Mathematics Colloquia and Seminars

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Description

Rafael Nunez and I are finishing up a draft of a book (now at 600 pages) called Where Mathematics Comes From: How the Embodied Mind Creates Mathematics. The talk is an overview of the book.

The book is an application of cognitive science to mathematics. Given what we know about the nature of conceptual systems, the book asks, How is mathematics conceptualized using the basic cognitive mechanisms that have been discovered: image-schemas, conceptual metaphors, conceptual blends, and so on? How could mathematics have arisen, given the limited innate mathematical mechanisms (subitizing, baby addition, etc.)? This includes such questions as, What is the cognitive source of the laws of arithmetic? What is the conceptual structure of set theory and logic? How is e-to-the-1/4i conceptualized in terms of basic cognitive mechanisms? How do we conceptualize infinity in all its forms? What are real numbers?

The book argues that the fundamental mechanisms extending our miniscule innate mathematics are conceptual metaphor and conceptual blending. We analyze the metaphorical mappings used to conceptualize arithmetic, set theory, logic, analytic geometry, trigonometry, exponentials, and imaginary numbers. e1/4i is shown to be conceptualized via a blend of many metaphors. We also argue that there is a single basic metaphor for most (if not all) forms of infinity in mathematics, with special cases covering points at infinity, infinite sets, infinite unions, mathematical induction, infinite decimals, infinite sums, limits, least upper bounds, real numbers, transfinite numbers, infinitesimals, and infinite objects. We argue on the basis of these results that the real numbers do not exhaust the continuum, that space-filling curves do not fill space, that the sum of an infinite series is not always equal to its limit, and that Weierstrass continuity does not characterize conceptual continuity.

The main philosophical thrust of the book is that mathematics is a creation of the embodied human mind; that it is not an arbitrary creation but rather is structured by aspects of the mind, brain, and bodily experience; that it is largely metaphorical; and that it has no objective existence. At the same time, the cognitive properties of mathematics explain why arithmetic works where it does, why mathematics is effective in scientific descriptions of the world, why mathematics is stable, and why theorems that are proved stay proved.

We further argue that the principal philosophies of mathematics - Platonism, formalism, and constructivism - are all inadequate and that a new philosophy of mathematics, one responsive to scientific knowledge about the mind, is required.

The talk will try to give the flavor of these results and arguments.

Coffee and Tea before the talks at 3:30 in the Fifth Floor Commons Room, Kerr Hall.