Sunday, September 18, 2011

Where mathematics comes from

The above is the title of a book by Lakoff & Nunez (Basic Books, 2000). It's come up again in recent discussions at IPS due to mathematical notions of infinity and how this reflects in notions of the absolute. The Preface is available at this link and here are some excerpts from it:

"In the course of our research, we ran up against a mythology that stood in the way of developing an adequate cognitive science of mathematics. It is a kind of “romance” of mathematics, a mythology that goes something like this.

• Mathematics is abstract and disembodied—yet it is real.
• Mathematics has an objective existence, providing structure to this universe and any possible universe, independent of and transcending the existence of human beings or any beings at all.

• Human mathematics is just a part of abstract, transcendent mathematics.

• Hence, mathematical proof allows us to discover transcendent truths of the universe.

• Mathematics is part of the physical universe and provides rational structure to it. There are Fibonacci series in flowers, logarithmic spirals in snails, fractals in mountain ranges, parabolas in home runs, and p in the spherical shape of stars and planets and bubbles.

• Mathematics even characterizes logic, and hence structures reason itself—any form of reason by any possible being.

• To learn mathematics is therefore to learn the language of nature, a mode of thought that would have to be shared by any highly intelligent beings anywhere in the universe.

• Because mathematics is disembodied and reason is a form of mathematical logic, reason itself is disembodied. Hence, machines can, in principle, think.

But the more we have applied what we know about cognitive science to understand the cognitive structure of mathematics, the more it has become clear that this romance cannot be true. Human mathematics, the only kind of mathematics that human beings know, cannot be a subspecies of an abstract, transcendent mathematics. Instead, it appears that mathematics as we know it arises from the nature of our brains and our embodied experience. As a consequence, every part of the romance appears to be false, for reasons that we will be discussing."

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