Wednesday, 21 August 2013

The Modal Status of Semantic Facts

I think this is probably one of the fundamental issues in metasemantics: the modal status of semantic facts. Consider the following thought experiment:
Thought Experiment 1:
Agent A lives in a certain valley and speaks a language $L_1$ in which "snow" refers to snow.
Agent B lives in neighbouring valley, and speaks a language $L_2$ in which "snow" refers to coal.
In this scenario, it seems to me that $L_1$ and $L_2$ are distinct. Consider the following temporal thought experiment:
Thought Experiment 2:
Agent A speaks a language $L_1$ in which "snow" refers to snow.
One day, agent A is hit by lightning, and afterwards, speaks a language $L_2$ in which "snow" refers to coal.
In this scenario, it seems to me that $L_1$ and $L_2$ are also distinct. Consider the following counterfactual thought experiment:
Thought Experiment 3:
Agent A speaks a language $L_1$ in which "snow" refers to snow. Agent A narrowly avoided being hit by lightning one day. But if agent A had been hit by a lightning, A would have spoken a language $L_2$, in which "snow" refers to coal.
In this scenario, it seems to me that $L_1$ and $L_2$ are distinct.

So, in all these cases, $L_1 \neq L_2$. Changing a language, even a tiny bit, yields a new language. The argument is admittedly based on thought experiments and therefore appeals to intuitions. But even so, if it is right, then a semantic fact like
(1) The intension of $\sigma$ in $L$ is $m$ 
is a necessity. It could not have been otherwise. If that is right, languages are individuated very finely.

On the other hand, in the philosophy of language and metasemantics, we're interested in "U-facts", such as,
(2) Agent A uses string $\sigma$ in a certain way.
and how they're related to other facts connected to meaning. (2) is contingent. How an agent uses a string can presumably vary in all kinds of ways. And also we interested in "C-facts", such as,
(3) Agent A cognizes language $L$.
This is contingent too. Which language an agent cognizes is a contingent matter, and can presumably vary in all kinds of ways. It seems that there must be a very intimate relation between facts of the kind (2) and facts of the kind (3).

Sunday, 18 August 2013

The Abstract Structure of a Binary Relational Model

Generalizing the previous M-Phi post on the abstract structure of an $\omega$-sequence, suppose that $\mathcal{A} = (A, R)$ is any (set-sized) binary relational model (i.e., $A$ is the domain/carrier set and $R \subseteq A^2$). Let $\kappa = |A|$. Let $L$ be a second-order, possibly infinitary, language, with $=$ (but no non-logical primitive symbols), which allows compounds over $\kappa$-many formulas and allows quantifier prefixes to be a set $V$ of variables of cardinality $\kappa$. For each $a \in A$, let $x_a$ be a unique variable that "labels" $a$. Let the second-order unary variable $X$ label the domain $A$ and let the second-order binary variable $Y$ label the relation $R$.

The (possibly infinitary) diagram formula $\Phi_{\mathcal{A}}(X,Y)$ is then:
$\exists V[\bigwedge_{a,b \in A; a \neq b} (x_a \neq x_b) \wedge \bigwedge_{a \in A} Xx_a \wedge \forall x(Xx \to \bigvee_{a \in A} (x = x_a))$ $\wedge \bigwedge_{a,b \in A} (\pm_{ab} Yx_a x_b) ]$
where $V = \{x_a \mid a \in A\}$ and $\pm_{ab}Yx_ax_b$ is $Yx_ax_b$ if $(a,b) \in R$ and $\neg Yx_ax_b$ otherwise.

On the Diagram Conception of Abstract Structure,
The abstract structure of $\mathcal{A}$ is the proposition $\hat{\Phi}_{\mathcal{A}}$ expressed by the formula $\Phi_{\mathcal{A}}(X,Y)$. 
Categoricity ensures that, for any $\mathcal{B} = (B,S)$ (a relational model, with a single binary relation $S \subseteq B^2$), we have:
$\mathcal{B} \models \Phi_{\mathcal{A}}(X,Y)$ if and only if $\mathcal{B} \cong \mathcal{A}$.

Saturday, 17 August 2013

The Abstract Structure of an Omega Sequence

Suppose that $(\omega, <)$ is the usual $\omega$-sequence where $\omega$ is the set of finite von Neumann ordinals and $<$ is their order (i.e., $\alpha < \beta$ iff $\alpha \in \beta$). On the Diagram Conception, the abstract structure of $(\omega, <)$ is the proposition expressed by the following infinitary diagram formula $\Phi(X,Y)$:
$\exists V[\bigwedge_{i \neq j} (x_i \neq x_j) \wedge \bigwedge_{i \in \omega} Xx_i \wedge \forall x(Xx \to \bigvee_{i \in \omega} (x = x_i)) \wedge
 \bigwedge_{i,j \in \omega} \pm_{ij} Yx_i x_j ]$
where $V = \{x_0, x_1, \dots\}$ and $\pm_{ij}Yx_ix_j$ is $Yx_ix_j$ if $i < j$ and $\neg Yx_ix_j$ otherwise.

Categoricity ensures that, for any $(A,R)$ (a relational model, with a single binary relation $R \subseteq A^2$), we have:
$(A,R) \models \Phi(X,Y)$ if and only if $(A,R) \cong (\omega,<)$.
On a certain version of structuralism, "ante rem structuralism", there would be an abstract, "ante rem", $\omega$-sequence: and this would itself be an $\omega$-sequence, say
$\Omega = (N, <_{N})$,
whose domain $N$ contains "nodes". But this approach seems to face the following problem, which is a modification of Einstein's "Hole argument". Let
$\pi : N \to N$
be any non-trivial permutation of the nodes and let $\pi_{\ast}[<_N]$ be the resulting ordering and let
$\pi_{\ast}\Omega = (N, \pi_{\ast}[<_N])$. 
Then,
$\pi_{\ast}\Omega \neq \Omega$
$\pi_{\ast}\Omega \cong \Omega$
So, which one is the "real" abstract, ante rem, $\omega$-sequence?

Instead, I think the abstract structure of any $\omega$-sequence is not itself an $\omega$-sequence; rather, it's a purely structural, infinitary, second-order proposition, whose models are the $\omega$-sequences.


[UPDATE (18th August 2013):
Robert Black in the comments below alerted me to this mini-"Hole"/permutation argument appearing before, in Geoffrey Hellman's article,
Hellman, G. 2007: "Structuralism", in Shapiro 2007 (ed.), Oxford Handbook of the Philosophy of Mathematics and Logic
Geoffrey puts it like this (Here $SGS$ is Shapiro's sui generis "ante rem" structuralism):
In fact, $SGS$, seems ultimately subject to the very objection of Benacerraf ["What Numbers Could Not Be"] that helped inspire recent structuralist approaches to number systems in the first place. Suppose we had the ante rem structure for the natural numbers, call it $\langle N, \varphi, 1 \rangle$, where $\varphi$ is the privileged successor function, and $1$ the initial place. Obviously, there are indefinitely many other progressions, explicitly definable in terms of this one, which qualify equally well as referents for our numerals and are just as “free from irrelevant features”; simply permute any (for simplicity, say finite) number of places, obtaining a system $\langle N, \varphi^{\prime}, 1^{\prime} \rangle$, made up of the same items but “set in order” by an adjusted transformation, $\varphi^{\prime}$.
Why should this not have been called “the archetypical ante rem progression”, or “the result of Dedekind abstraction”?
We cannot say, e.g., “because 1 is really first”, since the very notion “first” is relative to an ordering; relative to $\varphi^{\prime}$, $1^{\prime}$, not $1$, is “first”. Indeed, Benacerraf, in his original paper, generalized his argument that numbers cannot really be sets to the conclusion that they cannot really be objects at all, and here, with purported ante rem structures, we can see again why not, as multiple, equally valid identifications compete with one another as “uniquely correct”. Hyperplatonist abstraction, far from transcending the problem, leads straight back to it. (Hellman, "Structuralism", pp. 11-12 in the preprint).
My $\Omega$ above corresponds to Hellman's $\langle N, \varphi, 1 \rangle$ and my $\pi_{\ast}\Omega$ corresponds to Hellman's $\langle N, \varphi^{\prime}, 1^{\prime} \rangle$.]

Friday, 16 August 2013

Conservativeness of PAV

This post continues on the theme of the argument against deflationism, concerning the conservativness of truth theories. I want to discuss the role of induction.

It turns out that this is a very complicated issue, and people have slipped up a bit over it. But there are some points that aren't too complicated, and one concerns the result of extending $\mathsf{PA}$ with new vocabulary, and therefore new induction instances, but without new axioms for the new vocabulary. If there are no new axioms, then the result is conservative.

Suppose we let $L$ be the usual first-order language of arithmetic (non-logical vocabulary is $\{0,^{\prime},+,\times\}$) and let
$V := \{P_1, \dots\}$
be some new vocabulary. Each $P_i$ is a predicate symbol (say, of arity $k_i$). Let $L_V$ be the extended language. Let $\mathsf{PA}$ be the usual system of Peano arithmetic in $L$, with the induction scheme,
$\phi^x_0 \wedge \forall x(\phi \to \phi^x_{x^{\prime}}) \to \forall x \phi$
(here $\phi$ is a formula (possibly with parameters), and $\phi^x_t$ is the result of substituting the term $t$ for all free occurrences of $x$, relabelling bound variables inside $\phi$ if necessary to avoid collisions).

Definition: $\mathsf{PAV}$ is the result of extending $\mathsf{PA}$ with all instances of induction for the extended language $L_V$.

Theorem: $\mathsf{PAV}$ conservatively extends $\mathsf{PA}$.

Proof. The "proof idea" is to give a translation that maps every proof (derivation) in the "bigger" theory into a proof in the "smaller" theory. We define a translation
$^{\circ}: L_V \to L$
as follows. For any $L$ symbol, including variables and terms, the translation is the identity mapping. For compounds, we assume $^{\circ}$ just commutes. For a $n$-ary predicate symbol $P \in V$, we translate an atomic formula $P(t_1, \dots, t_n)$ as follows:
$(P(t_1, \dots, t_n))^{\circ} := (P)^{\circ}(t_1, \dots, t_n)$.
where $(P)^{\circ}(x_1, \dots, x_n)$ is any $L$-formula. This translation is crazy, of course. But we have no axioms constraining the symbol $P$, so we can translate it any way we like. (If we did have such axioms, say $Ax_{P}$, we would need to try and translate $Ax_{P}$ as a theorem of $\mathsf{PA}$.)

The formula $(P(t_1, \dots, t_n))^{\circ}$ is then an $L$-formula. This means that, for any $\phi \in L_V$, $(\phi)^{\circ}$ is equivalent to some $L$-forrnula, say $\theta$. Now consider the induction axiom for any $\phi \in L_V$:
$\phi^x_0 \wedge \forall x(\phi \to \phi^x_{x^{\prime}}) \to \forall x \phi$
Its translation under $^{\circ}: L_V \to L$ is equivalent (because $^{\circ}$ commutes with connectives and quantifiers) to
$\theta^x_0 \wedge \forall x(\theta \to \theta^x_{x^{\prime}}) \to \forall x \theta$
And this is an induction axiom in $L$. Hence, it is an axiom of $\mathsf{PA}$.
Finally, suppose that
$\mathsf{PAV} \vdash \psi$, 
where $\psi \in L$. Then applying the translation $^{\circ}$ to all the formulas in the derivation of $\psi$ converts the derivation into a derivation of $\psi$ in $\mathsf{PA}$ (in $L$), as each induction instance is translated to an induction instance in $L$ (which is an axiom of $\mathsf{PA}$) and the translation preserves $\vdash$ too. So,
$\mathsf{PA} \vdash \psi$.
QED.

This may be applied to the case where the new vocabulary is a predicate $T(x)$, perhaps to be thought of as a truth predicate, but we do not include any new axioms for $T(x)$ itself. Halbach 2011 calls the corresponding theory $\mathsf{PAT}$. So, $\mathsf{PAT}$ conservatively extends $\mathsf{PA}$.

Thursday, 15 August 2013

Is the Role of Truth Merely Expressive?

I worked (a bit) on truth theories in my PhD. I was interested in the question of whether the role of truth is "merely expressive" (as deflationists claimed) or whether it might be "explanatory" (as non-deflationists might claim). One day in 1997, the thought
"I wonder if the T-scheme is conservative over $\mathsf{PA}$?"
popped into my head. (I remember the incident quite well!) If so, there might be an interesting analogy between three kinds of instrumentalism:
(i) Instrumentalism about infinitary objects in mathematics (Hilbert)
(ii) Instrumentalism about mathematical objects in connection with physics (Field).
(iii) Instrumentalism about semantic properties (Field, Horwich).
Debates surrounding the first two eventually turned on whether an extended "ideal" theory $T_I$ was conservative with respect to the underlying "real" one $T_R$. And I knew that, in some cases, the extended theory is not conservative: for Gödelian reasons.

For the case of Hilbertian finitism/instrumentalism, the aim of Hilbert was to show that "Cantor's Paradise" (of infinite sets) was a convenient, but dispensable, instrument for proving finitary combinatorial facts. The hope was to prove Cantorian set theory consistent using just finitary assumptions, perhaps as encoded in $\mathsf{PRA}$ or $\mathsf{PA}$:
In the early 1920s, the German mathematician David Hilbert (1862-1943) put forward a new proposal for the foundation of classical mathematics which has come to be known as Hilbert's Program. It calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent. The consistency proof itself was to be carried out using only what Hilbert called “finitary” methods. The special epistemological character of finitary reasoning then yields the required justification of classical mathematics. (Richard Zach, 2003, "Hibert's Program", SEP.)
But, it follows from Gödel's results that a consistent finitary theory $T$ of arithmetic can be extended with a suitable amount of Comprehension (one extends induction too), and the result, $T^{+}$, proves theorems in the language of $T$ that $T$ doesn't. The most well-understood examples involve a theory $T$ and its impredicative second-order extension $T^{+}$. For example, if we pass from $\mathsf{PA}$ to second-order arithmetic $\mathsf{Z}_2$. But there are other kinds of example.

It is this insight about increasing power which in part leads to the idea of an "Interpretability Hierarchy", an idea pursued vigorously by Harvey Friedman for several decades. The idea is that, with certain exceptions, mathematical theories form an ascending linear hierarchy of interpretability strength, with weak systems at the bottom (e.g., $\mathsf{Q}$ and $\mathsf{AST}$ (Visser's Adjunctive Set Theory)) and very powerful set theories, with large cardinal axioms, much higher up.

Similarly, for the case of Fieldian nominalism. Consider, for example, Hartry Field's nominalistic reformulation of Newtonian mechanics and gravity in his brilliant monograph,
Field, H. 1980: Science Without Numbers.
(Tragically out of print.) This theory $T$ can be extended with a suitable amount of Comprehension (one extends certain schemes in $T$ too), and the result, $T^{+}$, proves theorems in the nominalistic language that $T$ doesn't.

These Gödelian objections were raised by John Burgess, Yiannis Moschovakis, and Field mentions them in the Appendix of his monograph; I believe similar objections were raised but not published by Saul Kripke; and shortly after, the objections were spelt out in a bit more detail by Stewart Shapiro in his:
Shapiro, S. 1983: "Conservativeness and Incompleteness" (Journal of Philosophy)
So, knowing much of this, I began to see whether the same situation held for truth/semantics, and eventually it became clear that the analogy did hold quite well (I got some help here from John Burgess). In some cases, truth axioms are conservative: e.g., disquotational axioms, $T(\ulcorner \phi \urcorner) \leftrightarrow \phi$, with the sentence $\phi$ used restricted to the object language. And in some cases they are not (e.g., Tarski-style compositional truth axioms). At the time, I had no idea that these technical issues had been investigated by other authors (Feferman, Friedman, Sheard, Cantini, Halbach), though I found out quite quickly after my paper on the topic appeared! My paper on this is:
Ketland, J. 1999: "Deflationism and Tarski's Paradise" (Mind)
("Tarski's Paradise" is a joke, alluding to the analogy mentioned above with "Cantor's Paradise".)

Around the same time, Stewart Shapiro developed more or less the same argument, in his
Shapiro, S. 1998: "Proof and Truth - Through Thick and Thin" (Journal of Philosophy)
This kind of objection---the "conservativeness argument", as it's been called---to deflationism argues that if deflationism is understood as claiming that the role of truth is "merely expressive", then it must require conservativeness of one's truth theory; but then this is inconsistent with the manifest non-conservativeness of truth axioms, in certain cases. More recently, some authors have followed Shapiro and myself in calling truth theories "deflationary" or "non-deflationary" depending on whether they are conservative or not, and explicitly restricting to conservative truth theories if they wish to defend deflationism. This issue is complicated and usually depends on whether one allows inductive reasoning using the truth predicate.

The argument that Shapiro and I gave can be summarized like this (see this M-Phi post "Reflective Adequacy and Conservativeness" (17 March 2013)):
(P1) A truth theory is deflationary only if conservative over suitably axiomatized theories $B$.
(P2) A truth theory is reflectively adequate only if it combines with $B$ to prove "all theorems of $B$ are true".
(P3) For many cases of $B$, reflective adequacy implies non-conservativeness.
-----------------------------------------------------------------------
(C) So, deflationary truth theories are reflectively inadequate.
For a similar formulation and discussion, see also:
Armour-Garb, B. 2012: "Challenges to Deflationary Theories of Truth" (Philosophy Compass)
The second premise (P2) corresponds to Hannes Leitgeb's adequacy condition (b) in his:
Leitgeb, H. 2007: "What Theories of Truth Ought to be Like (But Cannot Be)" (Philosophy Compass)
Leitgeb formulated (P2) as follows:
"(b) If a theory of truth is added to mathematical or empirical theories, it should be possible to prove the latter true" 
Leitgeb adds that this is "uncontroversial".

Paul Horwich has an interesting interview with 3am Magazine, titled "Deflationism and Wittgenstein". In it, he makes a couple of crucial points concerning his own form of deflationism, which make it clear that he endorses some form of instrumentalism:
Thus truth is not as profound a phenomenon as has often be assumed. Its role, even in philosophy, must be merely expressive rather than explanatory.
The idea here is that the single sentence:
(1) for any proposition $x$, if $A$ asserts $x$, then $x$ is true 
"re-expresses" the scheme:
(2) if $A$ asserts that $p$, then $p$. 
It is certainly the case that recursively enumerable theories $S$ in some language $L$ meeting certain conditions can be re-axiomatized as a finite set of axioms, by introducing a satisfaction predicate $Sat(x,y)$. This result was originally given by S.C. Kleene (see here). See,
Craig and Vaught 1958: "Finite Axiomatizability using Additional Predicates" (JSL)
for a strengthened version of Kleene's results.

In philosophy of logic, this point appeared quite a while later, in W.V. Quine's Philosophy of Logic (1970) and then
Leeds, S. 1978: "Theories of Reference and Truth" (Erkenntnis). 
Quine was probably not advocating the deflationary view, and was instead endorsing Tarski's semantic conception of truth. It is a serious error to insist that Tarski was a "deflationist". He argued strongly against the deflationary view of this time, "the redundancy theory". Most contemporary arguments against deflationism are due, in fact, to Tarski. E.g., both the problem of generalizations and the non-conservativeness of axiomatic truth are there in his classic:
Tarski, A. 1936: "Der Wahrheitsbegriff in den formalisierten Sprachen" (Studia Philosophica)
But Leeds was putting forward the deflationary view, and many others have followed suit, including Horwich and Field: the deflationary claim is that the reason that languages contain a truth predicate is so that schemes like (2) can be "re-expressed" as single sentences, like (1). Hence the slogan that
"Truth is merely a device for expressing schematic generalizations"
The presence of "merely" is crucial. Remove the "merely", and the above becomes a theorem of mathematical logic! (As mentioned above.) But with "merely" added, it becomes an interesting philosophical claim.

It is not entirely clear what "re-expresses" means. If "re-expresses" means that one can infer one from the other, then although, in fact, one can infer all instances of (2) from (1) (in some fixed language, assuming disquotation sentences $T(\ulcorner \phi \urcorner) \leftrightarrow \phi$ with $\phi$ in the $T$-free language), one cannot infer (1) from all instances of (2), for compactness reasons. There has been an interesting mini-literature on this topic, and important papers on the topic are:
Halbach, V. 1999: "Disquotationalism and Infinite Conjunctions" (Mind)
Heck, R. 2005: "Truth and Disquotaiton" (Synthese)
Even if the "re-expression" component of deflationism could be clarified and sustained, we still need to understand what "explanatory" comes to, and whether the principles governing truth (or a truth predicate) are never explanatory. In mathematics, there are some proofs considered non-explanatory and some considered explanatory. In mathematical logic, semantic reasoning is sometimes used. After all, one makes use of notions like
"$\phi$ is true in the structure $\mathcal{A}$".
The following example is of the kind given in Stewart Shapiro's "Proof and Truth ..." cited above:
Question: If $G$ is a Godel sentence for $\mathsf{PA}$, it is true. Why is it true?
Answer: Because each axiom of $\mathsf{PA}$ is true, and derivations preserve truth; therefore all theorems of $\mathsf{PA}$ are true. In particular, $G \leftrightarrow \neg Prov_{\mathsf{PA}}(G)$ is a theorem of $\mathsf{PA}$, and is therefore true. So, $G$ is true if and only if $G$ is not a theorem of $PA$. But since all theorems of $\mathsf{PA}$ are true, it follows that if $G$ is a theorem of $\mathsf{PA}$, then $G$ is true. So, $G$ is not a theorem of $\mathsf{PA}$. And therefore $G$ is true.
I should note that one can respond to this by insisting that consistency (rather than soundness) is sufficient to obtain the conclusion that $G$ is true. Then the question turns into one about why one should think $\mathsf{PA}$ is consistent.

Furthermore, one can formalize the truth theory within some object language, containing a good theory of truth bearers and a truth predicate governed by certain axioms, and examine its properties and behaviour. We might extend a theory we accept, such as $\mathsf{PA}$, with a truth predicate $T(x)$, with compositional truth axioms. The result is what Volker Halbach calls $\mathsf{CT}$ in his monograph
Halbach, V. 2011:  Axiomatic Theories of Truth.
It seems to me that, within $\mathsf{CT}$, the role of truth is not merely "expressive". For
  • $\mathsf{CT}$ proves "All theorems of $\mathsf{PA}$ are true";
  • $\mathsf{CT}$ proves new arithmetic facts beyond what $\mathsf{PA}$ does (in particular, coded consistency facts). 
This means that the assumption, "All theorems of $\mathsf{PA}$ are true", used in the above reasoning for the truth of $G$ for $\mathsf{PA}$ can be proved by extending $\mathsf{PA}$ with the compositional truth axioms (with full induction).

Any such truth theory is therefore non-conservative, at least with respect to a large class of of "base theories" ("object language theories", in Tarski's terminology). From this, surely the role of the truth predicate is not "merely expressive". And, if that's right, then the resulting truth theory is non-deflationary.

Later in the interview, Horwich suggests:
Absent a demonstration of this – absent some evidently good theory in which truth is not just a device of generalization – then to speak of deflationism robbing us of a valuable tool is to beg the question.
But we do, at least plausibly, have a demonstration of this: the compositional theory $\mathsf{CT}$.

[Updates (16,17 August). I added some more material related to the reflective adequacy condition on truth theories (Hannes Leitgeb's condition (b) in his article about adequacy conditions on truth theories). I added a link to the paper by Richard Heck on T-sentences and the issue of the role of disquotational truth axioms in the "re-expression"of schematic generalizations.]

Monday, 12 August 2013

101 Dalmatians

Here is an example of applied mathematical reasoning, similar to a couple of examples in a paper "Some More Curious Inferences" (Analysis, 2005), about the phenomenon (discovered by Gödel 1936) of proof speed-up. It's a modification of the kind of arithmetic examples given in the work of Putnam and Field (Science Without Numbers, 1980).
(1) There are exactly 100 food bowls
(2) There are exactly 101 dalmatians
(3) For each dalmatian $d$, there is exactly one food bowl $b$ that $d$ uses.
--------------------------------------
(C) So, there least two dalmatians $d_1, d_2$ that use the same food bowl.
To apply mathematics, we apply Comprehension and Hume's Principle, which allow us to reformulate the premises (1)-(3) and conclusion (C), referring to two mixed sets, $B$ and $D$ and one mixed function $u$:
(1)* $|B| = 100$
(2)* $|D| = 101$
(3)* $u : D \to B$.
--------------------------------------
(C)* So, there are $d_1, d_2 \in D$, with $d_1 \neq d_2$ such that $u(d_1) = u(d_2)$.
To show that this is correct, note first that $100 < 101$. So, $|B| < |D|$. Next, note that the Pigeonhole Principle implies that if $u : D \to B$ and $|B| < |D|$, then $u$ is not injective. So, $u$ is not injective. So, there are distinct $d_1, d_2 \in D$ such that $u(d_1) = u(d_2)$. This is the required conclusion (C)*, which takes us back to (C).

Saturday, 10 August 2013

Humpty Dumptyism

For anyone interested in a rambling dialogue about metasemantics, language individuation, the modal status of semantic (syntactic, etc.) facts, the notion of "cognizing" a language, micro-idiolects, etc., a version of my Humpty Dumptyist dialogue "There's Glory for You!" will appear in the journal Philosophy.

Friday, 9 August 2013

Spacetime Groupoid in GR

Yet another post about equivalence of spacetime models. Apologies. I've been thinking about this since I learnt GR in 1985 (Part III Mathematics at the Other Place). In particular, learning that spacetimes related by "diffeomorphisms" are (are?) the same physical world. "But how can that be?", I thought. If they're different, how can they be the same? It just doesn't make sense ...

But part of the answer is quite simple: spacetime models are spacetime models, mathematical representations: and what they represent is different. So, we need to understand what "representation" means, and what the represented thing---i.e., a physical world---is. So, if $\mathcal{M}$ is a spacetime model and $w$ is a world, we need to understand what
(1) $\mathcal{M}$ represents $w$
means. And we wish to understand it in a way that explains why suitably related $\mathcal{M}_1$ and $\mathcal{M}_2$ represent the same $w$.

I think I know an answer to this, using the diagram conception of structure and worlds. (1) should first be rephrased to include the physical relations $\mathsf{R}_1, \dots$ which "interpret" the model, as
(2) $\mathcal{M}$ represents $w$ relative to $\mathsf{R}_1, \dots$.
This relationship can then be defined as,
(3) $w = \hat{\Phi}_{\mathcal{M}}[\mathsf{R}_1, \dots]$
where $\hat{\Phi}_{\mathcal{M}}$ is the propositional diagram of $\mathcal{M}$. It is an infinitary, categorical, propositional function which defines the isomorphism type of the model $\mathcal{M}$. It is the best candidate for the abstract structure of the model $\mathcal{M}$, since it is what all isomorphic copies "have in common". That is, the mapping
$\mathcal{M} \mapsto \hat{\Phi}_{\mathcal{M}}$
satisfies Leibniz Abstraction:
Leibniz Abstraction
$\hat{\Phi}_{\mathcal{M}_1} = \hat{\Phi}_{\mathcal{M}_2}$ iff $\mathcal{M}_1 \cong \mathcal{M}_2$.
And, from this framework, one can then prove Leibniz Equivalence. So, Leibniz Equivalence is not a peculiarity of a particular physical theory, Einstein's GR. Rather, GR helps to motivate the "propositional diagram conception" of what worlds are, and how they are "represented" by "models", and then Leibniz Equivalence comes out as consequence, not an assumption. And an interesting further consequence of this is that worlds do not have domains. For if one picks a domain, $D_w$, of "things existing at $w$", one can apply a "Hole argument" (an argument quite analogous to the one given in philosophy of physics literature: see, e.g, Norton 2011, "The Hole Argument", SEP) against any such choice, generating indiscernible duplicates, violating Leibniz Equivalence. The selection of a domain is something like a "gauge choice".

I keep wishing to see how this approach to abstract structure, Leibniz equivalence, anti-haecceitism, etc., fits together with the way that category theorists talk, as I don't know enough category theory beyond the basics and basic constructions. In model theory, models look like
$\mathcal{A} = (A, \dots)$, 
and one has some associated language $L$ whose signature matches the signature of $\mathcal{A}$. One can then define $\mathcal{A} \models \phi$ and talk about definable sets in $\mathcal{A}$, etc.; in addition, we may consider the language $L(\mathcal{A})$, obtained by adding to $L$ a constant $\underline{a}$ for each $a \in A$ (where $A$ is the domain of $\mathcal{A}$). This lets one do things a bit more easily: e.g., to prove results about $\mathcal{A}$ by considering the elementary diagram of $\mathcal{A}$.

A "model" is, in other contexts, sometimes called a "mathematical structure" or a "structured set": examples are orderings, graphs, groups, monoids, fields, ordered fields, algebras, lattices, etc. For example:
an ordering $(A, \leq)$ with certain conditions on $\leq$.
a graph $(A, E)$ with certain conditions on $E$ (the edge relation).
monoid $(A, \circ)$ with certain conditions on $\circ$.
group $(A, \circ)$ with certain conditions on $\circ$.
field $(A, 0,1,+, \times)$ with certain conditions on $0$,$1$,$+$, $\times$, etc.
In each case, $A$ is the domain, or carrier set, of the model (structure). And $\leq$, $E$, $+$, etc., are called distinguished relations and operations. A model (structure) is called an algebraic structure just when all of its distinguished parts are operations (except identity, which is taken as primitive). Not every relation/operation on the domain $A$ is distinguished or special, obviously. The special ones are singled out somehow. E.g., when we say,
"a set $A$ with (or equipped with) a binary operation $\circ : A \times A \to A$, etc., ...", 
we mean that this relation/operation $\circ$ is distinguished in the signature of the model. It follows from this that models are individuated as follows:
Individuation for Models/Structures
$(A, R_1, \dots) = (B, S_1, \dots)$ iff $A = B$, and $R_i = S_i$, for each $i$.
This individuation condition explains why models are represented as ordered n-tuples. Models are identical (literally identical: the same "thing") when they have the same carrier set and the same distinguished relations. I.e., extensionally identical. For example, for $n$-ary relations $R, S \subseteq A^n$, we have
$R = S$ iff for any $n$-tuple $\vec{a} \in A^n$, $\vec{a} \in R$ iff $\vec{a} \in S$.
With a bit more effort, one can think of vector spaces (over a field) as models or structured sets, but with modifications. A vector space $\mathbb{V}$ is a 2-sorted model:
$\mathbb{V} = ((V,F); 0, 1, +, \times, +_v, \cdot)$
where there are two carrier sets, $V$ and $F$, and the underlying field is $\mathbb{F} = (F, 0,1,+, \times)$, and
$\cdot : F \times V \to V$
is scalar multiplication, and
$+_v : V \times V \to V$
is vector addition. (Usually, in linear algebra, one just uses the same symbol for the field addition operation, and the vector addition operation.) Passing from the 2-sorted vector space $\mathbb{V}$ to the 1-sorted underlying field $\mathbb{F}$ would be what model theorists call a reduct. Category-theoretically this corresponds to a "forgetful functor". The "inverse" of this is called by model theorists an expansion: i.e., adding new distinguished relations and operations, usually to interpret new symbols in an extended language.

One can then write down axioms $\mathsf{Vect}$ in a 2-sorted language $L$ such that
$\mathbb{V} \models \mathsf{Vect}$ iff $\mathbb{V}$ is a vector space. 
[Here it's not so clear to me how to write down first-order axioms for a topological space $\mathbb{T} = (X, \mathcal{T})$ as a 2-sorted model, because $\mathcal{T} \subseteq \mathcal{P}(X)$ is required to be closed under arbitrary unions, and this requires quantification over sets of sets of points.]

A class of models isomorphic to some given one is called an isomorphism class or an isomorphism type. One asks questions, usually cardinality questions, about isomorphism types. E.g.,
  • how many unique--up-to-isomorphism countable models does $PA$ have?
(The answer to this is $2^{\aleph_0}$, by considering complete extensions of $PA$.)

Let $\mathsf{DLO}$ be the theory of Dense Linear Orders without endpoints. Then:
  • how many unique-up-to-isomorphism countable models does $\mathsf{DLO}$ have?
(The answer to this is $1$. All such orders are isomorphic to $(\mathbb{Q}, \leq)$, a result due originally to Cantor and a proof is here.)

My interest in this, however, comes from General Relativity and Leibniz Equivalence. In GR, one considers spacetime models which are usually presented something like this:
$\mathcal{M} = (M, g, T, \psi, \dots)$
where $M$ is understood to be a differentiable manifold, $g$ and $T$ symmetric $(0,2)$ tensors, and perhaps $\psi$ is some other tensorial field on $M$. We needn't worry about the conditions that these models have to satisfy, and when a model $\mathcal{M}$ is a solution to Einstein's equations. But, given the setup, one can then consider questions like:
"What are the properties of a massless scalar charged field $\phi$ on a 26-dimensional curved space-time $(M,g)$?"
Is this mathematics? Or is it physics? The sense in which it is mathematics is obvious. For
  • You know what a manifold $M$ is.
  • You know that a symmetric $(0,2)$ tensor is.
  • You know what a $\mathbb{C}$-valued function on $M$ is.
  • You know what the differential equations involved are (Einstein's equation, the Klein-Gordon equation).
So, we can, if we work hard, work out properties of such models.

These models are not in fact given in the usual first-order (and possibly many-sorted) way. They involve further structure (definable, though usually second-order definable), related to various vector spaces which appear as values of tensorial fields on $M$. If one tried to make all this explicit---to make explicit everything that is being quantified over---one should probably start with the notion of a principle bundle.

Now, for the philosophy: what does "represent" mean? We have a model $\mathcal{M} = (M, g, \dots)$ and this somehow "represents" physical worlds. The models are themselves just models. The models are not spacetime! They "represent" spacetime. What is the relation between these models when they are to be considered as representing the same physical situation/world? That relationship, it turns out, is simply: isomorphism. That is,
Leibniz Equivalence (see Wald 1984: 428)
If $\mathcal{M}_1 \cong \mathcal{M}_2$, then $\mathcal{M}_1$ and $\mathcal{M}_2$ are physically equivalent.
Some authors suggest that this reflects a kind of "gauge freedom" in GR. Well, if "gauge freedom" is the right word, then gauge freedom in GR is simply selecting one model from an "isomorphism type" of isomorphic spacetime models. This is not a gauge freedom in the ordinary sense of the word, as it appears in, e.g., Yang-Mills gauge theories, where a gauge transformation is applied to a field $\phi$, $A_{\mu}$, etc., living on $M$: e.g., with the vector potential for electromagnetism,
$A \mapsto A^{\prime} = A + d \Lambda$. 
We then have two (mathematically) distinct fields $A$ and $A^{\prime}$, both on the same manifold $M$, but the "physical quantities" they correspond to are identical. One can easily see that the field strengths $F$ and $F^{\prime}$ are identical (because $F = dA$ and $d^2 = 0$). On the other hand, an isomorphism between a pair of different models is quite different from this!

Now, finally, the category theory connection: here I am very grateful to John Baez for a bit of help in the comments to the earlier M-Phi post. In algebra and model theory, we have isomorphism types; in category theory, an isomorphism type is a category, which is a groupoid. It is a connected groupoid, because for any pair objects $x,y$ of the type, there is an (iso)-morphism between them.

So, suppose that we pick some spacetime model $\mathcal{M}$. The collection of all isomorphic "copies" of $\mathcal{M}$ is the collection of all the physically equivalent representations of some (physical) world. This is a connected groupoid whose objects are these models, and whose morphisms are the isomorphisms. I want to call it "the spacetime groupoid" and denote it $\mathsf{Iso}(\mathcal{M})$. (In an earlier post I used a different notation.) That is,
$\mathsf{Iso}(\mathcal{M})$ is the groupoid of all spacetime models isomorphic to $\mathcal{M}$.
Such a groupoid is called a concrete category: all of its objects are "structured sets". But, they are all isomorphic. So, in a sense, what is being "abstracted away" are simply the irrelevant carrier sets.

For  GR, we may consider some solution $(M, g, T, \dots)$ of the relevant differential equations (e.g., famously the Schwarzchild solution one learns about in a GR course and Robertson-Walker models one learns about in a cosmology course). Then the spacetime groupoid $\mathsf{Iso}(\mathcal{M})$ somehow "represents" a single physical world. How does a whole groupoid of models represent a single world?

In the comments to the earlier M-Phi post, John mentions that the groupoid $\mathsf{Iso}(\mathcal{M})$ is equivalent (in the category theory sense) to $\text{Aut}(\mathcal{M})$, the automorphism group of $\mathcal{M}$.

I haven't quite got this clear in my mind. One puzzle is this. Suppose that $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid spacetime models, but not isomorphic to each other. (One can consider, e.g., a pair of non-isomorphic models $(M, g)$ for which there are no isometries: i.e., no Killing fields $v$ such that $\mathcal{L}_vg = 0$). In this case, the automorphism groups are trivial:
$\text{Aut}(\mathcal{M}_1) = 1 = \text{Aut}(\mathcal{M}_2)$.
where $1$ is the trivial group with one element.

So, though this is probably down to my own lack of knowledge of category theory, it seems that if $\mathcal{M}_1$ and $\mathcal{M}_2$ are rigid but non-isomorphic spacetime models, the categories $\mathsf{Iso}(\mathcal{M}_1)$ and $\mathsf{Iso}(\mathcal{M}_2)$---which I'd expected to be distinct---are both equivalent to $1$. And this is very puzzling (the puzzlingness is probably due to my own confusion!).

[Update, 10 August: I changed the title from "Isomorphism Groupoid in GR" to "Spacetime Groupoid in GR", as "Isomorphism Groupoid" is kind of redundant.]

Wednesday, 7 August 2013

Jacquette's Argument for the Inconsistency of Type Theory

Several years ago, Dale Jacquette wrote a piece in Analysis called "Grelling's Revenge" (2004), arguing that type theory (or higher-order logic, it makes not too much difference) is inconsistent. His argument is that Grelling's paradox reappears in type theory. I won't give the argument except to write down his "definition" of $\mathsf{H}^n$:
(D) $\forall F^n \forall F^{n - 1}[\mathsf{H}^n(F^{n-1}) \leftrightarrow \neg F^n(F^{n -1})]$ 
There were several submissions to Analysis replying to it, and the editor showed them to me. These analyses were mistaken; so the editor asked me to write a short reply explaining Jacquette's error. My reply was "Jacquette on Grelling's Paradox" (2005, Analysis).

The error is this: (D) is not a definition, as the definiens contains two free variables, $F^n$ and $F^{n-1}$ while the definiendum, $\mathsf{H}^n(F^{n-1})$, contains only one.

Let L be a language and suppose we introduce a new unary predicate P. And let us set down this "definition":
(D*) $P(x) \leftrightarrow F(x, y)$
where $F(x,y)$ is some formula from $L$. Now this is not an acceptable definition, because the free variable $y$ in the definiens $F(x,y)$ does not appear in the definiendum, $P(x)$.

In some cases, this will lead to inconsistency. For example, consider adding such a "definition" to arithmetic. E.g.,
(D**) $P(x) \leftrightarrow x < y$
Then, since we can prove $1 < 2$, we can prove
$P(1)$
And since we can prove $\neg(1 < 0)$, we can prove
$\neg P(1)$
This is an inconsistency. Does this imply that $\mathsf{PA}$ is inconsistent? Well, no. It is simply caused by the presence of extra free variables in (D**).

Jacquette for some reason does not accept the error and continues to believe that type theory is inconsistent, and that these constraints on definitions are not important. His article is here:
Make of it what you will!

[Update (8th Aug): At the risk of belabouring the point, in his article, Jacquette writes:
$(1) \hspace{5mm} \forall x \forall y(P(x + y) \leftrightarrow x < y) \hspace{2cm}$ Definition
Obviously, (1) is not a definition: it contains both $P$ and the function symbol $+$.]

Tuesday, 6 August 2013

Ehrenfest's Theorem

From Ladyman & Ross 2007 "Every Thing Must Go" (p. 95: review by Cian Dorr here):
Similarly, the equation (known as Ehrenfest’s theorem), $gradV(\langle r \rangle) = m(d^2\langle r \rangle /dt^2)$, where V is the potential and r is the position operator, exhibits continuity between classical and quantum mechanics.
This formula is not quite correct. The expectation value brackets $\langle . \rangle$ are misplaced.

There are many ways of putting it, but in this context, Ehrenfest's Theorem states:
$\frac{d}{dt} \langle p \rangle = - \langle \nabla V \rangle$.
This is very similar though to the usual law from classical mechanics, for the rate of change of momentum of a particle in a field $V$:
$\frac{dp}{dt} = - \nabla V$.

Does the Truth of a Ramsey Sentence Reduce to its Observational Consequences?

From Ladyman, "Structural Realism" (SEP 2007; revision 2009):
As Demopoulos and Friedman point out, if $\Pi$ is consistent, and if all its purely observational consequences are true, then the truth of the corresponding Ramsey sentence follows as a theorem of second-order logic or set theory (provided the initial domain has the right cardinality—and if it does not then consistency implies that there exists one that does).
The wording is nearly the same as that of Demoupolos & Friedman 1985, "Bertrand Russell's Analysis of Matter: Its Historical Context and Contemporary Interest":
... if our theory is consistent, and if all its purely observational consequences are true, then the truth of the Ramsey-sentence follows as a theorem of set theory or second-order logic, provided our initial domain has the right cardinality--if it doesn't, then the consistency of our theory again implies the existence of a domain that does.
But in any case this is not quite right (though it can sort of be modified and fixed: see below). Let $\Theta$ be a finitely axiomatized theory in a language with an O/T distinction. Suppose that $\Re(\Theta)$ is its Ramsey sentence. Suppose that all observational consequences of $\Theta$ are true. It does not follow that "the truth of [$\Re(\Theta)$] follows as a theorem of second-order logic".

A counterexample to this is given in Ketland 2004, "Empirical Adequacy and Ramsification" (BJPS). That counter-example involved the Friedman-Sheard truth theory $FS$. But a simpler counter-example, which I learnt from reading a 1978 paper by van Benthem ("Ramsey Eliminability") around the time when my paper was published, is this. Suppose we think of numbers in $\mathbb{N}$, along with their properties definable in terms of $0$ and $S$, as observational. Let $I = (\mathbb{N}, 0, S)$ be the intended interpretation. So, $(L, I)$ is the interpreted observational language. So, $\phi$ is true if and only if $I \models \phi$. Let our observational theory be $Th_L((\mathbb{N}, 0, S))$.

Let us introduce a new theoretical predicate $F$. Let our additional axioms be
$F(0)$
$\forall x(F(x) \to F(s(x)))$
$\exists x \neg F(x)$ 
This theory is not finitely axiomatizable, but it has a finitely axiomatized conservative extension by adding a new satisfaction predicate. Call the extended language $L^{+}$. Let $T$ be the resulting theory in $L^{+}$. Note two things:
  • $T$ is consistent (using a compactness argument).
  • $T$ is $\omega$-inconsistent. 
So, $T$ has no model whose observational part is isomorphic to $(\mathbb{N}, 0, S)$. All of $T$'s models are non-standard. Let $\Re(T)$ be its Ramsey sentence. Then
(i) All observational consequences of $T$ are true.
(ii) But the truth of $\Re(T)$ does not follow. In fact, $\Re(T)$ is not true.
Given that the intended interpretation of $L$ is $I = (\mathbb{N}, 0, S)$, it follows that
$\Re(T)$ is true if and only if $I$ can be expanded to a model of $T$. 
But $T$ is $\omega$-inconsistent, and therefore no expansion of $(\mathbb{N}, 0, S)$ is a model of $T$. So, $\Re(T)$ is not true. QED.

I noticed this around 2000 or 2001, and worried about how to fix it for quite a while, so as to get something that approximates the claim in Demopoulos & Friedman 1985.

There is a fix, which is the one I used in 2004, "Empirical Adequacy and Ramsification". It involved changing the structure of the language used to formulate scientific theories to a 2-sorted language, including a separate range of variables for unobservable objects. In this setting, even given a fixed observational interpretation $I$, one can extend the domain (of a model) representing the unobservable objects and define various relations etc., as need be. But even so, one still requires that the original theory $T$ must have an empirically correct model $\mathcal{A}$: i.e., a model whose purely observational part is isomorphic to the intended one $I$. It is not sufficient that merely all theorems of $T$ in the observational language be true in $I$.

Conservativity for Compositional Truth

For those interested in axiomatic truth theories, conservativity/conservativeness, deflationism, etc., here is a new article:
Graham Leigh (Oxford): "Conservativity for theories of compositional truth by cut-elimination"
giving a proof-theoretic proof of conservativity of the theory $\mathsf{CT}$ of compositional truth over certain theories of arithmetic.

(People vary in the word used: conservation, conservativeness, conservativity? All mean the same in these debates.)

Sunday, 4 August 2013

Sean Carroll on Leibniz Equivalence

Below is a formulation of Leibniz Equivalence from Sean Carroll's (online) Lecture Notes on General Relativity (Sc. 5, "More Geometry"):
Let's put some of these ideas into the context of general relativity. You will often hear it proclaimed that GR is a "diffeomorphism invariant" theory. What this means is that, if the universe is represented by a manifold $M$ with metric $g_{\mu \nu}$ and matter fields $\psi$, and $\phi : M \to M$ is a diffeomorphism, then the sets $(M, g_{\mu\nu},\psi)$ and $(M, \phi_{\ast}g_{\mu\nu},\phi_{\ast}\psi)$ represent the same physical situation.
This principle might be called Weak Leibniz Equivalence, a version of the frequent claim that the "gauge freedom" of GR is $\text{Diff}(M)$. This version is fairly standard. For example, it is the same as John Norton's in his SEP article "The Hole Argument" (from note 5):
For any spacetime model $(M, O_1, \dots, O_n)$ and any diffeomorphism $h$ on $M$, Leibniz equivalence asserts that the two models
$(M, O_1, \dots, O_n)$ and $(hM, h^{\ast}O_1, \dots, h^{\ast}O_n)$
represent the same physical system.
And here is the formulation by Urs Schreiber in n-Lab:
In physics, the term general covariance is meant to indicate the property of a physical system or model (in theoretical physics) whose configurations, action functional and equations of motion are all equivariant under the action of the diffeomorphism group on the smooth manifold underlying the spacetime or the worldvolume of the system.
These formulations are weaker than Robert Wald's formulation (from General Relativity 1984: 438):
Leibniz Equivalence (LE)
If spacetimes $(M, g_{\mu\nu},\psi)$ and $(N, h_{\mu\nu},\theta)$ are isomorphic, then they represent the same physical situation.
(This is equivalent to Wald's wording, which can be read here.) Wald's formulation refers to pairs of spacetime models, rather than a single spacetime model.

(LE) implies the weaker principle. For suppose that $(M, g,\psi)$ is our spacetime model, and $\phi : M \to M$ is a diffeomorphism. (I ignore tensor indices.) Then, given definitions of a "pushforward" $\phi_{\ast}$, the models $(M, g,\psi)$ and $(M, \phi_{\ast}g, \phi_{\ast}\psi)$ are isomorphic. Then, by (LE), they represent the same physical situation.

However, (LE) is stronger than the weaker principle. Let $(M, g,\psi)$ be our spacetime model and let us identify the internal structure of the manifold $M = (X, \mathcal{C})$, where $X$ is the carrier set and $\mathcal{C}$ is a maximal atlas on $X$ ensuring that $M$ is a differentiable manifold. Now let
$\pi : X \to X$
be an arbitrary permutation of the carrier set $X$. We may pushforward the charts in $\mathcal{C}$ to $\pi_{\ast}\mathcal{C}$. The resulting space $(X, \pi_{\ast}\mathcal{C})$ is, by construction, isomorphic to the original. Now let $N = (X, \pi_{\ast}\mathcal{C})$. It follows that
$\pi : M \to N$
is a diffeomorphism.

Apply the diffeomorphism $\pi$ to the metric and matter field(s). The result $(N, \pi_{\ast}g, \pi_{\ast}\psi)$ is again isomorphic, by construction, to $(M, g, \psi)$. So, by (LE), $(M, g, \psi)$ and $(N, \pi_{\ast}g, \pi_{\ast}\psi)$ represent the same physical situation.

So, (LE) is much stronger than the claim that GR is "diffeomorphism invariant". (LE) says that General Relativity is "permutation invariant": one can permute the base set $X$ of a spacetime model $(M, g, \psi)$ any way one likes, so long as one pushes forward everything else along the permutation. It tells us that the identity of the points in the base set $X$ does not matter. Only the "overall structural pattern"---including the pattern involving the topology (i.e., the collection of open sets), the metric and matter fields---matters.

Carroll goes on to give an informal intuitive explanation.
Since diffeomorphisms are just active coordinate transformations, this is a highbrow way of saying that the theory is coordinate invariant. Although such a statement is true, it is a source of great misunderstanding, for the simple fact that it conveys very little information. Any semi-respectable theory of physics is coordinate invariant, including those based on special relativity or Newtonian mechanics; GR is not unique in this regard. When people say that GR is diffeomorphism invariant, more likely than not they have one of two (closely related) concepts in mind: the theory is free of "prior geometry", and there is no preferred coordinate system for spacetime. The first of these stems from the fact that the metric is a dynamical variable, and along with it the connection and volume element and so forth. Nothing is given to us ahead of time, unlike in classical mechanics or SR. As a consequence, there is no way to simplify life by sticking to a specific coordinate system adapted to some absolute elements of the geometry. This state of affairs forces us to be very careful; it is possible that two purportedly distinct configurations (of matter and metric) in GR are actually "the same", related by a diffeomorphism. In a path integral approach to quantum gravity, where we would like to sum over all possible configurations, special care must be taken not to overcount by allowing physically indistinguishable configurations to contribute more than once. In SR or Newtonian mechanics, meanwhile, the existence of a preferred set of coordinates saves us from such ambiguities. The fact that GR has no preferred coordinate system is often garbled into the statement that it is coordinate invariant (or "generally covariant"); both things are true, but one has more content than the other.
If the argument given above is correct, one should add to this that purportedly distinct configurations (of topology, matter and metric) for spacetime models $\mathcal{M}_1$ and $\mathcal{M}_2$ may be related by a permutation of the carrier set, even a completely wild permutation (e.g., one that simply transposes two distinct points). If so, they represent the same physical situation.