair more complicated than the usual one;
*Taking ordered pairs as primitive.
The Romance of Mathematics
The "Romance of Mathematics" is ''WMCF''s light-hearted term for a perennial philosophical viewpoint about mathematics which the authors describe and then dismiss as an intellectual myth:
* Mathematics is transcendent, namely it exists independently of human beings, and structures our actual physical
universe
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...
and any possible universe. Mathematics is the language of nature, and is the primary conceptual structure we would have in common with extraterrestrial aliens, if any such there be.
*
Mathematical proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proo ...
is the gateway to a realm of transcendent truth.
*
Reasoning
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, lang ...
is
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, and logic is essentially mathematical. Hence mathematics structures all possible reasoning.
* Because mathematics exists independently of human beings, and reasoning is essentially mathematical, reason itself is disembodied. Therefore,
artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech re ...
is possible, at least in principle.
It is very much an open question whether ''WMCF'' will eventually prove to be the start of a new school in the
philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
. Hence the main value of ''WMCF'' so far may be a critical one: its critique of
Platonism and romanticism in mathematics.
Critical response
Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews of ''WMCF'' by mathematicians in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the ''WMCF''s philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings. For example,
Fermat's Last Theorem means exactly what it meant when
Fermat initially proposed it 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
and the conceptual strategy are not the same as the formal
definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definitio ...
which mathematicians employ. However, ''WMCF'' points out that formal definitions are built using words and symbols that have meaning only in terms of human experience.
Critiques of ''WMCF'' include the humorous:
and the physically informed:
Lakoff made his reputation by linking
linguistics
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
to
cognitive science and the analysis of
metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
. Núñez, educated in
Switzerland
). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Luzern, Neuchâtel ...
, is a product of
Jean Piaget's school of
cognitive psychology
Cognitive psychology is the scientific study of mental processes such as attention, language use, memory, perception, problem solving, creativity, and reasoning.
Cognitive psychology originated in the 1960s in a break from behaviorism, which ...
as a basis for logic and mathematics. Núñez has thought much about the foundations of
real analysis, the
real and
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, and the Basic Metaphor of Infinity. These topics, however, worthy though they be, form part of the superstructure of mathematics. Cognitive science should take more interest in the
foundations of mathematics. And indeed, the authors do pay a fair bit of attention early on to
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
,
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
and the
Zermelo–Fraenkel axioms, even lingering a bit over
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. But neither author is well-trained in
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, the philosophy of set theory, the
axiomatic method
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains ...
,
metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the ter ...
, and
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
. Nor does ''WMCF'' say enough about the derivation of
number systems (the
Peano axioms go unmentioned),
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
,
equivalence
Equivalence or Equivalent may refer to:
Arts and entertainment
*Album-equivalent unit, a measurement unit in the music industry
*Equivalence class (music)
*''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre
*'' Equival ...
and
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
relations,
mereology
In logic, philosophy and related fields, mereology ( (root: , ''mere-'', 'part') and the suffix ''-logy'', 'study, discussion, science') is the study of parts and the wholes they form. Whereas set theory is founded on the membership relation bet ...
,
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, and
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
.
Lakoff and Núñez tend to dismiss the negative opinions mathematicians have expressed about ''WMCF'', because their critics do not appreciate the insights of cognitive science. Lakoff and Núñez maintain that their argument can only be understood using the discoveries of recent decades about the way human brains process language and meaning. They argue that any arguments or criticisms that are not grounded in this understanding cannot address the content of the book.
It has been pointed out that it is not at all clear that ''WMCF'' establishes that the claim "intelligent alien life would have mathematical ability" is a myth. To do this, it would be required to show that intelligence and mathematical ability are separable, and this has not been done. On Earth, intelligence and mathematical ability seem to go hand in hand in all life-forms, as pointed out by
Keith Devlin among others. The authors of ''WMCF'' have not explained how this situation would (or even could) be different anywhere else.
Lakoff and Núñez also appear not to appreciate the extent to which
intuitionists and
constructivists have anticipated their attack on the Romance of (Platonic) Mathematics.
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'.
Brouwer
* Adriaen Brouwer (1605–1638), Flemish painter
* Alexander Brouwer (b. 1989), Dutch beach volleyball player
* Andries Bro ...
, the founder of the
intuitionist/
constructivist point of view, in his dissertation ''On the Foundation of Mathematics'', argued that mathematics was a mental construction, a free creation of the mind and totally independent of logic and language. He goes on to upbraid the formalists for building verbal structures that are studied without intuitive interpretation.
Symbolic language should not be confused with mathematics; it reflects, but does not contain, mathematical reality.
Summing up
''WMCF'' (pp. 378–79) concludes with some key points, a number of which follow. Mathematics arises from our bodies and brains, our everyday experiences, and the concerns of human societies and cultures. It is:
*The result of normal adult cognitive capacities, in particular the capacity for conceptual metaphor, and as such is a human universal. The ability to construct
conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...
s is neurologically based, and enables humans to reason about one domain using the language and concepts of another domain.
Conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...
is both what enabled mathematics to grow out of everyday activities, and what enables mathematics to grow by a continual process of analogy and abstraction;
*
Symbol
A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
ic, thereby enormously facilitating precise calculation;
*Not transcendent, but the result of human
evolution
Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
and
culture
Culture () is an umbrella term which encompasses the social behavior, institutions, and norms found in human societies, as well as the knowledge, beliefs, arts, laws, customs, capabilities, and habits of the individuals in these groups.Tyl ...
, to which it owes its effectiveness. During experience of the world a connection to mathematical ideas is going on within the human mind;
*A system of human concepts making extraordinary use of the ordinary tools of human cognition;
*An open-ended creation of human beings, who remain responsible for maintaining and extending it;
*One of the greatest products of the collective human imagination, and a magnificent example of the beauty, richness, complexity, diversity, and importance of human ideas.
The cognitive approach to
formal system
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system.
A form ...
s, as described and implemented in ''WMCF'', need not be confined to mathematics, but should also prove fruitful when applied to formal logic, and to formal philosophy such as
Edward Zalta
Edward Nouri Zalta (; born March 16, 1952) is an American philosopher who is a senior research scholar at the Center for the Study of Language and Information at Stanford University. He received his BA at Rice University in 1975 and his PhD fro ...
'
theory of abstract objects Lakoff and Johnson (1999) fruitfully employ the cognitive approach to rethink a good deal of the
philosophy of mind
Philosophy of mind is a branch of philosophy that studies the ontology and nature of the mind and its relationship with the body. The mind–body problem is a paradigmatic issue in philosophy of mind, although a number of other issues are addre ...
,
epistemology
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Episte ...
,
metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
, and the
history of ideas.
See also
*
Abstract object
In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
*
Cognitive science
*
Cognitive science of mathematics
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes ...
*
Conceptual metaphor
In cognitive linguistics, conceptual metaphor, or cognitive metaphor, refers to the understanding of one idea, or conceptual domain, in terms of another. An example of this is the understanding of quantity in terms of directionality (e.g. "the pr ...
*
Embodied philosophy
*
Foundations of mathematics
* ''
From Action to Mathematics per Mac Lane''
*
Metaphor
A metaphor is a figure of speech that, for rhetorical effect, directly refers to one thing by mentioning another. It may provide (or obscure) clarity or identify hidden similarities between two different ideas. Metaphors are often compared wit ...
*
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people's ...
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Footnotes
References
* Davis, Philip J., and
Reuben Hersh, 1999 (1981). ''The Mathematical Experience''. Mariner Books. First published by Houghton Mifflin.
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George Lakoff
George Philip Lakoff (; born May 24, 1941) is an American cognitive linguistics, cognitive linguist and philosopher, best known for his thesis that people's lives are significantly influenced by the conceptual metaphors they use to explain comple ...
, 1987. ''Women, Fire and Dangerous Things''. Univ. of Chicago Press.
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Mark Johnson, 1999. ''Philosophy in the Flesh''. Basic Books.
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Rafael Núñez, 2000, ''Where Mathematics Comes From''. Basic Books. {{ISBN, 0-465-03770-4
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John Randolph Lucas, 2000. ''The Conceptual Roots of Mathematics''. Routledge.
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Saunders Mac Lane, 1986. ''Mathematics: Form and Function''. Springer Verlag.
External links
WMCF web site.* Reviews of ''WMCF''.
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Joseph Auslanderin ''American Scientist'';
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Bonnie Gold
Bonnie Gold (born 1948) is an American mathematician, mathematical logician, philosopher of mathematics, and mathematics educator. She is a professor emerita of mathematics at Monmouth University.
Education and career
Gold completed her Ph.D. i ...
''MAA Reviews'' 2001*
to Gold's MAA review.
Books about philosophy of mathematics
Infinity
Linguistics books
2000 non-fiction books
Mathematics books
Books about metaphors
Cognitive science literature