[FOM] natural language and the Fof M

[Harvey Friedman]

Reply to Slater 4/11/03  11:49AM.

Slater hints at a theory, following Bunt, with possibly several properties.

1. Set theory is included as a special case. There is a more general 
concept of "ensemble", which may or may not be made up of elements.

2. There is a new conditional separation axiom which avoids inconsistency.

3. The theory is some variant of mereology.

*Rather than have FOM readers go find Bunt's book to see if they are 
interested, it would be much better if you could briefly sketch the 
basics of the theory.*

I am familiar with mereology and David Lewis' idea of interpreting 
set theory in mereology. The trouble is, of course, that one has to 
augment merology in some significant way in order to do this.

In particular, I think that it follows from the Tarski proof of the 
decidability of the first order theory of Boolean algebras given in 
Chang/Keisler, Model Theory, that even the first order theory of 
successor cannot be interpreted in any consistent extension of the 
axioms of Boolean algebra. This almost rules out the development of 
any serious amount of mathematics in mereology alone - although one 
could conceivably work in some form of mereology which violates one 
or more of the axioms of Boolean algebra. It is not clear if this can 
be done in any natural or productive way.

In fact, it would be interesting to see just how much of mereology 
can be preserved in a theory in the one binary relation symbol 
(part/whole) that interprets ZFC.

In order to get around the weakness of mereology from the point of 
view of the foundations of mathematics, Lewis introduced a singleton 
relation "x is the singleton of y". I have preferred to call it the 
naming relation: "x is a name of y", and this has an analogous effect.

A second way to get around the weakness of mereology from the point 
of view of the foundations of mathematics is to instead introduce 
families of objects (every object is called a "part" my 
mereologists), also due to (or at least suggested by) Lewis and 
developed by Burgess and others. One can get away with families of 
objects rather than relations between objects.

It is well known that one can follow either approach and rather 
slavishly write down axioms in this framework that amount to a 
translation of the axioms of set theory.

It is not clear what is to be gained by this. One possibility is to 
take a different view of this second way. Instead of actually 
introducing the families of parts, one can think of this instead as 
using monadic predicate calculus with comprehension, so that instead 
of objects, one views this as a kind of "logic". This can also be 
thought of as "plural quantification" instead of monadic logic. This 
is a line taken by Boolos and later Burgess. One can view this as 
taking a cue from natural language philosophers to make things more 
natural language friendly.

At my visit to Princeton Philosophy Dept last fall, I discovered a 
number of surprisingly simple reflection principles that alleviate 
any requirement that one slavishly transcribe axioms of set theory 
into such contexts, or even in the most bare bones context of all - 
just a binary relation epsilon.

My original formulation was in terms of standard mereology with the 
"naming relation" - really just Lewis' singleton relation.

This work led to the papers these papers of Burgess:

 From Frege to Friedman: A Dream Come True? 
http://www.princeton.edu/~jburgess/anecdota.htm

E Pluribus Unum
http://www.princeton.edu/~jburgess/anecdota.htm

Burgess continues to emphasize plural quantification. I haven't yet 
taken a view as to how much is gained, philosophically, by this, and 
want to look into it further.

I thought it would be simpler to start with just the binary relation 
epsilon, and worked hard after returning from Princeton to make 
further simplifications. This led to the work reported in my postings 
158-161, of 3/31/03. These postings are now in pdf format at 
http://www.mathpreprints.com/math/Preprint/show/

One feature of this work in 158-161 is the avoidance of assuming any 
form of comprehension. Any form of comprehension that is needed has 
to be derived. In the plural quantification approaches comprehension 
appears as part of the apparatus.

The earlier work along these lines that I did in the summer of 2002 is in

A Way Out, http://www.mathpreprints.com/math/Preprint/show/

However, this early form of the work isn't nearly as thematic. The 
new work is based entirely on various forms of reflection.

I plan to make a posting on the mereological formulation(s). In light 
of the new formulations involving epsilon only, in the postings of 
3/31/03, the mereological approach might be somewhat simpler than 
what I had at Princeton last fall.

In any case, all of this tends to go into the direction of making the 
foundations of mathematics more philosophy of language friendly - at 
least to some extent.

>   Shaughan Lavine, for instance, has produced an axiomatisation 
>equivalent to von Neumann's 'which seemed to me to be more natural 
>and conceptually based than the usual ones - though I doubt it is 
>the last word on the subject' ('Understanding the Infinite', Harvard 
>U.P. Cambridge MA 1994, p320, see also pp141-53).

Again, rather than have FOM readers go find Lavine's book to see if 
they are interested, it would be much better if you could briefly 
sketch it and indicate what is more natural.
>
>Specifically, for instance, y is water not because it is a member of 
>a certain set, but because it is part of a certain aggregate (the 
>totality of all water), while the formal possibility of a non-set 
>equivalent has previously been taken to be a matter of y still being 
>a member, but a member of a different type of thing to a set - a 
>proper class.  The representation of stuff in terms of membership is 
>on account of the influence of late nineteenth century thinking 
>about number.


The unanswered question is whether anything is to be gained by this 
mereological approach to water, for the foundations of mathematics. I 
believe that it is philosohically interesting.