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Mar5-05, 08:51 PM
Sci Advisor
HW Helper
P: 1,593
We invent Mathematics.

Numbers don't have a separate existence from the mind: The earth and sun were here long before us but the equations which describe the orbit of the earth were not.

Von Neumann considers the natural numbers as a play on empty sets by the mind: The mind ponders the empty set. Conceiving the empty set generates the number 1. Conceiving the set containing the empty set and the empty set generates the number 2 and so on. Thus the natural numbers from his perspective are created independently from any physical existence.

Mathematics fits nature so well because the geometry of Mathematics itself and that of nature are very similar: The real numbers system is dense: Between any two real numbers lies another. Nature appears dense as well: no smallest small nor largest large. Nature is "nested" (plans within plans). So too is math: Think of the chain rule. Nature is non-linear, so too is the geometry of math: Make one small mistake in solving a problem and it's not just a small error in the results.

It makes perfect Darwinian sense for a human mind to conceive of a math which fits nature from the perspective of survival: When in New York, act like a New Yorker. When the human brain finds itself in a non-linear world, a successful survival strategy it seems to me would be to devise a system of metaphors (mathematics) which fit that non-linearity.