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Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor. George P. Lakoff (, born 1941) is a professor of linguistics (in particular, cognitive linguistics) at the University of California, Berkeley, where he has taught since 1972. ...
In linguistics and cognitive science, cognitive linguistics (CL) refers to the currently dominant school of linguistics that views the important essence of language as innately based in evolutionarily-developed and speciated faculties, and seeks explanations that advance or fit well into the current understandings of the human mind. ...
Rafael E. Núñez is a professor of Cognitive science at the University of California, San Diego and is well known for promoting the idea of embodied cognition. ...
A psychologist is a scientist who studies psychology, the systematic investigation of the human behavior and mental processes. ...
The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. ...
Embodied philosophy (also known as the embodied mind thesis, embodied cognition or the embodied cognition thesis) usually refers to a set of beliefs promoted by George Lakoff and his various co-authors (including Mark Johnson, Mark Turner, and Rafael E. Núñez), which suggest that the mind can only be...
Conceptual metaphor: In cognitive linguistics, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain; for example, using one persons life experience to understand a different persons experience. ...
WMCF definition of mathematics Mathematics makes up that part of the human conceptual system that is special in the following ways: - "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, [ranging from] sports, to building, business, technology, and science." (WMCF, pp. 50, 377)
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 metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path. In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
WMCF builds on the important earlier books of Lakoff (1987) and Lakoff and Johnson (1980, 1999). While these books are good academic writings, their probing analyses of metaphor, Image Schemata, and other concepts from second-generation cognitive science are not for the faint of heart. Some of the riches of these earlier books, such as the interesting technical ideas in Lakoff (1987), are absent from WMCF. In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
Rendering of human brain based on MRI data Cognitive science is usually defined as the scientific study either of mind or of intelligence (e. ...
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 which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to them. The idea analysis is distinct from mathematics, and cannot be performed by mathematicians unless they are well trained in cognitive science. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ...
Lakoff and Núñez start by reviewing the psychological literature, concluding that humans appear to have an innate ability, called subitizing, to count, add, and subtract up to about 4 or 5. They describe experiments with infants, conducted since 1980 or so. 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. Introduction Kaufman et al. ...
The authors argue that mathematics goes far beyond this very elementary level thanks to a large number of metaphorical constructions. For example, they argue that 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. In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
Pythagoreanism is a term used for the esoteric and metaphysical beliefs held by Pythagoras and his followers, the Pythagoreans, who were much influenced by mathematics and probably a main inspirational source for Plato and platonism. ...
Much of WMCF wrestles with the important concept of infinity and of limit processes, seeking to explain how finite humans living in a finite world could eventually conceive of the actual infinite. Consequently, much of WMCF is, in effect, a study of the epistemological foundations of the calculus. Lakoff and Núñez conclude that while the potential infinite is not metaphorical, the actual infinite is. Moreover, all manifestations of actual infinity are instances of what they call the "Basic Metaphor of Infinity." The word infinity comes from the Latin infinitas or unboundedness. It refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. ...
Wikipedia does not yet have an article with this exact name. ...
It has been suggested that this article or section be merged with Knowledge. ...
Calculus is a central branch of mathematics, developed from algebra and geometry. ...
Wikipedia does not yet have an article with this exact name. ...
WMCF emphatically reject the Platonistic philosophy of mathematics. They emphasize that all we know and can ever know is human mathematics, the mathematics arising from our brains. Whether a transcendent mathematics, independent of human thought, can be said to exist is an unanswerable and perhaps meaningless question. Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ...
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WMCF (p. 81) likewise criticizes the emphasis mathematicians place on the concept of closure. Lakoff and Núñez argue that the demand for closure is an artifact of the human mind's ability to relate fundamentally different concepts via metaphor. In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
Educators have found WMCF interesting for its suggestions about how mathematics is learned, and about why students find some elementary concepts more difficult than others.
Examples of mathematical metaphors Conceptual metaphors described in WMCF, in addition to the Basic Metaphor of Infinity, include: Conceptual metaphor: In cognitive linguistics, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain; for example, using one persons life experience to understand a different persons experience. ...
- Arithmetic 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 axiomatic algebraic structure;
- Functions are sets of ordered pairs, curves in the Cartesian plane;
- Geometric figures are objects in space;
- Logical independence is geometric orthogonality;
- Numbers 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 of Algebra. In rhetoric and cognitive linguistics, metonymy (in Greek μετά (meta) = after/later and όνομα (onoma) = name) (pronounced //) is the use of a single characteristic to identify a more complex entity. ...
A technical example WMCF (p. 151) includes the following example. Take the set A={{∅},{∅,{∅}}}. Then recall two bits of standard elementary set theory: In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In abstract mathematics, naive set theory1 was the first development of set theory, which was later to be framed more carefully as axiomatic set theory. ...
- The recursive construction of the ordinal natural numbers, whereby 0 is ∅ is 0, and is n is n-1∪{n-1}.
- The ordered pair (a,b), defined as {{a},{a,b}}.
By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the "unordered pair" {1,2} are fully distinct concepts. Lakoff and Johnson term this situation "metaphorically ambiguous." This very elementary example calls into question any Platonistic foundations for mathematics. See: Recursion Recursively enumerable language Recursively enumerable set Recursive filter Recursive function Recursive set Primitive recursive function This is a disambiguation page — a list of pages that otherwise might share the same title. ...
Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
An ordered pair is a collection of two objects such that one can be distinguished as the first element and the other as the second element. ...
While (1) and (2) above are admittedly canonical, especially within the prevailing set theory consensus 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, Principia Mathematica, and New Foundations (a body of axiomatic set theory begun by Quine in 1937) define cardinal and ordinal numbers as equivalence classes under the relations of equinumerosity 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 complicated variant set-theoretic definition of the ordered pair, or simply taking such pairs as primitive. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. ...
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910-1913. ...
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. ...
This article or section is in need of attention from an expert on the subject. ...
In computing, a quine is a program (a form of metaprogram) that produces its complete source code as its only output. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
In mathematics, a finitary relation is defined by one of the formal definitions given below. ...
Two sets A and B are said to be equinumerous if they have the same cardinality, i. ...
Several equivalence relations in mathematics are called similarity. ...
The Romance of Mathematics The "Romance of Mathematics" is WMCF's light-hearted term for a perennial philosophical viewpoint about mathematics the authors describe, then dismiss as an intellectual myth: - Mathematics is transcendent, namely it exists independently of human beings, and structures our actual physical universe 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 is the gateway to a realm of transcendent truth.
- Reasoning is logic, 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 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. Hence the main value of WMCF so far may be a critical one: its critique of Platonism, and its having exposed and demolished the Romance of Mathematics. The deepest visible-light image of the cosmos, the Hubble Ultra Deep Field. ...
In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
Hondas intelligent humanoid robot AI redirects here. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ...
Critical response Working mathematicians have not infrequently resisted the approach and conclusions of Lakoff and Núñez. Reviews by mathematicians of WMCF 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 objective meanings. For example, Fermat's last theorem means exactly what it meant when Fermat initially proposed it 1664. Reviewers have also pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person. The metaphor and the conceptual strategy are not the same as the formal definition which mathematicians employ. Pierre de Fermat Problem II.8 in the Arithmetica of Diophantus, annotated with Fermats comment which became Fermats Last Theorem (edition of 1670). ...
Pierre de Fermat Pierre de Fermat (August 17, 1601 – January 12, 1665) was a French lawyer at the Parliament of Toulouse and a mathematician who is given credit for the development of modern calculus. ...
In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
A definition delimits or describes the meaning of a concept or term by stating the essential properties of the entities or objects denoted by that concept or term. ...
Mathematicians have also complained that Lakoff and Núñez have misunderstood some basic mathematical notions. The authors reply that the errors found in earlier printings of WMCF are now corrected.
Neither Lakoff nor Núñez is a trained mathematician. Lakoff made his reputation by linking linguistics to cognitive science and the analysis of metaphor. Núñez, educated in Switzerland, is a product of Jean Piaget's school of cognitive psychology as a basis for logic and mathematics. Núñez has thought much about the foundations of real analysis, the real and complex numbers, 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, Boolean algebra and the Zermelo-Fraenkel axioms, even lingering a bit over group theory. But neither author is well-trained in logic (there is no index entry for "quantifier" or "quantification"), the philosophy of set theory, the axiomatic method, metamathematics, and model theory. Nor does WMCF say enough about the derivation of number systems (the Peano axioms go unmentioned), abstract algebra, equivalence and order relations, mereology, topology, and geometry. Linguistics is the scientific study of human language, and someone who engages in this study is called a linguist. ...
Rendering of human brain based on MRI data Cognitive science is usually defined as the scientific study either of mind or of intelligence (e. ...
In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
Jean Piaget (August 9, 1896 – September 16, 1980) was a Swiss natural scientist and developmental psychologist, well known for his work studying children and his theory of cognitive development. ...
Cognitive Psychology is the school of psychology that examines internal mental processes such as problem solving, memory, and language. ...
Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...
Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Wikibooks has more about Boolean logic, under the somewhat misleading title Boolean Algebra For a basic intro to sets, Boolean operations, Venn diagrams, truth tables, and Boolean applications, see Boolean logic. ...
The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...
In language and logic, quantification is a construct that specifies the extent of validity of a predicate, that is the extent to which a predicate holds over a range of things. ...
In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...
In mathematics, a number system is a set of numbers, or number-like objects, together with one or more operations, such as addition or multiplication. ...
In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...
In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
Mereology is a collection of axiomatic formal systems dealing with parts and their respective wholes. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
The authors tend to dismiss the negative comments by mathematicians on the grounds that the latter do not take the view of cognitive science. In a book whose subject matter is mathematics and philosophy as well as cognitive science, this strategy cuts both ways. What mathematics is about and where (in terms of cognitive science) it comes strike many as distinct questions.
WMCF would be a major addition to the philosophy of mathematics and the theory of abstract objects, were it not that its authors say little about those subjects and cite little of the relevant literatures. They do not seem all that familiar with Davis and Hersh (1981), even though WMCF warmly acknowledges Reuben Hersh's support. Lakoff and Núñez do invoke the authority of Saunders MacLane (1986) (the inventor, with Samuel Eilenberg, of category theory) in support of their position. For example, MacLane includes a remarkable table relating various mathematical concepts to ordinary human activities, mostly interactions with the physical world. Reuben Hersh (December 9, 1927 - ) is an American mathematician, now an emeritus professor of the University of New Mexico. ...
Saunders Mac Lane (born 4 August 1909) is a US mathematician. ...
Samuel Eilenberg (September 30, 1913-January 30, 1998) was a Polish mathematician. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
Lakoff and Núñez do not appreciate the extent to which intuitionists and constructivists have anticipated their attack on the Romance of (Platonic) Mathematics. Brouwer, the founder of the intuitionist/constructivist point of view, wrote "Mathematics is a free construction of the human mind." Hence at least one person writing before Lakoff and Núñez were born concluded that mathematics emerged to serve human purposes and has no existence apart from this fact. Finally, WMCF is silent about a finding that strongly supports their argument: Godel's monumental 1931 result that, even for a realm as simple as the arithmetic of the natural numbers, truth necessarily outruns proof. In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ...
In education, constructivism is a learning theory which holds that knowledge is not transmitted unchanged from teacher to student, but instead that learning is an active process of learning. ...
Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, measure theory and complex analysis. ...
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. ...
In education, constructivism is a learning theory which holds that knowledge is not transmitted unchanged from teacher to student, but instead that learning is an active process of learning. ...
Kurt Gödel Kurt Gödel [ kurt gøːdl ], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics, whose biography lists quite a few nations, although he is usually associated with Austria. ...
In mathematical logic, Gödels incompleteness theorems are two theorems about the limits of formal systems, proved by Kurt Gödel in 1931. ...
In mathematics, the Peano axioms (or Peano postulates) are a set of second-order axioms proposed by Giuseppe Peano which determine the theory of arithmetic. ...
La Vérité by the French painter Jules Joseph Lefebvre Common dictionary definitions of truth mention some form of accord with fact or reality. ...
In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
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 metaphors is neurologically based, and enables humans to reason about one domain using the language and concepts of another domain. Conceptual metaphor 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;
- Symbolic, thereby enormously facilitating precise calculation;
- Not transcendent, but the result of human evolution and culture, to which it owes its effectiveness. The connection between mathematical ideas and our experience of the world occurs within human minds;
- 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 systems, 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's theory of abstract objects. Lakoff and Johnson (1999) fruitfully employ the cognitive approach to rethink a good deal of the philosophy of mind, epistemology, metaphysics, and the history of ideas. Conceptual metaphor: In cognitive linguistics, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain; for example, using one persons life experience to understand a different persons experience. ...
Conceptual metaphor: In cognitive linguistics, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain; for example, using one persons life experience to understand a different persons experience. ...
A hypothetical phylogenetic tree of all extant organisms, based on 16S rRNA gene sequence data, showing the evolutionary history of the three domains of life, bacteria, archaea and eukaryotes. ...
The word culture, from the Latin colo, -ere, with its root meaning to cultivate, generally refers to patterns of human activity and the symbolic structures that give such activity significance. ...
Edward N. Zalta is a Senior Research Scholar at the Center for the Study of Language and Information. ...
A Phrenological mapping of the brain. ...
It has been suggested that this article or section be merged with Knowledge. ...
Plato and Aristotle, by Raphael (Stanza della Segnatura, Rome). ...
The history of ideas is a field of research in history and in related fields dealing with the expression, preservation, and change of human ideas over time. ...
Mathematics has grown into an extremely powerful toolbox for the mind, one whose potential applications extend well beyond those traditional bastions of mathematical application, science and technology. For example, logic and abstract algebra have much to offer to the social sciences and humanities. But communicating these riches to the wider community of nonmathematicians has proved difficult, and the problem is worsening. Even formal systems as basic as first order logic and axiomatic set theory are nowadays learned only by the more technical philosophy majors, and by a small fraction of mathematics students. Hence only a few specialists learn any mathematics beyond calculus, applied statistics, differential equations, and a bit of linear algebra. Just how many persons with a university education know what an equivalence class, partial order, or morphism are? What it means for a collection of axioms to have a model? If the cognitive approach to mathematics suggests improvements to the basic mathematical toolbox and better ways to communicate that toolbox to nonspecialists, it will move humanity closer to the fulfillment of Leibniz's great dream of a universal symbolistic. Logic, from Classical Greek λόγος (logos), originally meaning the word, or what is spoken, (but coming to mean thought or reason) is most often said to be the study of criteria for the evaluation of arguments, although the exact definition of logic is a matter of controversy among philosophers. ...
Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ...
First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...
This article or section is in need of attention from an expert on the subject. ...
Calculus is a central branch of mathematics, developed from algebra and geometry. ...
A graph of a bell curve in a normal distribution showing statistics used in educational assessment, comparing various grading methods. ...
Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...
Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...
In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...
In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...
In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...
In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. ...
Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...
Characteristica Universalis from Latin is commonly interpreted as Universal Character in English. ...
See also Rendering of human brain based on MRI data Cognitive science is usually defined as the scientific study either of mind or of intelligence (e. ...
The cognitive science of mathematics is the study of mathematical ideas using the techniques of cognitive science. ...
To meet Wikipedias quality standards, this article or section may require cleanup. ...
Embodied philosophy (also known as the embodied mind thesis, embodied cognition or the embodied cognition thesis) usually refers to a set of beliefs promoted by George Lakoff and his various co-authors (including Mark Johnson, Mark Turner, and Rafael E. Núñez), which suggest that the mind can only be...
In language, a metaphor (from the Greek: metapherin) is a rhetorical trope defined as a direct comparison between two or more seemingly unrelated subjects. ...
Conceptual metaphor: In cognitive linguistics, metaphor is defined as understanding one conceptual domain in terms of another conceptual domain; for example, using one persons life experience to understand a different persons experience. ...
The Unreasonable Effectiveness of Mathematics in the Natural Sciences, published by physicist Eugene Wigner in 1960, argues that the capacity of mathematics to successfully predict events in physics cannot be a coincidence, but must reflect some larger or deeper or simpler truth in both. ...
Foundations of mathematics is a term sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
References - Davis, Philip J., and Reuben Hersh, 1999 (1981). The Mathematical Experience. Mariner Books. First published by Houghton Mifflin.
- George Lakoff, 1987. Women, Fire and Dangerous Things. Univ. of Chicago Press.
- ------ and Mark Johnson, 1999. Philosophy in the Flesh. Basic Books.
- ------ and Rafael Núñez, 2000, Where Mathematics Comes From. Basic Books.
- John Randolph Lucas, 2000. The Conceptual Roots of Mathematics. Routledge.
- Saunders Mac Lane, 1986. Mathematics: Form and Function. Springer Verlag.
Reuben Hersh (December 9, 1927 - ) is an American mathematician, now an emeritus professor of the University of New Mexico. ...
George P. Lakoff (, born 1941) is a professor of linguistics (in particular, cognitive linguistics) at the University of California, Berkeley, where he has taught since 1972. ...
Mark Johnson may refer to: Mark Johnson (professor), philosophy professor Mark Johnson (footballer) (born 1978), Australian rules footballer Mark Johnson (film producer) Mark Johnson (umpire), baseball umpire Mark Johnson (hockey player) (born 1957) Mark Johnson (rugby) Mark Johnson (baseball analyst) Mark Johnson (musician) Mark Johnson (football club director), director of...
Rafael Núñez (politician) Rafael E. Núñez - cognitive scientist Categories: Disambiguation ...
John Randolph Lucas (born 18 June 1929) is a British philosopher. ...
Saunders Mac Lane (4 August 1909, Taftville, Connecticut - 14 April 2005, San Francisco) was an American mathematician who cofounded category theory with Samuel Eilenberg. ...
External links - WMCF web site.
- Auslander, Joseph, "Review of WMCF." American Scientist
- Gold, Bonnie, MAA Review.
- Lakoff's response to Gold's MAA review.
- List of reviews.
- Lakoff and Núñez: Response to the mathematical community.
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