FOM: conservative extensions of nominalistic theories

[Jeffrey John Ketland]

First, I would like to thank Professor Field for joining in with this 
discussion of his extremely important and interesting work. For non-
philosophers out there, Field’s work is easily the most important 
nominalist approach within contemporary philosophy of 
mathematics.

If Field is right, then his program solves about seven of the most 
basic metaphysical problems since Plato! (Do abstract objects 
exist? If so, how do we know about them? Why is mathematics 
useful in describing the physical world? Does the notion of truth 
apply to mathematical discourse? What is objective about 
mathematical practice? Etc.).

I strongly recommend that any mathematician interested in 
philosophy of mathematics have a look at Field’s approach. 
Furthermore, discussions of Field’s program also contains what 
look like interesting open research problems as well (which I’ll 
come to below).

In my posting, I wrote:
 
>Adding set theory to certain synthetic descriptions of spacetime is
> non-conservative. This is closely connected to Godel’s theorems. 

I then referred to a paper by Stewart Shapiro (1983), which 
develops a point about Godel sentences which Field himself 
mentions in the last chapter of his book. (Apparently this objection 
to Field's program originated from comments by John Burgess and 
Yiannis Moschovakis).

Professor Field replied (correctly)

>It depends what you mean by 'adding'. If you don't allow set-
>theoretic vocabulary into the comprehension axioms used in the 
>synthetic physical theory then the extension IS conservative.

I agree with this technical point. (I hope I didn’t misrepresent 
Field’s position here). Strictly speaking, you can get a conservative 
extension of your nominalistic theory even if you add the 
whole of ZFC with ur-elements. (I’d like to know whether logicians 
out there all agree that this technical claim is right, and also 
whether they think it has the foundational significance that Field 
thinks it does).

Let me explain a bit, for anyone who is interested in Field's 
program (John Burgess calls it “geometrical nominalism” or 
“synthetic physics”). We begin with a certain (first-order) 
mathematical platonistic theory P of spacetime (this is a 
mathematical theory which says that spacetime is a flat Euclidean 
manifold [and also says that the gravitational potential satisfies 
Poisson's equation, but that's not important here]). The main idea, 
roughly, is to find a nice nominalistic spacetime theory N such that 
P is a conservative extension of N. Then, we say, in light of the 
conservativeness, that all the set-theoretical mathematical 
apparatus in P is just a "convenient instrument for proving 
theorems" that are already theorems of N. (You know: lengths of 
proofs are shorted in P, etc.).

Building on work on axiomatic geometry that goes back to Pasch, 
Hilbert, Tarski and others, Field proposes, as this replacement, a 
certain first-order nominalistic theory N of spacetime, containing an 
comprehension axiom scheme,

(If phi(x) is satisfied by at least one point), there is a region R such 
that, for any point p, p is a part of R iff phi(x)

(within N, regions are "mereological aggregates" of points, rather 
than sets of points. So, the relation of a point p to a region R is 
that p is a *part* of R, rather than that p is an *element* of R. Field 
argues that mereological aggregates of spacetime points are 
physical objects, rather than abstract mathematical objects).

Field insists that when adding our favourite set theory S to N, we 
must not allow the defining condition phi(x) to contain set-
theoretical vocabulary. But why not? Surely this just begs the 
question. Indeed, the mathematical realist (influenced by the Quine-
Putnam argument) would insist that the extra strength (of being 
able to talk about these extra set-theoretically definable regions of 
spacetime) is part of the reason for being a realist about their 
favourite set theory S to begin with.

In any case, Field does agree that adding the new set-theoretical 
instances of the comprehension axiom scheme can lead to new 
theorems about the spacetime points and regions. Field also 
admits that these are theorems of the original (platonistic) 
spacetime theory P that N is meant to replace, but that these 
new theorems are NOT theorems of N. In short, the original theory 
P is NOT a conservative extension of its proposed replacement N, 
after all.

As Shapiro explained in his 1983 article, a witness to this non-
conservativeness is a godel sentence Con(N), which codes (within 
the language of N) the consistency of N. This sentence Con(N) is 
provable in P, but not in N. 

It may seem like nit-picking to claim that a major program in the 
philosophy of mathematics (that could solve “seven major problems 
in metaphysics since Plato”) all hinges on a technical point 
concerning what we mean by “adding” one theory to another. But I 
think that the connection with godelian incompleteness is an 
important reason to examine this issue more carefully. For, given 
the analogous history concerning the provability of mathematically 
interesting facts about the numbers (Paris-Harrington result about 
Ramsey’s Theorem) using set theory, but which are not provable in 
PA, it is surely no good just to say that these theorems provable in 
the original spacetime theory P but not provable in N are not 
“relevant to physics” and can perhaps be forgotten about. (This is 
part of Field’s response to the godelian argument). We just don’t 
know. It’s an open research problem whether our best platonistic 
theory of spacetime P contains theorems which its Field-style 
nominalistic replacement N doesn’t prove, AND which also have 
some “instrinsic physical significance” (whatever that means!).

Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel:    0115 951 5843
Fax:    0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>