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Posted by Harvey on September 15, 2003 17:13:04 UTC

Mike,

Paul and I had this nearly identical discussion a while ago. We are talking about the epistemological issues surrounding math (i.e., knowledge and meaning of mathematics as it pertains to us as humans), whereas Paul is talking about it in terms of the education of mathematics as an abstract science of patterns. If thought in terms of math education, then abstractions should be thought of as independent of 'real' things, however in the context of how mathematics is discussed here (i.e., in epistemological terms), then you are quite right that mathematics is about reality since what else could it be about? There is a whole history that can trace how things become abstract and that this forms a chain of references that ultimately reduce any abstract thing to a particular thing. I think, as you said, as humans age (or as a mathematical society ages) they tend to forget the old references and just treat them as assumptions (or axioms). Of course, the reason that they are concerned about those axioms versus some ridiculous axioms is because of the chain of references that establish those axioms as meaningful and the ridiculous ones as not meaningful. This is why we don't see modern mathematicians postulating ridiculous axioms. They aren't meaningful to entertain for their profession.

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