[FOM] naive continuum metaphysics

Alexander M Lemberg sandylemberg at juno.com
Thu Mar 4 23:46:22 EST 2004

In response to your question, the following is information I have put on
this list before in another context. There is a subject known as "Smooth
Infinitesimal Analysis" or "Synthetic Differential Geometry" which treats
the continuum as "non punctate". I think this may address your question
fairly directly. My response has something in common with that of
Spitters in that the law of the excluded middle fails in these systems.
They are an approach to infinitesimals alternative to the nonstandard
analysis of Robinson. In these systems, the infinitesimals are non
punctiform and nilpotent whereas in the Robinson approach, infinitesimals
are invertible.


1. A Primer of Infinitesimal Analysis by John Lane Bell. (introductory)
2. Models for Smooth Infinitesimal Analysis by Ieke Mordeijk and Gonzalo
3. Synthetic Differential Geometry by Anders Kock  (out of print, but
should be available at your library)

This does not bear on CH because the continuum is not viewed as composed
of points as conventionally understood.


On Tue,  2 Mar 2004 08:14:57 -0700 William.Piper at colorado.edu writes:
> Following up on Steven's question regarding the continuum as a 
> non-set:
> I was wondering if anyone on the list has seen mathematical or 
> philosophical
> work on the continuum being a fundamental unity? To consider the 
> continuum as a
> pointless entity where the reals are in some sense indiscernable 
> from one
> another was proposed to me by a friend. This person regards the CH 
> as a
> fundamentally meaningless question and claims that we make the 
> mistake of
> thinking of the continuum as composed of objects (points or reals) 
> when it is
> really a single, "solid" entity. Has anyone seen any writing on this 
> or perhaps
> written something on this themselves?
> Everett
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