Editing Where Mathematics Comes From (section)

== Human cognition and mathematics ==
[[Image:Complex conjugate picture.svg|right|thumb|The complex plane: a visual metaphor of the abstract idea of a [[complex number]], which allows operations on complex numbers to be visualized as simple motions through ordinary space]]

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]]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]] 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]] 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]].

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. 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]]ical constructions. For example, the [[Pythagoreanism|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]] and of limit processes, seeking to explain how finite humans living in a finite world could ultimately conceive of the [[actual infinite]]. Thus much of ''WMCF'' is, in effect, a study of the [[epistemology|epistemological]] foundations of the [[calculus]]. Lakoff and Núñez conclude that while the [[potential infinite]] 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 [[Platonism|Platonistic]] [[philosophy of mathematics]]. 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.&nbsp;81) likewise criticizes the emphasis mathematicians place on the concept of [[closure (mathematics)|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 [[Reuben Hersh|Hersh]] (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]]) in support of their position. ''[[Mathematics, Form and Function]]'' (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.<ref>See especially the table in Mac Lane (1986), p. 35.</ref>