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Posted by Harvey on July 26, 2004 12:15:08 UTC

The term 'infinite' is a concept that we don't know if it even exists in reality. However, with regard to mathematics, I think you are referring to interesting mathematics, and if you are, then my answer is no. If we include non-interesting math or trivial math, then of course there's an infinite number of mathematics (i.e., there's an infinite number of ways to say the same thing or an infinite number of things to say about things that don't say very much).

In terms of interesting or non-trivial math where basic standards of consistency must be met, then I think that this forces the hand of mathematicians (whereever they may live or could potentially live), and limits the number of axioms that can possibly be used. For example, the axiom of identity (e.g., 0=0) is not selected from millions of other possibly identity axioms, rather the identity axioms that lead to interesting math looks to be very few. Perhaps our thought processes are just limited in this regard, but once you say two things are not 'identical' in math, you are no longer really talking about the concept of identity, hence it is hard to see how you can get infinite diversity just from this axiom alone. This is just an example, but enough for me to conclude that interesting mathematics by its very nature is a finite discipline.

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