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Where Mathematics Comes From


WMCF definition of mathematics


Human cognition and mathematics


WMCF builds on earlier books by Lakoff (1987) and Lakoff and Johnson (1980, 1999), whose 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.




WMCF emphatically rejects the Platonistic philosophy of mathematics. They emphasize that all we know and can ever know is human mathematics, the mathematics arising from the human intellect. Whether a transcendent mathematics, independent of human thought, can be said to exist is probably an unanswerable question, and perhaps even a meaningless one.



Lakoff and Núñez cite Saunders MacLane (the inventor, with Samuel Eilenberg, of category theory) in support of their position. MacLane (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. See From Action to Mathematics per Mac Lane.


Examples of mathematical metaphors



Example of metaphorical ambiguity

WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set A={{empty},{empty,{empty}}}. Then recall two bits of standard elementary set theory:

  1. The recursive construction of the ordinal natural numbers, whereby 0 is empty, and n is n-1 cup {n-1}.
  2. The ordered pair (a,b), defined as {{a},{a,b}}.



  • 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 the authors describe, then dismiss as an intellectual myth:

  • 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.


Critical response


Critiques of WMCF include the humorous:


and the physically informed:


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.



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. 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.

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." 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.

Summing up


  • 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;



See also


  • 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.

External links

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