# Rules and Meaning in Quantum Mechanics

Iulian D. Toader Institute Vienna Circle, University of Vienna,
<EMAIL_ADDRESS>

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it
pursues an investigation at an intersection of the philosophy of physics and
the philosophy of semantics, and it offers a critical analysis of rival
explanations of the semantic facts of standard QM. Such facts, about the
meaning of its formalism, are typically taken to be grounded in language-world
relations or in the rule-based linguistic practices of scientific communities.
These relations and practices raise a series of important questions, two of
which are taken up and analyzed at length. The problem of completeness,
understood as categoricity, is one focus of the book, discussed not only as a
problem for standard QM, but also for quantum logic (QL). The problem of
permanence, understood as perseverance of rules in the transition from
classical physics to QM, is another focus. Both problems are discussed, as
metasemantic questions, in their historical contexts, but it should be clear
that their philosophical significance transcends these contexts.

Anyone whose research partly overlaps with the span of these domains might be
interested in my analysis of these problems. The new results of this analysis
include a reconstruction of Einstein’s incompleteness argument, which
concludes that a local, separable, and categorical QM cannot exist; a
reinterpretation of Bohr’s principle of correspondence, grounded in permanence
as a metatheoretical principle; an improved meaning-variance argument for QL,
which follows a line of critical reflections initiated by Weyl; and an
argument for semantic indeterminacy leveled against inferentialism about QM,
inspired by Carnap’s work in the philosophy of classical logic. Some of these
arguments are drawn from my previous papers, as noted in the right places in
the book. Two papers, on QL, have been co-authored with Sebastian Horvat, who
also read a full draft of the book and made numerous suggestions for
improvement. Most of the material from previous papers has been rewritten
here, so any remaining errors are my responsibility entirely.

###### Contents

  1. 1 The metasemantics of quantum mechanics
    1. 1.1 Hilbert’s question
    2. 1.2 Representationalism vs. non-representationalism
  2. 2 The problem of categoricity
    1. 2.1 Intended limitations of descriptive completeness
      1. 2.1.1 Hertz on illegitimate questions
      2. 2.1.2 Weyl’s universal boundary
    2. 2.2 The categoricity of quantum mechanics
      1. 2.2.1 Preliminary considerations
      2. 2.2.2 The Stone-von Neumann theorem
    3. 2.3 Einstein completeness as categoricity
      1. 2.3.1 Misconstruals of Einstein completeness
      2. 2.3.2 Reconstruction of Einstein’s argument
  3. 3 The problem of permanence
    1. 3.1 Historical roots and metasemantic implications
      1. 3.1.1 Peacock’s principle of permanence
      2. 3.1.2 Hankel’s principle of permanence
    2. 3.2 Bohr’s correspondence principle
      1. 3.2.1 The nature of the correspondence principle
      2. 3.2.2 Correspondence as permanence of rules
      3. 3.2.3 Howard on Bohr’s essential equivalence
    3. 3.3 The permanence of the rule of modularity
      1. 3.3.1 Birkoff and von Neumann’s quantum logic
      2. 3.3.2 Weyl’s objections to quantum logic
  4. 4 The logic of quantum mechanics
    1. 4.1 Rational adjudication against classical logic
      1. 4.1.1 Putnam’s logical revisionism
      2. 4.1.2 Rumfitt’s rule-circularity objection
      3. 4.1.3 Revising the eigenstate-eigenvalue link
    2. 4.2 Quantum logic and meaning
      1. 4.2.1 Abstract formulation of quantum logic
      2. 4.2.2 Bivalence and metaphysical realism
      3. 4.2.3 Hellman’s meaning-variance argument
    3. 4.3 Applied classical mathematics and quantum logic
      1. 4.3.1 Williamson’s attack on deviant logics
      2. 4.3.2 Quantum models and idealized tests
      3. 4.3.3 Withstanding the attack on quantum logic
  5. 5 Quantum inferentialism
    1. 5.1 Non-representationalist approaches to physics
      1. 5.1.1 Global inferentialism
      2. 5.1.2 Healey’s quantum inferentialism
    2. 5.2 Two preliminary objections to quantum inferentialism
      1. 5.2.1 Reconstructions of quantum mechanics
      2. 5.2.2 Extended Wigner’s friend scenarios
    3. 5.3 The categoricity problem for inferentialism
      1. 5.3.1 Carnap’s categoricity problem
      2. 5.3.2 A categoricity problem for quantum inferentialism
  6. 6 The logical perfection of physical theories
    1. 6.1 Categoricity as a metasemantic ideal
    2. 6.2 Imperfect knowledge of logical perfection
  7. 7 References

##  1 The metasemantics of quantum mechanics

One cannot fully understand a theory – any theory – without knowing how its
meaning is determined, without figuring out the purported relationship between
the language of the theory and its semantic attributes. Explaining this
relationship is the main task of metasemantics. After introducing the task, in
general terms, this chapter presents the overall theme of the book: the
potential contribution of metasemantics to an understanding of quantum
mechanics (QM). The focus is on problems for both representationalism and non-
representationalism, as rival metasemantic views that have been adopted in the
historical development of QM.

###  1.1 Hilbert’s question

The metasemantics of QM, as understood in this book, is concerned with
clarifying the relationship between the formalism of the physical theory and
its semantics, quite apart from the nature or existence of the reality that
the theory might be supposed to describe. In this sense, it is a particular
case of a general question formulated by David Hilbert in his “Axiomatic
Thinking”, a lecture delivered in 1917 at the ETH in Zurich. He described
there two development strategies that he took to characterize the axiomatic
method: a progressive strategy, which leads from a system of axioms to what
can be logically derived from them, and a regressive strategy, which leads to
deeper axioms, uncovered by a critical examination of the system. Then Hilbert
mentioned a series of “difficult epistemological questions”, such as the
questions about the internal and external consistency, the independence, and
the decidability of an axiomatic system, but also

> the question of the relationship between content and formalism in
> mathematics and logic. (Hilbert 1918, 1113)

Although this expresses a particular concern with classical mathematics and
logic, Hilbert’s question can be generalized, and extended to physics as well.
Indeed, Hilbert himself would later become very much concerned with the
relationship between content and formalism in QM (Hilbert et al. 1928).

In the present book, I pursue an investigation of the metasemantics of
standard QM and its quantum logic. This should extend to the interpretations
and the information-theoretical reconstructions of QM, but this extension will
not be considered here, although the relation between interpretations and
reconstructions will be briefly discussed towards the end of the book, when I
consider several arguments that reconstructionists have proposed against the
rational acceptability of realist (or so-called $\psi$-ontic) interpretations
of QM, such as the many-worlds, hidden variables, or spontaneous collapse
interpretations.111For an useful overview of interpretations, with a special
metaphysical knack, see Lewis 2016. For all things interpretational, see
Freire Jr. et al. 2022.

What is metasemantics? Roughly, just as metaethics is the philosophy of
ethics, and metaphilosophy is the philosophy of philosophy, and perhaps just
as metaphysics is (or ought to be, at least in part) philosophy of physics, so
metasemantics is the philosophy of semantics. As David Kaplan characterized
it:

> The fact that a word or phrase has a certain meaning clearly belongs to
> semantics. On the other hand, a claim about the basis for ascribing a
> certain meaning to a word or phrase does not belong to semantics. “Ohsnay”
> means snow in Pig-Latin. That’s a semantic fact about Pig-Latin. The reason
> why “ohsnay” means snow is not a semantic fact; … because it is a fact about
> semantics, as part of the Metasemantics of Pig-Latin (or perhaps, for those
> who prefer working from below to working from above, as part of the
> Foundations of semantics of Pig-Latin). (Kaplan 1989, 573sq)

Working from below, Robert Stalnaker considered the explanation of semantic
facts as an essentially foundational investigation:

> First, there are questions of what I will call “descriptive semantics”. A
> descriptive‐semantic theory is a theory that says what the semantics for the
> language is, without saying what it is about the practice of using that
> language that explains why that semantics is the right one. … Second, there
> are questions … of “foundational semantics”, about what the facts are that
> give expressions their semantic values, or more generally, about what makes
> it the case that the language spoken by a particular individual or community
> has a particular descriptive semantics.222Cf. Stalnaker 1997, 903. Both
> quotations, from Kaplan and Stalnaker, are given in García-Carpintero 2012a
> (see also García-Carpintero 2012b) and in Burgess and Sherman 2014, which
> also describe other possible approaches to metasemantics.

Quite generally, semantics assigns semantic attributes, i.e., content,
meaning, and truth values, to expressions in a language (singular terms,
sentences, etc.). To put it differently, semantics states or describes
semantic facts, such as the fact that “oshnay” means snow, or for that matter
the fact that “snow” means snow. Metasemantics is the part of philosophy
concerned with what explains such assignments, with what determines semantic
facts. It is concerned, more exactly, with understanding the nature of the
relationship between linguistic expressions and their semantic attributes,
with uncovering the grounds of semantic facts. While metasemantics concerns
semantics in general, I will be especially interested in the semantics of QM
and its logic, and more particularly, in the constraints that ought to be
imposed on that relationship such that a precise explanation of semantic
attributes, a precise determination of semantic facts, becomes possible.

Typical questions in metasemantics ask for reasons why expressions have the
meaning they have: what exactly makes it the case that they have that meaning,
or as I prefer to put it, in virtue of what do they have that meaning? Two
general answers to such questions are formulated in terms of a language-world
relation of representation or in terms of rule-based linguistic practices.
According to one answer, expressions have meaning in virtue of their
representational properties. Why does “Findus is on the mat” mean that Findus
is on the mat? It is because “Findus is on the mat” represents a state of
affairs in which Findus is on the mat. Semantic facts are determined by
representation relations. This answer characterizes what I will call a
representationalist metasemantics. According to the other answer, expressions
have meaning independently of their representational properties (if they have
any), only in virtue of their use properties. Why does “Findus is on the mat”
mean that Findus is on the mat? It is because “Findus is on the mat” is used
in accordance with the communication protocols and practices of a community
that shares the relevant language. Semantic facts are determined by
communicational uses. This answer characterizes what I will call a non-
representationalist metasemantics.

The split between representationalist and non-representationalist
metasemantics can also be explicated in terms of rules, if one takes
representation relations to be specified by semantic rules that postulate
correspondences between linguistic expressions and extra-linguistic states of
affairs, while communicational uses are specified by non-semantic rules which
state conditions for the proper employment of expressions within a community
of speakers. Note that a non-representationalist metasemantics, according to
which non-semantic rules do all the metasemantic work that is to be done, need
not require that there be no semantic rules specifying the representational
properties of expressions. A semantic fact, e.g., “Findus is on the mat” means
that Findus is on the mat, can be determined by use in accordance with
communication protocols and practices, even though “Findus is on the mat” does
represent a state of affairs in which Findus is on the mat.

###  1.2 Representationalism vs. non-representationalism

These rival metasemantic views are easily identified in the historical
development of QM. In a Schrödinger’s cat scenario, “Findus is both dead and
alive” could mean that Findus is both dead and alive, and this semantic fact
may be understood in virtue of that expression’s representing a superposition
of my cat’s states.333With thanks and apologies to Sven Nordqvist. A
representationalist metasemantics of QM, just like that of the natural
language, explains the relationship between expressions and their semantic
attributes via representational properties. Semantic rules specifying
representational properties for quantum expressions are naturally adopted in
QM, as in its standard axiomatization due to John von Neumann (see below
section 2.2.1), and have been adopted by most interpretations of the theory,
especially the $\psi$-ontic interpretations.

For example, Tim Maudlin has maintained the following: “The mathematical
object [i.e., the $\psi$-function] must be supplemented with a physical
ontology and semantic rules specifying how the physical ontology is
represented by the wavefunction.” (Maudlin 2016, 8). Similar statements can be
found in his recent book on quantum theory: “a wavefunction is a mathematical
item – as ‘function’ testifies – and the quantum state is whatever real
physical feature of an individual system (if any) obtains iff the system is
represented by a given wavefunction.” (Maudlin 2019, 81) On this view,
semantic rules postulate a correspondence between physical states and
$\psi$-functions, between physical magnitudes and self-adjoint operators, etc.
But what explains the semantic attributes of these mathematical expressions?
In virtue of what do they have any meaning? One way in which the
representationalist can attempt to answer this question is by pointing to the
representational properties specified by their semantic rules. These rules are
then taken to do double work: they both assign semantic attributes to
expressions and explain in virtue of what these expressions have such
attributes.444Another way to answer the question, which assumes however a
problematic distinction between theoretical and observational terms that has
been long considered as such by philosophers of science, is suggested by the
view held by Howard Stein, who maintained that quantum concepts have “the
status of theoretical terms, whose empirical application has to be given by
some sort of ‘correspondence rules’.” (Stein 1970, 96) Correspondence rules
can do metasemantic work in the following way: they explain the semantic
attributes of quantum concepts via the representational properties of the
observational terms, rather than their own representational properties (if
they have any). For more recent accounts of the representational capacities of
quantum models, see e.g. van Fraassen 2008, Hughes 2010.

A non-representationalist metasemantics of QM, like that of the natural
language, understands the relationship between expressions and their semantic
attributes via use properties. In the Schrödinger’s cat scenario, the fact
that “Findus is both dead and alive” means that Findus is both dead and alive
is determined in virtue of the communicational use of this expression, rather
than in virtue of its representing a superposition of cat states. A version of
non-representationalism, dispensing with any semantic rules for quantum
concepts, was arguably endorsed by Niels Bohr and then adopted by neo-
Bohrians, implicitly or explicitly, ever since. For example, Richard Healey
has recently declared: “All our concepts, including those of physics, derive
their content from use in human communication.” (Healey 2017, 203) On this
view, the semantic attributes of mathematical expressions like
$\psi$-functions and self-adjoint operators are determined, as we will see, in
virtue of their inferential properties, rather than any purported
representational capacities of quantum models. Semantic rules are simply idle:
they neither assign semantic attributes to expressions, nor explain in virtue
of what these expressions have such attributes.

These rival explanations of the semantic facts of QM raise important
philosophical questions, which I think should be carefully and extensively
analyzed. A good grasp of these explanations, their features, their problems,
and their implications might prove beneficial, perhaps even indispensable, to
our understanding of QM. Two problems that I will be concerned with in this
book are the problem of categoricity and the problem of permanence.

The problem of categoricity is the main focus of chapter 2. Categoricity has
been extensively discussed in philosophy of mathematics and logic, though not
so much in philosophy of science, and almost not at all in philosophy of
physics. As we will see, Hermann Weyl understood categoricity as the
expression of a certain limitation of the descriptive completeness of a
theory, intended to prevent epistemic commitment to non-relational properties
of objects. A categorical theory, i.e., a theory the models of which belong to
the same isomorphism class, constrains the relationship between its linguistic
expressions and their semantic attributes such that the latter can be
precisely determined. Thus, I will say that a theory has a categoricity
problem when its language is semantically indeterminate, i.e., the semantic
attributes of its vocabulary cannot be precisely determined, because the
theory allows non-isomorphic models.

I will adopt a distinction between global and local models that a scientific
theory can admit of (Hughes 2010), and I will address the question whether,
for any physical system, standard QM can admit of local models, i.e., models
that are assigned to that individual system relative to a given physical
location, such that all of them belong to the same isomorphism class. This
question is, thus, significantly different than the question about the
categoricity of mathematical theories like, say, arithmetic. To address the
question about QM, I will first discuss some worries about whether the
question is well posed or well posedness requires an interpretation of QM, or
a rational reconstruction, i.e., a re-axiomatization in a formal (ideally,
first-order) language. Then I will provide a critical discussion of the
intuitive claim that the Stone-von Neumann theorem can be understood as a
categoricity result, focused on the nature of, and the assumptions behind, the
relation of unitary equivalence described by the theorem. Taking up a
suggestion by Don Howard (1990; 1992) that one could read Einstein
completeness as categoricity, I will give a rigorous reconstruction of
Einstein’s incompleteness argument that clarifies the assumptions behind that
suggestion and shows that his no-go result may be interpreted as claiming that
a local, separable, and categorical QM cannot exist. Against
representationalist metasemantics, this implies that the semantic attributes
of a local and separable QM cannot be precisely determined by semantic rules,
because this theory allows non-isomorphic models. This, of course, does not
imply that nonlocal QM is categorical.

The problem of permanence is the main focus of chapter 3. It is concerned with
the application of the principle of permanence of rules (forms or formal
laws). This principle has been almost exclusively a topic for the history and
philosophy of mathematics, where it is typically regarded as the precursor of
the modern notion of conservativity. It is arguably also the precursor of the
notion of open-endedness, much debated in contemporary philosophy of logic.
But it has been largely ignored in the philosophy of physics. Nevertheless, as
we will see, Bohr understood his correspondence principle as grounded in a
version of the principle of permanence that had been crucial to the
development of 19th century mathematics, e.g., in the work of Hermann Hankel.
As a problem in metasemantics, the application of the principle of permanence
concerns the explanation of the semantic attributes of a theory developed in
accordance with this principle. What does the preservation of rules imply with
respect to semantics?

My analysis of the principle of permanence, and the emphasis on its
metasemantic implications, will help us explain why Bohr believed that QM
should be regarded as a “rational generalization” of classical physics, and
that the meaning of QM is to be determined by its non-semantic rules
(especially the Born rule) – a non-representationalist metasemantics tersely
expressed in his reply to the EPR paper. I will explain what Bohr understood
by this, and how it exposed his related doctrine of the necessity of classical
concepts to criticism. The principle of permanence was also applied by von
Neumann to the transition from finite- to infinite-dimensional QM, in his work
with Garrett Birkhoff leading to the introduction of quantum logic (QL).
Although partly justified by this principle, their preservation of the logical
rule of modularity (rather than distributivity) was not taken to have any non-
representationalist implications: they did not hold, for instance, that the
semantic attributes of QL terms are determined by its logical rules. Rather,
they took these attributes to be precisely determined by semantic rules.

However, just like the problem of permanence, the problem of categoricity
arises not only for mathematical and physical theories; it arises also for
logics. Here, instead of asking whether the models of a theory are all
mutually isomorphic, the issue is whether a logical calculus admits of truth
valuations that are all mutually isomorphic (or, more precisely, whether all
valuations are homomorphisms from the set of sentences to the two-element set
$\\{0,1\\}$ of possible truth values). The fact that this is not the case for
the standard system of classical logic (CL) was proved by Carnap (1943), who
solved this categoricity problem by introducing a rejection rule and allowing
valid arguments with multiple conclusions. But the fact that it is not the
case for the QL introduced by Birkhoff and von Neumann had been already shown
by Weyl (1940): the semantic attributes of QL connectives are not precisely
determined by their semantic rules.

I will further pursue the line of critical reflection initiated by Weyl, in
chapter 4, where I will focus on arguments developed after Hilary Putnam
(1968) enrolled QL in the service of realist metaphysics, which was intended
to oust what he took to be some unpalatable metaphysical hypotheses, e.g., the
existence of hidden variables. One of these arguments, leveled against
Putnam’s logical revisionism, is based on an assumption about the meaning of
connectives and their truth-functionality (Hellman 1980). I will explain how,
on the same assumption, one can significantly improve the argument and thus
finally bury QL revisionism for metasemantic reasons: the semantic attributes
of the QL connectives are indeed different than the semantic attributes of
their CL counterparts. This deflates Ian Rumfitt’s argument to the effect that
QL revisionism is impossible on rational grounds because the rule-circularity
of the proof against distributivity in QM shows that one cannot rationally
adjudicate against CL (Rumfitt 2015). But I will show that this argument could
be resisted even if CL and QL connectives were semantically equivalent, by
appropriately revising the eigenstate-eigenvalue link of standard QM. I will
then explain how to withstand Timothy Williamson’s renewed attack on deviant
logics (Williamson 2018), in the particular case of QL, by recalling the
seemingly forgotten point that QL is consistent with the application of
classical mathematics in QM.

In chapter 5, I will return to non-representationalist metasemantics. For
natural languages, non-representationalism was defended by Wilfried Sellars,
who proposed a global inferentialist metasemantics: “There is nothing to a
conceptual apparatus that isn’t determined by its rules.” (Sellars 1953, 337)
Drawing on Robert Brandom’s further development of Sellars’ view (Brandom
1994; 2000), Healey has embraced inferentialism, as already noted, as a
metasemantics for standard QM. After comparing his version of quantum
inferentialism with the more encompassing expressivist view defended by Huw
Price for the whole of science (Price 2013), I will present two preliminary
objections to Healey’s view, one from recent information-theoretical
reconstructions of QM, and the other from just as recent, relativistic
extensions of Wigner’s friend scenario. The first objection points out that
the metasemantics of reconstructed QM is not inferentialist, since without
correspondence rules that assign informational-theoretical properties to
quantum concepts the semantic attributes of the latter cannot be explained.
The second objection shows that Healey’s attempt to make the relativistic
extensions of Wigner’s friend scenario compatible with the objectivity of QM
weakens his inferentialism, since it adds a semantic constraint on truth
valuations, rather than one based on inferential rules.

Inspired by the categoricity problem identified by Carnap (as mentioned
above), and rightly regarded today as a challenge for logical inferentialism
about CL, I will then formulate a categoricity problem for quantum
inferentialism. This problem, I will argue, is raised by the failure of the
rules of material inference in standard QM to determine, relative to a system
and a physical location, a quantum state assignment – a local model – that is
unique up to isomorphism. Against inferentialist metasemantics, this implies
that the semantic attributes of the standard formalism of QM cannot be
precisely determined by its inferential rules. For the scientific practice,
this may seem inconsequential, just like it seems negligible for our daily
argumentative practice that logical connectives are semantically
indeterminate. But for inferentialism as a metasemantics of QM, it is a
serious problem since it points out that the meaning of quantum concepts fails
to be articulated in just the way the inferentialist claims it is in fact
articulated. It shows that the objection that inferentialism is “unable to
nail down a claim’s content sufficiently to explain how this may be
unambiguously communicated in public discourse” (Healey 2017, 211) has not
been successfully addressed. In order to successfully address this objection,
the quantum inferentialist should solve the categoricity problem in a way that
is fully compatible with inferentialism.

In the concluding chapter, I will articulate two suggestions for further work.
First, categoricity can be conceptualized as the expression of an ideal of
logical perfection. Historically, this draws a line from Einstein’s
characterization of principle theories (1919) to a recent development due to
Boris Zilber (2014). The latter illustrates the view that first-order
languages should not be discounted in the metasemantic analysis of QM, despite
their Löwenheim-Skolem properties. Secondly, the seemingly unworkable project
of a rational reconstruction of QM can be the basis of metasemantic analysis.
But since, due to Gödel’s second incompleteness theorem, the consistency of
any viable rational reconstruction cannot be proved in an informative way, any
such analysis may necessarily yield only imperfect knowledge of logical
perfection.

##  2 The problem of categoricity

This chapter introduces the categoricity problem for standard quantum
mechanics (QM). An intuitive reading of the Stone-von Neumann theorem as a
categoricity result is critically discussed. Drawing on this, a rigorous
reconstruction of Einstein’s incompleteness argument – the argument that he
thought did not make it into the EPR paper – is then proposed. Since
categoricity is a type of completeness, some intended limitations of the
descriptive completeness of physical theories are presented, which are
motivated by the view, advocated by Hertz and Weyl, that some things are
beyond what a complete theory ought to tell us about its domains of
application.

###  2.1 Intended limitations of descriptive completeness

Various types of completeness were formulated, before and after the early
1930s, when von Neumann’s axioms for QM emerged. For instance, Russell and
Whitehead had stated that a system of axioms for pure mathematics, like that
presented in their Principia Mathematica, is complete just in case it is able
to capture “as much as may seem necessary” of the domain that it aims to
describe, or more precisely, just in case it logically derives the whole class
of theorems of ordinary mathematics. Similarly, von Neumann maintained that a
system of axioms for QM, like the standard system presented in his
Mathematical Foundations of Quantum Mechanics, is complete just in case it is
able to derive all statistical formulas of QM. But there is no doubt that
there are limits to such descriptive completeness that are inevitable and
clearly unintended by the proponent of an axiomatic system. On some arguably
reasonable assumptions, Gödel proved that there are statements in the class of
theorems which cannot be derived in the system of Principia Mathematica, and
he took this to show that system necessarily incomplete.555For a recent
discussion of Russell and Whitehead’s notion of descriptive completeness, and
an argument that Gödel’s own understanding of completeness, as negation-
completeness, is however relevantly different than their descriptive variety,
see Detlefsen 2014. Likewise, von Neumann noted that the statistical nature of
QM formulas could be thought to betray “an ambiguity (i.e., incompleteness) in
our description of nature” and to motivate an attempt to eliminate this
incompleteness by adding statements about hidden dynamical variables to the
theory. But he rejected such attempts as inconsistent with his system of
axioms.666See Acuña 2021 and Mitsch 2022 for recent re-evaluations of (the
debate on) von Neumann’s proof of the descriptive completeness of QM.

In any case, there appears to be no doubt that there are intended limitations
of descriptive completeness as well. These concern statements that are
considered epistemically defective or illegitimate, rather than statements
that are provably impossible to derive in a system. One argument to this
effect was given by Hermann Hertz, who justified an intended limitation of
descriptive completeness on the basis of a notion of epistemic comfort.
Another argument to a similar effect was offered by Hermann Weyl, who thought
that descriptive completeness should be constrained by an invariantist notion
of objectivity. They both concern the representation relation between the
statements of a scientific theory and the phenomena within its domain of
applicability. Even a cursory look at these arguments will help us appreciate,
when we later turn to QM, the significance of the question about its
categoricity.

####  2.1.1 Hertz on illegitimate questions

Hertz famously diagnosed a certain tendency to ask what he called “illegimate
questions” in science, e.g., questions asked in mechanics about the essence
(Wesen) of force or electricity, expressed his skepticism that they can be
answered, and offered a solution as to how to avoid asking them in the first
place. Here is the full passage:

> Weighty evidence seems to be furnished by the statements which one hears
> with wearisome frequency, that the essence of force is still a mystery, that
> one of the chief problems of physics is the investigation of the essence of
> force, and so on. In the same way electricians are continually attacked as
> to the essence of electricity. Now, why is it that people never in this way
> ask what is the essence of gold, or what is the essence of velocity? Is the
> essence of gold better known to us than that of electricity, of the essence
> of velocity better than that of force? Can we by our conceptions, by our
> words, completely represent the essence of any thing? Certainly not. I fancy
> the difference must lie in this. With the terms “velocity” and “gold” we
> connect a number of relations to other terms; and between all these
> relations we find no contradictions which offend us. We are therefore
> satisfied and ask no further questions. But we have accumulated around the
> terms “force” and “electricity” more relations than can be reconciled
> amongst themselves. We have an obscure feeling of this and want to have
> things cleared up. Our confused wish finds expression in the confused
> question as to the essence of force and electricity. But the answer which we
> want is not really an answer to this question. It is not by finding out more
> and fresh relations and connections that it can be answered; but by removing
> the contradictions existing between those already known, and thus perhaps by
> reducing their number. When these painful contradictions are removed, the
> question as to the essence of force will not have been answered; but our
> minds, no longer vexed, will cease to ask illegitimate questions.777Cf.
> Hertz 1899, 7sq. I have replaced the term “nature” in the translation of
> “Wesen” by “essence”.

Contradictions are identified when the relations between the statements of
science, including statements about force, are carefully attended to. They are
perceived as painful because, as Hertz assumed, they create mental vexation.
But since, as seems natural to believe, one desires epistemic comfort, and
since introducing more relations by adding statements about the essence of
force cannot remove those contradictions, one should instead focus primarily
on removing the contradictions. One should continue to remove contradictions
until epistemic comfort is achieved, to the extent that the very tendency to
ask illegitimate questions is eliminated.888See Eisenthal 2021, for a
discussion of a particular ambiguity in Newton’s laws of motion, due to the
use of a slightly different notion of force in the third law than that used in
the first and second laws, which Hertz thought created mental vexation with
respect to the meaning of “force”.

This argument, it seems to me, can be best understood on the background of a
19th century view according to which the removal of contradictions is a
necessary and sufficient goal of scientific inquiry. Peirce expressed this
view when he characterized consciousness as perceived consistency: “Again,
consciousness is sometimes used to signify the I think, or unity in thought;
but the unity is nothing but consistency, or the recognition of it.” (Peirce
1868, CP 5.313) He argued that scientific inquiry lacks proper motivation when
there is no doubt that is caused by inconsistencies: “When doubt ceases,
mental action on the subject comes to an end; and, if it did go on, it would
be without a purpose.” (Peirce 1877, CP 5.376) Once contradictions are
removed, mental vexation ceases, belief settles, and purpose-driven inquiry
stops.

Unlike Peirce, Hertz thought that epistemic comfort was not the end of
scientific inquiry, but a precondition for asking only legitimate questions.
It is worth emphasizing how Hertz believed one may achieve such epistemic
comfort in science. His proposal is not to remove contradictions by reducing
the number of relations between the statements of a theory, but rather the
other way around, to reduce the number of such relations by eliminating
contradictions.999On this point, one commentator writes: “The solution is to
remove the contradictions by reducing the number of relations in which the
term ‘force’ stands – in effect by determining a single meaning for the term
‘force’.” (Kremer 2012, 204sq) But it seems to me that this gets Hertz’s
“solution” backwards. In any case, the removal of contradictions requires a
delicate consideration of essential and inessential relations (cf. Hüttemann
2009). By eliminating contradictions, the number of relations between the
statements about force and other statements in the axiomatic system of
mechanics would be reduced to an extent that a precise meaning for the term
“force” can be determined. Having determined a precise meaning for all
scientific terms, illegitimate questions about essences will not arise any
more.

####  2.1.2 Weyl’s universal boundary

Weyl justified a similar limitation of descriptive completeness. He gave an
argument to the effect that, in science, statements about the essence (Wesen)
of any objects are epistemically insignificant, to such an extent that science
does not really care about such statements at all. Like Hertz, Weyl further
expressed skepticism that questions about essence could actually be answered
in scientific terms, and then offered a solution as to how science could stop
or avoid asking such questions:

> A science can determine its domain only up to an isomorphic mapping. In
> particular it remains entirely indifferent as to the ‘essence’ of its
> objects. That which distinguishes the real points in space from number
> triples or other interpretation of geometry one can only know by immediate,
> living intuition. … The idea of isomorphism designates the natural
> insurmountable boundary of scientific cognition. This thought has
> clarificatory value for the metaphysical speculations about a world of
> things in themselves behind the phenomena. … Thus, even if we do not know
> the things in themselves, still we have just as much cognition about them as
> we do about the phenomena. (Weyl 1949, 26)

Unlike Hertz, Weyl did not construct his argument on a notion of epistemic
comfort, but on an invariance-based conception of scientific objectivity,
which he imposed as a constraint on a theory’s descriptive completeness.
According to him, a scientific theory is objective only if it does not ask
questions about the essence of objects, about their intrinsic or non-
relational properties. All possible answers to such questions would require
immediate intuition, but this violates the objectivity constraint. This is
because such answers can be true in one interpretation of a theory, but false
in another interpretation of the same theory. Thus, if objectivity is taken as
the primary epistemic ideal of science, and so if objectivity gives the
measure of epistemic significance, all statements ascribing intrinsic
properties to objects are epistemically insignificant. While questions about
the essence of objects may not be illegitimate, they are definitely
unanswerable by objective science. In a lecture titled “Scientific Method”,
Weyl remarked:

> Before any closer discussion I should like to point to that universal
> boundary of all theories as revealed in the idea of isomorphism. … With
> science thus proclaiming its own barriers, we must always be afraid that
> some high-sounding soothsayers may come along and attempt to settle with
> their pseudo-science beyond the border. They believe they have the arcanum
> disclosing to them the essence of things, which science is admittedly unable
> to grasp. However, the border which we have pointed out here is not a border
> towards a different brand of knowledge; it is rather a border beyond which
> there is nothing. (Weyl 1953)

Since relations can remain invariant across the interpretations of a theory,
science can offer objective knowledge only about the structure of relations,
i.e., about the structure of relational properties of all objects in a domain.
This intended limitation of descriptive completeness led Weyl to identify the
latter as categoricity:

> One might have thought of calling a system of axioms complete if the meaning
> of the basic concepts present in them were univocally fixed through the
> requirement that the axioms be valid. But this ideal cannot be realized, for
> the isomorphic mapping of a contentual interpretation is surely just another
> contentual interpretation. The final formulation is therefore this: a system
> of axioms is complete, or categorical, if any two contentual interpretations
> of it are necessarily isomorphic. (Weyl 1949, 25)

One must ask, however, whether Weyl’s boundary can be reached at all. Can any
science describe the relations between the objects in its domain uniquely up
to isomorphism? Can QM, more particularly, be characterized by categoricity?
On Weyl’s view, non-categoricity, that is the existence of mutually non-
isomorphic models of a theory, has primarily epistemological implications. For
if QM failed to be categorical, then it would fail to be objective, since
there would be statements in its language that are true in one model of the
theory, but false in another model. This failure of objectivity is what Weyl
thought obtained in the case of quantum field theory. Towards the end of his
life, he wrote:

> Objective Being, reality, becomes elusive; and science no longer claims to
> erect a sublime, truly objective world above the Slough of Despond in which
> our daily life moves. … the objective Being that we hoped to construct as
> one big piece of cloth each time tears off; what is left in our hands are –
> rags.101010Cf. Weyl 1954, 627. For my analysis of what motivated this view,
> based on the failure in QFT of the Stone-von Neumann theorem (further
> discussed below, in section 2.2.2), see Toader 2018.

The remainder of this chapter deals with the categoricity problem for standard
QM, assumed to have a representationalist metasemantics, that is one in which
semantic rules are taken to do metasemantic work. Chapters 3 and 4 will
discuss the categoricity problem for modular and orthomodular QL, on a similar
assumption. Chapter 5 will then formulate a categoricity problem for standard
QM, assumed to have a non-representationalist metasemantics, where all
metasemantic work is done by non-semantic rules.

###  2.2 The categoricity of quantum mechanics

A theory is categorical with respect to an isomorphism class of models if and
only if all models of the theory are in that class. A model is a relational
structure whose elements can be assigned to the vocabulary of the theory. The
vocabulary, together with the rules for manipulating its symbols, constitute
the syntax of the theory, while models constitute its semantics. Thus, we will
say that a categorical theory has a unique semantics up to a suitable
isomorphism. In this sense, for example, first-order arithmetic is not
categorical because it allows both standard and non-standard models. Likewise,
classical logic is not categorical, since it allows both normal and non-normal
models, or more exactly, non-normal valuations that make, for example,
disjunction and negation non-truth-functional.111111See Skolem 1955 for the
construction of a non-standard model of arithmetic, and Carnap 1943 for the
construction of non-normal valuations of classical logic. For a recent
presentation of these ideas, see Button and Walsh 2018, esp. chapters 7 and
13.

In the history of QM, categoricity concerns appear to have been raised in at
least two particular contexts: Einstein’s argument for the incompleteness of
QM, and Weyl’s argument for the incompleteness of quantum logic. An analysis
of the latter will be offered in section 3.3.2. A rigorous reconstruction of
the former, which identifies the assumptions behind an understanding of
Einstein completeness as categoricity, and draws on an analysis of the Stone-
von Neumann theorem, will be given in section 2.3.2. But I shall start with
some preliminary considerations about the kind of categoricity that may be
attributed to QM, if at all, and some remarks about its desirability and
indispensability.121212Section 2.2 draws on Toader 2021a.

####  2.2.1 Preliminary considerations

The general problem of categoricity, formulated independently of any
particular theory, has been suggested by Tarski in the following words:

> A non-categorical set of sentences (especially if it is used as an axiom
> system of a deductive theory) does not give the impression of a closed and
> organic unity and does not seem to determine precisely the meaning of the
> concepts contained in it. (Tarski 1934, 311)

Thus, a theory has a categoricity problem when the meaning of expressions in
its language is not determined precisely by its axioms because the latter
admit of non-isomorphic models. As I see it, the problem is, thus, a
metasemantic one. Whatever Tarski might have meant by “closed and organic
unity”, such an impression would be given, I take it, by a precisely
determined semantics. But the language of a non-categorical theory is, or at
least seems to be, semantically indeterminate, i.e., indeterminate with
respect to the semantic attributes of its terms. It is also semantically
indeterminate with respect to the truth values of its sentences: some
sentences are true in some models, but false in others. In contrast, a
categorical theory determines (or seems to determine) the meaning of its
concepts precisely, and its models agree on all theorems in its language.
Thus, for example, the meaning of arithmetical expressions is precisely
determined if arithmetic is categorical with respect to the isomorphism class
of an omega sequence, that is if it allows only models in which arithmetical
vocabulary is used in a natural way, i.e., in agreement with the natural
numbers. Likewise, the meaning of classical logical connectives (such as
disjunction and negation) is precisely determined if classical logic is
categorical with respect to the isomorphism class of a two-element Boolean
algebra, that is if it allows only models in which the connectives are used in
the normal way, i.e., in agreement with the normal truth tables.

Categoricity appears, therefore, to be a desirable property of theories
because it blocks semantic indeterminacy by allowing one to pin down the
intended unique semantics of the language of a theory, like the natural
numbers for arithmetic and the normal truth tables for classical logic. In the
case of first-order theories with infinite models, however, metatheoretical
properties like compactness or Löwenheim-Skolem, as well as Gödel
incompleteness, frustrate any attempt to establish categoricity. Therefore,
any success in this respect requires either augmenting the language of a
theory or adjusting its logic. Full second-order logic proves the categoricity
of arithmetic, although there are strong and well-known arguments against such
proofs, e.g., based on the fact that second-order logic is not a deductively
complete logic. Furthermore, since second-order quantification is non-
categorical, in this case the categoricity problem just gets pushed up a
level: in order to pin down the intended model of arithmetic up to
isomorphism, one must first be able to pin down the intended model of the
second-order quantifiers up to isomorphism. It also turns out that any attempt
to prove the categoricity of arithmetic by means of a logic weaker than
second-order, but stronger than first-order, similarly presupposes the
categoricity of that underlying logic (cf. Read 1997).

Such considerations suggest that although categoricity appears to be a
desirable property, it is beyond reach for most theories: one can never
actually pin down the intended unique semantics of a theory without first
pinning down the intended unique semantics of another background theory. But
is categoricity really indispensable? Some argued that the existence of non-
standard models of first-order arithmetic is no real concern, because any two
non-isomorphic models of it are arithmetically indiscernible, in the sense
that the application of induction within the language of the theory yields
arithmetical results that hold in any model. Thus, all models of arithmetic
agree on all theorems in its language (cf. Resnik 1996). Of course, the fact
that provability obliterates the differences between non-isomorphic models
does not seem enough to guarantee that one pins down the intended model of
arithmetic. For there exist unprovable sentences in the language of the theory
on which non-isomorphic models do in fact disagree, but one would have to
simply dismiss them as surplus structure.

Furthermore, it has also been noted that the non-categoricity of other
mathematical theories, like the theories of fields, rings, groups, algebras,
etc., is actually a result of design and acumen. Their non-categoricity is not
considered a failure, but a theoretical success. For no unique semantics is
typically supposed to exist for such theories, so no unique semantics can be
intended. Normally, these theories have a variety of distinct non-isomorphic
models, and they may all be thought of as intended. Of course, this does not
mean that categoricity, if it can be obtained, would necessarily be a
disadvantage. Some categorical algebraic theories have some rather nice
features like quantifier elimination. But semantic indeterminacy does not seem
to be considered a problem for such theories, although the reasons for this
are not always clear. This view, which has been extended also to arithmetic
and set theory, stems from considerations put forth by logicians and
mathematicians in the early part of the 20th century, who believed that non-
categoricity is a theoretical virtue, rather than a liability, and one should
formulate axioms that are weak enough to admit a multitude of non-isomorphic
models (see e.g., Zermelo 1930).

The problem of categoricity does not loom large in the philosophy of science,
at least by comparison to the philosophy of mathematics and philosophy of
logic. Is categoricity a desirable feature of scientific theories? Is it
perhaps indispensable? Is the existence of non-isomorphic models of a physical
theory a reason to be concerned about the semantic indeterminacy of its
language? On the one hand, categoricity is sometimes rejected by philosophers
of science as a kind of “rigidity or inflexibility” that should have no place
in physics (Bunge 1973). I will return to this idea below. On the other hand,
non-categorical scientific theories are sometimes considered more abstract
than the categorical ones, in the sense that their class of models is too
large and diverse to allow a simple mental image of a typical model (like the
natural numbers, say) despite dropping some of its complex properties, but
such level of abstractness should arguably be avoided in science (Suppes 1993,
74). In the particular case of QM, if this is a categorical theory, we should
say that its standard formalism has a unique semantics up to isomorphism. But
there are two immediate questions that I need to address, before I move on to
a discussion of whether the Stone-von Neumann theorem can be read as a
categoricity result in this context.

First, what does it mean to say that QM has a unique semantics up to
isomorphism? Does it mean to say that it has a unique interpretation, where
this notion is understood as in the foundations of QM? Since there are so many
mutually non-isomorphic interpretations of QM, such as $\psi$-ontic and
$\psi$-epistemic interpretations, the categoricity problem would turn out to
be really trivial: QM is a non-categorical theory, admitting models as
distinct as, e.g., the pilot wave model, dynamical reduction models, etc. The
answer to this question is that it is actually misleading to consider any of
the $\psi$-ontic and $\psi$-epistemic interpretations of QM as models, since
such interpretations are typically extensions of QM, extensions that add, for
example, extra dynamical laws to the Schrödinger equation. But the extensions
of a theory are surely not its models, although these extensions – as theories
in their own right – may admit as models some suitable expansions of the
models of QM. The pilot wave model, dynamical reduction models, etc., are
precisely such expansions. They are models of QM’s dynamical extensions.

Secondly, is the categoricity problem well-posed in the case of standard QM?
Is it possible to even ask whether its Hilbert space formalism can give what
Tarski saw as an impression of a closed and organic unity? Shouldn’t this
require interpretation? One might insist that categoricity is a well-posed
problem even though QM is only “a very effective and accurate recipe for
making certain sorts of predictions” (Maudlin 2019, 2), not a full-fledged
theory, but rather “a general method, a framework in which many theories can
be developed” (Laloë 2019, 13). This view of QM is typically motivated by the
belief that a physical theory must provide an ontology, which implies that QM
can be considered a theory in its own right, and the problem of its
categoricity can then be well-posed, only if QM has been given a $\psi$-ontic
interpretation. But as several neo-Bohrian approaches have contended, it is
false that QM can be considered as a theory only after an interpretation has
been given (see also Fuchs and Perez 2000). It would therefore be contentious
to say that only after an ontology has been given can the categoricity problem
be well posed for QM.131313Indeed, in chapter 5, I will formulate a
categoricity problem for a neo-Bohrian approach.

The categoricity problem, as understood here, concerns the standard axioms or
rules of QM, which can be stated as follows:

1\. The state space of a physical system corresponds to an infinite-
dimensional complex Hilbert space $\mathcal{H}$, such that the quantum state
of a physical system is a mathematical function $\psi:T\rightarrow\mathcal{H}$
(or an equivalence class of unit-norm vectors of $\mathcal{H}$), defined at
each time instant $t\in T$, where $T\subseteq\mathbb{R}$ denotes a time
interval.

2\. The set $\mathcal{A}$ of dynamical quantities of a physical system (e.g.,
position, momentum, spin, polarization, etc.) corresponds to the set of self-
adjoint operators acting on $\mathcal{H}$, such that the possible values of
any variable are contained in the spectrum of its corresponding operator.

3\. The unitary time evolution of a physical system is described by the
Schrödinger equation $i\partial_{t}\ket{\psi}_{t}=H\ket{\psi}_{t}$, for some
operator (the Hamiltonian) $H\in\mathcal{A}$.

4\. For each unit-norm vector $\ket{\phi}\in\mathcal{H}$, $\mu_{\phi}$ is a
measure on the set of subspaces of $\mathcal{H}$ such that for an arbitrary
subspace $K\subseteq\mathcal{H}$: $\mu_{\phi}(K)=\bra{\phi}P_{K}\ket{\phi}$,
where $P_{K}$ is the projector on $K$.

The language of these axioms includes terms for physical concepts like “state
space”, “quantum state”, “dynamical quantities”, etc., as well as symbols for
their associated mathematical objects like “Hilbert space”, “functions”,
“self-adjoint operators”, etc. Since the axioms explicitly formulate semantic
rules, their implicit metasemantics is clearly assumed to be representational,
i.e., the semantic attributes of all mathematical expressions are supposed to
be determined in virtue of the representational capacities of their quantum
models. But what exactly are these models?

A common view in the philosophy of quantum physics is the following: “Quantum
mechanics, we may say, uses the models supplied by Hilbert spaces.” (Hughes
1989, 79) Similarly, and more recently: “The models of NRQM [i.e., non-
relativistic quantum mechanics] are Hilbert spaces, along with a suitable
subalgebra of the bounded operators on that Hilbert space.” (Weatherall 2019,
7) This view suggests that one should take a model of QM to be a
representation on the Hilbert space $\mathcal{H}$ of the algebra generated by
the canonical commutations relations holding between the operators in the set
$\mathcal{A}$, which represent the dynamical quantities of an individual
physical system. The fact that such representations are taken to describe
individual systems justifies the view that Hilbert space representations are
local, rather than global, models (Hughes 2010).

Two further objections can now be raised against the claim that QM can be a
categorical theory. The first objection goes as follows:

> [A] corollary of Gödel’s first incompleteness theorem asserts that any
> theory as powerful as or more powerful than Peano arithmetic – in first-
> order formulation – will be not only deductively incomplete but also non-
> categorical, which is to say that it will have models that are not
> isomorphic to one another. And since any moderately sophisticated theory in
> physics will incorporate a mathematical apparatus as powerful as or more
> powerful than arithmetic, the same will be true of our physics. (Howard
> 2012)

According to this objection, even if the question of categoricity is well-
posed for QM, one can object that the latter cannot be categorical because,
like most physical theories, it incorporates first-order arithmetic, which
allows non-isomorphic models and is, thus, non-categorical. More generally,
the categoricity of QM would seem to require that all mathematical theories,
including its background logic, incorporated as components in the physical
theory, must be categorical. But this objection assumes a questionable
transfer of non-categoricity from the mathematical (and the logical)
components to the physical theory; or conversely, a questionable transfer of
categoricity from the physical theory to its mathematical (and logical)
components. For it is doubtful that a categorical physical theory could entail
the categoricity of its mathematics (or its logic): in the case of QM, this
would require that the isomorphism that holds between the local models of QM,
relative to a physical system, could define an isomorphism that holds between
the models of its mathematics (as well as one that holds between the models of
its logic). Indeed, this requirement obviously fails even within mathematics:
there are categorical theories, like second-order arithmetic, that incorporate
first-order, and so non-categorical, arithmetic. In other words, it fails
because the isomorphism that holds between second-order models cannot define
one that holds between the first-order models of arithmetic.

The second objection states that if QM were categorical, then it would be
incapable of application to the wide variety of physical systems to which in
fact it can be, and has been, successfully applied. This would be so because
the Hilbert space representations that describe such systems are far from
being mutually isomorphic: for example, even if all representations are
constructed on infinite-dimensional Hilbert spaces, which are mutually
isomorphic, the algebras of operators corresponding to the system’s dynamical
magnitudes may have widely different structures. This objection is in line
with the claim noted above that a categorical theory would be too rigid to be
acceptable as a physical theory. But this rigidity merely calls for a more
exact specification of the notion of categoricity that might be attributed to
QM.

The specification is as follows: we will say that QM is categorical with
respect to an isomorphism class of models if and only if all models of the
theory, relative to an individual physical system, are in that class. The
relativity clause is justified by the fact, already emphasized above, that any
representation of canonical commutation relations on the Hilbert space – any
algebra of operators – always describes the dynamical magnitudes of an
individual physical system. This is why in QM, as already noted, the relevant
notion of a model is that of a local model, which always specifies the algebra
of operators and the Hamiltonian for an individual system.141414We could also
introduce a new notion, call it absolute categoricity, to dispense with the
relativity clause. Let $\mathcal{QM}$ be an abstract quantum theory, which
leaves the algebra of operators and the Hamiltonian unspecified. Let
QM($\mathcal{A},H$) be a concrete quantum theory, which specifies the algebra
of operators and the Hamiltonian for any individual system. Then we can say
that $\mathcal{QM}$ is absolutely categorical if and only if, for any algebra
of operators and any $H\in\mathcal{A}$, QM($\mathcal{A},H$) is categorical.
Thus, obviously, if categoricity fails, then absolute categoricity fails as
well.

Now, we are ready to address the main question: for any physical system, is
there an isomorphism class of Hilbert space representations such that all
representations of the canonical commutation relations associated to that
individual system are in that class? The existence of such an isomorphism
class is usually taken to be established by the Stone-von Neumann theorem,
first conjectured by Marshall Stone in 1930 and then proved by von Neumann in
the following year.

####  2.2.2 The Stone-von Neumann theorem

In what sense can the Stone-von Neumann theorem be read as a categoricity
result for standard QM, if at all? More precisely, is the relation of unitary
equivalence asserted by the theorem sufficient for constraining all local
models of QM, relative to an individual physical system, into one isomorphism
class? The question is if the kind of isomorphism that underlies unitary
equivalence is enough for categoricity. This question is raised here for
standard QM, but I will also consider the point that the very consideration of
the question would actually require a rational reconstruction of QM, i.e., a
reformulation of the theory in a formal language. We will see that, on the one
hand, the Stone-von Neumann theorem can be intuitively read as establishing
categoricity – in the sense specified – only if all its assumptions, including
the problematic assumption of regularity, can be properly justified. On the
other hand, if one insists on rational reconstruction, the theorem can be
rigorously read as establishing categoricity only if all components of
standard QM could be properly formalized, ideally within a first-order
language. However, this is at least prima facie questionable.

The Stone-von Neumann theorem states that any irreducible, faithful, and
regular Hilbert space representation of the Weyl algebra, which describes a
quantum mechanical system (or, more generally, any system with a finite number
of degrees of freedom), is uniquely determined up to a unitary
transformation.151515Cf. Stone 1930 and von Neumann 1931. For discussion, see
Ruetsche 2011, esp. section 2.3. A Weyl algebra $\mathfrak{A}$ is a C∗-algebra
generated by the Weyl form of the canonical commutation relations. A Hilbert
space representation $(\mathcal{H},\pi)$ is a ∗-homomorphism
$\pi:\mathfrak{A}\rightarrow B(\mathcal{H})$, where $B(\mathcal{H})$ is the
set of bounded linear self-adjoint operators on an infinite-dimensional
complex Hilbert space $\mathcal{H}$. A representation is irreducible if no
(nontrivial) subspace of $\mathcal{H}$ is invariant under the operators in
$\pi(\mathfrak{A})$. If $\pi$ is a ∗-isomorphism, then the representation is
also faithful. A faithful and irreducible representation is also regular if
the operators in $\pi(\mathfrak{A})$ are weakly continuous. In the
representation theory of C∗-algebras, the Stone-von Neumann theorem entails
that any two irreducible, faithful, and regular representations
$(\mathcal{H}_{1},\pi_{1})$ and $(\mathcal{H}_{2},\pi_{2})$ of $\mathfrak{A}$
are unitarily equivalent if and only if there is an element $U\in\mathfrak{A}$
which acts as an operator $U:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$ such
that $UU^{*}=U^{*}U=1$ and $\pi_{1}(A)$ = $U\pi_{2}(A)U^{*}$ for all elements
$A\in\mathfrak{A}$. If $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are separable,
infinite-dimensional spaces, so their orthonormal bases are both countably
infinite, $U$ intertwines all bounded linear self-adjoint operators on the two
spaces. Thus, there is an isometric isomorphism between $\mathcal{H}_{1}$ and
$\mathcal{H}_{2}$ that underlies the unitary equivalence of
$(\mathcal{H}_{1},\pi_{1})$ and $(\mathcal{H}_{2},\pi_{2})$.

Unitary equivalence is typically taken to entail physical equivalence, in the
sense that the quantum states described as density matrices in unitarily
equivalent representations assign the same expectation values to corresponding
physical observables (Weyl 1930, 407; Ruetsche 2011, 24sq). This is, for
example, the sense in which one has come to speak of the physical equivalence
of the Schrödinger and the Heisenberg representations of a quantum mechanical
system: time evolution on states, in the Schrödinger representation, is the
same as time evolution on observables, in the Heisenberg
representation.161616See Muller 1997 and Perovic 2008 for discussion of the
sense and scope of the equivalence before von Neumann’s 1931 proof of unitary
equivalence. For this reason, the Stone-von Neumann theorem has been sometimes
intuitively read as proving the categoricity of QM.171717Cf. Stöltzner 2002,
45. More recently, I also maintained that the Stone-von Neumann theorem “can
be naturally read as a categoricity result.” (Toader 2018, 21)

This intuitive reading, however, hides several assumptions regarding the
Hilbert space representations. There exist representations of the Weyl
algebra, e.g. the position representation (with position eigenstates, but no
operator for momentum) and the momentum representation (with momentum
eigenstates, but no operator for position), such that there is no intertwiner,
and thus no isometric isomorphism, between them. These are non-regular
representations, which are unitarily inequivalent to the Schrödinger
representation, and to one another. The expectation values of corresponding
physical observables are different in such representations than in the
Schrödinger representation. The existence of non-regular representations also
suggests that the Stone-von Neumann theorem cannot be understood as a
categoricity result for standard QM: if non-regular representations are
allowed by the standard formalism, and they are considered as non-isomorphic
local models of the theory, this would be enough to make QM non-categorical.

Can non-regular representations be eliminated in standard QM? Arguably, non-
regular representations have physical significance, and Hans Halvorson used
them to rigorously articulate Bohr’s notion of complementarity (cf. Halvorson
2004). Despite their non-regularity, position and momentum representations can
be taken to describe the behavior of a physical system with a finite number of
degrees of freedom. Against this claim, the objection has been recently raised
that the dynamics of this physical system cannot be the unitary dynamics
governed by Schrödinger’s equation, since non-regular states, i.e., the states
on the Hilbert spaces of these representations in which positions or momenta
have determinate values, are unitarily inaccessible from regular states (as
well as mutually inaccessible). The claim that non-regular representations are
not compatible with a unitary dynamics builds on an unpublished result by
David Malament, according to which if a free dynamics is assumed in the
position representation, then exact localizability is violated (Feintzeig et
al. 2019, 127). Thus, one either changes the standard formalism of QM by
replacing the Weyl algebra with a different mathematical structure that does
not allow non-regular representations,181818Cf. Feintzeig and Weatherall 2019.
The introduction of a different structure than the Weyl algebra may allow the
recovery of non-regular states as approximations or idealizations of regular
ones: “non-regular quantum states should be considered unphysical for
essentially the same reasons that classical states at infinity are considered
unphysical.” (Feintzeig 2022, 472) or one eliminates non-regular
representations in standard QM as physically insignificant on account of their
incompatibility with the unitarity of quantum dynamics. If the latter option
turns out to be unproblematic, then we have a sense in which the Stone-von
Neumann theorem can be intuitively read as a categoricity result for standard
QM.

The intuitive reading is, however, questionable if one insists that addressing
the categoricity problem requires a formal axiomatization of QM, or what
Carnap called a rational reconstruction, i.e., a reformulation of the theory
in an appropriate formal language with a rigorously defined consequence
relation. For suppose that one demands that before the problem can be even
properly discussed, one should provide a rational reconstruction of standard
QM as a formal system in which not only a consequence relation is defined, but
both logical rules and QM rules are formally expressed in its language. The
trouble with this is, of course, that there is no such reconstruction of QM,
and moreover that no such reconstruction seems possible. I will get back to
the question whether a rational reconstruction of QM is possible, in chapter
6. Here I want to discuss only one reason why a viable rational reconstruction
of standard QM in a first-order language does not seem possible. Briefly, this
is because whereas the unitary equivalence of Hilbert space representations
can be formalized as a first-order relation, representations cannot be
formalized as first-order models.

Let’s develop this argument in a bit more detail, starting with the claim that
unitary equivalence makes the Heisenberg representation of the Weyl algebra of
a quantum system intertranslatable with its Schrödinger representation. What
this may be taken to mean is that “[a]ny data which elements of the [one
representation] accommodate, counterpart elements of the [other
representation] accommodate as well – and mutatis mutandis for falsifying
data”.191919Cf. Ruetsche 2011, 45. Note that Ruetsche speaks of Heisenberg and
Schrödinger theories, rather than representations. Their intertranslatability
is understood in the more general sense that had been articulated by Clark
Glymour: “[The intertwiner of representations] provides the translation manual
[Glymour] is after.” (loc. cit.) Glymour’s manual helps translating between
theories and it “guarantees that all and only theorems of [one theory] are
translated as theorems of [another theory], and conversely.” (Glymour 1970,
279) But the data accommodated by unitarily equivalent representations are the
expectation values assigned to corresponding physical observables in states
described as density matrices in those representations. Therefore,
representations are intertranslatable in this sense only if one takes
statements about expectation values as theorems (Clifton and Halvorson 2001,
430). Representations are intertranslatable in the sense that they agree on
all theorems. More exactly, they are intertranslatable only if they have a
“common definitional extension” (Glymour 1970, 279), that is they have
definitional extensions that derive the same theorems. Since we consider the
Hilbert space representations as models of QM, the semantic counterpart of
intertranslatability holds only if the representations have definitional
expansions that satisfy the same theorems. But, of course, the converse does
not hold, since non-isomorphic structures can satisfy the same theorems.
Therefore, categoricity cannot be the semantic counterpart of definitional
equivalence.

Furthermore, and more importantly, intertranslatability understood as
definitional equivalence is a first-order relation, which means that whatever
its semantic counterpart is, it holds between Hilbert space representations
qua models only in the sense that they satisfy all and only first-order
theorems. But representations are not first-order models, on account of the
metric completeness of Hilbert spaces, a property that cannot be expressed in
a first-order language. If some higher-order statements have any physical
consequences, and there is no apparent reason to think that they cannot have
such consequences, then the unitary equivalence established by the Stone-von
Neumann theorem is insufficient for physical equivalence.

One might, alternatively, define intertranslatability as Morita equivalence:
“The ‘modern’ approach to the Stone-von Neumann Theorem, which is somewhat
more algebraic, is due to Rieffel [1972, 1974]. The key observation of Rieffel
is that the theorem is really about an equivalence of categories of
representations, or in the language of ring theory, a Morita equivalence.”
(Rosenberg 2004, 342) The Stone-von Neumann theorem, as formulated by Rieffel,
states the following: “Every irreducible Heisenberg G-module is unitarily
equivalent to the Schrödinger G-module.” (Rieffel 1972) Unitary equivalence is
constructed as the Morita equivalence of G-modules, where the G-modules are
Hilbert space representations together with the collection of all intertwining
operators between them. Then, the semantic counterpart of Morita equivalence
holds between G-modules considered as models only if they have Morita
expansions that satisfy the same theorems. However, just as above, the
converse does not hold. Moreover, Morita equivalence is a generalized
definitional equivalence relation (Barrett and Halvorson 2016), and thus it is
also first-order, which makes it inadequate for present purposes, since just
like the Hilbert space representations, Rieffel’s G-modules cannot be
formalized as first-order models.

Such considerations suggest that while the Stone-von Neumann theorem might be
intuitively read as a categoricity result for standard QM, this reading can
hardly be maintained if one requires a rational reconstruction of QM in a
first-order language. But as in the case of mathematics, one might be inclined
to respond by beefing up logic. Here, one might point again to second-order
quantification as problematic, or one might emphasize that the categoricity of
QM is not really indispensable. Indeed, one might be inclined to adopt the
attitude that logicians and mathematicians from Zermelo to Hamkins have
adopted in the case of set theory and arithmetic, and learn to appreciate non-
categoricity as a theoretical virtue of QM and to ignore or otherwise minimize
its metasemantic consequences. Although he considered the categoricity of set
theory a desirable property (von Neumann 1925, 412), von Neumann himself never
considered the Stone-von Neumann theorem as one that established the
categoricity of QM. Moreover, when he later developed his axiomatic theory of
games, for example, he expressly noted that this is intended as non-
categorical (von Neumann and Morgenstern 1944, section 10.2).

Following von Neumann’s view, Miklós Rédei speaks, very aptly, of “intended
non-categoricity” to characterize an intention to construct physical theories
that allow for multiple non-isomorphic models (Rédei 2014, 80). Nevertheless,
as he identifies it, this intention is typically exemplified by theories like
quantum field theory, which invalidate the Stone-von Neumann theorem. This
suggests that what is really meant to be described here is an intention to
construct theories that allow for unitarily inequivalent representations, so
that, as in QFT, one can have representations of the C∗-algebra on a free
field, but also representations on an interacting field. The failure of the
Stone-von Neumann theorem for quantum systems with an infinite number of
degrees of freedom has been correctly, I think, considered sufficient to
establish the non-categoricity of a theory describing such systems, like
quantum statistical mechanics (in the thermodynamic limit) and QFT:

> Our best current theory is QFT. It is a relativistic theory (in the sense of
> special, not general relativity), and it is a theory of systems with an
> infinite number of degrees of freedom. As such, in its most natural
> algebraic form, it can be shown to possess representations that are, of
> necessity, unitarily inequivalent. This is the algebraist’s way of saying
> that the theory is not categorical, that it does not constrain the class of
> its models up to the point of isomorphism. (Howard 2011, 231)

True, but I think that this needs to be qualified in the following way: if the
Stone-von Neumann theorem should not be rigorously read as a categoricity
result until a rational reconstruction of standard QM has been provided and
subjected to analysis, then the theorem should not be taken to make any
metasemantic difference between QM and QFT. In particular, one cannot say that
its validity for QM, but not QFT, is enough to justify one’s belief that the
semantic attributes of the standard formalism of QM are precisely determined.

I will discuss the demand for a rational reconstruction of QM, and some of the
main obstacles that seem to make it impossible, in the concluding chapter of
the book. I will suggest there that, in order for a viable reconstruction to
be possible, one would need to drop any semantic or correspondence rules and
adopt non-representationalism for the language of the reconstruction. Until
then, in the next section, drawing on my analysis so far, I will provide a
reconstruction of Einstein’s incompleteness argument, which is one of the few
historical contexts in which categoricity concerns appear to have been raised
about QM. As I will explicate it, the argument aims at establishing that a
local, separable, and categorical QM cannot exist.

###  2.3 Einstein completeness as categoricity

Einstein’s argument for the incompleteness of QM, which did not make it into
the EPR paper (Einstein, Podolski, and Rosen 1935) in the way Einstein thought
it should have, was clearly formulated in letters to Schrödinger and Popper,
as well as in several publications (e.g. Einstein 1936). After Arthur Fine
brought it to philosophical attention (Fine 1981), Don Howard suggested that
the argument might be understood as deploying a notion of completeness known
as categoricity (Howard 1990). This suggestion was motivated by Einstein’s
claim that QM fails to assign a unique wavefunction to the real state of one
subsystem of an EPR system, since the assignment depends on the measurement
that could be performed on the other subsystem. If multiple wavefunctions can
be assigned to the same subsystem, and if one is justified in considering them
as (parts of) non-isomorphic models, then this would be enough to show QM non-
categorical. If Howard’s suggestion is taken seriously, then Einstein
completeness turns out to be a rather different type of completeness than the
one articulated in the EPR paper.

In this section, I will provide a reconstruction of Einstein’s argument, which
I think can clarify the assumptions underlying an understanding of Einstein
completeness as categoricity. The key idea to be rigorously articulated is the
following: “if one understands a theoretical state as, in effect, a model for
a set of equations plus boundary conditions …, then Einstein’s conception of a
completeness requirement should really be understood as a categoricity
requirement.” (Howard 1992, 208) To stay as close as possible to Einstein’s
own argument, I will focus on the original EPR state, with observables having
a continuous spectrum, suitably defined within an algebraic framework (Arens
and Varadarajan 2000, Werner 1999), and I will explain under what conditions
one would be justified to read Einstein completeness as categoricity. On my
reconstruction, the argument assumes that representations on (tensor products
of) Hilbert spaces are the models of QM of (composite) systems.202020For more
on this assumption, and on the relevant notion of model, see section 2.2.1
above. It assumes as well that the unitary equivalence of such representations
is a necessary (though possibly not sufficient) condition for categoricity.
The argument then points out that there are representations of the (tensor
product of) algebras describing a subsystem of an EPR system that are not
unitarily equivalent. As such, Einstein’s argument concludes that categoricity
is inconsistent with separability and locality.

It is, of course, difficult to say that the reconstruction I propose is
actually entirely faithful to Einstein’s own thought. One worry one might
raise is that while my reconstruction works for the original EPR state, it
does not work for entangled spins, since on finite-dimensional Hilbert spaces
there are no unitarily inequivalent representations. But Einstein, as is well
known, never cared much about Bohm’s version of the EPR argument.212121The
only place where Einstein formulated a spin version of his argument appears to
be in a late manuscript from around 1955: for discussion, see Sauer 2007.
Focusing on the infinite-dimensional case is thus historically reasonable.
Moreover, for Einstein’s objection to stand, it is of course sufficient that
his argument goes through in one case; it is not required that it should do so
in all cases. In any event, I will argue that my reconstruction is preferable
to others, according to which Einstein’s argument should be taken to establish
“overcompleteness” (Lehner 2014) or unsoundness (Gömöri and Hofer-Szabó 2021),
rather than non-categoricity.

Furthermore, I will suggest that my reconstruction sheds some new light on the
Bohr-Einstein controversy. As mentioned in the previous section, Bohr’s
doctrine of complementarity has been rigorously interpreted in terms of the
unitary inequivalence of non-regular Hilbert space representations, which
vindicates the view that Bohr’s notion of completeness was significantly
distinct from the descriptive completeness articulated in the EPR
paper.222222For an expression of this view, see, e.g., Norsen 2017, 148. On my
reconstruction, Einstein completeness fails precisely due to this unitary
inequivalence. From an algebraic point of view, it appears that the sense in
which Bohr thought QM was complete is exactly the sense in which Einstein
argued it wasn’t. From a metasemantic perspective, their views would be
precisely antithetical: whereas Einstein deplored QM’s non-categoricity and
the ensuing semantic indeterminacy, Bohr embraced non-categoricity as a
theoretical asset and arguably ignored semantic indeterminacy as
insignificant.232323Section 2.3 draws on Toader 2023a.

####  2.3.1 Misconstruals of Einstein completeness

Recall the completeness condition that the EPR paper purported to argue it is
not satisfied by QM: “Whatever the meaning assigned to the term complete, the
following requirement for a complete theory seems to be a necessary one: every
element of the physical reality must have a counterpart in the physical
theory.” (1935, 777) This condition has typically been understood to formulate
a type of descriptive completeness, since it applies to a physical theory just
in case that theory is able to describe all of the physical reality that it
aims to describe. The EPR paper argued that QM does not satisfy the
completeness condition because in the case of a system in an EPR state there
are elements of physical reality, i.e., properties of a subsystem of that
system, that the theory aims to describe, but fails to do so. As is well
known, crucial to the EPR argument is the following criterion of reality: “If,
without in any way disturbing a system, we can predict with certainty (i.e.,
with probability equal to unity) the value of a physical quantity, then there
exists an element of physical reality corresponding to this physical
quantity.” (1935, 777; italics removed) The role and character of this
criterion, as well as the formal structure of the argument, have been often
addressed.242424Perhaps the most detailed formal reconstruction of the EPR
argument has been given in (McGrath 1978) where the following is also noted:
“Regrettably EPR equate two notions of completeness: ‘complete representation
by a wave function’ and ‘complete theory’ are used interchangeably.” (560) See
also (Gömöri and Hofer-Szabó 2021) for a nice discussion of the EPR criterion
of reality, its indispensable role within the EPR argument for incompleteness,
and its absence from Einstein’s own argument. What has been often ignored is
the fact that EPR completeness and Einstein completeness are not the same type
of descriptive completeness.252525The fact that EPR completeness and Einstein
completeness are not the same type of completeness has been, to my knowledge,
first explicitly noted by Arthur Fine: “Einstein does not give [the latter] a
catchy name, but … [we can] call this more technical conception bijective
completeness.” (Fine 1981, 72)

In one widely quoted passage from his letter to Schrödinger, Einstein wrote:

> In the quantum theory, one describes a real state of a system through a
> normalized function, $\psi$, of the coordinates (of the configuration-
> space). … Now one would like to say the following: $\psi$ is correlated one-
> to-one with the real state of the real system. … If this works, then I speak
> of a complete description of reality by the theory. But if such an
> interpretation is not feasible, I call the theoretical description
> ‘incomplete’. (Letter to Schrödinger, 19 June 1935; translated in Howard
> 1985, 179)

Einstein completeness can thus be attributed to QM if and only if there exists
a one-to-one correlation between a $\psi$-function and the real state of a
system the theory aims to describe. Einstein’s point that a one-to-one
correlation does not exist in the case of an EPR system is meant to be
supported by his separability and locality assumptions, so his argument can be
formulated in the following way:

1\. Spacelike separated physical systems have real states, which cannot
causally influence one another.

2\. Consider a system with two subsystems, A and B, in an EPR state.

3\. Thus, each subsystem has a real state, no matter what measurements can be
carried out on its other subsystem.

4\. QM assigns different $\psi$-functions to A, depending on which observable
one can choose to measure on B.

5\. But if QM is complete, these $\psi$-functions should be identical.

6\. Thus, QM is incomplete.

Most of my discussion below will be focused on premises 4 and 5. It should be
clear that the notion of completeness in premise 5 is directly implied by that
defined in the letter to Schrödinger. In a letter to Pauli, Schrödinger put it
in the following metaphorical terms: “He [i.e., Einstein] has a model of that
which is real consisting of a map with little flags. To every real thing there
must correspond on the map a little flag, and vice versa.” (Schrödinger to
Pauli, July 1935, translated in Howard 1990, 106) It immediately follows that
a map with little flags is an Einstein complete description of physical
reality only if the map is essentially unique, in the sense that there is a
bijection between any two maps describing that reality. In the case of QM, by
analogy, Einstein completeness requires that the $\psi$-function representing
the real state of a system be essentially unique.

Howard’s suggestion as to how to properly understand Einstein completeness,
first offered in a footnote to his presentation of Einstein’s argument, is the
following:

> A complete theory assigns one and only one theoretical state to each real
> state of a physical system. [Footnote: This is a curious conception of
> completeness, more akin to what is called in formal semantics
> “categoricity.”] But in EPR-type experiments involving spatio-temporally
> separated, but previously interacting systems, A and B, quantum mechanics
> assigns different theoretical states, different ‘psi-functions,’ to one and
> the same real state of A, say, depending upon the kind of measurement we
> choose to carry out on B. Hence quantum mechanics is incomplete. (Howard
> 1990, 64)

The suggestion is then uplifted from footnote to the main text of a later
paper:

> Let me conclude with one really outrageous suggestion. … the operative
> criterion of completeness in Einstein’s thinking was this: a theory is
> complete if and only if it assigns a unique theoretical state, such as a
> psi-function, to every unique real physical state. But if one understands a
> theoretical state as, in effect, a model for a set of equations plus
> boundary conditions …, then Einstein’s conception of a completeness
> requirement should really be understood as a categoricity requirement. In
> other words, Einstein is saying that a ‘complete’ (read ‘categorical’)
> theory is one that determines a unique (eindeutige) model for the reality it
> aims to represent. (Howard 1992, 208; my emphasis)

The conception of completeness that Howard thought should be credited to
Einstein is that according to which a theory is Einstein complete if and only
if all its models belong to one and the same isomorphism class. In order to
intuitively grasp this suggestion, starting from my reconstruction of
Einstein’s argument, it will be enough to motivate, informally, an algebraic
requirement for categoricity, and then argue that Einstein can be understood
to have doubted that QM satisfies that requirement. It will then follow that
what Einstein argued for is that QM is descriptively incomplete in the sense
that, roughly, even though it may be able to describe all of the physical
reality that it aims to describe, it ends up describing a lot more besides
that as well.

The reconstruction of Einstein’s argument, in an algebraic framework, will be
given further below. That will require a preliminary consideration of premise
2, in particular, a properly algebraic definition of the original EPR state,
for continuous observables. It will also require a discussion of premise 4,
that is an algebraic account of the difference between the $\psi$-functions
assigned to susbsystem A. The outcome of all that will, I hope, be three-fold:
(i) a good understanding of the assumptions behind Howard’s suggestion to read
Einstein completeness as categoricity, (ii) an explanation of what makes
Einstein completeness different from EPR completeness as a type of descriptive
completeness, and (iii) an unequivocal sense of where some of the misreadings
of Einstein completeness have gone wrong.

Although my real emphasis will be on (i), let me briefly address (iii) right
away, in order to candidly raise the reader’s interest in what is to come
later. Einstein’s argument for incompleteness is sometimes considered too
confused to take seriously. Klaas Landsman, for example, maintained that it is
a “muddled” argument (Landsman 2006, 234) for the following reason:

> Unfortunately, Einstein (and EPR) insisted on a further elaboration of this
> disjunction [i.e., completeness or separability], namely the idea that there
> exists some version of quantum mechanics that is separable … at the cost of
> assigning more than one state to a system (two in the simplest case). It is
> this unholy version of quantum mechanics that Einstein (and EPR) called
> ‘incomplete’. Now, within the formalism of quantum mechanics such a multiple
> assignment of states (except in the trivial sense of wave functions
> differing by a phase factor) makes no sense at all, for the entanglement
> property lying at the root of the non-separability of quantum mechanics is
> so deeply entrenched in its formalism that it simply cannot be separated
> from it. (op. cit., 227)

However, if it implies that Einstein’s argument requires that entanglement be
relinquished, this is simply missing the point. One would have thought it
obvious that, quite the contrary, the argument is essentially based on
entanglement. Clearly, Landsman’s criticism fails insofar as it neglects the
fact that Einstein separability (expressed in premise 1 above) and
entanglement (implicit in premise 2) are compatible.262626See (Murgueitio
Ramírez 2020) for the point that Einstein’s argument would be trivially
unsound, if separability and entanglement were not compatible. Landsman argues
that, as a consequence of Raggio’s theorem, Einstein separability is
mathematically equivalent to Bohr’s doctrine of the necessity of classical
concepts. But his argument takes Einstein separability as state
decomposability. The alleged reason why multiple assignment of states is
nonsensical doesn’t stand: there is no “unholy version” of QM, i.e., a version
of the theory without entanglement, and Einstein did not think otherwise.
Furthermore, and this is a more significant problem, Landsman seems to
misconstrue Einstein’s view on the difference between the $\psi$-functions
assigned to subsystem A as a trivial phase difference. As we will see further
below, the multiple assignment of $\psi$-functions that Einstein actually
considered can be aptly interpreted algebraically in terms of the unitary
inequivalence of non-regular Hilbert space representations. Were this
nonsensical, Bohr’s complementarity doctrine would appear to be nonsensical as
well, but this is a doctrine that Landsman professes to defend.

Einstein completeness, and Einstein’s argument that this cannot be attributed
to QM, have been characterized by Christoph Lehner in these terms: “Einstein …
concludes that the quantum mechanical description is not biunique because it
is incomplete. … This conclusion is not warranted logically.” (Lehner 2014,
334) A description that is not biunique is one that assigns to subsystem A a
$\psi$-function which is not correlated one-to-one with its real state. On
this reconstruction, Einstein’s second half of the argument would be as
follows:

4L. QM assigns different $\psi$-functions to A, depending on which observable
one can choose to measure on B.

5L. Thus, QM is incomplete.

6L. Thus, QM is not biunique.

Lehner maintained that the inference from 5L to 6L is not justified. But this
clearly puts Einstein’s cart before his horse: the one-to-one correlation does
not fail because of incompleteness; rather, its failure is the actual reason
for incompleteness, just as the reason why the one-to-one correlation fails is
provided by the possible assignment of different $\psi$-functions to A
(expressed by 4L). Moreover, since a $\psi$-function is just a theoretical
description and such “a description that is not invariant is not necessarily
incomplete … it is ‘overcomplete’ or nonabsolute” (loc. cit.), i.e., it
assigns to A a theoretical state that is correlated many-to-one with its real
state, then what Einstein should have allegedly argued is the following
(statements 1-3 in the argument are, again, unchanged):

4L. QM assigns different theoretical states to A, depending on which
observable one can choose to measure on B.

5L. Thus, QM is overcomplete.

6L. Thus, QM is not biunique.

The inference from 5L to 6L is immediate, but trivial. However, Lehner appears
to reduce Einstein’s incompleteness to a quite uninteresting case of empirical
underdetermination. Jos Uffink has recently pointed out that reading Einstein
incompleteness as overcompleteness is “surprising”, since “cases of
overcompleteness are ubiquitous in physics… Indeed one might wonder whether
overcompleteness is a worrisome issue at all in theoretical physics.” (Uffink
2020, 557) In QM, as Uffink emphasizes, such cases of overcompleteness as
illustrated by phase differences between $\psi$-functions are not worrisome at
all. But, of course, Einstein did not think otherwise. As already noted, we
will see below that the essential differences between $\psi$-functions that
Einstein did think worrisome can be aptly interpreted algebraically in terms
of the unitary inequivalence of non-regular Hilbert space representations. If
this is enough to conclude that QM is not categorical, then Lehner’s
reconstruction of Einstein’s argument turns out to conflate (the metasemantic
problem of) categoricity and (the epistemological problem of) empirical
underdetermination.

Along different lines, Márton Gömöri and Gábor Hofer-Szabó argue for a similar
conclusion, that Einstein’s argument should not really be taken to establish
incompleteness: “According to Einstein’s later [than the EPR] argument, the
Copenhagen interpretation is committed to the existence of elements of reality
that cannot be out there in the world –- under the assumptions of locality and
no-conspiracy. Hence, given these assumptions, the Copenhagen interpretation
is unsound –- as opposed to being incomplete.” (Gömöri and Hofer-Szabó 2021,
13453) Supposing it right that Einstein intended his argument against the
Copenhagen interpretation of QM, the second half of Einstein’s argument would
change as follows:

4G. QM assigns different $\psi$-functions to A, depending on which observable
one can choose to measure on B.

5G. Thus, at least some $\psi$-functions represent states of A that cannot
exist.

6G. Thus, QM is unsound.

Gömöri and Hofer-Szabó readily acknowledge that their reconstruction of
Einstein’s argument is in conflict with his own understanding of completeness.
Pursuing “logical reconstruction” in spite of what “Einstein actually thought
to argue”, they find his own notion “not quite apt”, in part because it is not
identical to the EPR notion of completeness (loc. cit.). But one would have
thought that the difference between EPR completeness and Einstein completeness
cannot be sufficient to deny the latter as an apt notion, since there is
nothing to indicate that EPR completeness is the only possible type of
descriptive completeness attributable to physical theories. Still, their main
reason for having Einstein’s argument conclude that QM is unsound, rather than
incomplete, is the incompatibility of the different $\psi$-functions assigned
to subsystem A of an EPR system. This is correct as an understanding of the
essential difference between $\psi$-functions, expressed by premise 4G. In the
next section, this incompatibility will be precisely interpreted algebraically
as the unitary inequivalence of non-regular Hilbert space representations. If
my interpretation is adequate, I do not see why that incompatibility would
make QM unsound, rather than sound but non-categorical. To argue that my
interpretation is adequate, I turn now to reconstructing Einstein’s argument
in the framework of C∗-algebras.

####  2.3.2 Reconstruction of Einstein’s argument

Taking Howard’s suggestion seriously requires that one consider the
theoretical states assigned to a system by QM as models of the physical
theory. But it’s not immediately clear how a $\psi$-function, and in
particular one that represents the EPR state of a system, can be considered in
this way. For one thing, such a $\psi$-function cannot be construed as a unit-
norm vector in an infinite-dimensional Hilbert space $\mathcal{H}$. This is
for multiple related reasons, such as that the $\psi$-function has infinite
norm, and the (operators associated to) observables (like position and
momentum) of the subsystems of an EPR system have continuous spectra, so their
probability distribution will have a probability density that cannot be
concentrated on a single point in $\mathcal{H}$. This is why the definition of
the EPR state for a system composed of subsystems A and B as a unit vector in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ for observables in subalgebras
$\textbf{B}(\mathcal{H}_{A})\otimes\mathcal{I}$ and
$\mathcal{I}\otimes\textbf{B}(\mathcal{H}_{B})$ is not entirely adequate
(Arens and Varadarajan 2000, 638).

But that definition has been generalized, as we will presently see, and the
EPR state has been identified with a normalized positive linear functional on
$\mathfrak{A}_{A}\otimes\mathfrak{A}_{B}$ for observables in the von Neumann
algebras $\pi(\mathfrak{A}_{A}\otimes\mathcal{I})^{\prime\prime}$ and
$\pi(\mathcal{I}\otimes\mathfrak{A}_{B})^{\prime\prime}$ (Werner 1999,
Halvorson 2000). The generalized definition will be used below to reformulate
premises 4 and 5 in Einstein’s argument. The essential difference between the
theoretical states assigned to subsystem A of the EPR system will be
interpreted in terms of the unitary inequivalence of non-regular
representations of $\mathfrak{A}_{A}\otimes\mathcal{I}$ on
$\mathcal{H}_{A}\otimes\mathcal{I}$. This can support a model-theoretical
approach to Einstein’s argument, as suggested by Howard, provided one takes
representations as models of QM. But this would not be at all unusual, and as
noted in the previous section it is the view I adopt here. These models are
furthermore assumed to be definable as formal structures in a suitable
rational reconstruction of QM, but for my purposes in this section no such
definitions will be needed.272727This assumption might be problematic,
however, if one dispensed with Hilbert space representations in the formalism
of QM, and constructed models entirely algebraically. For the ∗-automorphisms
of the Weyl algebra are unique up to unitary equivalence, so at the abstract
algebraic level there are no non-regular states. But I think this problem for
my reconstruction of the argument can be ignored, since Einstein, interested
as he was in local wavefunctions, was surely not an Algebraic Imperialist of
the kind introduced and described in Arageorgis (1995).

Let’s start by considering observables $O_{A}\otimes
I\in\textbf{B}(\mathcal{H}_{A})\otimes\mathcal{I}$ and $I\otimes
O_{B}\in\mathcal{I}\otimes\textbf{B}(\mathcal{H}_{B})$. As defined by Richard
Arens and Veeravalli S. Varadarajan, $\omega$ is an EPR state if and only if
the joint distribution of $O_{A}\otimes I$ and $I\otimes O_{B}$ is a measure
on $\mathbb{R}^{2}$ concentrated on the diagonal, $\mu_{\omega}^{O_{A}\otimes
I,I\otimes O_{B}}(\left\\{(x,x)|x\in\mathbb{R}\right\\})=1$. Such pairs of
observables are typically called EPR-doubles: the outcome of measuring one
predicts with certainty the outcome of measuring the other. These EPR-doubles
form type I factors (isomorphic to von Neumann algebras
$\textbf{B}(\mathcal{H}_{A})\otimes\mathcal{I}$ and
$\mathcal{I}\otimes\textbf{B}(\mathcal{H}_{B})$ respectively), so they have a
discrete spectrum (their distributions relative to $\omega$ take discrete
values only) (Arens and Varadarajan 2000, 647). Thus, this definition is not
enough to characterize the original EPR state – the state that Einstein was
concerned with.

In order to overcome this limitation, Reinhard F. Werner considered
$\mathfrak{A}_{A}\otimes\mathcal{I}$ and $\mathcal{I}\otimes\mathfrak{A}_{B}$
as the mutually commuting subalgebras of
$\mathfrak{A}_{A}\otimes\mathfrak{A}_{B}$, and took
$\pi(\mathfrak{A}_{A}\otimes\mathcal{I})^{\prime\prime}$ as a self-adjoint
unital subalgebra of $\textbf{B}(\mathcal{H}_{A})\otimes\mathcal{I}$, and
$\pi(\mathcal{I}\otimes\mathfrak{A}_{B})^{\prime\prime}$ as a self-adjoint
unital subalgebra of $\mathcal{I}\otimes\textbf{B}(\mathcal{H}_{B})$, both
closed in the weak operator topology. Then $\pi(O_{A}\otimes
I)\in\pi(\mathfrak{A}_{A}\otimes\mathcal{I})^{\prime\prime}$ and $\pi(I\otimes
O_{B})\in\pi(\mathcal{I}\otimes\mathfrak{A}_{B})^{\prime\prime}$ are EPR-
doubles and have continuous spectra, since
$\pi(\mathfrak{A}_{A}\otimes\mathcal{I})^{\prime\prime}$ and
$\pi(\mathcal{I}\otimes\mathfrak{A}_{B})^{\prime\prime}$ are type II1 factors
(cf. Werner 1999).

Now, consider the representations $(\pi,\mathcal{H}_{A}\otimes I)$ and
$(\pi,I\otimes\mathcal{H}_{B})$ of $\mathfrak{A}_{A}\otimes\mathfrak{A}_{B}$.
If $\omega$ is an original EPR state and $\pi$ is faithful, then there is a
state $\tau$ in each of these representations such that for any element
$O_{A}\otimes O_{B}\in\mathfrak{A}_{A}\otimes\mathfrak{A}_{B}$, we will have
$\omega(O_{A}\otimes O_{B})=\tau(\pi(O_{A}\otimes O_{B}))$ (Halvorson 2000,
327). This entails, on the same conditions, that for any two representations
$(\pi_{1},\mathcal{H}_{A}\otimes I)$ and $(\pi_{2},\mathcal{H}_{A}\otimes I)$
of $\mathfrak{A}_{A}\otimes\mathcal{I}$, there are different states $\tau_{1}$
and $\tau_{2}$ in these representations, respectively, such that for two
different elements $O^{1}_{A}\otimes I,O^{2}_{A}\otimes
I\in\mathfrak{A}_{A}\otimes\mathcal{I}$, we have the corresponding
restrictions of $\omega$, that is $\omega(O^{1}_{A}\otimes
I)=\tau_{1}(\pi_{1}(O^{1}_{A}\otimes I))$ and $\omega(O^{2}_{A}\otimes
I)=\tau_{2}(\pi_{2}(O^{2}_{A}\otimes I))$. All this suggests the following
reformulation of the second half of Einstein’s argument:

4T. QM assigns different states, $\tau_{1}$ or $\tau_{2}$, to subsystem A,
depending on which EPR-double, $\pi_{1}(I\otimes O^{1}_{B})$ or
$\pi_{2}(I\otimes O^{2}_{B})$, one can choose to measure on B.

5T. But if QM is complete, $\tau_{1}$ and $\tau_{2}$ should be identical.

6T. Thus, QM is incomplete.

How should we understand premise 4T? Within this framework, what makes
$\tau_{1}$ and $\tau_{2}$ different states? As we have seen above, some
commentators took Einstein’s argument to be confused on this very point, for
the reason that the only differences between such states allowed by standard
QM are “trivial” phase differences. But this is not what Einstein had in mind,
as Howard already pointed out:

> Might there not be situations in which the differences between two
> $\psi$-functions (phase differences, for example) are inessential from the
> point of view of the system whose real state they aim to describe?
> Einstein’s completeness condition would, indeed, be too strong if it
> required that literally every difference between $\psi$-functions mirror a
> difference in the real state of the system in question; but such was not
> Einstein’s intention. (Howard 1985, 181)

What are then the non-trivial differences between theoretical states that
Einstein did have in mind? Here is Howard, again:

> The kind of difference with which Einstein was concerned is clear from his
> argument: [$\tau_{1}$] and [$\tau_{2}$] differ in the predictions they yield
> for the results of certain objective, local measurements on A. … (For
> example, if [$\tau_{1}$] attributed a definite position to [A], but not a
> definite momentum, it would be incomplete in its description of [A]’s
> momentum; but, of course, Einstein’s argument does not require any such
> reference to specific parameters or ‘elements of reality’.) (loc. cit.,
> modified for uniform notation)

The essential differences between $\tau_{1}$ and $\tau_{2}$ concern their
predictions of the measurement outcomes for A’s observables. For instance,
$\tau_{1}$ and $\tau_{2}$ are essentially different if one of them “lives” in
$(\pi_{1},\mathcal{H}_{A}\otimes I)$, say a position representation, and the
other in $(\pi_{2},\mathcal{H}_{A}\otimes I)$, say a momentum representation.
More generally then, $\tau_{1}$ and $\tau_{2}$ are essentially different if
$(\pi_{1},\mathcal{H}_{A}\otimes I)$ and $(\pi_{2},\mathcal{H}_{A}\otimes I)$
are unitarily inequivalent representations of
$\mathfrak{A}_{A}\otimes\mathcal{I}$. Taking this into account, one obtains
the following reconstruction of the second half of Einstein’s argument:

4T. QM allows unitarily inequivalent representations of
$\mathfrak{A}_{A}\otimes\mathcal{I}$, depending on which EPR-double,
$\pi_{1}(I\otimes O^{1}_{B})$ or $\pi_{2}(I\otimes O^{2}_{B})$, one can choose
to measure on subsystem B.

5T. But if QM is complete, the representations
$(\pi_{1},\mathcal{H}_{A}\otimes I)$ and $(\pi_{2},\mathcal{H}_{A}\otimes I)$
should be unitarily equivalent.

6T. Thus, QM is incomplete.

I believe that there is a good chance that this captures exactly what Einstein
thought on the matter. But one immediate criticism against this reconstruction
concerns the apparent conflict between premise 4T and the Stone-von Neumann
theorem (discussed in the previous section), a theorem that Einstein was no
doubt fully aware of by 1935. This asserts, recall, that any irreducible,
faithful and regular Hilbert space representation of the Weyl algebra
describing a system is uniquely determined up to a unitary transformation, and
in fact unitarily equivalent to the Schrödinger representation. The Stone-von
Neumann theorem applies, of course, to EPR systems. But the conflict with my
reconstruction is merely apparent, since the representations mentioned in
premise 4T are non-regular and, therefore, not in the range of the Stone-von
Neumann theorem.282828There are other ways in which the conflict can be
avoided, for example, by turning to phase spaces with nontrivial topologies,
on which representations can be regular but unitarily inequivalent (Landsman
1990).

This reply, however, just seems to have opened the door to further criticism.
For if Einstein’s argument required that regularity be dropped, then one might
seem justified to consider it muddled after all, for it is not at all clear
what physical meaning can be given to non-regular representations.292929As
discussed in the previous section, if position and momentum representations
are taken to be physically significant despite their non-regularity, then
unitary dynamics must be given up (cf. Feintzeig et al. 2019). But I think
that this captures precisely the sense of incompatibility that Gömöri and
Hofer-Szabó, as we have seen above, associate with the multiple assignment of
$\psi$-functions to subsystem A. The dynamical incompatibility between
(regular and) non-regular representations not only does not muddle Einstein’s
argument, but rather helps clarify his justification for rejecting the
completeness of QM.

Now, even if my algebraic reconstruction is correct, one might insist that in
order to fully support Howard’s suggestion, a formalization in a suitable
language should be given. Until we consider more closely the challenges, as
well as the necessary adjustments for a rational reconstruction of QM, an
informal version of the reconstructed argument should be enough here:

4H. QM allows non-isomorphic models of its description of A, depending on
which EPR-double one can choose to measure on subsystem B.

5H. But if QM is categorical, these models should be isomorphic.

6H. Thus, QM is not categorical.

As noted already, this version of the argument assumes that unitarily
inequivalent representations (in premise 4T) can be construed as non-
isomorphic models of the quantum mechanical description of subsystem A (in
premise 4H). In other words, it assumes that unitary equivalence is a
necessary, though possibly not a sufficient, condition for categoricity. On
this assumption, which I consider unproblematic, if the validity of the
argument is to be preserved, premise 5T must be reformulated as premise 5H.
Thus, Einstein completeness can (and perhaps should) be understood as
categoricity. I think that this clarifies and supports, at least in part,
Howard’s suggestion. Einstein’s no-go result may be understood to say that a
local, separable, and categorical QM cannot exist.

Reconstructing Einstein’s argument in this way points to a series of general
consequences often discussed, in other theoretical contexts, throughout the
rich history of philosophical concerns with the categoricity of logic and
mathematics. For instance, as noted in the preliminary considerations above,
the meaning of arithmetical terms is precisely determined only if Peano
arithmetic is categorical with respect to the isomorphism class of an omega
sequence. And the meaning of classical logical connectives is precisely
determined only if classical logic is categorical with respect to the
isomorphism class of a two-element Boolean algebra. Quite similarly, the
meaning of concepts like quantum state, observable, etc. is precisely
determined only if QM is categorical with respect to the relevant isomorphism
class of models. Thus, Einstein’s argument, reconstructed as I proposed above,
attributes to QM a kind of semantic indeterminacy that may help explain the
difference between EPR completeness and Einstein completeness as types of
descriptive completeness. The question is what kind of descriptive failure is
the incompleteness that Einstein attributed to QM? As a non-categorical
theory, QM is descriptively incomplete, just not in the sense that it fails to
describe all of the physical reality that it aims to describe, but rather in
the sense that it describes more than just the physical reality that it aims
to describe. This does not necessarily imply that the theory describes states
that cannot exist; rather, it describes dynamically incompatible states,
construed as elements of non-isomorphic models.

Before I conclude this chapter, let me briefly note some implications I take
my reconstruction of Einstein’s argument to have for our understanding of his
debate with Bohr. Let us recall one important point in that debate, that is,
that Bohr’s reply to the EPR paper, as was well understood by Einstein, denies
separability (Howard 2007). Thus, it also applies to Einstein’s own argument
(by rejecting the first half of premise 1). Einstein famously commented on
this point as follows:

> By this way of looking at the matter it becomes evident that the paradox
> forces us to relinquish one of the following two assertions: (1) the
> description by means of the $\psi$-function is complete. (2) the real states
> of spatially separated objects are independent of each other. (Einstein
> 1949, 682)

Einstein thought that relinquishing assertion (2), i.e., denying that
spatially separated objects have real states, would make physics impossible.
But a second important point made by Bohr is that EPR-doubles are
complementary. What does this point amount to, when considered in the
algebraic framework described above?

As already noted above, Bohr’s complementarity has been rigorously interpreted
in terms of the unitary inequivalence of non-regular Hilbert space
representations. Since Einstein’s argument, on my reconstruction, points out
that there are representations describing subsystem A of an EPR system that
are unitarily inequivalent (premise 4T), Bohr’s reply can be understood as a
rejection of premise 5T: a complete QM does not require unitary equivalence.
Furthermore, if it is correct to reformulate premise 5T as premise 5H, then
one can see Einstein and Bohr holding opposite positions about the semantics
of QM: while Einstein deplored its non-categoricity, Bohr embraced it as a
theoretical asset. The metasemantic contrast here should be apparent: the
semantic indeterminacy of the standard formalism was considered a serious
problem by Einstein, but ignored as insignificant by Bohr.

In any case, the reconstruction I offered in this section clarifies, I hope,
at least some of the assumptions behind Howard’s suggestion to read Einstein
completeness as categoricity. It also reinforces criticism of some of the
misconstruals of Einstein completeness in the literature. Finally, it provides
an explanation of the sense in which Einstein completeness is different than
EPR completeness as a type of descriptive completeness, and it may shed some
metasemantic light on the Bohr-Einstein controversy.

At this point in my narrative, I want to go back to an even earlier stage in
the history of mathematical science – more precisely, to the development of
symbolic algebra in the 19th Century – in order to then return to QM and
explain the nature and metasemantic implications of another central piece in
its development: Bohr’s correspondence principle.

##  3 The problem of permanence

This chapter offers a reinterpretation of Bohr’s correspondence principle as
grounded in a version of the principle of permanence, which played a crucial
role in the development of 19th century mathematics. This elucidates his view
on the meaning of quantum mechanics (QM), but also his contention that QM is a
rational generalization of classical physics, as well as a seemingly strange
criticism against his doctrine of the necessity of classical concepts. The
chapter further discusses von Neumann’s use of the principle of permanence in
his generalization of QM to infinite dimensions, and Weyl’s critical analysis
of quantum logic.

###  3.1 Historical roots and metasemantic implications

The problem of permanence concerns the applicability of the so-called
principle of permanence and its metasemantic implications. This
metatheoretical principle is the stipulation that the forms of certain
statements, especially the statements expressing the rules or laws of a
mathematical or a scientific theory, should be preserved when extending or
generalizing the theory, or when changing its domain of objects. It expresses
a fundamental feature of science, a perseverance that has long been
presupposed and often tacitly applied by the practitioners of science. In the
history of mathematics, this feature has played a crucial role in the
development of algebra, for example in the works of De Morgan, Boole, and
Dedekind. It has then been instrumental in articulating formalism – a cluster
of philosophical views on mathematics, including views expressed in the 19th
century by George Peacock and Hermann Hankel, but also that defended by
Hilbert in the 20th century. As I will show in this chapter, the principle of
permanence is centrally connected with the development of modern physics as
well, especially in the work of Bohr, where it was taken to support the
transition from classical physics to QM, and von Neumann, where it was used in
further generalizations of the latter.303030Section 3.1 draws on Toader 2021b
and 2023b.

####  3.1.1 Peacock’s principle of permanence

While the actual roots of the principle of permanence may lie deep into the
philosophy and science of the 18th century, or even earlier ones, its first
explicit formulation – as the principle of permanence of equivalent forms
(PPF) – is due to Peacock, mathematician and theologian at Cambridge:

> Direct proposition: Whatever form is algebraically equivalent to another
> when expressed in general symbols, must continue to be equivalent whatever
> those symbols denote.
>
> Converse proposition: Whatever equivalent form is discoverable in
> arithmetical algebra considered as the science of suggestion, when their
> symbols are general in their form, though specific in their value, will
> continue to be an equivalent form when the symbols are general in their
> nature as well as in their form. (Peacock 1833, 198sq, emphasis removed)

Peacock’s arithmetical algebra, or what had also been called “specious or
universal arithmetic”, is a theory the language of which includes not only
constants and signs for operations, like common arithmetic, but also variables
ranging over the domain of positive integers. He contrasted this theory with
symbolical algebra, the language of which further includes variables that are
allowed to range over any domain of objects whatsoever, but most importantly
negative, rational, and imaginary numbers such as the “impossible” roots of
equations of second or higher degree.

The direct proposition of the PPF was taken to be rather self-evident: “The
direct proposition must be true, since the laws of combination of symbols by
which such equivalent forms are deduced, have no reference to the specific
values of the symbols.” (op. cit., 199) But the converse proposition
stipulates that some equivalent forms, i.e., at least the ones that can be
discovered and expressed in the language of arithmetical algebra, will be
preserved as equivalent forms when expressed in the language of symbolical
algebra. A prime example of such forms are the “laws of combination”, which
Peacock indiscriminately also considered as the principles or the rules of a
theory. For instance, the rule of distributivity in arithmetical algebra,
which he wrote as $ma+na=(m+n)a$, is to be preserved in symbolical algebra as
such. He seems to have intended the notion of equivalent forms to be more
general, so what is to be preserved in passing from one theory to another are
not only such rules, but theorems as well. In any case, he clearly rejected
the universal validity of the PPF, for he saw that those equivalent forms of
arithmetical algebra that are “essentially arithmetical” cannot be preserved
when passing to symbolical algebra.313131For this point, see Toader 2021b, 80.
Similarly, the rule of commutativity will be later considered an essentially
classical rule, since it cannot be preserved as a relation between operators
corresponding to physical variables when passing from classical physics to QM.
For discussions of Peacock’s algebra and his PPF, see e.g. Pycior 1981, and
more recently, Lambert 2013. Whenever the PPF could be applied, however, it
was so useful for solving problems and proving theorems that other
mathematicians, such as De Morgan, came to regard it not merely as a heuristic
principle, but as a necessary mathematical truth.323232For De Morgan’s 1837
formulation of the principle of permanence, and for his discussion of this
with Ada Lovelace, who questioned the validity of the principle and his
insistence on “the necessity of its truth”, see Hollings et al. 2017.

The sense in which arithmetical algebra can be considered as a “science of
suggestion” seems, however, rather obscure, so a brief explanation is in
order. Peacock’s main concern in his “Report” was the clarification of the
nature of the principles of symbolical algebra, which he thought had been
misunderstood as being founded on, i.e., deduced from, arithmetical principles
– a misunderstanding that he took to be responsible for the view of symbolical
algebra as a mere generalization of arithmetical algebra. The PPF, and in
particular the notion of a science of suggestion, was meant to help with
Peacock’s refutation of this view. On his own view, the principles of
symbolical algebra are suggested by, rather than deduced from, the principles
of arithmetical algebra and, as a consequence, the former was something else
than a mere generalization of the latter. To say otherwise was, to him, an
abuse of terminology, for the following reason:

> The operations in arithmetical algebra can be previously defined, whilst
> those in symbolical algebra, though bearing the same name, cannot: their
> meaning, however, when the nature of the symbols is known, can be generally,
> but by no means necessarily, interpreted. The process, therefore, by which
> we pass from one science to the other is not an ascent from particulars to
> generals, which is properly called generalization, but one which is
> essentially arbitrary, though restricted with a specific view to its
> operations and their results admitting of such interpretations as may make
> its applications most generally useful. (op. cit., 194)

Peacock dismissed the view according to which one passes from arithmetical
algebra to symbolical algebra by a properly called generalization, for this
would entail that the operations in symbolical algebra must admit of an
interpretation over the domain of arithmetical algebra. On his view, one
passes from the latter to the former by suggestion, which does not entail that
those operations must admit of such an interpretation. Further, he added that
when one passes from arithmetical algebra to symbolical algebra, the former
“necessarily suggests its [i.e., the latter’s] principles”. But this does not
mean that arithmetical algebra is indispensable as a science of suggestion for
symbolical algebra, for Peacock considered the former simply the “most
convenient”, though in principle not the only possible, science of suggestion
for the latter.

What he meant, I take it, is that the intertheoretical relation of suggestion
is necessary. In principle, any “arbitrary assumptions” that are logically
consistent, or as he put it, “as far as they can exist in common”, might be
stipulated as principles of symbolical algebra. But since useful application
of the results obtained in this theory requires their interpretability over
the domain of application, those assumptions must be further restricted such
that the operations can admit of interpretation over that domain. This
restriction is carried out by assuming some subordinate science of suggestion,
like arithmetical algebra. Thus, if the principles of symbolical algebra are
not entirely arbitrary, but they are both consistent and suggested by the
principles of arithmetical algebra, then the operations of symbolical algebra
are interpretable over the positive integers.

Peacock’s view, as I see it, is that when arithmetical algebra is actually
taken as a science of suggestion, although it is not the case that the
operations in symbolical algebra must admit of an arithmetical interpretation,
it must be the case that they can admit of an arithmetical interpretation.
This explains the general sense of PPF taken as the “proper guide” for the
development of symbolical algebra: an arithmetical interpretation is not
necessary; it is only necessarily possible. This necessary interpretability is
dictated by the requirement of usefulness in applications. But it allows that
other sciences of suggestion might be considered instead, and consequently
that other interpretations of the symbolic operations are admissible.
Peacock’s point is that some such particular interpretation must be possible
and, as clear from the direct proposition of the PPF, any such interpretation
is admissible. Moreover, if one takes the PPF as the proper guide for the
development of symbolical algebra, then its principles, albeit not deducible
from the principles of arithmetical algebra, are deducible from the
conjunction of the principles of arithmetical algebra and the PPF. This is why
Peacock said the PPF is the “real foundation” of symbolical algebra. But this
also helps explain why the latter’s principles are considered arbitrary
assumptions, but also “necessary consequences” of the PPF.

What are the metasemantic implications of Peacock’s principle? Could one
recover, from his reflections on the PPF, an answer to Hilbert’s question
about metasemantics (mentioned at the outset, in section 1.1), that is the
question about the relationship between the language of a theory and its
semantic attributes? If the operations in symbolical algebra cannot be
“previously defined”, if their meaning cannot be determined before its
principles or rules are determined, then what is it that gives meaning to the
operations afterwards? Peacock’s view is as follows:

> In arithmetical algebra, the definitions of the operations determine the
> rules; in symbolical algebra, the rules determine the meaning of the
> operations, or more properly speaking, they furnish the means of
> interpreting them (op. cit., 200)

I take this answer as an expression of the metasemantic revolution brought
about by the PPF, a revolution in the metasemantics of the mathematics of the
19th century. I believe it is a metasemantic revolution insofar as it
represents a radical change from a traditional view, according to which the
meaning of mathematical operations is determined by their definitions relative
to a particular domain, as is the case in arithmetical algebra, to a novel
view, according to which the meaning of operations is determined by their very
rules, as in the case of Peacock’s symbolical algebra.

One way to understand his claim that the rules of symbolical algebra determine
the meaning of its operations, in the more proper sense that these rules
“furnish the means of interpreting” the operations, is to say that the rules
determine the interpretation of the operations. But if arithmetical algebra is
taken as a science of suggestion, and if one understands this notion as I
proposed above, then an interpretation over the positive integers of the
operations of symbolical algebra is necessarily possible, without being