[FOM] Re: naive continuum metaphysics

Andrej Bauer Andrej.Bauer at andrej.com
Thu Mar 4 04:16:44 EST 2004

William.Piper at colorado.edu writes:
> 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?

What you are looking for is called synthetic differential geometry. In
it the continuum appears as an object which is "coherent" in a precise
way (cannot be detachted into two non-trivial parts). _Some_ reals are
still discernible, of course, since 0 is discernible from 1. But
_general_ reals are indiscernable (for logicians "general = free
variable", for category-theorists "general = generic", for
philosophers "general = general", I think).

On this subject I very much recommend John Bell's booklet (it's thin!)
"A Primer of Infinitesimal Analysis". The book is easily readable by
non-topos-theorists and it really does get across very nicely the main
ideas of synthetic differential geometry. The book is sensitive to
typical brainwashed-by-classical-set-theory mathematician's biases.

Andrej Bauer

P.S. I always wondered why the book wasn't typeset in TeX.

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