Where Mathematics Comes From

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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.

Contents

[edit] 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, [ranging from] sports, to building, business, technology, and science." (WMCF, pp. 50, 377)

[edit] 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.

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.

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. Idea analysis is distinct from mathematics and cannot be performed by mathematicians unless they are trained in cognitive science.

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.

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.

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."

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.

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.

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.

[edit] Examples of mathematical metaphors

Conceptual metaphors described in WMCF, in addition to the Basic Metaphor of Infinity, include:

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.

[edit] A technical example

WMCF (p. 151) includes the following example. Take the set A={{∅},{∅,{∅}}}. Then recall two bits of standard elementary set theory:

  1. The recursive construction of the ordinal natural numbers, whereby 0 is ∅ is 0, and is n is n-1∪{n-1}.
  2. 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.

While (1) and (2) above are admittedly canonical, especially within the consensus set theory 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.

[edit] 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 in mathematics, and the Romance of Mathematics.

It can be argued that only the first of the above quotes really describes Platonism. The second is rather vague, the third is obviously true (logic is a subset of mathematics), while the fourth is not necessarily associated with Platonism. For example, the Platonists Kurt Godel and Roger Penrose deny that artificial intelligence is possible.[citation needed]

[edit] Critical response

Many working mathematicians resist 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 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 and the conceptual strategy are not the same as the formal definition 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:

"It’s difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I’d sure like to see it." Joseph Auslander

and the physically informed:

"But their analysis leaves at least a couple of questions insufficiently answered. For one thing, the authors ignore the fact that brains not only observe nature, but also are part of nature. Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known. It's something else for that math to tell of phenomena never previously suspected. When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac's math told him they must exist. If math is a human invention, nature seems to know what was going to be invented." (Tom Siegfried, The Dallas Morning News, 3/5/2001)

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.

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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.

The authors tend to dismiss the negative comments by mathematicians on the grounds that the latter do not take the view of cognitive science. Their position is that what they are saying can only be understood using the prerequesite insights that have emerged in 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 do not actually 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.[citation needed] The authors of WMCF have not explained how this situation would (or even could) be different anywhere else. From this point of view, whatever one's views on Platonism (right, wrong, meaningless), the 'invention' of mathematical concepts such as number would be impossible since they are hard-wired into our brains from the moment we are born. Also, the word "invention" insinuates that things could somehow be different, so that we could have invented a number theory where 1+1=3, or prime decomposition is false. However, any such number theory immediately falls apart in the face of simple reasoning. In reality, 1+1=3 is obviously false, and Euclid and the ancient Indians stumbled upon prime decomposition rather than inventing it.

Another criticism is that WMCF does not explain where arithmetic comes from (if that is even possible or makes sense). Rather, it merely concluded that humans possess innate arithmetical ability. Some argue that WMCF is entirely consistent with the Platonic philosophy which it rejects.

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. Some point out that 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 (1981), even though WMCF warmly acknowledges Reuben Hersh's support. Lakoff and Núñez 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.[citation needed]

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, the founder of the intuitionist/constructivist point of view, wrote "Mathematics is a free construction of the human mind."[citation needed] 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.

[edit] 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.

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 a 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.

[edit] See also

[edit] 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. ISBN 0465037704
  • John Randolph Lucas, 2000. The Conceptual Roots of Mathematics. Routledge.
  • Saunders Mac Lane, 1986. Mathematics: Form and Function. Springer Verlag.

[edit] External links

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