Carlo Rovelli
email: rovelli@cpt.univ-mrs.fr

'Space is blue and birds fly through it'

Keywords: quantum physics relational quantum mechanicsŠ, interpretations of quantum mechanics, quantum state, relations, wave function Q1

Quantum mechanics is not about 'quantum states': it is about values of physical variables. I give a short fresh presentation and update on the relational perspective on the theory, and a comment on its philosophical implications.

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A misleading notion: quantum state

In his celebrated 1926 paper [START_REF] Schrödinger | Quantisierung als Eigenwertproblem (Erste Mitteilung)[END_REF], Erwin Schrödinger introduced the wave function ψ and computed the Hydrogen spectrum from first principles. But the theory we call 'quantum mechanics' (QM) was born 1 year earlier, in the work of Werner Heisenberg [START_REF] Heisenberg | Uber quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen[END_REF], and had already evolved into its current full set of equations in a spectacular series of articles by Born et al. [START_REF] Born | Zur Quantenmechanik[END_REF][START_REF] Born | Zur Quantenmechanik II[END_REF]. Dirac, following Heisenberg's breakthrough, got to the same structure independently, in 1925, the year before Schrödinger's work, in a work titled 'The fundamental equations of quantum mechanics' [START_REF] Dirac | The fundamental equations of quantum mechanics[END_REF]. (See [START_REF] Fedak | The 1925 Born and Jordan paper 'On quantum mechanics[END_REF][START_REF] Van Der Waerden | Sources of quantum mechanics[END_REF] for a historical account.) Even the Hydrogen spectrum had been computed by Pauli in [START_REF] Pauli | Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik [On the hydrogen spectrum from the standpoint of the new quantum mechanics[END_REF], using the language of Heisenberg, Born and Jordan, based on the equations language of the theory, unfamiliar at the time, into a familiar one: differential equations. This brought ethereal quantum theory down to the level of the average theoretical physicist.

The conceptual step was to introduce the notion of 'wave function' ψ, soon to be evolved into the notion of 'quantum state' ψ, endowing it with heavy ontological weight. This conceptual step was wrong, and dramatically misleading. We are still paying the price for the confusion it has generated.

The confusion got into full shape in the influential second paper of the series [START_REF] Schrödinger | Quantisierung als Eigenwertproblem (Zweite Mitteilung)[END_REF], where Schrödinger stressed the analogy with optics: the trajectory of a particle is like the trajectory of a light ray: an approximation for the behaviour of an underlying physical wave in physical space. That is: ψ is the 'actual stuff', like the electromagnetic field is the 'actual stuff' underlying the nature of light rays.

Notice that this step is entirely 'interpretational'. It does not add anything to the predictive power of the theory, because this was already fully in place in the previous work of Heisenberg, Born and Jordan, where the 'quantum state' does not play such a heavy role. Schrödinger's conceptual step provided only a (misleading) way of reconceptualizing the theory.

The idea that the quantum state ψ represents the 'actual stuff' described by QM has pervaded later thinking about the theory. This is largely due to the toxic habit of introducing students to quantum theory beginning with Schrödinger's 'wave mechanics': thus betraying at the same time history, logic and reasonableness.

The founders of QM saw immediately the mistakes in this step. Heisenberg was vocal in pointing them out [START_REF] Kumar | Quantum: Einstein, Bohr and the great debate about the nature of reality[END_REF]. First, Schrödinger's basis for giving ontological weight to ψ was the claim that quantum theory is a theory of waves in physical space. But this is wrong: already the quantum state of two particles cannot be expressed as a collection of functions on physical space. Second, the wave formulation misses the key feature of atomic theory: energy discreteness, which must be recovered by additional ad hoc assumptions, because there is no reason for a physical wave to have energy related to frequency. Third, and most importantly, if we treat the 'wave' as the real stuff, we fall immediately into the horrendous 'measurement' problem. In its most vivid form (due to Einstein): how can a 'wave', spread over a large region of space, suddenly concentrate on a single place where the quantum particle manifests itself? All these obvious difficulties, which render the ontologicization of ψ absurd, were rapidly pointed out by Heisenberg. But Heisenberg lost the political battle against Schrödinger, for a number of reasons. First, all this was about 'interpretation' and for many physicists this was not so interesting after all, once the equations of QM began producing wonders. Differential equations are easier to work with and sort of visualize, than non-commutative algebras. Third, Dirac himself, who did a lot directly with non-commutative algebras, found it easier to make the calculus concrete by giving it a linear representation on Hilbert spaces, and von Neumann followed: on the one hand, his robust mathematical formulation of the theory brilliantly focused on the proper relevant notion: the non-commutative observable algebra, on the other, the weight given to the Hilbert space could be taken by some as an indirect confirmation of the ontological weight of the quantum states. Fourth, and most importantly, Bohr-the recognized fatherly figure of the community-tried to mediate between his two brilliant bickering children, by obscurely agitating hands about a shamanic 'wave/particle duality'. To be fair, Schrödinger himself realized soon the problems with his early interpretation, and became one of the most insightful contributors to the long debate on the interpretation; but the misleading idea of taking the 'quantum state' as a faithful description of reality stuck.

If we want to get any clarity about QM what we need is to undo the conceptual confusion raised by Schrödinger's introduction of the quantum state ψ.

The abstract of the breakthrough paper by Heisenberg reads: 'The aim of this work is to set the basis for a theory of QM based exclusively on relations between quantities that are in principle observable.' Only relations between variables, not new entities. The philosophy is not to inflate ontology: it is to rarefy it.

Felix Bloch reports an enlightening conversation with Heisenberg [START_REF] Bloch | Heisenberg and the early days of quantum mechanics[END_REF]: 'We were on a walk and somehow began to talk about space. I had just read Weyl's book Space, Time and Matter, and under its influence was proud to declare that space was simply the field of linear operations. 'Nonsense,' said Heisenberg, 'space is blue and birds fly through it.' This may sound naive, but I knew him well enough by that time to fully understand the rebuke. What he meant was that it was dangerous for a physicist to describe Nature in terms of idealized abstractions too far removed from the evidence of actual observation. In fact, it was just by avoiding this danger in the previous description of atomic phenomena that he was able to arrive at his great creation of QM. In celebrating the 15th anniversary of this achievement, we are vastly indebted to the men who brought it about: not only for having provided us with a most powerful tool but also, and even more significant, for a deeper insight into our conception of reality.'

What is thus this 'deeper insight into our conception of reality' that allowed Heisenberg to find the equations of QM, and that has no major use of the quantum state ψ?

Quantum theory as a theory of physical variables

Classical mechanics describes the world in terms of physical variables. Variables take values, and these values describe the events of nature. Physical systems are characterized by sets of variables and interact. In the interaction, systems affect one another in a manner that depends on the value taken by their variables. Given knowledge of some of these values, we can, to some extent, predict more of them.

The same does QM. It describes the world in terms of physical variables. Variables take values, and these values describe the events of nature. Physical systems are characterized by sets of variables and interact, affecting one another in a manner that depends on the value taken by their variables. Given knowledge of some values, we can, to some extent, predict more of them.

The basic structure of the two theories is therefore the same. The differences between classical and QM are three, interdependent:

(a) There is fundamental discreteness in nature, because of which many physical variables can take only certain specific values and not others. (b) Predictions can be made only probabilistically, in general. (c) The values that a variables of a physical system takes are such only relative to another physical system. Values taken relatively to distinct physical systems do not need to precisely fit together coherently, in general.

I discuss with more precision these three key aspects of quantum theory, from which all the rest follows, below. The first-discreteness-is the most important characteristic: it gives the theory its name. It is curiously disregarded in many, if not most, philosophers' discussions on quantum theory. The third is the one with heavy philosophical implications, which I shall briefly touch upon below. This account of the theory is the interpretative framework called 'Relational QM'. It was introduced in 1996 in [START_REF] Rovelli | Relational quantum mechanics[END_REF] (see also [START_REF] Laudisa | Stanford Encyclopedia of Philosophy[END_REF][START_REF] Laudisa | The EPR argument in a relational interpretation of quantum mechanics[END_REF][START_REF] Smerlak | Relational EPR[END_REF][START_REF] Rovelli | An argument against the realistic interpretation of the wave function[END_REF]). In the philosophical literature it as been extensively discussed by Bas van Fraassen [START_REF] Van Fraassen | Rovelli's world[END_REF] from a marked empiricist perspective, by Michel Bitbol [START_REF] Bitbol | Physical relations or functional relations? A non-metaphysical construal of Rovelli's relational quantum mechanics[END_REF][START_REF] Bitbol | De l'intérieur du monde[END_REF] who has given a neo-Kantian version of the interpretation, by Mauro Dorato [START_REF] Dorato | Rovelli' s relational quantum mechanics, monism and quantum becoming[END_REF] who has defended it against a number of potential objections and discussed its philosophical implication on monism, and recently by Laura Candiotto [START_REF] Candiotto | The reality of relations[END_REF] who has given it an intriguing reading in terms of (Ontic) Structural Realism. Metaphysical and epistemological implications of relational QM have also been discussed by Matthew Brown [START_REF] Brown | Relational quantum mechanics and the determinacy problem[END_REF] and Daniel Wolf (né Wood) [START_REF] Wood | Everything is relative: Has Rovelli found the way out of the woods? Q5 24[END_REF].

(a) Discreteness I find it extraordinary that so many philosophical discussions ignore the main feature of quantum theory: discreteness.

Discreteness is not an accessory consequence of quantum theory, it is its core. Quantum theory is characterized by a physical constant: the Planck constant h = 2π h. This constant sets the scale of the discreteness of quantum theory, and thus determines how bad is the approximation provided by classical mechanics. Several 'interpretations' of quantum theory seem to forget the existence of the Planck constant and offer no account of its physical meaning.

Here is a more detailed account of discreteness: A physical system is characterized by an ensemble of variables. The space of the possible values of these variables is the phase space of the system. For a system with a single degree of freedom, the phase space is two dimensional. Classical physics assumes that the variables characterizing a system have always a precise value, determining a point in phase space. Concretely, we never determine a point in phase space with infinite precision-this would be meaningless-, we rather say that the system 'is in a finite region R of phase space', implying that determining the value of the variables will yield values in R. Classical mechanics assumes that the region R can be arbitrarily small. Now, the volume Vol(R) of a region R of phase space has dimensions Length 2 × Mass/Time, for each degree of freedom. This combination of dimensions, Length 2 × Mass/Time, is called 'action' and is the dimension of the Planck constant. Therefore what the Planck constant fixes is the size of a (tiny) region in the space of the possible values that the variables of any system can take. Now: the major physical characterization of quantum theory is that the volume of the region R where the system happens to be cannot be smaller that 2π h:

Vol(R) ≥ 2π h, (2.1)
per each degree of freedom. This is the most general and most important physical fact at the core of quantum theory. This implies that the number of possible values that any variable distinguishing points within the region R of phase space, and which can be determined without altering the fact that the system is in the region R itself, is at most

N ≤ Vol(R) 2π h , (2.2)
which is a finite number. That is, this variable can take discrete values only. If it was not so, the value of the variable could distinguish arbitrary small regions of phase space, contradicting (2.1).

In particular: any variable separating finite regions of phase space is necessarily discrete. QM provides a precise way of coding the possible values that a physical quantity can take. Technically: variables of a system are represented by (self-adjoint) elements A of a (C * ) algebra A. The values that the variable a can take are the spectral values of the corresponding algebra element A ∈ A.

(b) Probability

Mechanics predicts the values of some variables, given some information on the values that another set of variables has taken. In QM, the available information is coded as a (normalized positive linear) functional ρ over A. This is called a 'state'. The theory states that the statistical mean value of a variable A is ρ(A). Thus values of variables can be, in general, predicted only probabilistically.

In turn, the state ρ is computed from values that variables take. I and so on.) The value of a quantity is sharp when the probability distribution is concentrated on it (ρ(A 2 ) = (ρ(A)) 2 ). For a non-commutative quantum algebra, there are no states where all variables are sharp. Therefore, the values of the variables can never determine a point in phase space sharply. This is the core of quantum theory, which is therefore determined by the non-commutativity of the algebra. The Planck constant h is the dimensional constant on the right-hand side of The non-commutativity between variables is Heisenberg breakthrough, understood and formalized by Born and Jordan, who were the first to write the celebrated relation [q, p] = ih and to recognize this non-commutativity as the key of the new theory, in 1925.

The non-commutativity of the algebra of the variables (measured by h) is the mathematical expression of the physical fact that variables cannot be simultaneously sharp, hence there is an (h-size) minimal volume attainable in phase space, hence predictions are probabilistic.

The fact that values of variables can be predicted only probabilistically raises the key interpretational question of QM: when and how is a probabilistic prediction resolved into an actual value?

(c) The relational aspect of quantum theory

When and how a probabilistic prediction about the value of a variable a of a physical system S is resolved into an actual value?

The answer is: when S interacts with another physical system S . Value actualization happens at interactions because variables represent the ways systems affect one another. Any interaction counts, irrespectively of size, number of degrees of freedom, presence of records, consciousness, degree of classicality of S , decoherence, or else, because none of these pertain to elementary physics.

In the course of the interaction, the system S affects the system S . If the effect of the interaction on S depends on the variable a of S, then the probabilistic spread of a is resolved into an actual value, or, more generally, into an interval I of values in its spectrum. Now we come to the crucial point. The actualization of the value of a is such only relative to the system S . The corresponding state ρ determined by the actualization is therefore a state relative to S , in the sense that it predicts only the probability distribution of variables of S in subsequent interactions with S . It has no bearing on subsequent interactions with other physical systems. This is the profoundly novel relational aspect of QM. Why are we forced to this extreme conclusion? The argument, detailed in [START_REF] Rovelli | Relational quantum mechanics[END_REF], can be summarized as follows.

We must assume that variables do take value, because the description of the world we employ is in terms of values of variables. However, the predictions of QM are incompatible with all variables having simultaneously a determined value. A number of mathematical results, such as the Kochen-Specker [24] theorem, confirm that if all variables could have a value simultaneously, the predictions of QM would be violated. Therefore, something must determine when a variable has a value.

The textbook answer is 'when we measure it'. This obviously makes no sense, because the grammar of Nature certainly does not care whether you or I are 'measuring' anything. Measurement is an interaction like any other. Variables take value at any interaction.

However, (this is the key point) if a system S interacts with a system S , QM predicts that in, a later interaction with a further system S , a variable b of the S ∪ S system is not determined by ρ . Rather, it is determined by the joint dynamical evolution of the S ∪ S quantum system. In physicists' parlance: quantum theory predicts interference effects between the different branches corresponding to different values of the variable a, as if no actualization had happened.

We have thus to combine the presence of these interference effects (which pushes us to say that a had no value) with the fact that the variable a does take a value. The answer of relational QM is that the variable a of the system S actualized in the interaction with S takes value with respect to S , but not with respect to S . This is the core idea underlying the 'relational' interpretation of QM.

Relationality is no surprise in physics. In classical mechanics, the velocity of an object has no meaning by itself: it is only defined with respect to another object. The colour of a quark in strong-interaction theory has no meaning by itself: only the relative colour of two quarks has meaning. In electromagnetism, the potential at a point has no meaning, unless another point is taken as reference; that is, only relative potentials have meanings. In general relativity, the location of something is only defined with respect to the gravitational field, or with respect to other physical entities; and so on. But quantum theory takes this ubiquitous relationalism, to a new level: the actual value of all physical quantities of any system is only meaningful in relation to another system. Value actualization is a relational notion like velocity.

What is the quantum state?

The above discussion shows that the quantum state ρ does not pertain solely to the system S. It pertains also to the system S , because it depends on variables' values, which pertain only to S . The idea that states in QM are relative states, namely states of a physical system relative to a second physical system is Everett's lasting contribution to the understanding of quantum theory [START_REF] Everett | Relative state formulation of quantum mechanics[END_REF].

A moment of reflection shows that the quantum states used in real laboratories where scientists use QM concretely is obviously always a relative state. Even a radical believer in a universal quantum state would concede that the ψ that physicists use in their laboratories to describe a quantum system is not the hypothetical universal wave function: it is the relative state, in the sense of Everett, that describes the properties of the system, relative to the apparata it is interacting with.

What precisely is the quantum state of S relative to S ? What is ψ (or ρ)? The discussion above clarifies this delicate point: it is a theoretical device we use for bookkeeping information about the values of variables of S actualized in interactions with S , values which can in principle be used for predicting other (for instance future, or past) values that variables may take in other interactions with S .

Charging ψ with ontological weight is therefore like charging with ontological weight a distribution function of statistical physics, or the information I have about current political events: a mistake that generates mysterious 'collapses' anytime there is an interaction. More specifically, in the semiclassical approximation ψ ∼ e iS where S is a Hamilton-Jacobi function. This shows that the physical nature of ψ is the same as the physical nature of a Hamilton-Jacobi function. Nobody in their right mind would charge S with ontological weight, in the context of classical mechanics: S is a calculational device used to predict an outcome on the basis of an input. It 'jumps' at each update of the calculation.

QM is thus not a theory of the dynamics of a mysterious ψ entity, from which mysteriously the world of our experience emerges. It is a theory of the possible values that conventional physical variables take at interactions, and the transition probabilities that determine which values are likely to be realized, given that others are [START_REF] Groenewold | Objective and subjective aspects of statistics in quantum description[END_REF].

The fact that the quantum state is a bookkeeping device that cannot be charged with ontological weight is emphasized by the following observation [START_REF] Rovelli | An argument against the realistic interpretation of the wave function[END_REF]. Say I know that at time t a particle interacts with a x-oriented Stern-Gerlach device. Then I can predict that (if nothing else happens in between) the particle has probability 1 2 to be up (or down) spinning, when interacting with a z-oriented Stern-Gerlach device at time t . Key point: this is true irrespectively of which between t and t comes earlier. Quantum probabilistic predictions are the same both forth and back in time. So: what is the state of the particle in the time interval between t and t ? Answer: it depends only on what I know: if I know the past (respectively, future) value, I use the state to bookkeep the future (respectively, past) value. The state is a coding of the value of the x spin that allows me to predict the z spin, not something that the particle 'has'. We can be realist about the two values of the spin, not about the ψ in between, because ψ depends on a time orientation, while the relevant physics does not.

A coherent ontology for QM is thus sparser than the classical one, not heavier. A good name for the actualization of the value of a variable in an interaction is 'quantum event'. The proper ontology for QM is a sparse ontology of (relational) quantum events happening at interactions between physical systems.

(a) Information Equation (2.1) can be expressed by saying that (P1) The amount of information that can be extracted from a finite region of phase space is finite.

'Information' means here nothing else than 'number of possible distinct alternatives'. The step from ρ to ρ determined by an actualization modifies the predictions of the theory. In particular, the value of a, previously spread, is then predicted to be sharper. This can be expressed in information theoretical theorems by saying that (P2) An interaction allows new information about a system to be acquired.

There is an apparent tension between the two statements (P1) and (P2). If there is a finite amount of information, how can we keep gathering novel information? The tension is only apparent, because here 'information' quantifies the data relevant for predicting the value of variables. In the course of an interaction, part of the previously relevant information becomes irrelevant. In this way, information is acquired, but the total amount of information available remains finite. 2It is the combination of (P1) and (P2) that largely characterizes quantum theory (for the case of qubit-systems, see [START_REF] Hoehn | Toolbox for reconstructing quantum theory from rules on information acquisition[END_REF]). These two statements were proposed as the basic 'postulates' of QM in [START_REF] Rovelli | Relational quantum mechanics[END_REF]. The apparent contradiction between the two capturing the counterintuitive character of QM in the same sense in which the apparent contradiction between the two Einstein's postulate for Special Relativity captures the counterintuitive character of relativistic space-time geometry. Very similar ideas were independently introduced by Zeilinger and Brukner [START_REF] Zeilinger | A foundational principle for quantum mechanics[END_REF][START_REF] Brukner | Information and fundamental elements of the structure of quantum theory[END_REF].

An attempt to reconstruct the full formalism of quantum theory starting from these two information-theoretic postulated was initiated in [START_REF] Rovelli | Relational quantum mechanics[END_REF] (see also [START_REF] Grinbaum | Information-theoretic princple entails orthomodularity of a lattice[END_REF]). Recently, a remarkable reconstruction theorem along these lines has been successfully completed for the case of finitedimensional systems in [START_REF] Hoehn | Toolbox for reconstructing quantum theory from rules on information acquisition[END_REF][START_REF] Höhn | Quantum theory from questions[END_REF], shedding considerable new light on the structure of the theory and its physical roots.

The role of information at the basis of quantum theory is a controversial topic. The term 'information' is ambiguous, with a wide spectrum of meanings ranging from epistemic states of conscious observers all the way to simply counting alternatives, à la Shannon. As pointed out, for instance, by Dorato, even in its weakest sense information cannot be taken as a primary notion from which all others can be derived, because it is always information about something. Nevertheless, information can be a powerful organizational principle in the sense of Einstein's distinction between 'principle theories' (like thermodynamics) versus 'constructive theories' (like electromagnetism) [START_REF] Spekkens | The invasion of Physics by Information Theory, talk at Perimeter Institute[END_REF]. The role of the general theory of mechanics is not to list the ingredients of the world-this is done by the individual mechanical theories, like the Standard Model of particle physics, general relativity, of the harmonic oscillator. The role of the general theory of mechanics (like classical mechanics or QM) is to provide a general framework within which specific constructive theories are realized. From this perspective, the notion of information as a number of possible alternatives may play a very useful role.

It is in this sense that the two postulates can be understood. They are limitations on the structure of the values that variables can take. The list of relevant variables, which define a physical system, and their algebraic relations, are provided by specific quantum theories.

There are several objections that come naturally to mind when one first encounters relational QM, which seem to render it inconsistent. These have been long discussed and have all been convincingly answered; see in particular the detailed arguments in van Fraassen [START_REF] Van Fraassen | Rovelli's world[END_REF] and Dorato [START_REF] Dorato | Rovelli' s relational quantum mechanics, monism and quantum becoming[END_REF] and the original paper [START_REF] Rovelli | Relational quantum mechanics[END_REF]; I will not rediscuss them here. Relational QM is a consistent interpretation of quantum theory.

However, like all other consistent interpretations, it comes at a price.

Philosophical implications (a) Every interpretation has a cost

Every interpretation of quantum theory comes with a 'cost'.

Examples from some interpretations popular nowadays are the following. The cost of the Many World interpretations is the hugely inflated ontology of a continuous family of equally existing 'worlds', of which we basically know nothing, and an awkward difficulty of rigorously recovering the actual values of the variables in terms of which we describe the world, from the pure-ψ picture taken as fundamental. The cost of the Physical Collapse interpretations, such as the Ghirardi-Weber-Rimini theory, is to be based on physics which is so far unobserved and that many physicists view as not very plausible. The cost of the Bohmian interpretations is to postulate the existence of stuff which is unobservable in principle and which, in the view of most physicists, violates too badly what we have learned about Nature in the last century. The cost of Quantum Informational interpretations (partially inspired by relational QM [START_REF] Fuchs | Quantum foundations in the light of quantum information[END_REF]) is to be tied to a basically idealistic stance where the holder of the information is treated as an unphysical entity, a priori differently from all other physical systems, which cannot be in superpositions. The so-called Copenhagen Interpretations, which are held by the majority of real physicists concretely working with QM, postulate the existence of a generally ill-explained 'classical world', whose interactions collapse quantum states. And so on. . .. Do not take these criticisms badly: they are not meant to dismiss these interpretations; they are simply the reasons commonly expressed for which each interpretation does not sound plausible to others: the point I am making is that there is no interpretation of QM that is not burdened by a heavy cost of some sort, which appears too heavy a price to pay to those who do not share the passion for that particular interpretation. Many discussions about quantum theory are just finger-pointing to one another's cost.

The evaluation of these costs depends on wider philosophical perspectives that we explicitly or implicitly hold. Attachment to full fledged strong realism leads away from Quantum Informational interpretations and towards Bohm or Many Worlds. Sympathy for empiricism or idealism leads in opposite directions, towards Copenhagen or Quantum Information. And so on; the picture could be fine-grained.

The beauty of the problem of the interpretation of QM is precisely the fact that the spectacular and unmatched empirical success of the theory forces us to give up at least some cherished philosophical assumption. Which one is convenient to give up is the open question.

The relational interpretation does not escape this dire situation. As seen from the reactions in the philosophical literature, relational QM is compatible with diverse philosophical perspectives. But not all. How strong is the philosophical 'cost' of relational QM?

Its main cost is a challenge to a strong version of realism, which is implied by its radical relational stance.

(b) Realism

'Realism' is a term used with different meanings. Its weak meaning is the assumption that there is a world outside our mind, which exists independently from our perceptions, beliefs or thoughts.

Relational QM is compatible with realism in this weak sense. 'Out there' there are plenty of physical systems interacting among themselves and about which we can get reliable knowledge by interacting with them; there are plenty of variables taking values, and so on. There is nothing similar to 'mind' required to make sense of the theory. What is meant by a variable taking value 'with respect to a system S ' is not S to be a conscious subject of perceptions-it just the same as when we say that the velocity of the Earth is 40 km s -1 'with respect to the sun': no implication of the sun being a sentient being 'perceiving' the Earth. In this respect, quantum theory is no more and no less compatible with realism (or other metaphysics) than classical mechanics. I myself think that we, conscious critters, are physical systems like any other. Relational QM is anti-realist about the wave function, but is realist about quantum events, systems, interactions. . .. It maintains that 'space is blue and birds fly through it', and space and birds can be constituted by molecules, particles, fields or whatever. What it denies is the utility-even the coherence-of thinking that all this is made up by some underlying ψ entity.

But there is a stronger meaning of 'realism': to assume that it is in principle possible to list all the features of the world, all the values of all variables describing it at some fundamental level, at each moment of continuous time, as is the case in classical mechanics. This is not possible in relational QM. Interpretations of QM that adhere to strong realism, like Many Worlds, or Bohm, or other hidden variables theories, circumvent the Kochen-Specker theorem, which states that in general there is no consistent assignment of a definite values to all variables, by restricting the set of elementary variables describing the world (to the quantum state itself, or to Bohmian trajectories, or else). Relational QM assumes seriously the Kochen-Specker theorem: variables take value only at interactions. The stronger version of the realist credo is therefore in tension with QM, and this is at the core of relational QM. It is not even realized in the relatively weaker sense of considering a juxtaposition of all possible values relative to all possible systems. The reason is that the very fact that a quantity has value with respect to some system is itself relative to that system [START_REF] Rovelli | Relational quantum mechanics[END_REF].

This weak realism of relational QM is in fact quite common in physics laboratories. Most physicists would declare themselves 'realists', but not realists about ψ. As one of the two (very good) referees of this paper put it: 'In physicists' circles, Schrödinger's ψ is mostly regarded as a mere instrument'. Relational QM is a way to make this position coherent.

There are three specific challenges to strong realism that are implicit in relational QM. The first is its sparse ontology. The question of 'what happens between quantum events' is meangless in the theory. The happening of the world is a very fine-grained but discrete swarming of quantum events, not the permanence of entities that have well-defined properties at each moment of a continuous time. This is the way the relational interpretation circumvents results like the Pusey-Barrett-Rudolph theorem [START_REF] Pusey | On the reality of the quantum state[END_REF]. Such theorems assume that at every moment of time all properties are well-defined. (For a review, see [START_REF] Leifer | Is the quantum state real? An extended review of ψ-ontology theorems[END_REF]) They essentially say that if there is a hidden variable theory, the hidden variables must contain at least the entire information which is in the quantum state. But the assumption is explicitly denied in relational QM: properties do not exist at all times: they are properties of events and the events happen at interactions.

In the same vein, in [START_REF] Laudisa | Open problems in relational quantum mechanics[END_REF] Laudisa criticizes relational QM because it does not provide a 'deeper justification' for the 'state reduction process'. This is like criticizing classical mechanics because it it does not provide a 'deeper justification' for why a system follows its equations of motion. It is a stance based on a very strong realist (in the narrow sense specified above) philosophical assumption. In the history of physics much progress has happened by realizing that some naively realist expectation was ill-founded, and therefore by dropping these kind of questions: How are the spheres governing the orbits of planet arranged? What is the mechanical underpinning of the electric and magnetic fields? Into where is the universe expanding? To some extent, one can say that modern science itself was born in Newton's celebrated 'hypotheses non fingo', which is precisely the recognition that questions of this sort can be misleading. When everybody else was trying to find dynamical laws accounting for atoms, Heisenberg's breakthrough was to realize that the known laws were already good enough, but the ontology was sparser and the question of the actual continuous orbit of the electron was ill-posed. I think that we should not keep asking what amounts to this same question over and over again: trying to fill in the sparse ontology of Nature with our classical intuition about continuity. On this, see the enlightening discussion given by Dorato in [START_REF] Dorato | Dynamical versus structural explanations in scientific revolutions[END_REF].

The second element of relational QM that challenges a strong version of realism is that values taken with respect to different systems can be compared [START_REF] Rovelli | Relational quantum mechanics[END_REF] (hence there no solipsism), but the comparison amounts to a physical interaction, and its sharpness is therefore limited by h. Therefore, we cannot escape from the limitation to partial views: there is no coherent global view available. Matthew Brown has discussed this point in [START_REF] Brown | Relational quantum mechanics and the determinacy problem[END_REF].

The third, emphasized by Dorato, is the related 'anti-monistic' stance implicit in relational QM. As the state of a system is a bookkeeping device of interactions with something else, it follows immediately that there is no meaning in 'the quantum state of the full universe'. There is no something else to the universe. Everett's relative states are the only quantum states we can meaningfully talk about. Every quantum state is an Everett's quantum state. A reason for rejecting relational QM, therefore, comes if we assume that the monistic idea of the 'state of the whole' must makessense and must be coherently given in principle. 3This relational stance of relational QM requires a philosophical perspective where relations play a central role. This is why Candiotto [START_REF] Candiotto | The reality of relations[END_REF] suggests to frame relational QM in the general context of Ontic Structural Realism. This is certainly an intriguing possibility. My sympathy for a natural philosophical home for relational QM is an anti-foundationalist perspective where we give up the notion of primary substance-carrying attributes, and recognize the mutual dependence of the concepts we use to describe the world. Other perspectives are possible, as we have seen in the strictly empiricist and neo-Kantian readings by van Fraassen and Bitbol, but strong realism in the strict sense of substance and attributes that are always uniquely determined is not.

(c) How to go ahead?

There are three developments that could move us forwards

The first is novel empirical information. Some interpretations of quantum theory lead to empirically distinguishable versions of the theory. Empirical corroboration of their predictions would change the picture; repeated failure to detect discrepancy from standard QM weakens their credibility. This is the way progress happens in experimental physics. So far, QM has been unquestionably winning for nearly a century, beyond all expectations.

The second is theoretical fertility. If, for instance, quantum gravity turned out to be more easily comprehensible in one framework than in another, then this framework would gain credibility. This is the way progress happens in theoretical physics.

My focus on relational QM, indeed, is also motivated by my work in quantum gravity [38,39]. In quantum gravity, where we do not have a background space-time where to locate things, relational QM works very neatly because the quantum relationalism combines in a surprisingly natural manner with the relationalism of general relativity. Locality is what makes this work. Here is how [40]: the quantum mechanical notion of 'physical system' is identified with the general relativistic notion of 'space-time region'. The quantum mechanical notion of 'interaction' between systems is identified with the general relativistic notion of 'adjacency' between space-time regions. Locality assures that interaction requires (and defines) adjacency. Thus quantum states are associated to three-dimensional surfaces bounding space-time regions and quantum mechanical transition amplitudes are associated to 'processes' identified with the space-time regions themselves. In other words, variables actualize at three dimensional boundaries, with respect to (arbitrary) space-time partitions. The theory can then be used locally, without necessarily assuming anything about the global aspects of the universe.

The third manner in which progress can happen is how it does in philosophy: ideas are debated, absorbed, prove powerful or weak, and slowly are retained or discarded. I am personally actually confident that this can happen for quantum theory.

The key to this, in my opinion, is to fully accept this interference between the progress of fundamental physics and some major philosophical issues, like the question of realism, the nature of entities and relations, and the question of idealism. Accepting the reciprocal interference means in particular to reverse the way general philosophical stances colour our preferences for interpretation. That is, rather than letting our philosophical orientation determine our reading of QM, be ready to let the discoveries of fundamental physics influence our philosophical orientations.

It woundn't certainly be the first time that philosophy is heavily affected by science. I believe that we should not try to understand the world rigidly in terms of our conceptual structure. Rather we should humbly allow our conceptual structure to be moulded by empirical discoveries. This, I think, is how knowledge develops best.

Relational QM is a radical attempt to directly cash out the initial breakthrough that originated the theory: the world is described by variables that can take values and obey the equations of classical mechanics, but products of these variables have a tiny non-commutativity that generically prevents sharp value assignment, leading to discreteness, probability and to the relational character of the value assignment.

The founders of the theory expressed this relational character in the 'observer-measurement' language. This language seems to require that special systems (the observer, the classical world, macroscopic objects. . .) escape the quantum limitations. But neither of this, and in particular no 'subjective states of conscious observers', is needed in the interpretation of QM. As soon as we relinquish this exception, and realize that any physical system can play the role of a Copenhagen's 'observer', we fall into relational QM. Relational QM is Copenhagen QM made democratic by bringing all systems onto the same footing.

  between physical values and eigenvalues, with no reference to ψ. So, what did Schrödinger do, in his 1926 paper? With hindsight, he took a technical and a conceptual step. The technical step was to change the algebraic 2018 The Author(s) Published by the Royal Society. All rights reserved. rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  (Technically: using the notation ρ(A) = Tr[ρA], a variable b taking value in the interval I of its spectrum, determines the state ρ = cP b I where P b I is the projector associated with I in the spectral resolution of B and c is the normalization constant fixed by ρ(1) = 1. If then a variable b takes value in I , ρ changes to ρ = cP b I P b I P b

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the commutator: it determines the amount of non-commutativity, hence discreteness, hence impossibility of sharpness of all variables.

1

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  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  rsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 2017.0312 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

In Many World interpretations, a takes a value indexically relative to a world; in Bohm-like theories only an (abelian) subset of variables has value, not all of them; in Quantum Information interpretations, a takes a value only when the interaction is with the idealistic holder of the information; in Copenhagen-like interpretations, when the interaction is with the 'classical world'; in Physical Collapse theories, when some not yet directly observed random physical phenomenon happens. . .

Here is a simple example: if a spin-1 2 particle passes through a z-oriented Stern-Gerlach apparatus and takes the 'up' path, we have one bit of information about the orientation of its angular momentum L. If it then crosses an x-oriented apparatus, we gain one bit of information (about L x ) and we lose one bit of information (about L z ).

This does not prevent conventional quantum cosmology to be studied, because physical cosmology is not the science of everything: it is the science of the largest scale degrees of freedom.

Data accessibility. This article has no additional data. Q2


Book review

"The Meaning of the Wave Function. In Search of the Ontology of Quantum Mechanics", by Shan Gao.

Reviewed by Carlo Rovelli

Shan Gao is a philosopher of physics at the Research Center for Philosophy of Science and Technology, in Shanxi, and a prolific writer, with nineteen books, both technical and for the large public, covering a variety of topics in fundamental physic. For some time, Gao has been developing an original perspective on quantum mechanics, based on the idea that the wave function describes the random discontinuous motion of real particles: a quantum particle jumps around discontinuously, and the wave function gives the density of its presence in a region during an arbitrary small time interval. The idea is supplemented by a model of physical collapse, which modifies the the quantum dynamics adding a stochastic, nonlinear evolution term, resulting from the random motion of the particles. Gao argues in favour of this view starting from the existence of protective measurements, a particular class of interactions allowed by quantum theory, where the expectation value of any observable of a system can be recorded by a second system, without altering the quantum state of the first. In the book "The Meaning of the Wave Function. In Search if the Ontology of Quantum Mechanics", he gives a detailed and meticulous illustration of this view, and the arguments supporting it, engaging in a careful discussion with many arguments present in the literature.

Up to discriminating empirical results, or proven logical inconsistencies, "interpretations" of quantum mechanics cannot be truly proven or disproven. They give us a way to think about the theory, and about nature in general, they can orient scientific research, and they interact strongly with our general philosophical orientations. The interaction is in both directions: on the one hand, they question or support general philosophical perspectives (realism/empiricism, for instance), showing these to be more or less capable of accounting for our present scientific knowledge; on the other, our preference for this or that interpretation is often strongly coloured by our inevitable philosophical prejudices.

The problem is hard because no interpretation of quantum mechanics is costless. Each has a cost, which is viewed by those who do not like it as too high a price to pay with respect to the gain. Examples of these "costs" are the multiplication of "worlds" in the many-worlds interpretation; the in principle un-observability and breaking of Lorentz invariance of postulated entities in Bohmiam interpretations; the perceived artificiality of physical collapse interpretations; the weakening of realism in the relational interpretation; the ambiguous status of the classical world in the common "Copenhagen" interpretation, which is still the popular view of quantum theory in physics labs; and so on. In whatever way we think about quantum theory, we always give up something cherished that many of our colleagues consider too high a price to pay for a coherent understanding of nature. Opinions do evolve, but so far no perspective has proven to be definitely more fruitful or widely convincing than the others.

Gao's random discontinuous motion interpretation adds itself to the list. Its strength is its realism and a certain physically intuitive concreteness. Among its weaknesses is the fact that it still needs a physical collapse mechanism to account for measurement. Perhaps the publication of the book will prompt a critical evaluation of its merits and difficulties; here I limit myself to a couple of very general considerations.

There are two general overall ways of understanding quantum theory. According to the first, the theory is about the values taken by variables of physical systems in interactions. This is Heisenberg's original view, which led to the actual discovery of the theory. According to the second, the theory is about an entity ψ, the wave function or, more in general, the "quantum state". This view was initiated by Schrödinger one year after Heisenberg, Born and Jordan had already developed the full mathematical apparatus of what we call quantum mechanics today. I myself find Heisenberg's view (values of variables) more coherent and I think that this view is going to be more fruitful in the future, but Schrödinger's view (entity ψ) is still very popular. Gao's perspective can be seen as an attempt at a compromise: a particle has position, but because of its random everywhere discontinuous motion position is spread during each arbitrary small time interval, and ψ describes this spread.

The argument the book offers to introduce such realistic interpretation of ψ is the existence of protective measurements. A protective measurement is an interaction between a system and a device, at the end of which the device records the expectation value of a physical quantity A of the system, and the state of the system remains unaltered. Contrary to naive expectation, this is indeed possible for any ψ and any A. At first sight, this seems to contradict the common idea that it is impossible to measure an arbitrary quantity A in a state ψ without altering ψ. The existence of such protective measurements is therefore often presented as good argument in favour of a realistic interpretation of the wave function. `If I can measure it without altering it, it must be real.' I am unconvinced by this interpretation of protective measurements. The catch, seems to me, is the ambiguity in the word measurement. To be able to concretely perform a protective measurement we need to have already measured ψ almost entirely: in fact, only up to a single number. The reason is that in order to perform a protective measurement we need to know preventively one of the observables, O, of which ψ happen to be an eigenstate of, and the `measuring' device must be built accordingly. But if we know that ψ is an eigenstate of O, we already know everything there is to know about ψ up to a single number (the eigenvalue); and this number, according to quantum theory, is precisely the number that can be measured without further altering ψ. This clarified, the existence of protective measurements is far less telling than what it looks at first sight: it just tells us that if we know almost everything about ψ we can then ascertain the last bit without altering it. It is then unsurprising that we can build a device that interacting with this largely known ψ can record any ψ-dependent quantity we wish. The same device would not work on a different ψ. It is a bit as if the information to be measured was already largely coded into the device itself. This consideration does not discount Gao's interpretation, of course. It exemplify how in the debate on the meaning of quantum theory often arguments that seem convincing to one appear less so to another. One of the best aspects of Gao's book, actually, is the punctilious account of many arguments and counterarguments given in the literature; as such, the book is also a useful source and an overview of number of debates around the interpretation of quantum theory.

After the great initial debates of the early days of the theory, and the long subsequent period of neglect, the grand debate on the meaning of quantum theory is now receiving ever-increasing attention. I think that this is healthy, and we need this debate to go ahead in science. I do not expect rapid convergence, but technical points are clarified, ideas are emerging and opinions evolve. At some points, the dust will begin to settle, as it has happened to similar great debates on the past, which appeared undecidable for a while. What have we precisely learned about nature, in discovering that this spectacular formalism is so effective in predicting its ways?


Carlo Rovelli

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1.

Against Philosophy is the title of a chapter of a book by one of the great physicists of the last generation: Steven Weinberg, Nobel Prize winner and one of the architects of the Standard Model of elementary particle physics1 . Weinberg argues eloquently that philosophy is more damaging than helpful for physics-although it might provide some good ideas at times, it is often a straightjacket that physicists have to free themselves from. More radically, Stephen Hawking famously wrote that "philosophy is dead" because the big questions that used to be discussed by philosophers are now in the hands of physicists2 . Similar views are widespread among scientists, and scientists do not keep them to themselves. Neil de Grasse Tyson, a well known figure in the popularisation of science in America, publicly stated in the same vein: "…we learn about the expanding universe, … we learn about quantum physics, each of which falls so far out of what you can deduce from your armchair that the whole community of philosophers … was rendered essentially obsolete."

3 I disagree with these opinions. I present here some counter-arguments, arguing that philosophy has always played an essential role in the development of science, physics in particular, and is likely to continue to do so. This is a long-standing debate. In this regard, a delightful chapter was played out in Athens during its classical period. At the time, the golden youth of the city was educated in famous schools. Among these, two stood out: the school of Isocrates, and the Academy, founded by a certain Plato.

The rivalry between the two was heated, like the rivalry between Oxford and Cambridge, but was not just about quality: the very approach to education was different in the two schools. Isocrates offered a high-level practical education, teaching the youth of Athens the kind of skills and knowledge required to become politicians, lawyers, judges, architects, and so on. The Academy on the other hand focused rather on discussing general questions about foundations: What is justice? What would be the best laws? What is beauty? What is matter made of? And Plato had invented a good name for this way of posing problems: "philosophy".

The disagreement between the two schools was deep-rooted. Isocrates' criticisms of Plato's approach to education and knowledge were direct:

"Those who do philosophy, who determine the proofs and the arguments … and are accustomed to enquiring, but take part in none of their practical functions, … even if they happen to be capable of handling something, they automatically do it worse, whereas those who have no knowledge of the arguments [of philosophy], if they are trained [in concrete sciences] and have correct opinions, are altogether superior for all practical purposes. Hence for sciences, philosophy is entirely useless." 4

This is remarkably like the claim by those contemporary scientists who argue that philosophy has no role to play in science.

As it happened, a brilliant young student in Plato's school wrote a short work in response to Isocrates' criticisms. This was the Protrepticus, a text that became famous in antiquity. It has survived only in part, and we know it only through reconstruction from extensive quotations by later writers. A group of classical scholars led by Doug Hutchinson and Monte Ransome Jonson have recently completed a novel reconstruction which can be accessed online. 5 The Protrepticus was probably written in the form of a dialog between characters defending opposing positions-dialog was of course the style preferred by Plato. What remains of the text is sufficient to understand the main arguments this young student put forward in his reply to Isocrates, in defence of philosophy.

The bright young fellow who authored the pamphlet later left Athens, but eventually returned to open his own school, and had quite a career. His name was Aristotle.

Two millennia of development of the sciences and philosophy have vindicated and, if anything, strengthened Aristotle's defence of philosophy against Isocrates' accusations of futility. His arguments are still remarkably relevant, and I take inspiration from them to respond here to the current claims that philosophy is useless to physics.

2.

The first of Aristotle's arguments is the fact that general theory supports and happens to be useful for the development of practice.

Let me separate two aspects of the issue: first, the relevance of philosophy to science in the past, and second, the question of whether philosophy has become irrelevant for science today. Let us start with the first.

Today, after a couple of millennia, during which both philosophy and science have developed considerably, historical evidence regarding the influence of philosophy on science has become overwhelming.

Here are some instances of this influence, drawn from astronomy and physics. I shall only mention a few examples. Ancient astronomy -that is, everything we know about the Earth being round, its size, the size of the moon and the sun, the distances to the moon and the sun, the motion of the planets in the sky, and the basis from which modern astronomy and modern physics have emerged-, is a direct descendent of philosophy. The key questions that motivated these developments were posed in the Academy and the Lyceum, and motivated by highly theoretical, rather than practical concerns. Although centuries later Galileo and Newton took great steps beyond previous physics and astronomy, and in particular, the Aristotelian world view, they relied heavily on what had come before. 6 Galileo and Newton extended previous knowledge, reinterpreting, reframing, and building upon it. Galileo's understanding in particular would have been inconceivable without Aristotelian physics.

More importantly, Galileo's work would have been inconceivable without the ideology he derived from Plato, namely, to search for the ideal mathematical order underlying appearances. Galileo was guided by an almost fanatic Platonism. In his work, Newton was explicit about his debt to ancient philosophy, Democritus in particular, for ideas that arose originally from philosophical motivations, such as the notions of empty space, atomism, and natural rectilinear motion. Furthermore, his crucial discussion about the nature of space and time built upon his discussions with (and against) Descartes.

But the direct influence of philosophy on physics is certainly not limited to the birth of modern physics. It can be recognised in every major step. Take the twentieth century. Both major advances made by twentieth century physics were strongly influenced by philosophy. They would have been inconceivable without the philosophy of the time. Quantum mechanics springs from an intuition due to Heisenberg, grounded in the strongly positivist philosophical atmosphere in which he found himself: one gets knowledge by restricting oneself to what is observable. The abstract of Heisenberg's 1925 milestone paper on quantum theory is explicit about this: "The aim of this work is to set the basis for a theory of quantum mechanics based exclusively on relations between quantities that are in principle observable." 7 The same distinctly philosophical attitude nourished Einstein's discovery of special relativity: by restricting to what is observable, we recognise that the notion of simultaneity is misleading. Einstein explicitly recognised his debt to the philosophical writings of Mach and Poincaré. Without these inputs, his special relativity would have been inconceivable. Although not the same, the philosophical influences on Einstein's conception of general relativity were even stronger. Once again, he was explicit in recognising his debt to philosophy, this time to the critical thinking of Leibniz, Berkeley, and Mach. Einstein's relation with philosophy was indeed complex: he claimed for instance that Schopenhauer had had a pervasive influence on him. This is less evident to detect in his physics, but Schopenhauer's ideas on time and representation are perhaps not so hard to recognise in Einstein's ideas leading to general relativity; this influence has also been studied. 8 Can it really be a coincidence that, in his younger days, the greatest physicist of the twentieth century should have had such a clear focus on philosophy, 9 reading Kant's three Critics when he was fifteen years old? Why this influence? Because philosophy can provide methods for producing new ideas, novel perspectives, and critical thinking. Philosophers have tools and skills that physics needs, but do not belong to the physicists training: conceptual analysis, attention to ambiguity, accuracy of , 1994, 111-138. expression, the ability to detect gaps in standard arguments, to devise radically new perspectives, to spot conceptual weak points, and to seek out alternative conceptual explanations. Nobody puts this better than Einstein himself: "A knowledge of the historic and philosophical background gives that kind of independence from prejudices of his generation from which most scientists are suffering. This independence created by philosophical insight is-in my opinion-the mark of distinction between a mere artisan or specialist and a real seeker after truth."10 Put more strongly, it is sometimes said that: "Scientists do not do anything unless they first get permission from philosophy."

Thus, if we read what the greatest scientists had to say about the usefulness of philosophy, physicists like Heisenberg, Schrödinger, Bohr, and Einstein, we find that they expressed totally opposite opinions to those of Hawking and Weinberg.

3.

Hers is a second argument due to Aristotle:

Those who deny the utility of philosophy, are doing philosophy.

This point is less trivial than it sounds at first. Let's examine what Weinberg and Hawking write. Both have obtained important scientific results. Weinberg, for instance, found the correct symmetry group to describe the interactions between the elementary particles, while Hawking discovered that black holes are hot, and computed their temperature. In doing this kind of thing, they were doing science. In writing things like "philosophy is useless to physics", or "philosophy is dead", they were not doing physics. So what were they doing? They were reflecting on the best way to develop science.

The issue here is the methodology of science. A central concern in the philosophy of science is of course precisely to ask how science is done and how it could be done to be more effective. Good scientists do reflect on their own methodology, and it is quite appropriate that Weinberg and Hawking have done so too. But how? They express a certain idea about the methodology of science. Is this the eternal truth about how science has always worked and should work? Is it the best understanding of science we have at present? It is neither. In fact, it is not difficult to trace the origins of this idea. It arises from the background of logical positivism, corrected by Popper and Kuhn. The current dominant methodological ideology in theoretical physics derives from their notions of falsifiability and scientific revolution, which are popular among theoretical physicists; they are often mentioned and are commonly used to orient research and evaluate scientific work.

Therefore, in declaring the uselessness of philosophy, Weinberg, Hawking, and other "antiphilosophical" scientists are in fact paying homage to the philosophers of science they have read, or whose ideas they have absorbed from their environment. The imprint is unmistakable. When viewed as an ensemble of pseudo-statements, words that resemble statements but have no proper meaning, of the kind recurrent for instance in the way Neil de Grasse Tyson mocks philosophy, these criticisms are easily traced to the Vienna Circle's anti-metaphysical stance.11 Behind these anathemas against "philosophy", one can almost hear the Vienna Circle's slogan of "no metaphysics!" Thus, when Weinberg and Hawking state that philosophy is useless, they are actually stating their adhesion to a particular philosophy of science. In principle, there's nothing wrong with that; but the problem is that it is not a very good philosophy of science.

On the one hand Newton, Maxwell, Boltzmann, Darwin, Lavoisier, and so many other major scientists worked within a very different methodological perspective, and did pretty good science as well. On the other hand, philosophy of science has advanced since Carnap, Popper, or Kuhn, recognising that the way science effectively works is richer and more subtle than the way it was portrayed in the analysis of these thinkers. Weinberg and Hawking's error is to mistake a particular, historically circumscribed, limited understanding of science for something like the eternal logic of science itself. The weakness of their position is the lack of awareness of its frail historical contingency. They present science as a discipline with an obvious and uncontroversial methodology, as if this had been the same from Bacon to the detection of gravitational waves, or as if it was completely obvious what we should be doing and how we should be doing it when we do science.

The reality is quite different. Science has repeatedly redefined its own understanding of itself, along with its goals, its methods, and its tools. This flexibility has played a major role in its success. Let us consider a few examples from physics and astronomy. In light of Hipparchus and Ptolemy's extraordinarily successful predictive theories, the proper goal of astronomy was to find the right combination of circles to describe the motion of the heavenly bodies around the Earth. But contrary to expectations, it turned out that Earth was itself one of the heavenly bodies. And indeed after Copernicus, the proper goal appeared to be to find the right combination of moving spheres that would reproduce the motion of the planets around the Sun. But once again contrary to expectations, it turned out that abstract elliptical trajectories were better than spheres. Then after Newton, it seemed clear that the aim of physics was to find the forces acting on bodies. Contrary to this program, it turned out that the world could be better described by dynamical fields rather than bodies. After Faraday and Maxwell, it was clear that physics had to find laws of motion in space, as time passes. Contrary to assumptions, it turned out that space and time are themselves dynamical. After Einstein, it finally became clear that physics must only search for the deterministic laws of Nature. Contrary to expectations, it turned out that we can at best give probabilistic laws. And so on. Here are some sliding definitions for what scientists have thought science to be: deduction of general laws from observed phenomena, finding out the ultimate constituents of Nature, accounting for regularities in empirical observations, finding provisional conceptual schemes for making sense of the world. (The last one is the one I like.) Science is not a project with a methodology written in stone, well circumscribed objectives, or a fixed conceptual structure. It is our ever evolving endeavour to better understand the world. In the course of its development, it has repeatedly violated its own rules and its own stated methodological assumptions.

A currently common description of what scientists do -the one we learn today at school-is collecting data (observations, experiments, measurements) and making sense of them in the form of theories. The relationship between data and theory is complex, and far from being uncontroversial, since it it not at all obvious how we get from data to theory, and it is not obvious how are theory-laden are the data themselves . But let's gloss over that. As time goes by, new data are acquired and theories evolve. In this picture scientists are depicted as rational beings who play this game using their intelligence, a specific language, and a well established cultural and conceptual structure.

The problem with this picture is that conceptual structures evolve as well. Science is not simply an increasing body of empirical information we have about the world and a sequence of changing theories. It is also the evolution of our own conceptual structure. It is the continuous search for the best conceptual structure for grasping the world, at a given level of knowledge. And the modification of the conceptual structure needs to be achieved from within our own thinking, rather as a sailor must rebuild his own boat while sailing, to use the beautiful simile of Otto Neurath so often quoted by Quine. 12 This intertwining of learning and conceptual change, this flexibility, and this evolution of methodology and objectives, have developed historically in a constant dialogue between practical science and philosophical reflection. This is a further reason why so much science has been deeply influenced by philosophical reflection. The views of scientists, whether they like it or not, are impregnated with philosophy.

And here we come back to Aristotle: "Philosophy provides guidance how research must be done."

Not because philosophy can offer a final word about the right methodology of science (contrary to the philosophical stance of Weinberg and Hawking). But because philosophers have conceptual tools for addressing the issues raised by this continuous conceptual shift. The scientists that deny the role of philosophy in the advancement of science are those who think they have already found the final methodology, that is, that they have already exhausted and answered all methodological questions. They are consequently less open to the conceptual flexibility needed to go ahead. They are the ones trapped in the ideology of their time.

4.

I think an even stronger case can be made. I believe that one reason for the relative sterility of theoretical physics over the last few decades may well be precisely that the wrong philosophy of science is held dear today by many physicists. Popper and Kuhn, so popular among theoretical physicists, have shed light on important aspects of the way good science works, but their picture of science is incomplete and I suspect that, taken prescriptively and uncritically, their insights have ended up misleading research. Let's see why.

Kuhn's emphasis on discontinuity and incommensurability has misled many theoretical and experimental physicists into disvaluing the formidable cumulative aspects of scientific knowledge. Popper's emphasis on falsifiability, originally a demarcation criterion, has been flatly misinterpreted as an evaluation criterion. The combination of the two has given rise to disastrous methodological confusion: the idea that past knowledge is irrelevant when searching for new theories, that all unproven ideas are equally interesting and all unmeasured effects are equally likely to occur, and that the work of a theoretician consists in pulling arbitrary possibilities out of the blue and developing them, since anything that has not yet been falsified might in fact be right. This is the current "why not?" ideology: any new idea deserves to be studied, just because it has not yet been falsified; any idea is equally probable, because a step further ahead on the knowledge trail there may be a Kuhnian discontinuity that was not predictable on the basis of past knowledge; any experiment is equally interesting, provided it tests something as yet untested.

I think that this methodological philosophy has given rise to mountains of useless theoretical work in physics and many useless experimental investments.

Arbitrary jumps in the unbounded space of possibilities have never been an effective way to do science. The reason is twofold: first, there are too many possibilities, and the probability of stumbling on a good one by pure chance is negligible; but more importantly, nature always surprises us and we, the limited critters that we are, are far less creative and imaginative than we may think. When we consider ourselves to be "speculating widely", we are mostly playing out rearrangements of old tunes: true novelty that works is not something we can just find by guesswork.

The most radical conceptual shifts and the most unconventional ideas that have actually worked have in fact always been strictly motivated, almost forced, either by the overwhelming weight of new data, or by a well-informed analysis of the internal contradictions within existing, successful theories. Science works through continuity, not discontinuity.

Examples of the first case -novelty forced by data-are Kepler's ellipses, and quantum theory. Kepler did not just "come out with the idea" of ellipses: nature had to splash ellipses on his face before he could see them. He was using ellipses as an approximation for the deferent-epicycle motion of Mars and was astonished to find that the approximation worked better than his model. 13 Similarly, atomic physicists of the early twentieth century struggled long and hard against the idea of discontinuities in the basic laws, doing everything they could to avoid accepting the clear message from spectroscopy, that is, that there was actually discontinuity in the very heart of mechanics. In both instances, the important new idea was forced by data.

Examples of the second case -radical novelty from old theories-are the heliocentric system and general relativity. Neither Copernicus nor Einstein relied significantly on new data. But neither did their ideas come out of the blue. They both started from an insightful analysis of successful wellestablished theories: Ptolemaic astronomy, Newtonian gravity, and special relativity. The contradictions and unexplained coincidences they found in these would open the way to a new conceptualisation.

In any case, it is not merely fishing out unfalsified theories and testing them that brings results. Rather, it is a sophisticated use of induction, building upon a vast and ever growing accumulation of empirical and theoretical knowledge, that provides the hints we need to move ahead. It is by focusing on empirically successful insights that we move ahead. Einstein's "relativity" was not a "new idea": it was Einstein's realisation of the extensive validity of Galilean relativity. There was no discontinuity: in fact it was continuity at its best. It was Einstein's insightful "conservatism" in the face of those who were too ready to discard the relativity of velocity just because of Maxwell's equations.

I think this lesson is missed by much contemporary theoretical physics, where plenty of research directions are too quick to discard what we have already found out about Nature.

Ironically, indeed, the recent momentous steps taken by experimental physics are all rebuttals of today's freely speculative attitude toward theoretical physics. Three major empirical results have marked recent fundamental physics: gravitational waves, the Higgs, and the absence of supersymmetry at LHC. All three are confirmations of old physics and disconfirmations of widespread speculation. In all three cases, Nature is telling us: do not speculate so freely. So let's look more closely at these examples.

The detection of gravitational waves, rewarded by the last Nobel Prize in fundamental physics, has been a radical confirmation of century-old general relativity. But it is more than this. The recent nearly simultaneous detection of gravitational and electromagnetic signals from the merging of two neutron stars (the event called GW170817) has improved our knowledge of the ratio between the speeds of propagation of gravity and electromagnetism by something like 14 orders of magnitudes in a single stroke. 14 One consequence of this momentous increase in our empirical knowledge has been to rule out a great many theories put forward as alternatives to general relativity, ideas that have been studied by a large community of theoreticians over the last few decades, confirming instead the century-old general relativity as the best theory of gravity available at present.

The well publicised detection of the Higgs particle at CERN has confirmed the Standard Model of particle physics (founded by Steven Weinberg, among others) as the best current theory for high energy physics, against scores of later alternatives that have long been receiving much attention.

CERN's emphasis on the discovery of the Higgs when the Large Hadron Collider became operational in Geneva has also served to hide the true surprise that emerged from this particular exploration of high energy physics: the absence of supersymmetric particles where a generation of theoretical physicists had been expecting to find them. Despite the rivers of ink and flights of fancy, the minimal supersymmetric standard model suddenly finds itself in deep trouble. So once again, Nature has seriously rebuffed the free speculations of a large community of theoretical physicists who ended up believing them.

I think that Nature's repeated snub of the current methodology in theoretical physics should encourage a certain humility, rather than arrogance, in our philosophical attitude. I suspect that part of the problem is precisely that the dominant ideas of Popper and Kuhn have misled current theoretical investigations. Physicists have been too casual in dismissing the insights of successful established theories. Misled by Kuhn's insistence on incommensurability across scientific revolutions, they fail to build on what we already know, which is how science has always moved forward. A good example of this is the disregard for general relativity's background independence in many attempts to incorporate gravity into the rest of fundamental physics. Furthermore, the emphasis on falsifiability has made physicists blind to a fundamental aspect of scientific knowledge: the fact that credibility has degrees and that reliability can be extremely high, even when it is not absolute certainty. This has a doubly negative effect: considering the insights of successful theories as irrelevant for progress in science (because "they could be falsified tomorrow"), and failing to see that a given investigation may have little plausibility even if it has not yet been falsified.

The scientific enterprise is founded on degrees of credibility, which are constantly updated on the basis of new data or new theoretical developments. Recent attention to Bayesian accounts of confirmation in science is common in the philosophy of science, but largely ignored in the theoretical physics community, with negative effects, in my opinion. 15 The worst episode of this misunderstanding is the confusion between the the (strong) commonsense notion of `confirmation' and the (weak) Bayesian notion of `confirmation' that has driven the controversy over Richard Dawid's work on non-empirical confirmation [R. Dawid, String Theory and What I intend here is not a criticism of Popper and Kuhn, whose writings are articulate and insightful. What I am pointing out is that a simple-minded version of their outlooks has been taken too casually by many physicists as the ultimate word on the methodology of science.

Far from being immune from philosophy, current physics is deeply affected by philosophy. But the lack of philosophical awareness needed to recognise this influence, and the refusal to listen to philosophers who try to make amends for it, is a source of weakness for physics.

5.

Here is one last argument from the Protrepticus: "More in need of philosophy are the sciences "where perplexities are greater".

Today fundamental physics is in a phase of deep conceptual change, because of the success of general relativity and quantum mechanics and the open "crisis" (in the sense of Kuhn, I would rather say "opportunity") generated by the current lack of an accepted quantum theory of gravity. This is why some scientists, including myself, working as I do on quantum gravity, are more acutely aware of the importance of philosophy for physics.

Here is a list of topics currently discussed in theoretical physics: What is space? What is time? What is the "present"? Is the world deterministic? Do we need to take the observer into account to describe nature? Is physics better formulated in terms of a "reality" or in terms of "what we observe", or is there a third option? What is the quantum wave function? What exactly does "emergence" mean? Does a theory of the totality of the universe make sense? Does it make sense to think that physical laws themselves might evolve?" It is clear to me that input from past and current philosophical thinking cannot be disregarded in addressing these topics.

In loop quantum gravity, my own technical area, Newtonian space and time are reinterpreted as a manifestation of something which is granular, probabilistic, and fluctuating in a quantum sense. Space, time, particles, and fields get fused into a single entity: a quantum field that does not live in space or time. The variables of this field acquire definiteness only in interactions between subsystems. The fundamental equations of the theory have no explicit space or time variables. Geometry appears only in approximations. Objects exist within approximations. Realism is tempered by a strong dose of relationalism. I think we physicists need to discuss with philosophers, because I think we need help in making sense of all this.

6.

To close, I would like to add a brief word on the opposite issue: the relevance of science for philosophy.

I do so only because some manifestations of anti-philosophical attitudes in scientific circles are only a reaction to anti-scientific attitudes in some areas of philosophy and other humanities.

the Scientific Method (Cambridge University Press, 2013).] An attempt to study the actual source of (possibly unjustified) confidence in a theory has been re-trumpeted by scientists as a proof of validity.

In the post-Heideggerian atmosphere that dominates some philosophy departments on the 'continent', ignorance of science is something to exhibit with pride. Science is not "authentic" knowledge; it misses true knowledge. "The botanist's plants are not the flowers of the hedgerow; the 'source' which the geographer establishes for a river is not the 'springhead in the dale'"16 implying in the context that the only thing that counts is the latter. And here is an example from another sector of the today's intellectual world, sociology: "…there is no obligation upon anyone framing a view of the world to take account of what twentieth-century science has to say."17 A comment which is either trivial ("there is no obligation to be intelligent") or misleading, in the etymological sense of leading in the wrong direction.

It seems clear to me that, just as the best science listens keenly to philosophy, so the best philosophy will listen keenly to science. This has certainly been so in the past: from Aristotle and Plato, to Descartes and Hume, Kant and Hegel, Husserl and Lewis, the best philosophy has always been closely tuned in to science. No great philosopher of the past would ever have thought for a moment of not taking seriously the knowledge of the world offered by the science of their times.

Science is an integral and essential part of our culture. It is far from being capable of answering all the questions we would like to ask, but it is nevertheless an extremely powerful tool, able to address innumerable problems, including those that concern ourselves and the **Universe as a whole**. Our general knowledge is the result of the contributions from vastly different domains, from science to philosophy, all the way to literature and the arts, and our capacity to integrate them. Those philosophers that discount science, and there are many of them, do a serious disservice to intelligence and civilisation, in my opinion. When they say that entire fields of knowledge are impermeable to science, and that they are the ones who know better, they remind me of two little old men on a park bench: "Aaaah," says one, his voice shaking, "all these scientists who claim they can study consciousness, or the beginning of the universe." "Ohh," says the other, "how absurd! Of course they can't understand these things. But we do!"

6C.

  Rovelli, "Aristotle's Physics: A Physicist's look", Journal of the American Philosophical Association, 1 (2015) 23-40, arXiv:1312.4057. 7 W. Heisenberg, "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen," Zeitschrift fur Physik 33 (1925) no. 1, 879-893. 8 D. Howard "A Peek behind the Veil of Maya: Einstein, Schopenhauer, and the Historical Background of the Conception of Space as a Ground for the Individuation of Physical Systems." In The Cosmos of Science: Essays of Exploration. John Earman and John D. Norton, eds. Pittsburgh-Konstanz Series in the Philosophy and History of Science, vol. 6. (Pittsburgh: University of Pittsburgh Press 1997) Konstanz: Universitätsverlag, 87-150. 9 D. Howard, "'A kind of vessel in which the struggle for eternal truth is played out'-Albert Einstein and the Role of Personality in Science." In The Natural History of Paradigms: Science and the Process of Intellectual Evolution. John H. Langdon and Mary E. McGann, eds. Indianapolis: University of Indianapolis Press

Steven Weinberg, Dreams of a Final Theory, Chapter VII(Vintage, 1994)

Stephen Hawking, The Grand Design(Bantam, 2012)

In https://www.youtube.com/watch?v=ltbADstPdek, at time 1:03.

A. Einstein. Letter to Robert A. Thornton, 7 December 1944. EA 61-574, in The Collected Papers of Albert Einstein (Princeton, NJ: Princeton University Press, 1986-present).

R. Carnap, "Überwindung der Metaphysik durch Logische Analyse der Sprache" in Erkenntnis, vol. 2, 1932 (English translation 'The Elimination of Metaphysics Through Logical Analysis of Language' in Sarkar, Sahotra, ed., Logical empiricism at its peak:Schlick, Carnap, and Neurath, New York : Garland Pub., 1996, pp. 10-31).

W. V. O. Quine, Word and Object. (Cambridge, Mass.: MIT Press, 2015).

Johannes Kepler, Astronomia Nova, translated by William H. Donahue, (Cambridge: Cambridge Univ. Pr., 1992).

Martin Heidegger, Sein und Zeit (Niemeyer, Tübingen 1967).

Andrew Pickering, Constructing quarks: a sociological history of particle physics. (Chicago, Illinois Chichester: University of Chicago Press Wiley, 1999).