[FOM] naive continuum metaphysics
Bas Spitters
spitters at cs.kun.nl
Wed Mar 3 02:38:50 EST 2004
On Tuesday 02 March 2004 16:14, William.Piper at colorado.edu wrote:
> 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?
You may want to have a look at the intuitionistic literature on the continuum.
Similar questions motivated Brouwer to develop his intuitionistic mathematics.
Heyting's book is a nice introduction, other standard reference include
Troelstra/van Dalen and Beeson.
For a more "modern" approach (consistent with classical mathematics) you may
want to look at pointfree (aka pointless) topology. For instance, the work on
formal topology (http://www.math.unipd.it/~logic/rgl/frame.html) or the
literature on locale theory (Johnstone's Stone spaces is a standard
reference).
Regarding CH, in intuitionistic mathematics on can look at it as follows:
It is not meanless (as your friend states), but only not formulated precisely
enough. In intuitionistic mathematics, there are several ways of making this
precise. Some are provable, some are not.
See, Gielen, de Swart and Veldman, W, The continuum hypothesis in
intuitionism, J. Symbolic Logic,46,1981,1,121--136
I hope this is of any help.
Bas Spitters
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Department of Computer Science, University of Nijmegen.
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