[FOM] Number Line/mereology

[Harvey Friedman]

On 11/16/05 6:55 PM, "Robert Tragesser" <thesavvydog at mac.com> wrote:

> At discussion at a talk at SUNY-Buffalo in the early 70's, Harvey
> Friedman referred to the conflict between our intuitive sense, when a
> line is split, that the parts have end points (termini) with no point
> is lost,  and our virtually paradigmatic representation of the line
> continua as composed of points (so that when split, either one part
> has an end-point and the other doesn't, or a point is removed and
> neither part has an endpoint), as "Gödel's Paradox of Geometric
> Intuition".  (So maybe Harvey has some idea of how Gödel understood
> this?)
> 
> Dana Scott remarked that this problem arose for Tarski when he was
> working on what became his paper on geometric solids (reprinted in
> Logic, Semantics, Metamathematics)...and he there found a way of
> resolving it.
> 

There is an extensive literature on this topic under the title "mereology".
One takes the part/whole relation as the fundamental primitive, and not
element/set. E.g., David Lewis (deceased, from Princeton Philosophy
Department) strongly promoted mereology as a branch of metaphysics. In
particular, Lewis and some of his colleagues (e.g., Burgess) studied
mereological interpretations of set theory.

There is the idea (e.g., from Lewis) that mereology is to be greatly
preferred to set theory. I.e., in commonsense reasoning, as opposed to
mathematics, part/whole seems to be more fundamental than element/set.

Hence the interest in providing mereological interpretations of set theory.
However, it does not appear that mereology is a good direct way to lay
foundations for mathematics.

Some of the most fundamentally interesting mereological systems are complete
and decidable, and so cannot recover any substantial amount of set theory.

I was involved in developing some systems of mereology that interpreted set
theory with large cardinals. However, this was obsoleted by more convincing
ways of interpreting set theory with large cardinals using other notions -
particularly that of ternary relations (with urelements), in my "Relational
System Theory". See manuscript 47 at

http://www.math.ohio-state.edu/%7Efriedman/manuscripts.html

However, I still have some inkling that one can develop a striking system of
mereology, surprisingly close to commonsense thinking, that is
bi-interpretable with set theory with large cardinals.

Harvey Friedman