Quantum mechanics; an empiricist view

Quantum Mechanics: An Empiricist View

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Quantum Mechanics: An Empiricist View

Bas C. van Fraassen

CLARENDON PRESS · OXFORD

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Que, touchant les choses que nos sens n'aperỗoivent point, il suffit d'expliquer comment elles peuvent être . . .
René Descartes, Principes, iv. 204

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Preface
Quantum theory grew up, from Planck to Heisenberg and Schroedinger, in response to a welter of new experimental
phenomena: measurements of the heat radiation spectrum, the photoelectric effect, specific heats of solids, radioactive
decay, the hydrogen spectrum, and confusingly much more. Yet this theory, emerging from the mire and blood of
empirical research, radically affected the scientific world-picture. If it did describe a world ‘behind the phenomena’, that
world was so esoteric as to be literally unimaginable. The very language it used was broken: an analogical extension of
the classical language that it discredits, and redeemed at best by the mathematics that it tries to gloss.
Interpretation of quantum theory became genuinely feasible only after von Neumann's theoretical unification in 1932.
Von Neumann himself, in that work, attempted to codify what he took to be the common understanding.
Astonishingly, the attempt led him to assert that in measurement something happens which violates Schroedinger's
equation, the theory's cornerstone. As he saw very clearly, interpretation enters a circle when its main principle is
Born's Rule for measurement outcome probabilities, while at the same time measurements are processes in the domain
of the theory itself. Behold the enchanted forest: every road leads into it, and none leads out—or does the hero's
sword cleave the wood by magic?
An empiricist bias will be evident throughout this book, but my own interpretation of quantum mechanics does not
begin until Chapter 9. The first three chapters provide philosophical background; though they overlap my Laws and
Symmetry, I have tried to make them interesting in their own right. The next four chapters mainly outline the
achievements of foundational research, though with an eye to the philosophical issues to come. The negative part is to
show that the phenomena themselves, and not theoretical motives, can suffice to eliminate Common Cause models of
the observable world. The positive part is the conclusion that there are adequate descriptions of

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measurement—in the sense required for Born's Rule—internal to quantum theory. To make the book relatively selfcontained, Chapters 6 and 7 introduce all the quantum mechanics needed for the philosophical discussions to come.

From a purely philosophical point of view, the most important clarification reached since 1925 concerns the criteria of
adequacy for interpretations of quantum mechanics. It appears at present that more than just one tenable
interpretation, already in process of development, can meet those criteria.
I regret that I may have done little justice to the promising interpretations now underway which differ from my own,
although I have tried to point to them as often as I could. I regard every interpretation as increasing our
understanding, and believe that an awareness of what rival interpretations may be tenable is crucial to clarity. But that
attitude already needs defence, for it involves views on what science is, and what philosophy can hope for.
I have also tried to take the philosophical debates somewhat further, into the fascinating cluster of problems that
concern quantum-statistical mechanics and identical particles. At every point, but here especially, I was acutely aware of
rapid progress in foundational research and of the kaleidoscopically changing philosophical debates. It is true that
interpretation focuses on a single theory at one more or less definite historical stage—and yet, what we try to interpret
is not static. Every time we understand a little more, we change what we are trying to understand. It is not surprising
that scientists often become impatient with philosophy: what is ever achieved if every generation has to face the same
questions again, with a new understanding of what is being asked, unable to rest on past answers? But philosophy does
not create our predicament. It is only a myth that modern science had arrived at a clear and well-integrated worldpicture, or that contemporary science has already effectively given us a new one. At best, we are in process of replacing
what never has existed by something that never will. It is only in this unendliche Aufgabe, this reaching for what we
cannot finally have or hold, that understanding consists.
The pleasures of acknowledgement are always accompanied by a good deal of soul-searching. Debts are subtle, and
always

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ix

so numerous that only a few can be avowed, for philosophy is a thoroughly historical and communal enterprise. For
the first part of the book, devoted to general philosophical background, my debts are largely acknowledged already in
my previous books. But I must thank above all my teachers Adolf Grünbaum, who led me into the intricacies of

determinism and indeterminism, and Wilfrid Sellars, who would not allow me to treat those or any other subjects in
isolation. To Henry Margenau I believe I am indebted in two ways, first through what I received from him through
Grünbaum, who was his student and my teacher, and then directly as he drew me into his quantum-mechanical
questioning during my two years at Yale. In the next year at Indiana University Wesley Salmon took me in, as it were, to
instil a preoccupation with causality, probability, and frequency. Salmon was the first to comment on my fledgling ideas
about identical particles. It was also around then that I participated in a symposium with Hilary Putnam, who
challenged me with a new way to see quantum logic. In the individual chapters I have tried as much as possible to
indicate my more specific debts, for example to Enrico Beltrametti and Gianni Cassinelli, whose book became one of
my bibles, to my frequent collaborator, R. I. G. Hughes, and to Jeffrey Bub, Nancy Cartwright, Roger Cooke, Maria
Luisa Dalla Chiara, Arthur Fine, Clifford Hooker, Simon Kochen, Pekka Lahti, James McGrath, Peter Mittelstaedt,
and Brian Skyrms, among others. Alan Hajek and R. I. G. Hughes read large parts of the manuscript and gave many
helpful comments. Almost every section of each chapter benefited from the close reading and comments by Sara
Foster. The National Science Foundation and Princeton University steadfastly supported my research, while Anne
Marie De Meo typed the results and helped me generously through many practical difficulties.
B.C.v.F.

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Summary Table of Contents
1. What Is Science?
PART I. DETERMINISM AND INDETERMINISM IN CLASSICAL PERSPECTIVE
2. Determinism
3. Indeterminism and Probability
PART II. HOW THE PHENOMENA DEMAND QUANTUM THEORY

4. The Empirical Basis of Quantum Theory
5. New Probability Models and their Logic
PART III. MATHEMATICAL FOUNDATIONS
6. The Basic Theory of Quantum Mechanics
7. Composite Systems, Interaction, and Measurement
PART IV. QUESTIONS OF INTERPRETATION
8. Critique of the Standard Interpretation
9. Modal Interpretation of Quantum Mechanics
10. EPR: When Is a Correlation Not a Mystery?
11. The Problem of Identical Particles
12. Identical Particles: Individuation and Modality
NOTES
BIBLIOGRAPHY
INDEX

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Contents
1. What Is Science?
1. Two views about science
2. Theories and models
3. Interpretation: science as open text
4. Models and scientific practice
5. More about empiricism
PART I. DETERMINISM AND INDETERMINISM IN CLASSICAL PERSPECTIVE
2. Determinism
1. How symmetry is connected to determinism
2. State-space models and their laws
3. Symmetry, transformation, invariance
4. Symmetries of time: classical (in)determinism
5. Conservation laws and covariance
3. Indeterminism and Probability
1. Pure indeterminism and the modalities
2. Probability as measure of the possible

3. Symmetry and a priori probability
4. Permutation symmetry: De Finetti's representation theorem
5. Ergodic theory: underlying determinism
6. A classical version of Schroedinger's equation
7. Holism: indeterminism in compound systems
PART II. HOW THE PHENOMENA DEMAND QUANTUM THEORY
4. The Empirical Basis of Quantum Theory
1. Threat of indeterminism

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2. Causality in an indeterministic world
3. Deduction of Bell's Inequalities
4. The experiments
5. General description of causal models
6. Does locality really play a role?
5. New Probability Models and their Logic
1. When are values indeterminate?
2. General and geometric probability models
3. Accardi's inequalities
4. The end of counterfactual definiteness
5. Models of measurement: a trilemma for interpretation
6. Introduction to quantum logic
7. Is quantum logic important?
PART III. MATHEMATICAL FOUNDATIONS
6. The Basic Theory of Quantum Mechanics
1. Pure states and observables
2. Pure states, observables, and vectors
3. Observables and operators
4. Mixed states and operators

5. Gleason's theorem and its implications
6. Symmetries and motion: Schroedinger's equation
7. Symmetries and conservation laws
8. The radical effect of superselection rules
7. Composite Systems, Interaction, and Measurement
1. Composition
2. Reduction
3. Interaction and the ignorance interpretation of mixtures
4. The quantum-mechanical theory of measurement
5. Preparation of state
PART IV. QUESTIONS OF INTERPRETATION
8. Critique of the Standard Interpretation
1. What is an interpretation?

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2. Two forms of indeterminism
3. What happens in measurement? von Neumann's answer
4. Von Neumann's first defence: consistency of measurement
5. Von Neumann's second defence: repeatable measurement
6. R. I. G. Hughes's argument from conditional probability
7. Two cat paradoxes and the macro world
8. Macroscopic character and superselection rules
9. Modal Interpretation of Quantum Mechanics

1. The modal interpretation
2. The modal account developed
3. What happens in a measurement?
4. Puzzle: how far does holism go?
5. Puzzle: is there chaos behind the regularities?
6. The resources of quantum logic
7. The modal interpretation, quantum-logically
8. Modal interpretation of composition and reduction
9. Consistency of the description of compound systems
10. Interpretation and the virtue of tolerance
10. EPR: When Is a Correlation Not a Mystery?
1. The paper by Einstein, Podolsky, and Rosen
2. Initial defence of the argument
3. The step to empirical testability
4. How are correlations explained?
5. Attempts at perfect explanation
6. Sinister consequences and spooky action at a distance
7. The end of the causal order?
11. The Problem of Identical Particles
1. Elementary particles: aggregate behaviour
2. Permutation invariance and the Dichotomy Principle
3. The Exclusion Principle
4. Blokhintsev's proof of the fermion–boson dichotomy
5. Permutations as superselection operators

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6. Quantum-statistical mechanics
7. Classical ‘reconstruction’ via Carnap and via De Finetti's Theorem
8. The modal interpretation applied to aggregate behaviour
9. Possible world-models for quantum mechanics
10. The Exclusion Principle in the modal interpretation
11. A fermion model with individuation
12. Bosons and genidentity
12. Identical Particles: Individuation and Modality
1. Are there individual particles?
2. Brief exposition of second quantization
3. Three parallel debates in metaphysics
4. The identity of indiscernibles
5. Conclusion: good-bye to metaphysics
NOTES
BIBLIOGRAPHY
INDEX

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1 What Is Science?
As painting, sculpture, and poetry turned abstract after the turn of the century; as Minkowski recast the
electrodynamics of moving bodies in four-dimensional geometry; as Hilbert and Russell turned geometry and analysis
into pure logic; philosophy of science too turned to greater abstraction. But philosophy of science is still philosophy,
and is still about science. Before broaching the philosophy of physics, and the foundations of quantum mechanics, I
shall locate those projects in the larger enterprise of philosophical reflection on science as a whole.

1. Two Views About Science
There is quite a difference between the questions ‘What is happening?’ and ‘What is really going on?’ Both questions
can arise for participants as well as for spectators, and usually no one has more than a fragmentary answer. To the
second question the answer must undeniably be more doubtful, because it has to be somewhat speculative in the
interpretation it puts on what happens. Yet both questions seem crucially important.
So far I might have been talking about war, a political movement, contemporary art, the Diaspora, the Reformation, or
the Renaissance—as well as about science, its current state or its historical development. Scientists, the participants in
this large-scale cultural activity, we can consult only about what is happening. Both these participants and the more
distant spectators cannot help but attempt some interpretation as well. Indeed, we are all to some extent both
participant and spectator, for science has become an activity of our civilization as a whole.

Philosophy of science has focused on theories as the main product of science, more or less in the way philosophy of
art has

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focused on works of art, and philosophy of mathematics on arithmetic, analysis, abstract algebra, set theory, and so
forth. Scientific activity is understood as productive, and its success is measured by the success of the theories
produced. In the most general terms, of course, the aim of an activity is success, and what that aim exactly is depends
on the ‘internal’ criterion of success. The personal aim of the chess-masters may well be fame and fortune, but the aim
in chess—as such, the ‘internal’ criterion of success in chess—is checkmate. To say what is really going on in science,
therefore, we try to determine its aim in this sense.
Both scientists and philosophers have come up with very different answers to this. To some extent, the differences may
reflect personal aim. Newton for example may have understood the aim of science as such as uncovering God's
design, or arriving at the truth about the most basic laws of nature. To some extent, also, the answers have reflected
current beliefs about just what it is possible to achieve in science. Newton believed that there was a unique derivation
of the laws of nature from the phenomena. If that is indeed so, it is of course reasonable to say that this is exactly what
science aims to do. When Mach answers instead in terms of economy of thought and organization of knowledge, or
when Duhem denies that the aim of science includes explanation, that is undoubtedly in part because they believed so
much less about what could be achieved in science, or any other way.
Can we find an internal criterion of success that characterizes scientific activity for all ages, and equally for
philosophical and unphilosophical participants? The first thing to do is see exactly what the product is whose success is
to be assessed by that criterion. When we focus on the scientific theory, as product of science, we turn this into a
question about theories. So, first, what sort of thing is a theory? A scientific theory must be the sort of thing that we
can accept or reject, and believe or disbelieve. Accepting a theory implies the opinion that it is successful; science aims to give
us acceptable theories. More generally, a theory is an object for the sorts of attitudes expressed in assertions of

knowledge and opinion. A typical object for such attitudes is a proposition, or more generally a body of putative
information, about what the world is like.

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Given this view about theories, we return to the question of aim.
First answer: realism. At this point we can readily see one very simple possible answer to all our questions, the answer we
call scientific realism. This philosophy says that a theory is just the sort of thing which is either true or false; and that the
criterion of success is truth. As corollaries, we have that acceptance of a theory as successful is, or involves, the belief
that it is true; and that the aim of science is to give us (literally) true theories about what the world is like.
That answer would of course have to be qualified in various ways to allow for our epistemic finitude and the tentative
nature of reasonable attitudes. Thus realists may add that, although it cannot generally be known whether or not the
criterion of success has been met, we may reasonably have a high degree of belief about this, and that the scientific
attitude precludes dogmatism. Whatever doxastic attitude we adopt, we stand ready to revise in face of further
evidence. These are all qualifications of a sort that anyone must acknowledge. They do not detract from the appealing
and, as it were, pristine clarity of the scientific-realist position. But that does not mean that it is right. Even a scientific
realist must grant that an analysis of theories—even one that is quite traditional with respect to what theories are—does not
presuppose realism. We may grant that theories are the sort of thing which can be true or false, that they say something
about what the world is like; it does not follow that we must be scientific-realists.
Second answer: empiricism. There are a number of reasons why I advocate an alternative to scientific realism (see van
Fraassen 1980b, 1985a). One concerns the difference between acceptance and belief; reasons for acceptance include
many which, ceteris paribus, detract from the likelihood of truth. This point was made very graphically by William James;
it is part of the legacy of pragmatism. The reason is that, in constructing and evaluating theories, we follow our desires
for information as well as our desire for truth. We want theories with great powers of empirical prediction. For belief
itself, however, all but the desire for truth must be ‘ulterior motives’. Since therefore there are reasons for acceptance

which are not reasons for belief, I conclude that acceptance is not belief. It is an elementary logical

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point that a more informative theory cannot be more likely to be true; therefore the desire for informative theories
creates a tension with the desire to have true beliefs.
Once we have driven the wedge between acceptance and belief, we can reconsider the ways to make sense of science.
As one such way, I wish to offer the anti-realist position I advocate, which I call constructive empiricism. It says that the
aim of science is not truth as such but only empirical adequacy, that is, truth with respect to the observable phenomena.
Acceptance of a theory involves as belief only that the theory is empirically adequate. But acceptance has a pragmatic
dimension; it involves more than belief.
The agreement between scientific realism and constructive empiricism is considerable and includes the literal
interpretation of the language of science, the concept of a theory as a body of information (which can be true or false,
and may be believed or disbelieved) and a crucial interest in interpretation, i.e. finding out what this theory says the
world is like.1 There is much that the two can explore together. Acceptance has a clear pragmatic aspect: besides belief,
it involves commitments of many sorts. Realists may hold that those commitments derive from belief, but they will
wish to join empiricists in exploring just what they are. Similarly, empiricists will wish to explore in the same way as
realists just what it is that our accepted theories say. Both agree, after all, that theories say something about what the
world is like. The content of a theory is what it says the world is like; and this is either true or false. The applicability of
this notion of truth value remains here, as everywhere, the basis of all logical analysis. When we come to a specific
theory, the question: how could the world possibly be the way this theory says it is? concerns the content alone. This is the
foundational question par excellence, and it makes equal sense to realist and empiricist alike.

2. Theories and Models
To formulate a view on the aim of science, I gave a partial answer to the question of what a scientific theory is. It is an

object of belief and doubt, so it is the sort of thing that could be

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true or false. That makes sense only for a proposition, or a larger body of putative information. It does not follow that
a theory is something essentially linguistic. That we cannot convey information, or say what a theory entails, without
using language does not imply that—after all, we cannot say what anything is without using language. We are here at
another parting of the ways in philosophy of science. Again I shall advocate one particular view, the semantic view of
theories. Despite its name, it is the view which de-emphasizes language.
Words are like coordinates. If I present a theory in English, there is a transformation which produces an equivalent
description in German. There are also transformations which produce distinct but equivalent English descriptions.
This would be easiest to see if I were so conscientious as to present the theory in axiomatic form; for then it could be
rewritten so that the body of theorems remains the same, but a different subset of those theorems is designated as the
axioms, from which all the rest follow. Translation is thus analogous to coordinate transformation—is there a
coordinate-free format as well?
The answer is yes (though the banal point that I can describe it only in words obviously remains). To show this, we
should look back a little for contrast. Around the turn of the century, foundations of mathematics progressed by
increased formalization. Hilbert found many gaps in Euclid's axiomatization of geometry because he rewrote the
proofs in a way that did not rely at all on the meaning of the terms (point, line, plane, . . . ). This presented philosophers
with the ideal: a pure theory is written in a language devoid of meaning (a pure syntax). A scientific theory should then
be conceived of as consisting of a pure theory (the mathematical formalism, ideally an exact axiomatic system in a pure
syntax) plus something that imparts meaning and so connects it with our real concerns.
This did help philosophical understanding of science in one crucial case. How could the geometry of the world be nonEuclidean? What sense does this make? Well, as pure theory, both Euclidean and non-Euclidean geometries are
intelligible. Suppose now we impart meanings by what Reichenbach called coordinative definitions. For example, say that a
light ray is the physical correlate of a straight line or geodesic. Then, at that point, it becomes an empirical question

whether physical

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geometry is Euclidean. For it is an empirical question how light rays behave.
For a little while it seemed that the meaning-restoring device would be very simple—a dictionary or a set of
‘operational definitions’. By its use, the language of theoretical physics would receive a complete translation into the
simple language of the laboratory assistant's observation reports. But the example of physical geometry already shows
that the physical correlates do not yield a complete translation. Any two points lie on a straight line, but not necessarily
on a light ray. If, on the other hand, we translate ‘straight line’ as ‘possible light ray path’, the theoretical element is not
absent, and criteria of application are indefinite; is this possibility relative to laws and circumstances which could be
stated without recourse to geometric language? The opposite view is to consider the mathematics in use an abstraction
from the science that uses it, and to leave the reconstruction of language along formalist lines to a different
philosophical enterprise.
Despite certain undoubted successes, the linguistic turn in analytic philosophy was eventually a burden to philosophy
of science. The first to turn the tide was Patrick Suppes, with his well-known slogan: the correct tool for philosophy of
science is mathematics, not meta-mathematics. This happened in the 1950s; bewitched by the wonders of logic and the
theory of meaning, few wanted to listen. Suppes's idea was simple: to present a theory, we define the class of its models directly,
without paying any attention to questions of axiomatizability, in any special language, however relevant or simple or
logically interesting that might be.2
This procedure is common in contemporary mathematics, where Suppes had found his inspiration. In a modern
presentation of geometry we find not the axioms of Euclidean geometry, but the definition of the class of Euclidean
spaces. Similarly, Suppes and his collaborators sought to reformulate the foundations of Newtonian mechanics, by
replacing Newton's axioms with the definition of a Newtonian mechanical system. This gives us, by example, a format
for scientific theories. In Ronald Giere's recent encapsulation, a theory consists of (a) the theoretical definition, which

defines a certain class of systems, and (b) a theoretical hypothesis, which asserts that certain (sorts of) real

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systems are among (or related in some way to) members of that class (see especially Giere 1985, 1988).
This new tradition is called the ‘semantic approach’ or ‘semantic view’. It has been developed, since the mid-1960s, by
a number of writers, some scientific realists (like Giere) and some not.3 In this approach the role of language (and
especially syntax or questions of axiomatization) is resolutely de-emphasized. The discussion of models treats them
mainly as structures in their own right, and views theory development as primarily model construction. Almost all
questions in philosophy of science take on a new form, or are seen in a new light, when asked again within the
semantic view. Two books have recently appeared, applying the semantic view in philosophy of biology (Lloyd 1988;
Thompson 1989). Most studies within this approach, however, have focused on philosophy of physics.
Families of structures, mathematically described, are something quite abstract. This is true even if we take an example
like ‘A Newtonian planetary system is a structure consisting of a star and one or more planets and is such that . . . ’.
The nouns are not abstract, and if a formalist says that we should deduce consequences from this definition only by
arguments which rely not at all on the meanings of ‘star’ and ‘planet’, that is just his or her predilection. The
abstractness consists rather in the fact that the same family of structures can be described in many ways; it is something
non-linguistic, but (banal as that is) we can still present it only by giving one specific verbal description. So concepts
relating to language must also retain a certain importance. Objectively, the conceptual anchor of informativeness is
logical implication: if T implies T′ then T is at least as informative as T′. But its informativeness for us depends on the
formulation we possess, and the extent to which we can see its implications. This brings us from semantics into the
area of pragmatics; our pragmatic reasons for accepting one theory rather than another may include that the former is
more easily processable by us.
How does this affect our conception of the aim of science? The empiricist takes this aim to be to give us empirically
adequate theories; the realist says that it is to give us true ones. Now, we identify a theory as a class of models. So is not

that aim at once satisfied, in either case, by someone who says: ‘I

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have a nice theory. It has as models exactly those structures which are isomorphic to the real world in the following
respects’?4 I think the problem arises equally for empiricist and realist. But I do not think that we need to revise the
statement of the aim of science. We should answer that person: ‘Of course we believe that your theory is true (or
adequate in the respects you specify). Obviously the real world, properly conceived, is one of the models of your
theory. But we have no reason to make the other commitments that go into acceptance—such as designing a research
programme for it, using it to answer why or how questions, attempting to redescribe phenomena in its terms, and the
like. The relevant pragmatic factors are missing because, however informative it is in a strictly objective or semantic
sense, it is not informative for us.’
It might be countered to this answer that the aim of science is not exactly to produce true or adequate theories, but to
produce acceptable ones. That seems wrong to me. Analogously, it is not the aim of mathematics to produce proofs
that we can follow; it is merely the case that we cannot know, or have good reason to believe, a given putative theorem
unless we can follow the proof. This concerns not the criteria of success, but our ability to see whether the criteria are
met. Nevertheless, this is a qualification of the description I gave in The Scientific Image of the theoretical virtues. My
gloss on ‘we want informative theories’ was that we want empirical strength, which I characterized as independent of
pragmatic factors. The qualification is that, as with other virtues characterizable semantically, whether they are
perceptible depends on the formulations of the theory that we actually possess.

3. Interpretation: Science as Open Text
The main project of the latter part of this book will be an interpretation of quantum mechanics. One other sort of
interpretation, following the lead given by von Neumann, will also be described in contrast, and still others will be
briefly discussed. But what sort of project is this? The semantic approach requires no axiomatization, nor a division

into pure syntax plus an interpretation in the sense of something that bestows meaning

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