The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations For a Mind-Based Mathematics

(M121-liter The Metaphorical Structure of Mathematics: Sketching Out Cognitive Foundations for a Mind-Based Mathematics George Lakoff Rafael E. Nunez University ofCalIIf0mia, Berke’/fly WARNING! This is an essay within a new field of study—-the cognitive science t'.)fXI“le,1[l’).t3.‘,~ matics. The contribution we seek ultimately to make is a new one: to characterize precisely what m.athemat2'z.ul ideas are. You might thirik that this enterprise would leave mathematics as it exists alone and simply add to it an account of the conceptual nature of mathematical tmderstancling. You could not be more wrong. Studying the nature of mathematical ideas changes what we titiderstaiid mathematics to be and it even changes the understanding of particular“ mathematical results. The reason is that a significant amount of Ztltlmtetiw tury mathematics rests on the assumption that matheinatics is not about minds and ideas, but rather about symbols and their niodel—theoretil:at interpretations. We call this 20th-century View mim1~free irzat/ievrizmfcs, where the substance of mathematics is assumed to be independent otany human minds. Our enterprise is to bring embodied human minds, as they hzztve come to be understood recently in cognitive science, back into mathemat~ ics, and to construct a precise min.d—based irial/mntmfcs. Mind-based matite-~ rnatics is not just inind—free matliematics with some cognitive amtiysis added. Rather, the introduction of mind changes mathematics itself. riot just mathematics education or the study of mathematical cognition. Mind—free mathematics obscures the full beauty of inatliematics. This essay is a first attempt to reveal that beauty through the cliaracterization 2.1