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Re: Infinitesimals

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On 02/19/2024 08:41 AM, root wrote:
> Paul N <gw7rib@aol.com> wrote:
>> Hi all
>>
>> Recently there was a discussion in comp.theory about infinitesimals. It seems
> I can't post to that group but can post to this one, so hopefully people will
> not mind too much and some kind person might even post a link there to my post
> here? > > I wanted to point out that Ian Stewart had written an article called
> "Beyond the vanishing point" in which he discusses the strange situation in
> which doing calculus by using very small numbers and then treating these numbers
> as zero after you've divided through by them is not valid but nevertheless seems
> to work. Here is some of the article as a taster:
>>
>
> I respect Ian Stewart because of his books, but here is being an alarmist.
>
> Continuous functions are defined as those for which the infinitesmal analysis
> works. Calculus applies to such functions.
>

The idea of there being a "continuous domain" for function theory,
and measure and the measure problem and so on, gets into why
"Dedekind's definition", of completeness and continuity, has
that there are others not just the same, like beads-on-a-string
and about Nyquist and Shannon and "super-sampling the
discontinuous dense".


Yaroslav Sergeyev's "Infinity Computer" was kind of an
interesting notion in terms of comp.theory, about the
ideas of infinity and "potential, practical, effective, actual",
infinity. I.e. for fixed-point arithmetic, a sufficiently
large value is the reciprocal of the multiplicative annihilator,
for implementing the usual "law of large numbers".



So, the idea that there are _three_ definitions of continuity,
three models of continuous domains, that all share the space
of real values, is rather rich. Of course it involves finding
in the foundations, of modern mathematics, how it can so be,
that [0,1] is split evenly, and complete, while the complete
ordered field is complete, or Dedekind's, and then that the
"discontinuous-dense super-sampling", complete, are three
different definitions of continuity, three different continuous
domains, and so on.


There are three definitions of continuous domains,
and various law(s) of large numbers, for various
models of what are mathematical infinities.


"Real-valued" is often enough how things are phrased,
with regards to "R" being the usual complete ordered field,
Archimedean.

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Infinitesimals Paul N <gw7rib@aol.com> - 2024-02-19 05:51 -0800
  Re: Infinitesimals root <NoEMail@home.org> - 2024-02-19 16:41 +0000
    Re: Infinitesimals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-07 10:12 -0800
  Re: Infinitesimals Ross Finlayson <ross.a.finlayson@gmail.com> - 2024-03-07 10:04 -0800

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