FOM: What is mathematics? was intuition and rigor was arbitrary objects
richman
richman at fau.edu
Tue Feb 12 21:58:36 EST 2002
Gordon Fisher wrote:
>let me say, in a preliminary way, that existence of infinitesimals
>_in_ the real number system (complete linearly ordered field,
>therefore archimedean, unique up to isomorphism) is not
>possible.
I think this statement misses the point. When someone asks whether
infinitesimals exist in the real number system, he is not asking whether
infinitesimals can exist in a complete ordered field. He is questioning the
orthodox view of the real number system. It is a foundational question, not a
mathematical question that can be decided in the accepted framework by a
conventional proof.
There are intuitive notions of the real number system that precede our models
of it and axioms for it. I think it's premature to say that we have formulated
the definitive rigorous idea of the real numbers so that no rival system may
legitimately be called "the real numbers".
--Fred
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