Some Remarks on the Logic of Quantum Gravity
aa r X i v : . [ g r- q c ] J un Some Remarks on the Logicof Quantum Gravity
Andreas D¨oring
Clarendon Laboratory, Department of Physics, University of Oxford [email protected]
13. June 2013
Abstract
We discuss some conceptual issues that any approach to quantumgravity has to confront. In particular, it is argued that one has to finda theory that can be interpreted in a realist manner, because theorieswith an instrumentalist interpretation are problematic for several well-known reasons. Since the Hilbert space formalism almost inevitablyforces an instrumentalist interpretation on us, we suggest that a the-ory of quantum gravity should not be based on the Hilbert spaceformalism. We briefly sketch the topos approach, which makes use ofthe internal logic of a topos associated with a quantum system andcomes with a natural (neo-)realist interpretation. Finally, we makesome remarks on the relation between system logic and metalogic.
Should storms, as well may happenDrive you to anchor a weekIn some old harbour-cityOf Ionia, then speakWith her witty scholars, menWho have proved there cannot beSuch a place as Atlantis:Learn their logic, but noticeHow its subtlety betraysTheir enormous simple grief;Thus they shall teach you the waysTo doubt that you may believe.
W.H. Auden, from
Atlantis (1941)1
Die Grenzen meiner Sprache bedeuten die Grenzen meiner Welt.”
Ludwig Wittgenstein,
Tractatus logico-philosophicus , Satz 5.6
Doing research in quantum gravity is a profoundly strange endeavour: neitherthe boundaries of the subject, nor the methods of inquiry, nor the goals ofthe search, nor the criteria of success are commonly agreed upon. Thisholds in particular when one considers not a specific approach with its oftenformidable technical apparatus and mathematical difficulties, but conceptualquestions that are common to the various approaches.There are no observable phenomena that unambiguously belong to therealm of quantum gravity, so our search is neither data-driven, nor would asuccessful theory of quantum gravity necessarily much expand the range ofnatural phenomena we can explain (or at least describe) conceptually andmathematically. A while ago, Chris Isham asked me rhetorically: “What ifsomeone came today, with a printout of three long articles in his or her hands,and claimed that these articles contain the Theory of Quantum Gravity? Howwould we judge if this person is right or wrong? Which criteria apply?”.In this article, I will consider some very general conceptual questions onthe way to quantum gravity. Whilst these questions may seem metaphysical(a word that is often used in a pejorative sense by working physicists), inthe end each technical approach will be confronted by such questions. I willmake some remarks on the following questions:1. Is quantum theory necessarily quantum? Is it adequate to (try to)expand concepts of quantum ideas to a theory of quantum gravity, withthe usual mathematical apparatus of quantum theory intact? Couldquantum gravity be an instrumentalist (or operational) theory, or doesit have to be a realist theory?2. If we aim at a realist form of quantum theory and theories ‘beyondquantum theory’ such as quantum gravity, what kind of logic couldwe potentially use in the face of no-go theorems such as the Kochen-Specker theorem?3. In an encompassing theory of the whole universe, which rˆole does thephysicist play – is she or he necessarily part of the description?Of course, I cannot hope to give full answers to these questions; I can merelysketch some ideas and recent technical developments using topos theory in2hysics which may become useful in finding answers to such questions.In section 2, the well-known argument why instrumentalist interpretationsare problematic in quantum gravity is presented. Yet, the conclusion wedraw from this is non-standard: a theory of quantum gravity should not bebased on Hilbert spaces. Section 3 gives an outline of some basic structuresin the topos approach to the formulation of physical theories, leading to aneo-realist interpretation of the new topos-based mathematical formalismfor quantum theory. The topos approach to the formulation of physicaltheories was initiated by Isham [28] and Isham/Butterfield [29, 30, 31, 32],was developed and substantially expanded by this author and Isham [16,17, 18, 19, 6, 7, 8, 9, 20, 21, 10, 11, 12], and further developed by Heunen,Landsman, Spitters and Wolters [25, 26, 39, 27, 40], Flori and collaborators[23, 3, 4] and others [36, 37]. In section 4, we briefly argue about the relationbetween ‘system logic’ and metalogic, and section 5 concludes.
This question may seem trivial at first sight: since quantum gravity is sup-posed to unify or reconcile quantum theory and general relativity, it will ofcourse be some sort of quantum theory (just as it will also be some sort oftheory of gravity). Yet, what is less clear is if a theory of quantum gravitynecessarily must be based on the Hilbert space formalism?Ever since von Neumann gave quantum mechanics its mathematical formin 1928, the Hilbert space formalism has been the mathematical underpinningof quantum theory. Further developments like quantum field theories addedmore mathematical structures, but the Hilbert space formalism remained thecore of the mathematical apparatus of quantum theory.
Quantum theory, like every physical theory, consists of a mathematical ap-paratus and an interpretation that links the mathematics to physical entitiesand processes. In the case of quantum theory, a plethora of interpretationsexists, and there is an ongoing debate about which of these interpretationsis to be preferred. Importantly, the debate is largely concerned with inter-pretations of the Hilbert space formalism, while the Hilbert space formalismitself is rarely questioned. Hence, the underlying mathematical apparatus ofquantum theory remains more-or-less fixed in this debate.
Instrumentalism.
Most interpretations of quantum theory, in particu-3ar the classical Copenhagen interpretation and operational interpretations,which have become popular again recently, posit a fundamental distinctionbetween quantum system and observer. Measurements are primitive notionsand hence are not in need of a definition in such an interpretation. Neces-sarily, observers and their measuring devices are not quantum systems, butclassical, which leads to many interpretational issues.An instrumentalist interpretation does not give rise to a picture of re-ality. It restricts itself to predicting outcomes of experiments (often in aprobabilistic sense) that an observer performs on the system. As such, aninstrumentalist interpretation does not tell us much about what the quan-tum system ‘does’ or ‘is’ if we don’t measure. Given a closed system, aninstrumentalist view is not informative.
Realism.
In contrast to an instrumentalist interpretation, a realist in-terpretation does not fundamentally depend on observers and measurements.Rather, such an interpretation gives a picture of reality, of ‘how things are’and what ‘is actually going on’. In a realist interpretation, observers andmeasurements are secondary concepts. Measurements can reveal what isgoing on, but they are not fundamentally adding anything. Of course, weidealise here and assume the case of non-disturbing measurements.Yet, as is well known, it is very hard to come up with a realist interpreta-tion of the Hilbert space formalism. The only two established examples arethe de Broglie-Bohm pilot wave formalism, which has massive problems withthe extension to special relativistic space-time, and the Everett many-worldsinterpretation. The latter posits that whenever a quantum experiment withseveral possible outcomes is performed, all outcomes are realised and the uni-verse splits up into corresponding branches. This (not very frugal) ontologymay be acceptable for some philosophers of physics, but it does not seemattractive to us. What is ‘real’ in many-worlds is the wave function of theuniverse, which of course is inaccessible in principle and does not undergomeasurement or collapse. It is doubtful if this can be seen as a ‘picture ofreality’ of the kind we are aiming at. An observer in many-worlds only hasexperience of one branch, but we also have to take the god’s-eye view of thewave function of the universe to make sense of the theory.
Born rule and instrumentalism.
In fact, the Hilbert space formalismalmost forces an instrumentalist interpretation on us. A key aspect of theHilbert space formalism, and the link to observable phenomena, is the Bornrule that allows calculating expectation values of observables when the systemis in a given state. Of course, the concepts of expectation values and proba-bilities presuppose the two-level ontology system-observer and are dependenton (repeated) measurements. Hence, there is a direct link between the usual4nterpretation of the Born rule and instrumentalist interpretations of quan-tum theory. It is much-debated whether a many-worlds interpretation, inwhich every possible outcome occurs and is equally ‘real’, can reproduce theprobabilistic predictions of quantum theory.
Relativistic quantum field theories.
It is commonly accepted thatthe interpretational problems of non-relativistic quantum theory are notsolved by going to relativistic quantum field theory. Instead of expectationvalues, one considers cross sections, scattering probabilities, etc., but alsothese arise from experiments performed by classical observers. This meansthat at least implicitly we use an instrumentalist interpretation also in QFT.
It is obvious that a formulation of quantum gravity based on Hilbert spaceswould inherit the interpretational problems associated with instrumentalism,since a mathematical apparatus based on Hilbert spaces most naturally com-bines with an instrumentalist interpretation to give a physical theory. Yet,there are strong reasons to doubt the usefulness of instrumentalist interpre-tations in quantum gravity.
Problems with instrumentalism in quantum gravity.
Firstly, ifwe assume that quantum gravity, like its classical counterpart, is a theoryof the whole universe, then there is no external observer who could performmeasurements on this system. As mentioned above, instrumentalist interpre-tations are not very useful for closed systems. The universe is the ultimate(and only true) closed system.A second reason to doubt the usefulness of instrumentalist interpretationsin quantum gravity is that the concept of measurements seems to presupposea space-time background, since measurements take place at some location atsome point (or during some period) in time. It has often been argued thatquantum gravity should be formulated in a background-independent way,but measurement does not seem to be background-free notion. In a theoryof quantum gravity, presumably space and time will be treated as quantumobjects, whatever that will mean in detail. What could a measurement ofquantum space or quantum time mean – where and when would such ameasurement take place?
Quantum gravity without Hilbert spaces.
These issues seem seri-ous enough to doubt that any instrumentalist (or operational) interpretationcould be useful in quantum gravity. If, moreover, we take into account thefact that any theory based on the mathematical apparatus of Hilbert spacespractically forces an instrumentalist interpretation on us, we come to the5ollowing conclusion:
The mathematical apparatus of a future theory of quantum gravity shouldnot be based on Hilbert spaces and hence should not be a quantum theory inthe standard sense.
Instead, we should try to formulate a theory of quantum gravity in such away that the mathematical apparatus can be combined with a realist inter-pretation, avoiding the serious conceptual issues with instrumentalist inter-pretations sketched above. This means we should strive for a mathematicallyand conceptually new form of theory of quantum gravity, departing from theparadigm of Hilbert spaces.Naturally, a good starting point for such a project is not quantum grav-ity, but much more modest non-relativistic quantum theory. What kind ofmathematical apparatus, replacing the Hilbert space formalism, is there thatwould allow a realist interpretation in a natural manner? We emphasise thatthis is not asking for yet another interpretation of the established formalism,but much more radically for a mathematical re-formulation of quantum the-ory, together with a conceptually new, realist kind of interpretation of thisnew mathematical formalism.
The topos approach.
The topos approach to the formulation of physi-cal theories, and in particular to quantum theory [20], is an attempt to pro-vide such a mathematical reformulation of quantum theory, together witha new, realist interpretation. The technical details are involved and can befound elsewhere. Here, we focus on some conceptual ingredients and partic-ularly focus on some logical aspects.
If we aim to be realists, the first question is: realists about what? The pro-totype of a realist theory is classical mechanics. At a very basic level, thisis a theory based on a state space, a space of values of physical quantities(which is the real numbers), and physical quantities as maps from the statespace to the space of values. A physical quantity A , for example position, isrepresented by a real-valued function f A from the state space, given mathe-matically by some set S (typically a symplectic or Poisson manifold), to thespace of values, given mathematically by the real numbers R .6f we consider a subset ∆ of the real line, then f − A (∆) is a subset of thestate space S . This subset represents a proposition “ A ε ∆”, that is, “thephysical quantity A has a value in the set ∆ ⊆ R ”. The subset f − A (∆)consists of all those states, i.e., elements of the state space S , for which theproposition is true . If the state s of the classical system is contained in f − A ( S ), then the physical quantity A has a value in the set ∆. Otherwise, if s / ∈ f − A (∆), then A does not have a value in ∆, and the proposition “ A ε ∆”is f alse .Different propositions such as “
A ε ∆”, “ Bε Γ”, “ Cε Ξ” about the val-ues of physical quantities correspond to (generally) different subsets f − A (∆), f − B (Γ), f − C (Ξ) of the state space S . The conjunction “ A ε ∆ and Bε Γ” corre-sponds to the intersection f − A (∆) ∩ f − B (Γ), the disjunction “ A ε ∆ or Bε Γ”corresponds to the union f − A (∆) ∪ f − B (Γ), and the negation “not A ε ∆”corresponds to the complement S\ f − A (∆).Hence, in classical physics there is an algebra of propositions, with con-junction, disjunction and negation, and this algebra is represented math-ematically by the Boolean algebra P ( S ) of subsets of the state space S .Moreover, states act as models for this propositional theory, i.e., they assigntruth values to propositions. Mathematically, each point s of the state spacegives a map t s : P ( S ) −→ ( f alse, true ) (1) X (cid:26) true if s ∈ Xf alse if s / ∈ X . (2)Clearly, ( f alse, true ) is a Boolean algebra itself, and t s is a morphism ofBoolean algebras, that is, t s ( X ∩ Y ) = t s ( X ) ∧ t s ( Y ) etc.Classical physics is a realist theory in the sense that(a) There is a space of states S whose subsets are interpreted as represen-tatives of propositions of the form “ A ε ∆”.(b) The subsets of S form a Boolean algebra P ( S ). The algebraic oper-ations ∩ , ∪ and S\ represent the logical operations of conjunctions,disjunctions and negations of propositions.(c) States s ∈ S provide models of the propositional theory representedby P ( S ), i.e., they are Boolean algebra morphisms from P ( S ) to theBoolean algebra ( f alse, true ) of truth values.(d) Every proposition X ⊆ P ( S ) has a truth value t s ( X ) in every givenstate s ∈ S , and every physical quantity A has a value f A ( s ) ∈ R in agiven state s . 7lassical physics is realist about propositions and their truth values. Thereis a ‘way things are’, independent from observers and measurements. Wewant to take this as the model for more general realist theories.Yet, the Kochen-Specker theorem [34, 5] seems to pose a strict limitationon any attempt at providing a realist form of quantum theory in this sense: itshows that under weak and natural assumptions, there is no way of assigningtruth values to all propositions like “ A ε ∆” in a consistent way. Mathemat-ically, there is no way of embedding the algebra representing propositionsabout a quantum system into a Boolean algebra.
The Kochen-Specker no-go theorem can be circumvented by relaxing theassumptions we make. In particular, we can (a) allow the representatives ofpropositions to form a weaker structure than a Boolean algebra, (b) allowmore truth values than just true and f alse , and (c) allow more general mapsthan Boolean algebra morphisms as states or models of our propositionaltheory.If, after relaxing assumptions in this way, we have a theory in which allpropositions have truth values in all given states, then we still regard thisas a mathematical formalism that can be interpreted in a realist way. Ofcourse, we have to show that quantum theory can be re-formulated in sucha way.
Some ingredients of the topos approach to quantum theory.
Asmentioned above, the topos approach to quantum theory provides such amathematical reformulation of quantum theory, together with a realist in-terpretation. We briefly sketch the main ingredients of the mathematicalapparatus.The topos approach gives a state space picture of quantum theory instrong analogy to the state space picture of classical physics. First of all,there is a notion of state space. Yet, this object is not assumed to be a set,but is a more general kind of object. Concretely, we use a presheaf , i.e., avarying set, as will be explained in more detail below. Also the space of valuesis a presheaf (of real intervals), not just the set of real numbers as in classicalphysics. In analogy to classical physics, physical quantities are representedby maps from the state presheaf to the value presheaf. These maps arenot mere functions, but maps between presheaves (natural transformations).Moreover, propositions such as “
A ε ∆” are represented by sub‘sets’ (in fact,subpresheaves) of the state presheaf. Finally, states are not represented by8oints of the state presheaf – it turns out the state presheaf has no pointsat all in a suitable technical sense! This is exactly equivalent to the Kochen-Specker theorem. Instead of points, one uses certain minimal (i.e., small)subobjects of the spectral presheaf to represent pure states.
Contexts and partial world views.
A fundamental feature of quan-tum theory is that experimentally, we only have partial access to the systemin the sense that only certain, compatible physical quantities can be mea-sured simultaneously. Mathematically, these are represented by commutingself-adjoint operators, forming a commutative subalgebra of the algebra ofphysical quantities. Such a subalgebra, and the partial perspective on thequantum system that it describes, is called a context .It was Bohr’s doctrinal view that one should only speak about quantumsystems in classical terms. This basically amounts to picking out a singlecontext by a measurement setup, after which it becomes impossible to speakmeaningfully about the values of physical quantities not contained in thiscontext. We follow a radically different route here: instead of singling out aparticular context, we consider all of them simultaneously and treat them onequal footing. We collect all the partial perspectives on a quantum system.Mathematically, we consider the set of all commutative subalgebras of anoncommutative algebra N of physical quantities, and we partially orderthis set by inclusion. This poset (partially ordered set) is called the contextcategory and is denoted V ( N ).While it may look very simple-minded to cut a noncommutative algebrainto commutative pieces, the context category contains a surprising amountof information about the original noncommutative algebra: for the case ofa von Neumann algebra N , one can show that the context category V ( N )determines the original algebra up to Jordan isomorphisms [24]. The formal-ism becomes powerful because we keep track of how contexts overlap, i.e.,intersect. The spectral presheaf, subobjects and propositions.
The stateobject for quantum theory is the so-called spectral presheaf
Σ. This is avarying set over the context category V ( N ). To each context V ∈ V ( N ), weassign the Gelfand spectrum Σ V of the algebra, which is a compact Hausdorffspace. This space can be seen as a ‘local state space’ for the physical quan-tities contained in the context V , where ‘local’ means ‘at this context withinthe global noncommutative algebra’. If V ′ ⊂ V is a smaller context, thenthere is a canonical continuous, surjective function from Σ V , the Gelfandspectrum of the bigger algebra, to Σ V ′ , the spectrum of the smaller algebra.In this way, we ‘glue together’ all the local state spaces into a global objectΣ, the spectral presheaf, which is the state object for quantum theory. One9an show that the spectral presheaf has no global elements [29, 31, 5], whichare the presheaf analogues of points. This lack of points is equivalent to theKochen-Specker theorem.Being a presheaf, Σ is an object in the topos Set V ( N ) op of presheaves over V ( N ). By a standard result, the subobjects, that is, subpresheaves, of anyobject in a topos form a Heyting algebra . Just as Boolean algebras mathe-matically represent classical Boolean propositional logics, Heyting algebrasrepresent intuitionistic logics, which are more general than Boolean logics,because the law of the excluded middle need not hold.In particular, the subobjects of the spectral presheaf Σ form a Heytingalgebra, and we use this structure to represent propositions about the valuesof physical quantities of our quantum system. There is a map from propo-sitions of the form “
A ε ∆” to subobjects of the spectral presheaf, called daseinisation of projections that was discussed in detail in [17, 9].
The value presheaf and representation of physical quantities.
Inthe topos approach, physical quantities do not take values in the usual realnumbers (which would run into trouble with the Kochen-Specker theorem).Instead, we allow more general, ‘unsharp’ values in the form of real intervals.The value presheaf is denoted R ↔ .Physical quantities are represented by arrows (natural transformations)from Σ to R ↔ . The eigenstate-eigenvalue link is preserved in the sense thatif one has an eigenstate of some physical quantity A , the value assignedto this physical quantity at all contexts that contain A is the one-pointinterval [ a, a ] that just contains the eigenvalue a . Details can be found in[18, 9, 13]. In order to represent propositions such as “ A ε ∆”, instead of usingdaseinisation of projections, one can also consider inverse images (technically,pullbacks along arrows in the topos) of subobjects of the presheaf R ↔ toobtain subobjects of the spectral presheaf Σ, see sections 13.8.7–8 in [20].This is structurally analogous to the situation in classical physics, where aproposition “ A ε ∆” is represented by the subset f − A (∆) of the state space. Topos logic and neo-realism.
Crucially, every topos comes with abuilt-in logic. This is a higher-order, typed, intuitionistic logic, often multi-valued. For the general theory, see [35, 33].In our case, the topos associated with a quantum system is
Set V ( N ) op ,presheaves over the context category. The available truth values in the logicof this topos are lower sets in the context category V ( N ): subsets T ⊆ V ( N )such that if V ∈ T and V ′ ⊂ V , then V ′ ∈ T . There are uncountably manytruth values available instead of just true and f alse , and the truth valuesform a Heyting algebra themselves.Recall that in classical physics, a proposition “ A ε ∆” is represented by10 subset f − A (∆) of the state space S , and a state is represented by a point s ∈ S . The truth value of the proposition in the state is the truth value of athe Boolean formula s ∈ f − A (∆), that is, v (“ A ε ∆”; s ) = ( s ∈ f − A (∆)) ∈ ( f alse, true ) . (3)In the topos approach, a proposition “ A ε ∆” is represented by a subobject δ o ( ˆ P ) of the state object, the spectral presheaf Σ. A pure state | ψ i is rep-resented not by a point of Σ (there are none), but by a certain minimalsubobject w ψ . We can interpret the formula w ψ ∈ δ o ( ˆ P ) in the internal logicof our topos using the so-called Mitchell-Benabou language, which gives atruth value, v (“ A ε ∆”; w ψ ) = ( w ψ ∈ δ o ( ˆ P )) . (4)In the classical case, a point s either lies in a subset f − A (∆) (giving true globally) or not (giving f alse globally). In the topos, we do not just get asingle true or f alse , but one such truth value for each context V ∈ V ( N ).The actual topos truth value is the collection of all these ‘local’ truth values.It is easy to show that if we have true at V and V ′ ⊂ V , then we also get true at V ′ . Globally, we get a lower set in V ( N ) and hence a truth value inthe intuitionistic, contextual and multi-valued logic of our topos.Just as in classical physics, every proposition has a truth value in anygiven state. There is no fundamental reference to observers, measurementsor other instrumentalist concepts. The topos approach provides a mathe-matical formalism for quantum theory that can naturally be given a realistinterpretation. The price to pay is that the logic employed is not classicaltwo-valued Boolean logic, but the intuitionistic, multi-valued logic providedby the topos of presheaves. For this reason, we usually speak of a neo-realist interpretation.Many other aspects of quantum theory can be described in the new topos-based mathematical formalism, such as mixed states [6, 8], time evolution[12], probabilities and the Born rule [6, 21, 22], etc. Extending to field theories and beyond.
Whilst we have consideredonly non-relativistic systems described by an noncommutative algebra ofphysical quantities, the general scheme can be extended straightforwardly toother kinds of theories. For example, instead of just taking the poset V ( N )of contexts, one can also consider systems where subalgebras of observablesare attached to space-time regions, as in algebraic quantum field theory. Asuitable context category would then also carry additional space-time labels.First steps in this direction were taken by Joost Nuiten in [37].11ven more generally, the basic scheme of a state space, a space of valuesand physical quantities as maps between them is so general that it appliesvirtually to all physical theories. By modelling state spaces and spaces ofvalues as objects in a topos, e.g. as presheaves or sheaves, and physicalquantities as arrows in the topos between them, many generalisations beyondthe usual set-based and Hilbert space-based mathematics become available.In each case, the topos comes with a built-in intuitionistic logic, and logicalformulas (e.g. about the values of physical quantities in a given state) can beinterpreted within the logic of the topos, in a manner completely analogousto the one sketched above for quantum theory.Crucially, all propositions have truth values in any given state, withoutthe need to invoke observers or measurements – there is a natural neo-realistinterpretation of the topos formalism, no matter what the specific topos isand what the choices of the state object and value object are. In this way,the topos approach to the formulation of physical theories avoids the seri-ous interpretational issues that instrumentalist interpretations of the Hilbertspace formalism have. In this short section, I present some quite speculative thoughts on the rˆoles ofthe topos-internal logic and the topos-external metalogic in which we definethe topos and structures within it.The internal logic of a topos generally is intuitionistic (only in particularcases, it is Boolean), which means that the law of the excluded middle doesnot hold. A topos can be seen as a generalised universe of sets and hence asan arena to do mathematics. Proofs in a topos with an intuitionistic internallogic must necessarily be constructive, since proof by contradiction is notavailable. Moreover, typically the axiom of choice is not available in a giventopos, though weaker forms like countable choice may hold.If we use a topos and its internal logic to argue about physical systems,it seems that we commit ourselves to using constructive mathematics. Thereis interesting work along these lines by Heunen, Landsman, Spitters [25, 26,39, 27] and Fauser, Raynaud and Vickers [22].Yet, when we do physics, we necessarily have to separate ourselves fromthe system to be described. We do not aim at providing some sort of inclu-sive report, but rather try to give an objective or at least inter-subjectivedescription of some system or phenomenon outside of us. There may well be12 logic adequate to the system in itself, but this system logic is not the logicin which we are thinking and arguing about the system. For this, we use ametalogic, typically Boolean, in which we define the mathematical structuresto describe the system. For this reason, in our description and mathematicalarguments about the system we are free to use the metalogic. Of course,we must take care not to mix metalogical arguments about the system witharguments within the logic of the system.In the topos approach, we define a topos and certain mathematical struc-tures in it, e.g. the state object. This definition takes place in a (typicallyBoolean) metalogic about which we do not reflect very much. As we showed,the internal logic of the topos is useful in quantum theory, in the sense thatpropositions about the values of physical quantities can be assigned truthvalues using this ‘system logic’. We cannot use the Boolean metalogic todo this assignment of truth values, but we have to reflect about the physicalinterpretation of such truth values from the external, metalogical perspective.In short, our metalogic for doing physics is not the same as the systemlogic provided by the topos in which we describe a given physical systemmathematically.This even applies if the physical system we consider is the whole universe,as in a theory of quantum gravity: if we argue physically, then we are outsideof the system we describe. Of course, even if we argue about the wholeuniverse, we actually only consider a very small number of degrees of freedom.There is no (meta)logical contradiction arising, we can describe the wholeuniverse but still ‘step out’ of it when doing so, since our description is veryfar from complete.
In this contribution, we gave some conceptual arguments concerning the‘logical shape’ of a future theory of quantum gravity. We first presentedthe well-known argument that quantum gravity will not be a theory whosemathematical apparatus comes with an instrumentalist interpretation, sincesuch an interpretation makes no sense for closed systems with no externalobservers. Moreover, we argued that any theory based on Hilbert spacesalmost automatically comes with an instrumentalist interpretation, whichled us to the radical conclusion that a theory of quantum gravity should notbe based on Hilbert spaces.Instead, we suggest to use a form of theory that is based on a state spacepicture, generalising from classical physics. Such theories more naturally lend13hemselves to realist interpretations. We sketched some aspects of the toposapproach to quantum theory and argued that by going from sets to presheavesand from Boolean logic to intuitionistic logic, we arrive at a mathematicalformalism for quantum theory that has a natural neo-realist interpretation.Moreover, the underlying scheme is general enough to allow generalisationsto field theories and beyond. Needless to say, much work remains to be done.Finally, we briefly argued that even when we commit ourselves to de-scribing the whole universe using structures in a topos, and if we use theinternal logic of the topos to assign truth values to propositions etc., we donot have to do all our proofs and mathematical arguments internally in thetopos, i.e., constructively. Doing physics necessarily means to separate one-self from the system to be described, even if this system is the whole universe.Since we have to ‘step out’ of the system, we have to argue using the (typ-ically Boolean) metalogic in which we define the mathematical structures,e.g. topoi and state objects, that we use in the mathematical description ofthe system at hand. It is this Boolean metalogic in which we do physics.
Acknowledgements.
Chris Isham has been the major influence on myphysical thinking over the last few years, and I very much thank him fornumerous discussions and his friendship. Dicussions with Harvey Brown,Steve Vickers, Klaas Landsman, Masanao Ozawa and Tim Palmer are grate-fully acknowledged. I thank my students Carmen Constantin, Rui SoaresBarbosa, Dan Marsden and Nadish de Silva for their questions and theirvaluable input. Finally, I thank Dean Rickles for inviting me to contributeto this volume.
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