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Those Axioms Are Insufficient

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Posted by Paul R. Martin on May 8, 2001 18:44:02 UTC

Hi Alex,

I don't think those axioms are enough to develop much mathematics at all. I think mathematicians agree that the Peano axioms are the bare minimum that can categorically generate arithmetic. Set theoretic axioms can also be used but they also go beyond your basic definitions of 0 and 1. To get any structure at least as complex as arithmetic, you need to choose some additional axioms, which are arbitrary. You get a different structure depending on what additional axioms you choose.

It turns out that the universe appears to have a great many correspondences with the system of arithmetic. It also has a great many correspondences with the system of analysis, which extends the arithmetic of natural numbers to the calculus of irrational and even complex numbers. However the correspondences with analysis break down in physical systems which are not classical: The smooth continuity of analysis is broken in the quantum world.

The same is true of geometry. The axioms of Euclid work very well as long as the physical system is nearly flat. If things get too big, or involve relativistic effects, Euclid's axioms no longer work and we have to choose Minkowski's or someone elses non-Euclidean geometry.

I don't think you can find any set of axioms that are natural and inviolate from which all of the mathematics that can describe the world can be derived. That's what Euclid, and even Pythagoras before him thought, and they were proved to be in error by Lobachevski and others in fairly recent times.

Warm regards,


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