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.

Lakoff and Núñez start by reviewing the psychological literature, concluding that human beings appear to have an innate ability, called subitizing, 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.

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.

Lakoff and Núñez cite Saunders Mac Lane (the inventor, with Samuel Eilenberg, of category theory) in support of their position.

For example, Frege, Principia Mathematica, and New Foundations (a body of axiomatic set theory begun by Quine in 1937) define cardinals and ordinals as equivalence classes under the relations of equinumerosity and similarity, so that this conundrum does not arise.

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.

When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter.

[3]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.

[4] 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.

[6] 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.

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.

[8] Second, the mathematics WMCF is concerned with is "almost entirely... standard utterances in textbooks and curricula",[8] which is the most-well established body of knowledge.

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.