FOM: What is mathematics? was intuition and rigor was arbitrary objects

richman richman at
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

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".


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