FOM: What is mathematics? was intuition and rigor was arbitraryobjects
Ayan Mahalanobis
amah8857 at fau.edu
Wed Feb 13 10:58:36 EST 2002
Gordon Fisher wrote:
> Prof. Davis would be the one to answer this question, but
> let me say,
I will love to hear from Prof. Davis speaking on this subject.
> In nonstandard analysis, one _extends_ the real number system,
> adjoining infinitesimals to get a larger field of which the
> real number system is a proper subfield, and the resulting
> extension field is non-archimedean.
Yes, that is a formal way of doing it another way probably is to use
synthetic differential geometry and the power of intuitionistic logic to
get infinitesimal. I had no formal system in mind when I was talking
about the infinitesimal.
Say, in my class I was able to give a intuitive definition of (uniform)
continuity as, a curve is continuous if it can be drawn without lifting
the chalk from the board. Then came the rigor part as I was going
intuitive I tried it too and it seemed that the existence of
infinitesimal is inevitable (intuitively?) to give a intuitive
justification. Of course I was not the first person to realize it. If as
Matt said we should nurture our intuition then we should nurture it
right. As Peter Shuster said that mathematics is proving, then it should
pay to have the right kind of idea in the background. So again we have
many informal idea about the continuum and we have some idea as to what
to expect from it. Based on that, Is the existence of the infinitesimal
intuitively clear? It is a vague question and probably ill posed but I
see perfect sense in that so I thought to ask.
--Ayan
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