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From: Leland McInnes on 16 Jun 2010 08:59
On Jun 15, 5:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jun 12, 8:24 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> > Transfer Principle <lwal...(a)lausd.net> writes:
> > > The response I'd like to see is one which defends classical analysis
> > > against these smooth infinitesimals
> > This idea, that classical analysis needs defending against smooth
> > infinitesimals, is bizarre.
>
> But there has to be a reason why most mathematicians use
> classical analysis and not smooth infinitesimal analysis.

I expect that relative newness of SIA is a big part. To make robust
foundations for SIA possible you need to ground things in topos theory
with its more flexible logics. That meant that SIA wasn't developed as
a theory until the 1980s. Compare that to classical calculus which has
more then a centurey of established history.

> I thought the fact that the latter contradicts the Law of
> the Excluded Middle was one reason to reject it. If not,
> then I'd like to see some of the real reasons that the
> classical analyis is more prevalent -- and once again,
> without the use of five-letter insults.

Because classical analysis is well established and has been around a
long time. SIA is very new and requires some deep mqathematics to
fully ground it, so it doesn't get taught. I think that's most of the
reasons.
From: Transfer Principle on 16 Jun 2010 20:38
On Jun 15, 3:08 pm, George Greene <gree...(a)email.unc.edu> wrote:
> On Jun 15, 5:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> > Smooth infinitesimal analysis is like non-standard analysis in that
> > (1) it is meant to serve as a foundation for analysis, and
> > (2) the infinitesimal quantities do not have concrete sizes
> > (as opposed to the surreals, in which a typical infinitesimal is 1/ω,
> > where ω is the von Neumann ordinal). However, smooth infinitesimal
> > analysis differs from non-standard analysis in its use of
> > nonclassical logic, and in lacking the transfer principle.
> That last line makes the whole discussion ironic.
> One would expect any contributor using THAT handle to be a supporter
> of the principle, not an advocate of a logic that doesn't use it.

I came up with the handle "Transfer Principle" right in the
middle of a thread about IST, which does use it. That
thread contained a heated debate between MoeBlee and
a new poster, Srinivasan, about IST. It might be possible
to search the Google archives to determine that the first
post in which I post under the name "Transfer Principle"
is the MoeBlee-Srinivasan IST thread.

> In any case, the question you need to be asking is why are people
> sticking with classical logic, not why aren't they using smooth
> infinitesimal analysis.
> Your question is not actually about analysis at all.

OK. The other posters in this thread helped answer this.
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