Lakoff and Núñez (2000) argues that mathematics arises from cognitive mechanisms that extend the structures of bodily experience into imaginary domains. This paper offers further examples in mathematics and further evidence in speech and gesture.

The author notes language that refers to movement in graphs that are by definition static. An exacting search is carried out to find the reason for this in the axioms of pure mathematics but none can be found. The author refers to the previous finding that

The structure of human mathematical ideas, and its inferential organization, is richer and more detailed than the inferential structure provided by formal definitions and axiomatic methods. Formal definitions and axioms neither fully formalize nor generalize human concepts.

and argues the point by contrasting the continuity in a particular continuous function that oscillates infinitely many times with a ‘natural continuity’.

Motion, in those examples, is a genuine and constitutive manifestation of the nature of mathematical ideas. In pure mathematics, however, motion is not captured by formalisms and axiomatic systems.

The author identifies a number of metaphors in the case of limits of infinite series, such as infinity as a single location that n (meaning the sequence of values) can ‘tend to’, these metaphors being in fact constitutive of the embodied ideas that eventually create mathematics. The author notes the difficulty that the difference between the formalism and the dynamic idea might pose for students.

The author then discusses how experience from everyday life is channeled into an understanding of a function through metaphorically supposing that numbers are locations in space and applying fictive motion. The author then examines gesture and speech to ascertain that these metaphors are not dead traces of previous thinking but in fact still play a part in constructing these mathematical objects.

Real numbers are metaphorical entities (with a very sophisticated inferential organization), and they do move, metaphorically

…and what’s more, their movement derives from their origins in bodily experience.

De Freitas and Sinclair refer to Lakoff and Núñez (2000) but criticise the dichotomy that it maintains between concept and physical gesture (one held to express the other, not to form it).

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Surely the reason that we use the language of motion when talking about graphs is because we all start off drawing tons of them with a pencil…

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