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If I have observed x, I deem a formal lan- P guage T which forbade the observation of x untenable. P con If, for instance, in two subsequent identical quantum measurements, one observed different values, quantum ′ fal P P mechanics, T qm = (Sqm, Rqm, P qm), would be contra- dicted: There are no quantum states and measurement operators that could, within the postulates of quantum mechanics, account for the corresponding result (x1, x2) qm con with x1 = x2. The elements in S together with rules Figure 1. Among all the syntactically correct sentences P ′ in Rqm do6 not allow to conclude that x may be observed about the experimental context, the sentences in P are con- qm tained in the formal language T =(S,P,R), whereas the ones for either there is no such in P or an associ- fal ated weight is zero. in P contradict T . Let T = (S,R,P ) be the formal language of some theory. In order to formally characterize the sen- tences that falsify T , we now consider statements re- depend on observations, qualified by a—subjective, as ar- stricted to a context: We assume a formal language gued below—verification function. A theory in this sense T con Scon, Rcon, P con is not statistical learning; thus, our approach differs from = ( ) whose syntactically correct 3 sentences p P con refer to possible results observable for the one in [10]. a given context,∈ i.e., experimental setup. Furthermore, Can the verification function be reduced to we assume that the formal language T can be restricted “atomic” (i.e., not further reducible) symbols and to some T ′ such that the corresponding sentences P ′ are thus gain a similar observer-independent status? Is the subset of P con of sentences that T can account for. there a “single language sufficient to state all the 4 For T to be falsifiable in a context T con, the set of sen- there are to state” [11]? In Section III, we address tences P ′ has to be a proper subset of P con. We can these questions. derive a formal language T fal from T ′ and T con with all sentences about the experimental context that con- tradict T as illustrated in Figure 1. If we considered a III. REDUCIBLE OR NOT? sentence in P fal true, then T would be falsified for the given context. In the previous example of two identical In the following, we argue that accounts of experience subsequent quantum measurements, the result (x1, x2) fal cannot be exhaustively expressed in a formal language. with x1 = x2 corresponds to a sentence in P . Falsification6 refers to theories, including semantic con- In a first step, we examine how limits on formal languages cepts beyond a formal language, rather than to the formal restrict a formalizability of the verification function (see language directly.2 The semantic concepts are delegated Section IIIA). Even if one removes the restriction to for- to the condition “I have observed x,” which we now turn mal languages: There remain arguments against the ex- to. istence of a privileged language that reveals the one Ultimately, sentences are to be gauged by experience (see Sections IIIB and IIIC). con 5 according to Characteristic (C2). Let vO : P In analogy to an observation by Putnam, it should be true, false be a function—the so-called verification→ noted that the subsequent arguments do not constitute { } function—, for which vO(p) = true if and only if the a of impossibility. We aim to show that the belief observer O has observed p. More specifically, if vO(p)= that there is one language that unveils the unique truth, true then O deems some apparatus fit to produce values in the context , and O is certain to have obtained the value p. TheM verification function establishes whether the condition “I have observed p” is satisfied or not. 3 The symbols in S and the rules R allow to deduce all sentences Thus, for O, T is falsified for the context P con if there in S that T accounts for independently of the prior observations fal true such as a history in the observer graph in [10]. exists p P with vO(p)= . 4 The reasoning∈ about a formal language T is indepen- In [11], Rorty characterizes reductionism as the attempt to find such a language. dent of the particular observer—it transcends single ob- 5 “Reichenbach, Carnap, Hempel, and Sellars gave principled rea- servers: Any observer, upon knowing the formal lan- sons why a finite translation of material-thing language into guage T , can conclude for all sentences p P con whether sense-datum language was impossible. Even if these fall they are in P fal or not and, hence, whether∈ they fal- short of a strict mathematical impossibility proof, they are enor- sify T or not. The rules R are, therefore, required not to mously convincing [. . . ]. In the same spirit, I am going to give principled reasons why a finite empirical definition of intentional relations and properties in terms of physical/computational re- lations and properties is impossible—reasons which fall short of a strict proof, but which are, I believe, nevertheless convincing.” 2 Popper in [9] refers to “empirical-scientific systems.” [12, §5] 3 is merely one of multiple philosophical stances.6 The with an incomplete formal language in which certain syn- readiness to associate symbols in a formal language with tactically correct sentences are excluded by assumption, accounts of experience, as is necessary in formulating the or (2) we can run into glitches in our experience that we measurement problem, stems from such a belief. Though cannot account for without contradiction, undermining not logically excluded, such a belief is in tension with an our confidence in experience as the appropriate norma- essential idea of empirical sciences7—and, as we argue tive authority (see also the reliability constraint in [16, below, an idea of how meaning comes about: No theory 2B]), or (3) we avoid the to sentences within can claim for itself to have exhaustively captured an ob- that language. Tarski follows the latter path: He con- server’s account of experience, while it draws legitimacy fines the definition of truth to languages that are not from experimental findings. Should not, instead, the role semantically closed—languages that do not refer to their of an ironic [13] appeal to the (quantum) theorist, always own sentences. The definition of a truth predicate then doubtful towards one’s final vocabulary? becomes part of an expanded metalanguage. Following Tarski’s path, we have to accept a dualism before ad- dressing the problematic dualism between states and the A. The limits of formal languages statistics of measurement results in quantum mechanics.

Self-reference imposes limitations on formal languages. Problems arising from self-reference crystallize, e.g., in B. Semantic holism the Liar’s antinomy: The argument of semantic holism has been turned This sentence is false. against the existence of any fundamental dualism. Quine turns against the dogma of reductionism—“the belief The sentence is self-contradictory because a statement is that each meaningful statement is equivalent to some true if and only if the claim that the statement be true logical construct upon terms which refer to immediate is true. This particular unquotation notion of truth— experience” [17]: In a first step, he argues for a “con- illustrated by: “Snow is white” is true if and only if snow firmation holism”—based on the observation “that ex- is white—is called Schema T [15]. For a truth predi- perience can confirm or refute only a whole system of cate T and with denoting the of a sentence, it 9 h·i knowledge” [18]. By means of a confirmation theory of can be written as meaning, this entails semantic holism: “If a statement cannot be confirmed in isolation, it does not have mean- φ T φ , sentences φ . ↔ h i ∀ ing in isolation either” [18]. Gonseth observes that in With the theorem on the undefinability of truth, Tarski light of semantic holism, no parts of language, not even 10 showed that any formal language extending first-order , can be exempt from revision. arithmetic8 with a truth predicate containing Schema T Wittgenstein’s Tractatus logico-philosophicus [14] (see allows for such a contradiction. Self-reference, together also [19]) shows similar holistic traits: If one joins the with a negation, thwarts the unification of different lin- dots, a dense network of relations between terms emerges, guistic contexts as a consistent truth predicate requires but no ultimate explanation. Wittgenstein exposes the a meta-language. “misunderstanding of the logic of our language” [14] in- The verification function above serves as a predicate. directly. He concludes: If it is formalized in a theory extending first-order arith- My propositions are elucidatory in this way: metics, we require it to contain the Schema T, and no he who understands me finally recognizes sentences of the formal language are excluded from aris- them as senseless, when he has climbed out ing from observation, then there is a liar’s antinomy. through them, on them, over them. (He must There are three escape routes: (1) Either nature mirac- so to speak throw away the ladder, after he ulously removes all problematic sentences, and leaves us has climbed up on it.) He must surmount these propositions; then he sees the world rightly. [14, 6.53] § 6 The arguments given here might not resonate easily: If one ad- So, how does meaning come about then? In [20], after heres to the belief that there exists a privileged language access- “surmounting these propositions,” Wittgenstein exam- ing the truth, then probably one does not believe the statement that there is no privileged language would be a true sentence of ines the root of first words considering language games that language. On the contrary: The latter statement is then easily dismissed. This shows the contingency of language—the incapability to step outside language, or into some meta-language that allows an innocent view on all languages [13, 14]. 9 The idea does not only originate from Quine; he rather develops 7 See (C2). Duhem’s holism further [18]. 8 A first-order arithmetic is an axiomatic system for the arithmetic 10 The principle of revision, one of four pillars of Gonseth’s open of natural numbers relying merely on statements of first-order philosophy, says “that every position and every scientific state- logic. ment, including statements of logic, can be revised” [18]. 4

account of experience ambiguities: Foucault distinguishes in his discussion of Magritte’s to a pipe [24] similitude and re- meaning semblance, and regards the painter to bring “the former into play against the latter” [24, 5].14 This leads to the question: “If lines in the sand,§ noises, etc., cannot Figure 2. The circular dependency of meaning and the ac- ‘in themselves’, represent anything, then how is it that count of experience, eventually, calls into question any reduc- thought forms can ‘in themselves’ represent anything?” tion to atoms. [23, 1]15 § The assumption of innate, non-contextual semantic structures, an underlying lingua mentis that allows to (“Sprachspiele”): For a child, that has no medium to deduce any if deciphered correctly is contested. establish meaning from explanation11, the acquisition of The existence of any one true and exhaustive language is language reduces to trimming12. This process, in turn, in doubt. And so are distinguished means to express the relies on observation. There is an issue of self-reference verification function vO, be they formal or not. The argu- at the very root of meaning—or at the very root of any ment is not against a world out there, but against a truth account of experience for that matter: Whenever one at- out there—a truth entirely beyond our creation [13]. tempts to reduce language to some atomic propositions or first words, one is faced with a problem of circular dependence. Meaning and experience are intricately in- tertwined (see Figure 2). IV. SYSTEMS, STATES, AND THE BORN RULE

The hope of escaping the perils of practical activity and C. No privileged language achieve ultimate certainty [26] is prevalent among quan- tum theorists.16 This has effects on the use and meaning Wittgenstein’s observations have been extended to ar- of terms.17 In the following, terms central to quantum gue against the existence of distinguished sentences that mechanics will briefly be discussed. directly correspond to sensory input—atomic sentences whose meaning cannot be analyzed further:13 Sellars ar- gues, “as a first step in a general critique of the entire A. State and system framework of givenness” [22, 1]: Sense-datum theories are faced with an inconsistent§ triad, stemming from a As Wittgenstein observes, the meaning of terms is spec- concept of sensation as “inner episodes [. . . ] without any ified and modified while—not prior to—being used. In prior learning or concept formation” [22, 7] and “inner this sense, the words “state” and “system” are not as episodes which are non-inferential knowings§ that certain clear and static as often insinuated in scientific prac- items are, for example, red or C# [the musical note] and tice:18 This foundation for explanatory constructs might that these episodes are the necessary conditions of em- be shakier than it appears at first sight. pirical knowledge as providing the evidence for all other Whereas in classical mechanics, a system and its state empirical propositions” [22, 7]. are both characterized by directly measurable quantities, Putnam argues against the§ existence of exhaustive cri- the matter gets more intricate in quantum mechanics: teria that determine reference or representation: Does an ant’s incidental “picture of Churchill” in the sand refer to Churchill? Similarity to the features of Churchill is nei- ther necessary nor sufficient to refer to Churchill (see [23, 14 1]). Magritte’s “Treachery of Images” exposes similar “Resemblance presupposes a primary reference that prescribes § and classes. [. . . ] Resemblance serves representation, which rules over it; similitude serves repetition, which ranges across it.” [24, §5] 15 “It seems perfectly clear, at least since Wittgenstein and Sellars, 11 “Man muss schon etwas wissen (oder k¨onnen), um nach der Be- that the ‘meaning’ of typographical inscription is not an extra nennung fragen zu k¨onnen. Aber was muss man wissen?” [20, ‘immaterial’ property they have, but just their place in a context §30] of surrounding events in a language-game, in a form of life. This 12 “Das Lehren der Sprache ist hier [f¨ur das Kind] kein Erkl¨aren, goes for brain-inscriptions as well.” [25, §1.2] sondern ein Abrichten.” [20, §5] 16 The recent commotion about statements that suggest that quan- 13 Similar thoughts have been expressed by Nietzsche before: “Ein tum mechanics is inconsistent exemplify this attitude [2, 27]. Maler, dem die H¨ande fehlen und der durch Gesang das ihm 17 “As Wittgenstein often pointed out, a philosophical problem is vorschwebende Bild ausdr¨ucken wollte, wird immer noch mehr typically generated in this way: certain assumptions are made bei dieser Vertauschung der Sph¨aren verrathen, als die empirische which are taken for granted by all sides in the subsequent dis- Welt vom Wesen der Dinge verr¨ath. Selbst das Verh¨altnis eines cussion.” [12, §2] Nervenreizes zu dem hervorgebrachten Bilde ist an sich noch kein 18 Van Fraassen, in [28], emphasizes the change of the use of words nothwendiges; [. . . ] [D]as Hart- und Starr-Werden einer Meta- like “atom”, “electron” or “field” to conclude: “Thus, scientific pher verb¨urgt durchaus nichts f¨ur die Notwendigkeit und auss- revolutions, and even evolutions, embarrass the standard scien- chliessliche Berechtigung dieser Metapher.” [21, p.18] tific realist” [28]. 5

Grete Hermann refers to quantum states as “new sym- states. Together with a model of the measurement - bols that express the mutual dependency of the deter- cess, the Born rule leads to issues such as the measure- minability of different measurements”19. It seems rea- ment problem. On a closer look, however, fundamental sonable to emphasize the semiotic character of the state: theories seem to require a formal framework that con- Symbols of a formal language T , though they might be nects to information: The core of Popper’s argument called states, are, a priori, not endowed with any further for a fundamental indeterminism in classical mechanics meaning than to serve as signs.20 Gleason’s theorem [31] (C) in [36] evolves around the observation that the only sheds a new light onto the quantum state and shifts the means of extracting information from a system in classi- semiotic character to measurements represented by sets cal mechanics—without going beyond classical mechan- of orthogonal projectors on a (finite-dimensional) Hilbert ics in the measurement process—involves introducing un- space: The theorem states that any measure µ for pro- traceable disturbances. The underlying assumption is: In jectors on a Hilbert space of dimension greater than two, order to extract any information, there has to be an inter- satisfying an additivity constraint for orthogonal projec- action.21 The problem in classical mechanics is usually tors, tamed by describing the measurement in another the- ory T ′: For instance, optical measurement devices may determine the position of a system otherwise described µ Pi = µ(Pi) by classical mechanics. The disturbance due to the inter- i ! i ′ X X action captured in T may be assumed to be small. Then ′ as well as the constraints the error one has to tolerate before falsifying C or T is small. (In contrast to this, Popper’s observation of µ(0)=0 µ(1)=1 a fundamental indeterminism relates to necessarily large errors.) In this light, the possibility of measuring an en- is of the form tity “without in any way disturbing a system” [38] is not a feature of classical mechanics alone—optics (or another µ (P ) = Tr(Pρ) external theory) is necessary to reduce the errors. with ρ being a positive trace-1 operator, i.e., a density In summary, a fundamental theory (not relying on matrix. A theory that represents measurements with sets other theories to capture an interaction necessary to of orthogonal projectors as above cannot bear a mea- extract information) has to model the emergence of sure µ with a range 0, 1 , as expected from a model information—potentially exposing it to a measurement assigning definite and{ non-contextual} values to its mea- problem. surement operators [32, 33]. The way back to a simple association of states and measurement results as in clas- sical mechanics is barred by non-locality [29, 34] and con- V. THE MEASUREMENT PROBLEM textuality [35]. The above considerations have repercussions on the mea- B. Born rule surement problem. In the proceeding, we illustrate the problem with the Wigner’s-friend experiment. A dis- cussion of Maudlin’s reading [1] of the problem can be As demanded in Section II, any physical theory has to found in Appendix A. Breuer’s approach [6] to self- yield assertions as to which results are not expected to reference issues in quantum mechanics in is examined in be observed. In classical mechanics, measurable entities Appendix B. have a one-to-one correspondence to states. The situ- The measurement problem roots in the irreconcilabil- ation in quantum mechanics is more intricate, as there ity of distinguishability of information and linearity in is no immediate correspondence between observed enti- quantum theory [2]. The problem can be illustrated ties and state symbols: The Born rule bridges between with a Wigner’s-friend experiment as depicted in Fig- these separate realms. At first sight, the need of such ure 3. Employing references to states—despite the above a bridge seems unfortunate as it manifests the separa- criticism—, the problem can be put as follows: A source tion of a realm of observable entities from the realm of

19 “Im Gegensatz dazu [Zust¨ande der klassischen Physik] braucht 21 Bohr anticipated this observation: “Indeed the finite interaction der quantenmechanische Formalismus zur Zustandsbeschreibung between object and measuring agencies conditioned by the very neuartige Symbole, die die gegenseitige Abh¨angigkeit in der Bes- existence of the quantum of action entails—because of the impos- timmbarkeit verschiedener Gr¨oßen zum Ausdruck bringt.” [29] sibility of controlling the reaction of the object on the measuring 20 “There is no quantum world. There is only an abstract quantum instruments if these are to serve their purpose—the necessity of a mechanical description. It is wrong to think that the task of final renunciation of the classical ideal of and a radical physics is to find out how Nature is. Physics concerns what we revision of our attitude towards the problem of physical reality” can say about Nature.” [Bohr as quoted in 30] [37]. 6

In a third with an objective collapse QOC , | ↑i|ui the joint system S F collapses, after the interaction |φi H ⊗ H S MF MW modelled by the isometry V above conditioned on the | ↓i|di obtained result. Importantly, this collapse is happening independently of the observer, not merely subjectively— that is, it has to be considered also by Wigner (simi- larly to “GRW” [41]). For Wigner, who does not know Figure 3. The Wigner’s-friend experiment. the result of the friend’s measurement, the entangled state V ( φ ) collapses to a mixture | i 1 emits a system in a state ρ = ( 0, 0 0, 0 + 1, 1 1, 1 ) , (1) 2 | ih | | ih | 0 + 1 φ = | i | i S . and Wigner measures either of two first vectors in the | i √2 ∈ H basis above with equal probability. So, the question is: Does a measurement within a This state is then measured by an observer, called closed system induce a collapse as modelled in QOC ? Wigner’s friend, and modelled by a quantum system with The Wigner’s-friend setup allows to empirically test this Hilbert space F , in the basis 0 , 1 . Finally, Wigner H {| i | i} question and destinguish QOC from Q/QC. himself measures the joint system S F in a basis H ⊗ H The precise meaning of the term “measurement” is cru- 0 S 0 F + 1 S 1 F 0 S 0 F 1 S 1 F cial in the above considerations: In Q, any measurement | i | i | i | i , | i | i − | i | i ,... . is primarily a unitary, entangling the observed system √2 √2   with a memory system which includes the observer into Within Everett’s relative-state formalism Q of quantum the framework. Even though a symbol in the formal lan- 22 mechanics [39] , the joint system S F after the guage is assigned a tag “the observer’s memory,” there friend’s measurement is in a state H ⊗ H is no simple correspondence to an “observer” in a dis- course of ordinary language surrounding an experiment. 1 V ( φ S )= V ( 0 S )+ V ( 1 S) The Born rule is then applied to the memory system of | i √2 | i | i the respective observer, after tracing out all other sys- 1  = 0 S 0 F + 1 S 1 F , tems. In QC , the measurement is a notion relative to the √2 | i ⊗ | i | i ⊗ | i system that is to be measured.23 Any interaction that  involves a part of the environment relative to this system where V is the isometry modelling the measurement of can be considered a measurement and induces a collapse. the friend [2]. Thus, Wigner’s final measurement yields On the one hand, a system is called isolated if there are the eigenvalue, corresponding to the first basis vector no such interactions: Then, the evolution is a unitary op- with probability 1. erator. As long as the perception of an “isolated system” Besides the relative-state formalism Q, one might con- on one side and a “measurement” or “interaction with sider the scenario in a formalism QC with a “collapse.” the environment” on the other are clearly separated and Within QC , a system is either isolated and evolves unitar- mutually exclusive, this is a consistent framework. Note ily according to the Schr¨odinger equation, or it interacts that the friend’s measurement is not a measurement rel- with its environment. A measurement is such an interac- ative to Wigner. tion and entails a collapse to the eigenvector associated If Q and QC were falsified in a Wigner’s-friend exper- with the measurement result. In QC , the measurement iment, and we were left with QOC , the situation would of the friend reduces to a unitary within the system un- be intricate: The relative character of the two other for- der consideration by Wigner: Environment is a subjec- malisms is lost, as there is an “objective collapse.” Then, tive concept, depending on what an observer considers an interaction with the environment or an associated dis- as the system to be measured. The system associated sipation of information is not a necessary requirement for with S F measured by Wigner is isolated from his H ⊗ H a collapse. An isolated system would then show a non- perspective as long as he measures the entire joint sys- unitary evolution. If the friend shared his result with tem: The friend’s “measurement” is not an interaction Wigner, he would dissipate information into Wigner’s with the environment relative to Wigner, but an inter- environment, and, thus, we would again be in a case action within an isolated system. Thus, QC yields pre- consistent with Q and QC . But can we establish that dictions on the outcome of Wigner’s final measurement a measurement has happened without revealing the re- identical to those of Q. sult? One might assume that the friend’s memory was

22 Here, we do not refer to the many-worlds interpretation of quan- 23 This relativity is reflected in the partial trace in Q, which is rela- tum mechanics. We want to emphasize not to confuse the for- tive to the memory of the respective observer, and in the unitary malism with an interpretation [40]. entangling the observed system with the observers memory. 7

in an initial state ∆ orthogonal to both 0 and 1 and nam’s reflections on reference and representation, and there was a unitary| operatori U with | i | i lead, eventually, to doubting the existence of a privileged language—a “semantic stage”—for accounts of experi- U : M S M S ence: Meaning arises along the discourse, not prior to it. H ⊗ H → H ⊗ H ∆ k k k k =1, . . . , k. There is, consequently, no objective language to exhaus- | i ⊗ | i 7→ | i ⊗ | i ∀ tively represent an observer’s account of experience. Reducing an “observer” to a “system” or a “measure- for an orthonormal basis k k. Then, a positive operator-valued measurement{| (POVM)i} could determine ment” as sensory perception to “measurement” as part of whether the memory was still in the initial state or not a formalism are light-footed extensions of the terms’ use without revealing the actual result. If we were to build in physics. In the face of the intricate relation between a consistent theory, then any other unitary of the same language and experience, as well as the normative au- form as U should induce a collapse. With this, quantum thority of experience, it is debatable to “interchangeably mechanics as we know it would eventually—collapse. use the words ‘experience’, ‘observation,’ and ‘state of the observer’” [10]. If an observer is measured, one can say In conclusion, the meaning of “measurement” is cru- something—but not necessarily reproduce his account of cial to both Q and QC: Different uses of the term— sensory evidence. formal ones on the level of the theory, ordinary ones, dif- In summary: If semantic reductionism has limits, ferent relative ones—are not to be simply equated. The then so does scientific reductionism. It is in the context measurement as an account of observation—and thus of of the measurement problem and issues of self-reference experience—does not have a direct correspondence to an that one runs into these limits. interaction expressed in the formal language of a theory: In light of the absence of an underlying privileged lan- The measurement problem is a semantic confusion.24 guage, physics is to shape meaning.

VI. CONCLUSION ACKNOWLEDGMENTS Leibniz’ reservation against space-time forming a fixed stage25 has its linguistic analogue: Wittgenstein’s ob- This work is supported by the Swiss National servation on the intricate and circular dependency of Foundation (SNF), the NCCR QSIT, and the Hasler meaning and experience lead him to conclude that mean- Foundation. We would like to thank Amin¨ Baumeler, ing derives from the use of language. These considera- Veronika Baumann, Cecilia Boschini, Paul Erker, Xavier tions on the plasticity of language have been extended Coiteux-Roy, Claus Beisbart, Manuel Gil, and Christian by, e.g., Quine’s arguments for semantic holism or Put- W¨uthrich for helpful discussions.

[1] Tim Maudlin, “Three measurement problems,” [3] Eugene P. Wigner, “The problem of measurement,” Topoi 14, 7–15 (1995) . American Journal of Physics 31, 6–15 (1963) . [2] Veronika Baumann, Arne Hansen, and Stefan [4] David Deutsch, “Quantum theory as a universal physical Wolf, “The measurement problem is the measure- theory,” International Journal of Theoretical Physics 24, ment problem is the measurement problem,” (2016), 1–41 (1985) . arXiv:1611.01111 [quant-ph] . [5] Eugene P. Wigner, “Remarks on the mind-body ques- tion,” in The Scientist Speculates, edited by I. J. Good (Heineman, 1961) . [6] Thomas Breuer, “The impossibility of accurate state 24 Wigner already remarked on the imprecise use of terms: “Most self-measurements,” Philosophy of Science 62, 197–214 importantly, he [the quantum theorist] has appropriated the (1995) . word ‘measurement’ and used it to characterize a special type [7] , Quantum Mechanics: An Empiricist of interaction by means of which information can be obtained on View, Clarendon paperbacks (Clarendon Press, 1991) . the state of a definite object. [. . . ] On the other hand, since he is unable to follow the path of the information until it enters his, or [8] Alfred Tarski, “Der Wahrheitsbegriff in den formal- the observer’s, mind, he considers the measurement completed isierten Sprachen,” Studia Philosophica 1, 261–405 as soon as a statistical relation has been established between the (1936) . quantity to be measured and the state of some idealized appara- [9] Karl Popper, Logik der Forschung, 11th ed. (Mohr tus. He would do well to emphasize his rather specialized use of Siebeck, T¨ubingen, 1934) . the word ‘measurement’ ” [3]. 25 [10] Markus P. M¨uller, “Could the physical world be emergent The debate about a fundamental absolute space-time [42] ap- instead of fundamental, and why should we ask? (full pears in a new light if Reichenbach’s principle is confronted with version),” (2017), arXiv:1712.01826 [quant-ph] . Bell non-locality, supporting the scepticism often attributed to [11] Richard Rorty, “Non-reductive physicalism,” in Leibniz and not shared by many physicists throughout history— Objectivity, Relativism, and Truth: Philosophical Papers most notably Mach. , 8

Vol. 1 (Cambridge University Press, 1990) p. 113125 . [36] Karl Popper, “Indeterminism in quan- [12] Hilary Putnam, Representation and Reality, A Bradford tum physics and in classical physics 1,” book (A Bradford Book, 1991) . Brit. Journ. for the Phil. of Sci. I, 117–133 (1950) [13] Richard Rorty, Contingency, Irony, and Solidarity (Cam- . bridge University Press, 1989) . [37] Niels Bohr, “Can quantum-mechanical description of [14] Ludwig Wittgenstein, Tractatus logico-philosophicus physical reality be considered complete?” Physical Re- (Routledge, 1922) . view 48, 696 (1935) . [15] Alfred Tarski, “The semantic conception of truth: and the [38] Albert Einstein, Boris Podolsky, and Nathan foundations of semantics,” Philosophy and Phenomeno- Rosen, “Can quantum-mechanical description logical Research 4, 341–376 (1944) . of physical reality be considered complete?” [16] , and Experience (Oxford Univer- Physical Review 47, 777–780 (1935) . sity Press, 2006) . [39] Hugh Everett III, ““Relative state” formulation of quan- [17] Willard V. O. Quine, “Two dogmas of empiricism,” tum mechanics,” Reviews of Modern Physics 29, 454 Philosophical Review 60, 20–43 (1951) . (1957) . [18] Michael Esfeld, “Gonseth and Quine,” Dialectica 55, [40] Veronika Baumann and Stefan Wolf, “On formalisms and 199–219 (2001) . interpretations,” Quantum 2, 99 (2018) . [19] Joachim Schulte, Wittgenstein, eine Einf¨uhrung [41] Giancarlo Ghirardi, Alberto Rimini, and Tullio Weber, (Reclams Universal Bibliothek, 1989) . “A model for a unified quantum description of macro- [20] Ludwig Wittgenstein, Philosophische Untersuchungen, scopic and microscopic systems,” in Quantum Probability edited by Joachim Schulte (Suhrkamp Verlag, 1953) . and Applications II, edited by Luigi Accardi and Wilhelm [21] Friedrich Wilhelm Nietzsche, Uber¨ Wahrheit und L¨uge von Waldenfels (Springer, Berlin, Heidelberg, 1985) pp. im aussermoralischen Sinne (1873) . 223–232 . [22] Wilfrid S. Sellars, “Empiricism and the philosophy of [42] Ezio Vailati, Leibniz and Clarke: A Study of Their Cor- mind,” Minnesota Studies in the Philosophy of Science respondence (Oxford University Press, 1997) . 1, 253–329 (1956) . [23] Hilary Putnam, , Truth and History, Philosophi- cal Papers (Cambridge University Press, 1981) . Appendix A: Maudlin’s reading [24] Michel Foucault, This is Not a Pipe, An art quantum (University of California Press, 1983) . [25] Richard Rorty, Philosophy and the Mirror of Nature Maudlin in [1] reads the measurement problem as the (Princeton University Press, 1979) . inconsistency of the following three “claims:” [26] John Dewey, The Quest for Certainty: A Study of the Relation of Knowledge and Action (Minton, Balch and 1.A The wave-function of a system is complete, i.e., the Company, 1929) . wave function specifies (directly or indirectly) all [27] Daniela Frauchiger and Renato Renner, “Quantum the physical properties of a system. theory cannot consistently describe the use of itself,” Nature Communications 9 (2018), 10.1038/s41467-018-05739-81.B The wave-function always evolves in accord with . a linear dynamical equation (e.g., the Schr¨odinger [28] Bas van Fraassen, “Structure: Its shadow and substance,” equation). British Journal for the Philosophy of Science 57, 275–307 (2006) . 1.C Measurements of, e.g., the spin of an electron al- [29] Grete Hermann, Die naturphilosophischen Grundlagen ways (or at least usually) have determinate out- der Quantenmechanik, Abhandlungen der Fries’schen comes, i.e., at the end of the measurement the mea- ¨ Schule (Verlag “Offentliches Leben”, 1935) . suring device is either in a state which indicates [30] Max Jammer, The philosophy of quantum mechanics: the spin up (and not down) or down (and not up). interpretations of quantum mechanics in historical per- spective, Wiley Interscience publication (Wiley, 1974) . To show the inconsistency, Maudlin considers, in addi- [31] Andrew Gleason, “Measures on the closed subspaces of a Hilbert space,” Indiana Univ. Math. J. 6, 885–893 (1957) tion to a two-dimensional observed system, a measure- . ment apparatus. The argument is then: By linearity, [32] Micheal Redhead, Incompleteness, Nonlocality, and Real- the apparatus should be in a superposition of its pointer ism: A Prolegomenon to the Philosophy of Quantum Me- states “up” and “down” if the measured system is in chanics, Clarendon Paperbacks (Clarendon Press, 1989) a superposition with respect to the measurement basis . (by (1.A) and (1.B)). According to (1.C), on the con- [33] Carsten Held, “The Kochen-Specker theorem,” in The trary, the pointer of the apparatus should be exclusively Stanford Encyclopedia of Philosophy, edited by Ed- in a state “up” or “down.” ward N. Zalta (Metaphysics Research Lab, Stanford Uni- The commitment to the above “claims” can, however, versity, 2016) fall 2016 ed. be questioned as discussed below. Thus, for running into [34] John S. Bell, “On the Einstein-Podolsky-Rosen para- dox,” Physics 1 (1964) . Maudlin’s contradiction, one has to take a specific stance, [35] Simon Kochen and Ernst Specker, “The problem of hid- namely assuming the sufficient representation of accounts den variables in quantum mechanics,” Journal of Math- by formal symbols possible. The resulting categoriza- ematics and Mechanics 17, 59–87 (1967) . tion of interpretations—by the choice of which claim to drop—is similarly subjective. 9

The completeness claim (1.A) does not specify what is where ρ is the density matrix in (1).26 The divergence of to be considered a “physical property.” Is it “what can the measurement for Wigner in QOC from be measured”? Or “what can be assigned a symbol in the the ones in Q and QC occurs already with a non-unitary, formal language of a physical theory”? It seems we are merely linear, evolution. faced with similar problems as with what is to be consid- In light of the discussion in the previous chapters, a ered a “physical system.” For the inconsistency to arise, “new physics” that provides a “real solution” [1], maybe accounts of experience are to be regarded as a “physical even a realistic solution, seems forlorn a hope. property.” The last claim, (1.C), does not only state that measurements have determinate results, but it specifies these in terms of states of a measuring device. If one Appendix B: Quantum self-reference takes the requirement of “determinate measurement re- sults” to be the capability to meaningfully assert “I have In [6], Breuer investigates problems of self-reference in observed ‘up’ and not ‘down’ ” (or vice versa)—that is, quantum mechanics. The approach relies on incorporat- to attribute “empirical content” to such a statement— ing propositions and semantics into the formal language it does not necessarily follow that the state of some ap- of quantum mechanics, and thus stands in contrast to paratus is exclusively in a correspondent state “up” or the argumentation in Section III above. Subsequently, “down.” One may include an apparatus into the for- the steps towards such an incorporation are examined in mal framework and assume that then, the apparatus is greater detail. being measured. Assuming the interaction between the In a first step, the framework of propositions P , corre- observed system and the apparatus to be an entangling sponding to sentences about an experimental context we unitary means that the result of a measurement of the discuss in Section II, is incorporated into T . system and the result of a measurement of the apparatus have correlated results. Propositions about physical systems can be reformulated by saying ‘The state of the sys- In a similar manner, further apparatuses “between” tem has this and that property.’ So instead of the observed system and the observer, who then reports speaking of propositions, we can equally well on his experience, might be included. One can even go speak about sets of states: to each proposi- as far as to assign a quantum state to the brain or mem- tion there corresponds the set of states for ory of the observer—as we do above in connection with which the proposition is true. [6] the relative-state formalism. There are, however, doubts as to whether the assigned state—or any symbol in any Propositions can be expressed in some formal language other formal language—exhaustively captures the mean- T con as done in Section II. As such, there is little to ing of an account as the one above. If one follows, e.g., object to incorporating T con and T into some T ′. The Quine’s arguments on semantic holism, or Putnam’s re- reduction of T con into T is, however, problematic. Let flections on reference, such a “representational complete- us take S to be the set of state symbols. Then, the ness [of the wave-function]” [1] seems out of reach: There reduction proposed above can be regarded as a func- is then a gap for any formal language due to issues of se- tion f : P con (S), assigning each proposition a set of mantics. This semantic gap is wider than what a Born states. To speak→P “equally well” about sets of states, we rule can bridge. require f to be injective. The set of falsifying proposi- − The second claim, (1.B), states the linearity of quan- tions, f 1( ), has then a cardinality of one. This poses a ∅ tum mechanics and refers to the Schr¨odinger equation, caveat to falsifiability: If single sentences have no inher- as an example for such an evolution. The Schr¨odinger ent meaning, then how can one meaningfully assert that equation does not hold unconditionally, but merely under the theory was contradicted? the restriction to closed or isolated systems. The claim One could allow for ordinary language to fill the void. that “the traditional theory did not [. . . ] state in clear This is not an option in the context of [6], as ordinary physical terms, the conditions under which the non-linear language is meant to be reduced to T as well. evolution takes place” [1] cannot be sustained if “isolated So good experiments serve to at least par- systems” is enough of a “clear physical term.” As ar- tially constitute the semantics of physical the- gued above, the distinction between closed systems and ories. In this sense, observation is a semantic systems interacting with their environment, e.g., during concept. [6] a measurement, is crucial in avoiding inconsistencies. There is yet another intricacy: If the system is iso- In light of the circular dependency of the account of ex- lated, then, according to the Schr¨odinger equation, the perience and semantics discussed above, observation is a system evolves unitarily. A system that interacts with its environment, i.e., that is not isolated, evolves linearly according to some CPTP map. This is the case for the 26 ′ The partial trace of an isometry V : |l, liSF 7→ |l,l,liSF G mod- formalism QOC above: There is a CPTP map so that elling a second measurement of the friend in the same basis with a memory outside the of Wigner, yields such a CPTP V ( φ S ) ρ , map. | i 7→ 10 semantic concept. An observation, however, is neither ory can be closed semantically: If apparatus to be equated with an experiment nor with a formalism and object system, as well as their interac- of “measurement” within T , for reasons discussed in Sec- tion, can be described by the theory, then the tion III. Thus, the formal language of quantum mechanics semantic concept of observation can be intro- is not semantically closed, as concluded in the following duced into the language of the theory. [6] statement. In conclusion: One may doubt whether the reduction Tarski (1956, 1969) calls a language seman- of the linguistic context, including the observer and his tically closed if it contains (1) semantic con- account of experience, to the formalism of quantum me- cepts and (2) expressions referring to its own chanics is feasible. Consequently, an important propositions. The language of a physical the- for the arguments in [6] is in question.