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

[Gordon Fisher]

Ayan Mahalanobis wrote:

> In the current ongoing debate, which started from considering "arbitrary
> objects" and now stands as a debate of intuition and rigor. I have a
> more basic and elementary question; What is mathematics? or rather What
> should be Mathematics?
>
> As I understand this understanding is subjective as there were/are
> mathematicians who were on opposite poles, and still there is no known
> understanding as "What is intuition?" or intuitively clear.
>
> Is the existence of infinitesimal in real number system intuitively
> clear? If so, should they be part of our mathematical rigor?
> I would like to ask this question to get some start on my earlier bigger
> question.
>
> Cheers
> --Ayan

Prof. Davis would be the one to answer this question, but
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

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.

Whether or not one wants to use such ARobinsonian
infinitesimals is not a question of rigor versus intuition,
but rather, I would say, a question of choice of fields,
together with certain matters of simplicity -- witness
the little use in universities of elementary calculus done
using a ARobinsonian extension field, or its little use
by physicists and the like.

Perhaps you have in mind a so-called "intuitive" use
of infinitesimals in the manner of many physicists,
who use their "intuition" to adjoin infinitesimals
to the real number system, rather than a formal
system of some sort?  One has to distinguish, I think,
between meanings of "intuitive" (and "intuition").
Admittedly, it is not easy to give mathematically
precise definitions, since we seem here to be dealing
with interfaces between people (mathematicians,
physicists, etc.) and whatever it is that mathematics
is about or arises from.

Gordon Fisher     gfisher at shentel.net