Interpretation & Reconstruction of Quantum Mechanics (contd.)

0:00 Thank you. Thank you. Thank you. I will talk about motivating for them. I will do it. Sorry. Now, there's a sufficiency. So, he's here. Yes, then you want to get a chair away. Oh, yes, it's... You need this one. Oh, good. Yes, that's much better. Yes, that's much better. OK, I'll move it up higher, at least. Let me first mention I have a partner in this work, a student, who did much of the detailed work that I wanted to talk about.

2:30 And here's a sort of outline. I will talk about the valley policies, obviously, and we'll get a short review of the continuing debate about what's really at stake in the violation of valid equalities, whether it's just the question of locality or whether there's something more assumed in the derivation of valid equality, maybe, of course, a substantive assumption about realism. And secondly, I will talk about the usual derivations for developing policies that rely on two main assumptions which Abdel and Shimoni call parameter independence and outcome independence. And I will focus particularly on outcome independence. And I will try to tell you in the purpose of this talk, that there is a motivation for outcome independence that's somewhat different from the usual approach in the literature. In particular, I would like to argue that it has a striking resemblance to a condition known in mathematical statistics or statistical inference, namely condition of sufficiency, And then one could give exactly the same motivation as one does there to outcome independence. And then, well, I will get to the main conclusions involving that exercise. So first, I guess we all know a typical PPR experiment, I've just sketched the basics of it here. There are two particles that emerge from a source and they travel to the left and the right and on both wings of the experiment one can do either a measurement of a quantity A or A prime

5:00 which is decided by the orientation alpha of a particular measurement device, Sternkeller, Apparatus, something like that. And the outcomes of the measurements will be denoted by small letters. And the same thing goes on in the right wing, except we use Bs and Betas to denote their quantities, parameters, and outcomes. Bell famously derived a bound on the correlations you can get for the measurements on these quantities, smaller than two, and yet quantum mechanics allow states, singlet states, such that for an appropriate choice of these measurements, these observables, you can get to a much larger value. And so Bell's inequalities are violated in quantum mechanics, and therefore the assumptions that are needed to derive the Bell's inequalities, one of them must surely be false. So what are the assumptions that go to a derivation of the Bell's inequality? Well, I'll focus on the two main culprits here, What I usually thought was the received view is that in order to derive the valid equality, you need both locality and a very substantial realistic subject, for example, the existence of hidden variables with particular properties. that's, for example, it's founded in Michael Redhead's excellent book, and as I said, I thought that was to receive you, but it seems that if you go to the recent literature, there are many voices that argue that it's really just about locality. It's a few that you actually find quite often in physics literature,

7:30 And in recent papers on quantum information, people easily say that value qualities apply something like quantum non-locality. And of course, physicists are known to be sloppy. But if you also find things like that in the philosophical literature, then it seems that there is something going on. And I have two more notes at the bottom of this slide. First of all, of course, apart from these two assumptions, there are further assumptions that go into the derivation. I'm just not highlighting them because they seem to be much less controversial. And secondly, there's also much talk talk about the experimental loopholes involved in experiments for testing Bellium qualities. And that's an issue that I will not touch on either because the experiment, the question whether nature experiments has actually tested Bellium qualities well enough to yield a definitive answer. but that's not a question that's relevant to my talk either. I'm just talking about the comparison between the theories, quantum mechanics on one hand, and the theories that allow the derivation of relative qualities on the other hand. So, I could give you some quotes to show that there's really this school in the literature that's in opposition to the received view. But let me just, for the moment, focus on the innovation of the Bell inequalities. And the usual way of doing that is to assume that there is a hidden variable lambda such that all the experimentally testable probabilities for outcomes a and b can be written in this this formula just an integral over

10:00 the conditional probability conditional lambda and And here are the three main assumptions. The distribution of lambda should be independent of the measurement settings, choices of the directions alpha and beta on the faraway side. That's called source independence. And then there is parameter and output independence, also in the usual way. the parameter independence says that if you just look at the marginal probability, the conditional probability for A, that should not depend on the choice of the parameter on the far away other particle, beta. And similarly, the conditional probability for B should not depend on the setting of the parameter alpha. Outcome independence, on the other hand, does not talk about the parameters, alpha and beta. It says that if you conditionalize not just on the hidden variable lambda, but also on the outcome B, then actually the conditional probability for A is going to be the same if you just conditionalize on the value of lambda. And from these assumptions, it's rather straightforward to derive the value and qualities. The motivations for the first two items on this list are also usually rather simple. To motivate source independence, people usually appeal to the idea that the experimenter has a free choice to choose his settings of alpha and beta independently of the state in which the particles were created. So even if they've already left the source, an experimenter, at least in principle, might have a choice to shift, change the values of alpha and beta, and that should not affect the distribution of the invariables. And similarly for parameter independence,

12:30 there's an appealing and well-known motivation. If this were not the case, if the conditional, if the marginal probability of A would depend on beta given the value of the variable, then in principle, this theory would allow instantaneous signaling of arbitrarily long distances because the second experimenter could by setting his choice of beta influenced the probabilities of A on the other side and that's something we would not want in our hidden variable view. So assumption two has a clear indefinite connection with locality and that's why locality is certainly at stake in the derivation of development. But the focus of my talk will be on the third assumption, outgoing things. What's the motivation for that? Well, originally, the first person to split up the assumptions for Bell inequalities in exactly this way was Garrett, and he called outcome independence completeness, and he We argue to motivate this assumption that if lambda provides a complete specification of the physical state of the pair of particles, then the additional knowledge of outcome B would provide redundant information about particle one. And so it would be enough to provide lambda and then the extra addition of the outcome B B would simply mean not provide any extra information at all, and that's why the conditional probability should not change if you provide B in addition to them. Shimoni, in a later paper, adopted Gerrit's split into two assumptions, but he proposed the present names of parameter independence in order to be more useful, as he claimed. However, if you look closer at this paper,

15:00 you will see that when he starts to try to give the motivation for outcome independence, he argues that it also some form of locality, just like parameter independence. And many later authors have thought the motivation for outcome independence, either in Reisenbach's principle of the common cause, or maybe of the common cause. We all heard an excellent talk about it yesterday by Adrian Guthrie, or other favorites are the principle of stochastic Einstein locality. And it seems that in all cases, authors who draw attention to these principles also see them as locality conditions. And it looks like today there are very few people who are still willing to follow Jarrett in his original motivation to think of outcome independence as an expression of completeness, in some sense. And in fact, Travis Norson has rather strongly condemned Jarrett's approach and said that it was all based on fundamental confusion. And so it looks like maybe perhaps the majority of authors of outcome independence, just like parameter independence, as some expression of locality. I want to take sides with Jared on this occasion. And I think that there is a veritable and honorable tradition in mathematical statistics that backs up this interpretation of outcome independence. and unfortunately, it has not been noticed very much in the literature on bell inequalities, and if there's any purpose of my talk, it is simply to draw attention to this tradition in mathematical statistics. The idea of sufficiency is very old already.

17:30 It is proposed in the early papers of Ronald Fisher of estimation. And I doubt that perhaps the idea is even older. There are people who claim that Fisher's teacher, a man called Johnston, who was also the teacher of Keynes and Harold Jeffries and the whole generation of Cambridge probabilists, is really the originator of the idea. I will give you a very short definition of the condition of sufficiency, although there are many equivalent definitions around. This one is from Savage's book on the foundations of statistics. We assume that there is some set of random variables, theta y1 to yn, and they're all subject to a given probability measure, and then we will call y1 sufficient for theta out of the collection of all the y's, just in case if this conditional probability for theta given the whole collection of y's is the same as the conditional probability of theta if you just give the total of y1. So again, in this case, knowledge of all the other values of y, y2, yn, would provide redundant information about theta. If you know y1 alone, that is sufficient to capture through all the relevant information about theta, and that's why it's called sufficient. Perhaps to give you a flavor of the usefulness of this idea, let me give you a simple example, perhaps the most famous example as well. Suppose I'm dealing with drawings from a random, from a normal probability distribution. given like this with an unknown mean value, theta. And of course, what statisticians do in that case is they perform many independent repeatless trials

20:00 from the same distribution. So I get a distribution of a collection of values, x1 to xn, and you can write down the product of the normal distribution in various ways. The curious point to note, of course, is that it splits into a product, one that depends on theta and only the average of the outcomes, and another factor that depends on all the data, but not on theta anymore. And in fact, that's another equivalent criterion for sufficiency if you can write a joint probability distribution in that form. It means that if you were to transform the outcomes x1 to xn by providing an independent functions of x, so there was another way of, you know, presenting the same data in terms of the y's instead of the x, and you choose the first function to be just the average of all the data. Then all the other data are completely irrelevant for your statistical inference about theta. That is to say, and this follows immediately, and that is to say the average outcome is sufficient in order to estimate the unknown mean. And all the The other information that's encapsulated in the data, there might be 100,000 drawings or so, is irrelevant for that particular purpose. It doesn't give you any relevant information simply because their distribution is not dependent on theta. That's another equivalent formulation of solutions. But all this discussion arises without mentioning locality at all, this could just be about the weight of tomatoes in a genetic experiment. That's what I'm saying, okay.

22:30 Okay. Well, I think I can be quick, and you will probably see that the point I'm trying to make. Outcome independence can likewise be interpreted as an expression of sufficiency. It says that, assuming that the directions alpha and beta are fixed, then lambda is supposed to be sufficient for any inference about outcome A in the sense that if you give the value of B in addition to lambda, then you are given no further relevant information for the prediction of A. Lambda is just sufficient for A. And this, of course, is exactly basically what I think Jarrett had in mind, it's just perhaps that he had chosen a different name to complete this instead of sufficiency, which he could have chosen and perhaps should have chosen, not only because it has this longer tradition, but also because I think it's a more modest term instead of complete this. So, here are my conclusions. Again, I think we can back up Jared's interpretation of how to mean independence by an appeal to the concept of efficiency in mathematical statistics, and that provides an alternative explanation, an alternative cue that does not need to appeal to Reichenbach's principle of common cause, or to the principles of Sylvester-Einstein locality. And again, it is silent about, it's mutual with respect to locality. Parameter independence, of course, has the motivation in terms of locality, but outcome independence is supposed to say something about how detailed the state description of lambda is going to be. It has to provide all the relevant information about the pair of particles in such a way that extra knowledge of the outcomes is no longer relevant for the probabilities.

25:00 And to come back then to the larger debate in the literature, of course this would mean that the failure of the bell inequalities in experiments if there is such a thing would not necessarily have to be due to a failure of locality when there is still another viable option and that is to say that maybe nature is local but somehow a sufficient is in exactly the same sense I've discussed is excluded by . But it's not necessarily locality . And that's the end of my talk. We have five, six minutes for discussion. Just to remark that there is not a big discrepancy between right and back's common cause, principle, and peace. In terms of efficiency of knowledge, because if you go to the right and back book, and especially to his examples providing the motivation for the principle, who will learn that it's exactly that kind of talk. You have a piece of knowledge and another piece of knowledge is not relevant. So this piece of knowledge is sufficient. I accept that, but I think there are important differences in flavor and in mathematical detail. For example, to mention just an obvious point, Reichenbach never discusses more than correlations or influences between two evils. In statistics, you can talk about 100,000 variables, you see. And secondly, and perhaps more importantly, in, for example, this statistical example I just presented with a normal distribution. It's not usual or even effective to think of an average value

27:30 as a cause for the distribution of 4 theta or something. So the cause of causality is causation is also a remove from those examples. And that's also contrary to the right of our intentions. OK, various questions. I think Michael was first. Just a short question. How is this statistical sufficiency related to typicality? I suppose that it's stronger, or is it? Typicality, as far as I know, is just not a word for probability in many senses. But sufficiency is a mathematical condition on certain statistical functions that might arise and that might not arise. In this example of the normal distribution, there is a sufficient estimator, as it's often called. But in many other distributions, there is no sufficient estimator. So it's a contingent property of certain statistical functions. And it's nice if you can find it. Unfortunately, very often it's not there. So dependent on the statistical function and of course the probability model you're working with, there might be such a sufficient function or there might not be. The typicality is something different at all. So the various questions, we are running short of time, so please be brief, Mauricio. Yeah. So very briefly, do you have any, can you think of any non-sufficient hidden variable theory? Because the hidden variable theories that we know distract me as sufficient. I mean, for instance, Bohmian mechanics obeys outcomes, independence, validates parameter independence, so that's a case of a non-local sufficient hidden variable theory. and it would be nice to have an example of a worked out here in variable theory that is not sufficient but locally to sense of self the term structure short answer, good question I will try to think yeah, I just have a quick remark

30:00 on one hand I'm sympathetic with this interpretation, on the other hand unlike in the mathematical example where you have, you know y2, etc., etc. From the point of view of physics, we of course don't treat lambda and B as homogeneous as being on the same level. So the reason to me why we're trying to formulate things in terms of B and not lambda is that in experiments we're interested in B and no one can measure lambdas. So there is a psychological reason I think why people prefer interpretations not in terms of lambdas, but in terms of what the real measured values are? Of course, they would prefer that if they can't get it. But the alternatives to the sufficiency... I mean, outcome independence is formulated in terms of lambda and B. There's no way I can't... And alternative interpretations of that condition, in terms of, say, Einstein locality, also have to deal with lambda. And so they don't succeed in removing lambda from the picture and just talking about the experimentally accessible probability. Yeah, but I'm saying as little as possible in the interpretation. I don't think that the alternative interpretations of outcome independence can do that. This is really interesting, but it makes something back up the question of why we would expect efficiency to hold here. And in particular, one could think that there are good reasons to expect efficiency to hold. They do have to do with locality, still. Because land is supposed to be a complete description or a sufficient description. And why should it be? characterizes the particles themselves, and in space-like separation, the particles themselves are moving within the backward-like form, or lower-like backward-like form. And B is outside of them. That's why B shouldn't be. Well, I tried not to say that the interpretations of outcome independence that do appeal to

32:30 that excluded or didn't mark me against it. I'm just providing an alternative point of view which I did not need to do that. But if you push me, I'm willing to go further and say, well, suppose that Lambert were not a complete explanation. So say we have, perhaps we do have a particular, Maybe that's also an example, but perhaps there is a specification. For example, we just have a disjunction. It's either lambda 1 or lambda 2. That's also a specification, but it's not sufficient to justify completely. In that case, an observation of the outcome B, even though it's outside, it's in the future space separately separated from the original production, it could still provide relevant information because it would, retrospectively, tell us about whether it's Lambda-1 or Lambda-2. And then it would still yield relevant information. So, I don't think locality can be adduced on its own to justify this particular outcome independence if the specification were not complete to begin with. Okay, so thank you very much again here. Thank you. Thank you.

35:00 Thank you very much. You see, this is a paper with many voters. Now, a theoretic part from Patton's theory to music might appear at first sight somewhat What is a natural bridge that connects these two powers, a holistic semantics that uses the characteristic holistic features of the quantum-theoretic form? What is the general logical thing? The general logical thing is a new form of quantum logic, call it quantum computational logic, that has been recently suggested by the theory of logical gains in quantum computation. That means sketch here only the basic intuitive idea of this new form of quantum logic. In this framework, formulas are interpreted as quantum information quantities, while the logical connectives are interpreted as quantum-logical gains that we briefly say gain. What is a gain? Gains are reversible dynamic logical operations, mathematically represented by unique operators in convenient inverse cases. So, for an intuitive point of view, a sequence of gates describes the time evolution of a quantum object, a quantum system, a quantum object that stores the given quantum information quantity. And what is a quantum information quantity? An atomic piece of quantum information is represented by a qubit. now the notion of qubit is very well known, qubit can be found as a quantum variant of the classical notion of bits. So generally a quantum superposition of

37:30 the two classical bits, zero and one. Mathematically it is well known that qubit is represented as a unit factor of the two-dimensional universe AC2, so generally it is as a superposition form. From the physical point of view, the qubit represents a possible pure state of a single particle, say a photon. How to represent a system of F qubits, also called Q register, the natural answer is as a pure state of a compound system consisting of n particles. So the running idea is to use the quantum theory formalism that is normally used to represent compound physical systems. So the n, the tensor product formalism, so we denote in this way the n-fold tensor product of C2, and I call simply n space, Hn, a space for a system of n particles, so the n-fold tensor product of C2. As is when known, pure states represent maximal pieces of information that cannot be consistently extended to a richer knowledge. However, in quantum information, one cannot help referring also to non-maximal pieces of quantum information that are represented by mixtures of Q-registers that are also shortly called in this framework Q-mixes. So we have Q-registers and Q-mixes. Now, our logic, quantum computational logic. The language of this logic is a sentential logic, so we will have atomic formulas with the privileged formula F representing the falsity, and then some logical connectives which correspond to some basic quantum logical gates. So we suppose we have at least three connectives, which correspond to three very important k's.

40:00 The k's not, the square root of not, and t, the top for k's, what is this t? T is a very important A that permits us to define a reversible conjunction. So the linguistic rule that reflects in this language the behavior of the top point A that is reversible, a unique operator, is the following. For any formulas, alpha and beta of the language, the ternary conjunction of alpha, beta, and the falsity is a formula. And here the falsity is dealt with as a kind of concealer. This point of note is a characteristic quantum computational gain, which has no counterpart either in classical semantics or in falsely semantics. and the basic property of this gate is of course the property that is the cause of the name applied twice the square root of the negation means negative and these gates can be generalized also to q mixes and what is interesting here is the following any formula alpha of this language describes a quantum circuit because the syntactical form of a formula alpha determines a sequence of gates and this is a quantum information process. Let me first define the notion of atomic complexity of the formula. This is simply the number of occurrences of atomic formulas in For instance, this formula that represents in our language the non-contradiction principle, this formula has atomic complexity 3 because we have 3 co-verses of atomic synthesis. So from the semantic point of view, the atomic complexity of the formula alpha is important

42:30 because it determines the dimension of the Hilbert space where a Q-register representing information about our formula alpha should live. So if our alpha has atomic complexity T, we call the Hilbert space T the semantic space of our formula alpha. Now, any formula can be naturally decomposed into its parts, giving rise to a special syntactical configuration called the syntactical tree. For instance, the idea is very simple. In the case of our non-contradiction principle, this is the syntactical tree of our formula, we suddenly drop, step by step, all the connectives of our formula. So we obtain, as you see, a sequence of levels. Each level sees a sequence of sub-formulas of our formula alpha. The bottom level is alpha itself. And the top level is the sequence of atomic appearances in alpha. Now, the syntactical tree of formula alpha is a pure syntactical object, but it determines a sequence of gates where each gate in this sequence is defined on the same space, the semantic space of alpha. And we call the sequence that is obtained in this way the qubit tree of our alpha. Natural, for instance, in the case of our non-contradictional principle, this was the syntactical three, and in the expected way, we obtain these sequels of three gates. We take the operators, the unitary operators that correspond to our connectives. This is the qubit of our formula. And this procedure can be generalized also to QNIXs. Now, the basic notion of our semantics is, of course, the notion of model, like in any semantics.

45:00 What is a model of the quantum computational language, a holistic model of our language? It is a Mach Hall that assigns to any formula alpha a meaning represented by a Q mix, density operator generally, living in the semantic space of alpha. And this Meck Hall works following the prescription of the syntactical tree of alpha and assigns a cumix to each level of the syntactical tree. I have no time to give the formal precise definition, this is only the intuitive idea. So generally, we will have a level of our syntactically, a sequence of some formulas of our formula alpha, a whole, a size, a q-mix, a meaning to this particular level. So any density operator rho i in this sequence represents a global information about the corresponding level. This might be also an intended statement, for instance. How to determine information about the parts of each level, about the sub-formulas in this sequence? And now it is, of course, very natural to apply the quantum theoretic formalism, which teaches us how to go from the whole to the whole. That is the notion of reduced state, and you can easily imagine how, because if Rodic that's the state of the compound system we know that all the term is the states of the two parts red one of rho red two of rho this is the normal quantum theoretic formalism that we can apply here in a very natural way the framework of our semantics and we obtain here and also in the semantics and typically holistic behavior, because it may happen that role is a pure state, the state of the compound system is a pure state, maximum information. Both the reduced states, the states of the parts, are proper mixtures.

47:30 So our information about the whole is more precise than our information about the parts. And the procedure cannot be inverted. Going from the parts to the whole gives rise to a different state. So, using the notion of reduced state, we can naturally define the contextual meaning with respect to the context, whole alpha, of any occurrence in alpha of a subformula beta. A whole alpha is the holistic meaning of a formula up of the language. Generally, a formula has different contextual meanings in different contexts. So we have a form not only of holistic semantics, but also of contextual semantics. And in this framework, one can then naturally define of logical consequence, so the quantum computational logic is, by definition, the logic characterized by the holistic quantum computational semantics. And we may have also an abstract version of the holistic quantum computational semantics, which is human space independent, so a true form of an abstract semantics. Now how to apply these ideas in order to obtain an abstract semantics for musical compositions? It is well known, the syntactical component of a musical composition is represented by a score. The question is, is it possible and interesting to look for a formal representation of musical scores. Are scores in a sense formalizable? And the answer to this question is yes.

50:00 Scores can be represented in a mathematical form. The basic idea will be the following. Any measure in the score can be mathematically represented as a matrix-like structure. in the expected way, with rows, columns, and so on. So the whole score can be represented as a sequence of matrices. And in this case, we have two kinds of syntactical combinations, vertical and horizontal combinations. And the vertical combinations represent, in a sense, the simpactical counterpart for the important phenomenon that is represented by musical polyphony, which is probably connected with the deep parallel structures of human brain. According to a happy metaphor of Antonio Damasio, human brain works like not established. In a score we have musical phrases which behave like well-formed expressions of a formal language, and they can be identified with special fragments of a score. And how to identify a possible meaning for a musical phrase occurring in an even score. As is well known, the sound word is a typically relational word. We cannot associate a well-determined meaning to a single note or to a single sound. Single notes are, in a sense, all semantically equivalent. And the meaning of a single note, of a chord, or a musical phrase is always determined by the context, so music requires a holistic and contextual semantics. So recalling what happens in the phase of our quantum holistic semantics, we can use the same kind of concepts and define a basic notion of musical interpretation of a score.

52:30 So musical interpretation of a score can be described as a whole that assigns to any syntactical musical phrase alpha, a meaning representing the semantic musical phrases. And what is a semantic musical phrase? Semantic musical phrases can be regarded as a special kind of intentional objects which reflect the linguistic form of the corresponding syntactical phrases. And so we can use here an abstract notion of reduced state that permits us to define contextual meanings. This kind of semantics can be used to analyze formally a number of important musical concepts. For instance, a crucial relation in music, the relation between a given text and even musical realization of that text. It is well known that music transforms a given text, for instance a poem, into a completely new global semantic object. There are many examples that confirm this thesis. I suggest you an interesting musical but also an interesting formal experience. leaders from greatest villainmeisters that have been set to music by different composers, for instance, in particular, Schubert and Schumann, and a very famous lead, and so on and so on, do you know the land where the lemon trees blossom, and so on and so on. If you look to the musical realisation of Schubert and of Schumann, they are completely different. Both leaders are very, very dramatic, but in Schubert there is also some somewhat consoling, for instance the Inchipit is almost a lullaby, it's a kind of lullaby, there is something consoling there.

55:00 On the contrary, Schumann is terribly dramatic, anxious, nervous, and from the musical point of view there are a lot of dissonant chords. Is it reasonable to ask which musical realization is more faithful with respect to Goethe's poem? No, it has no sense. In a sense, we can say that each musical realization determines a new poem. Now, yes, I am finishing. One is dealing with a semantic relation that can be analyzed according to the rules of our abster musical semantics. We think that the discovery of structural similarities in the semantics of physics and in the semantics of artistic languages might interestingly fill a gap between science and humanities and this is our very optimistic group. Thank you. Thank you. Well, thank you for this. Question? Comments? Just a very good discussion. The pool, the math pool, is defined to be taking these values in the set of density. That's part of the stipulation. Do we have completeness and soundness results with respect to that? In the abstract semantics, yes. In the Hilbert space semantics, no, because it's also for normal, for the historical quantum logic that we know very well. There is no completeness. We have a well-known completeness theorem for abstract quantum logic, but not for the Yes, Hilbert's day's logics are always terribly difficult from the logical point of view in order to obtain logical completeness results. We have still a lot of open problems in this direction.

57:30 I didn't exactly get you thought of a musical score as every measure was representable in the matrix. Could you repeat what is on the rows and the covers? It's very simple, perhaps too naive, you see. This is the idea. This is an instrument, for instance the first violin, and this row corresponds to what should be played by the and so on and so on. What are representing columns here, they are representing what should be played at the same time. This is the basic structure of a score which is a typically bidimensional syntactical object. On the contrary, we know very well that, with the exception of Frege, our formal languages are normally one-dimensional objects, and a score is typically a two-dimensional object, and this immediately reflects, at the syntactical level, the semantics of music. What has in mind a conductor during his activities? I think a conductor simply sees columns moving in time. This is my projector, I didn't know. So is there an analogy between the dimension of Hilbert space and the number of instruments Yes, yes, the syntactical complex, I have only an abstract semantics for music, not Hilbert space semantics. Anyway, also in the abstract semantics, there is something that corresponds to the dimensions of the corresponding Hilbert spaces in the case of the complete semantics.

1:00:00 And here, of course, the number of instruments, the number of notes, the number of the notes in a chord and so on, determine the complexity of the abstract space where all these musical notions in an intentional sense should be. Why do we need a semantics for music? You know there are many, many opinions in this field, there are many people who think and I said that we should simply listen to music and not advance. Anyway, in the recent times, there are a lot of investigations about the relation between music and science, music and mathematics, and so on. It seems to me that it is interesting to investigate the structure of music according to rational and scientific natures. Perhaps we don't understand music better from the point of view of our feelings and so on, but from the point of view of our understanding, yes. It's important to understand that there are some similar structures in our rational activity and in our artistic activities. What I would mean is that some 40 years ago there was an attempt to give, for instance, and generative grammar for music, and all these attempts were at least failures, so that's why I'd rather speak about it.

1:02:30 Alright, so, any more questions? Songs? If not, then let's thank you. Thank you.