Archive for ontology

Matter and forces in quantum field theory – an attempt at a philosophical elucidation

Posted in Maynooth, The Universe and Stuff with tags , , , on March 16, 2021 by telescoper

I thought the following might be of general interest to readers of this blog. It’s a translation into English of an MSc thesis written by a certain Jon-Ivar Skullerud of the Department of Theoretical Physics at Maynooth University. I hasten to add that Dr Skullerud hasn’t just finished his MSc. It just appears that it has taken about 30 years to translate it from Norwegian into English!

Anyway, the English version is now available on the arXiv here. There isn’t really an abstract as such but the description on arXiv says:

This is a translation into English of my Masters thesis (hovedoppgave) from 1991. The main topic of the thesis is the relation between fundamental physics and philosophy, and a discussion of several possible ontologies of quantum field theory.

I note from here that hovedoppgave translates literally into English as “main task”.

I haven’t read this thesis from cover to cover yet – partly because it’s in digital form and doesn’t have any covers on it and partly because it’s 134 pages long –  but I’ve skimmed a few bits and it looks interesting.

 

 

Kuhn the Irrationalist

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , , , , , , on August 19, 2012 by telescoper

There’s an article in today’s Observer marking the 50th anniversary of the publication of Thomas Kuhn’s book The Structure of Scientific Revolutions.  John Naughton, who wrote the piece, claims that this book “changed the way we look at science”. I don’t agree with this view at all, actually. There’s little in Kuhn’s book that isn’t implicit in the writings of Karl Popper and little in Popper’s work that isn’t implicit in the work of a far more important figure in the development of the philosophy of science, David Hume. The key point about all these authors is that they failed to understand the central role played by probability and inductive logic in scientific research. In the following I’ll try to explain how I think it all went wrong. It might help the uninitiated to read an earlier post of mine about the Bayesian interpretation of probability.

It is ironic that the pioneers of probability theory and its application to scientific research, principally Laplace, unquestionably adopted a Bayesian rather than frequentist interpretation for his probabilities. Frequentism arose during the nineteenth century and held sway until relatively recently. I recall giving a conference talk about Bayesian reasoning only to be heckled by the audience with comments about “new-fangled, trendy Bayesian methods”. Nothing could have been less apt. Probability theory pre-dates the rise of sampling theory and all the other frequentist-inspired techniques that many modern-day statisticians like to employ.

Most disturbing of all is the influence that frequentist and other non-Bayesian views of probability have had upon the development of a philosophy of science, which I believe has a strong element of inverse reasoning or inductivism in it. The argument about whether there is a role for this type of thought in science goes back at least as far as Roger Bacon who lived in the 13th Century. Much later the brilliant Scottish empiricist philosopher and enlightenment figure David Hume argued strongly against induction. Most modern anti-inductivists can be traced back to this source. Pierre Duhem has argued that theory and experiment never meet face-to-face because in reality there are hosts of auxiliary assumptions involved in making this comparison. This is nowadays called the Quine-Duhem thesis.

Actually, for a Bayesian this doesn’t pose a logical difficulty at all. All one has to do is set up prior probability distributions for the required parameters, calculate their posterior probabilities and then integrate over those that aren’t related to measurements. This is just an expanded version of the idea of marginalization, explained here.

Rudolf Carnap, a logical positivist, attempted to construct a complete theory of inductive reasoning which bears some relationship to Bayesian thought, but he failed to apply Bayes’ theorem in the correct way. Carnap distinguished between two types or probabilities – logical and factual. Bayesians don’t – and I don’t – think this is necessary. The Bayesian definition seems to me to be quite coherent on its own.

Other philosophers of science reject the notion that inductive reasoning has any epistemological value at all. This anti-inductivist stance, often somewhat misleadingly called deductivist (irrationalist would be a better description) is evident in the thinking of three of the most influential philosophers of science of the last century: Karl Popper, Thomas Kuhn and, most recently, Paul Feyerabend. Regardless of the ferocity of their arguments with each other, these have in common that at the core of their systems of thought likes the rejection of all forms of inductive reasoning. The line of thought that ended in this intellectual cul-de-sac began, as I stated above, with the work of the Scottish empiricist philosopher David Hume. For a thorough analysis of the anti-inductivists mentioned above and their obvious debt to Hume, see David Stove’s book Popper and After: Four Modern Irrationalists. I will just make a few inflammatory remarks here.

Karl Popper really began the modern era of science philosophy with his Logik der Forschung, which was published in 1934. There isn’t really much about (Bayesian) probability theory in this book, which is strange for a work which claims to be about the logic of science. Popper also managed to, on the one hand, accept probability theory (in its frequentist form), but on the other, to reject induction. I find it therefore very hard to make sense of his work at all. It is also clear that, at least outside Britain, Popper is not really taken seriously by many people as a philosopher. Inside Britain it is very different and I’m not at all sure I understand why. Nevertheless, in my experience, most working physicists seem to subscribe to some version of Popper’s basic philosophy.

Among the things Popper has claimed is that all observations are “theory-laden” and that “sense-data, untheoretical items of observation, simply do not exist”. I don’t think it is possible to defend this view, unless one asserts that numbers do not exist. Data are numbers. They can be incorporated in the form of propositions about parameters in any theoretical framework we like. It is of course true that the possibility space is theory-laden. It is a space of theories, after all. Theory does suggest what kinds of experiment should be done and what data is likely to be useful. But data can be used to update probabilities of anything.

Popper has also insisted that science is deductive rather than inductive. Part of this claim is just a semantic confusion. It is necessary at some point to deduce what the measurable consequences of a theory might be before one does any experiments, but that doesn’t mean the whole process of science is deductive. He does, however, reject the basic application of inductive reasoning in updating probabilities in the light of measured data; he asserts that no theory ever becomes more probable when evidence is found in its favour. Every scientific theory begins infinitely improbable, and is doomed to remain so.

Now there is a grain of truth in this, or can be if the space of possibilities is infinite. Standard methods for assigning priors often spread the unit total probability over an infinite space, leading to a prior probability which is formally zero. This is the problem of improper priors. But this is not a killer blow to Bayesianism. Even if the prior is not strictly normalizable, the posterior probability can be. In any case, given sufficient relevant data the cycle of experiment-measurement-update of probability assignment usually soon leaves the prior far behind. Data usually count in the end.

The idea by which Popper is best known is the dogma of falsification. According to this doctrine, a hypothesis is only said to be scientific if it is capable of being proved false. In real science certain “falsehood” and certain “truth” are almost never achieved. Theories are simply more probable or less probable than the alternatives on the market. The idea that experimental scientists struggle through their entire life simply to prove theorists wrong is a very strange one, although I definitely know some experimentalists who chase theories like lions chase gazelles. To a Bayesian, the right criterion is not falsifiability but testability, the ability of the theory to be rendered more or less probable using further data. Nevertheless, scientific theories generally do have untestable components. Any theory has its interpretation, which is the untestable baggage that we need to supply to make it comprehensible to us. But whatever can be tested can be scientific.

Popper’s work on the philosophical ideas that ultimately led to falsificationism began in Vienna, but the approach subsequently gained enormous popularity in western Europe. The American Thomas Kuhn later took up the anti-inductivist baton in his book The Structure of Scientific Revolutions. Initially a physicist, Kuhn undoubtedly became a first-rate historian of science and this book contains many perceptive analyses of episodes in the development of physics. His view of scientific progress is cyclic. It begins with a mass of confused observations and controversial theories, moves into a quiescent phase when one theory has triumphed over the others, and lapses into chaos again when the further testing exposes anomalies in the favoured theory. Kuhn adopted the word paradigm to describe the model that rules during the middle stage,

The history of science is littered with examples of this process, which is why so many scientists find Kuhn’s account in good accord with their experience. But there is a problem when attempts are made to fuse this historical observation into a philosophy based on anti-inductivism. Kuhn claims that we “have to relinquish the notion that changes of paradigm carry scientists ..closer and closer to the truth.” Einstein’s theory of relativity provides a closer fit to a wider range of observations than Newtonian mechanics, but in Kuhn’s view this success counts for nothing.

Paul Feyerabend has extended this anti-inductivist streak to its logical (though irrational) extreme. His approach has been dubbed “epistemological anarchism”, and it is clear that he believed that all theories are equally wrong. He is on record as stating that normal science is a fairytale, and that equal time and resources should be spent on “astrology, acupuncture and witchcraft”. He also categorised science alongside “religion, prostitution, and so on”. His thesis is basically that science is just one of many possible internally consistent views of the world, and that the choice between which of these views to adopt can only be made on socio-political grounds.

Feyerabend’s views could only have flourished in a society deeply disillusioned with science. Of course, many bad things have been done in science’s name, and many social institutions are deeply flawed. One can’t expect anything operated by people to run perfectly. It’s also quite reasonable to argue on ethical grounds which bits of science should be funded and which should not. But the bottom line is that science does have a firm methodological basis which distinguishes it from pseudo-science, the occult and new age silliness. Science is distinguished from other belief-systems by its rigorous application of inductive reasoning and its willingness to subject itself to experimental test. Not all science is done properly, of course, and bad science is as bad as anything.

The Bayesian interpretation of probability leads to a philosophy of science which is essentially epistemological rather than ontological. Probabilities are not “out there” in external reality, but in our minds, representing our imperfect knowledge and understanding. Scientific theories are not absolute truths. Our knowledge of reality is never certain, but we are able to reason consistently about which of our theories provides the best available description of what is known at any given time. If that description fails when more data are gathered, we move on, introducing new elements or abandoning the theory for an alternative. This process could go on forever. There may never be a “final” theory, and scientific truths are consequently far from absolute, but that doesn’t mean that there is no progress.

Deductivism and Irrationalism

Posted in Bad Statistics, The Universe and Stuff with tags , , , , , , , , , , , on December 11, 2010 by telescoper

Looking at my stats I find that my recent introductory post about Bayesian probability has proved surprisingly popular with readers, so I thought I’d follow it up with a brief discussion of some of the philosophical issues surrounding it.

It is ironic that the pioneers of probability theory, principally Laplace, unquestionably adopted a Bayesian rather than frequentist interpretation for his probabilities. Frequentism arose during the nineteenth century and held sway until recently. I recall giving a conference talk about Bayesian reasoning only to be heckled by the audience with comments about “new-fangled, trendy Bayesian methods”. Nothing could have been less apt. Probability theory pre-dates the rise of sampling theory and all the frequentist-inspired techniques that modern-day statisticians like to employ.

Most disturbing of all is the influence that frequentist and other non-Bayesian views of probability have had upon the development of a philosophy of science, which I believe has a strong element of inverse reasoning or inductivism in it. The argument about whether there is a role for this type of thought in science goes back at least as far as Roger Bacon who lived in the 13th Century. Much later the brilliant Scottish empiricist philosopher and enlightenment figure David Hume argued strongly against induction. Most modern anti-inductivists can be traced back to this source. Pierre Duhem has argued that theory and experiment never meet face-to-face because in reality there are hosts of auxiliary assumptions involved in making this comparison. This is nowadays called the Quine-Duhem thesis.

Actually, for a Bayesian this doesn’t pose a logical difficulty at all. All one has to do is set up prior probability distributions for the required parameters, calculate their posterior probabilities and then integrate over those that aren’t related to measurements. This is just an expanded version of the idea of marginalization, explained here.

Rudolf Carnap, a logical positivist, attempted to construct a complete theory of inductive reasoning which bears some relationship to Bayesian thought, but he failed to apply Bayes’ theorem in the correct way. Carnap distinguished between two types or probabilities – logical and factual. Bayesians don’t – and I don’t – think this is necessary. The Bayesian definition seems to me to be quite coherent on its own.

Other philosophers of science reject the notion that inductive reasoning has any epistemological value at all. This anti-inductivist stance, often somewhat misleadingly called deductivist (irrationalist would be a better description) is evident in the thinking of three of the most influential philosophers of science of the last century: Karl Popper, Thomas Kuhn and, most recently, Paul Feyerabend. Regardless of the ferocity of their arguments with each other, these have in common that at the core of their systems of thought likes the rejection of all forms of inductive reasoning. The line of thought that ended in this intellectual cul-de-sac began, as I stated above, with the work of the Scottish empiricist philosopher David Hume. For a thorough analysis of the anti-inductivists mentioned above and their obvious debt to Hume, see David Stove’s book Popper and After: Four Modern Irrationalists. I will just make a few inflammatory remarks here.

Karl Popper really began the modern era of science philosophy with his Logik der Forschung, which was published in 1934. There isn’t really much about (Bayesian) probability theory in this book, which is strange for a work which claims to be about the logic of science. Popper also managed to, on the one hand, accept probability theory (in its frequentist form), but on the other, to reject induction. I find it therefore very hard to make sense of his work at all. It is also clear that, at least outside Britain, Popper is not really taken seriously by many people as a philosopher. Inside Britain it is very different and I’m not at all sure I understand why. Nevertheless, in my experience, most working physicists seem to subscribe to some version of Popper’s basic philosophy.

Among the things Popper has claimed is that all observations are “theory-laden” and that “sense-data, untheoretical items of observation, simply do not exist”. I don’t think it is possible to defend this view, unless one asserts that numbers do not exist. Data are numbers. They can be incorporated in the form of propositions about parameters in any theoretical framework we like. It is of course true that the possibility space is theory-laden. It is a space of theories, after all. Theory does suggest what kinds of experiment should be done and what data is likely to be useful. But data can be used to update probabilities of anything.

Popper has also insisted that science is deductive rather than inductive. Part of this claim is just a semantic confusion. It is necessary at some point to deduce what the measurable consequences of a theory might be before one does any experiments, but that doesn’t mean the whole process of science is deductive. He does, however, reject the basic application of inductive reasoning in updating probabilities in the light of measured data; he asserts that no theory ever becomes more probable when evidence is found in its favour. Every scientific theory begins infinitely improbable, and is doomed to remain so.

Now there is a grain of truth in this, or can be if the space of possibilities is infinite. Standard methods for assigning priors often spread the unit total probability over an infinite space, leading to a prior probability which is formally zero. This is the problem of improper priors. But this is not a killer blow to Bayesianism. Even if the prior is not strictly normalizable, the posterior probability can be. In any case, given sufficient relevant data the cycle of experiment-measurement-update of probability assignment usually soon leaves the prior far behind. Data usually count in the end.

The idea by which Popper is best known is the dogma of falsification. According to this doctrine, a hypothesis is only said to be scientific if it is capable of being proved false. In real science certain “falsehood” and certain “truth” are almost never achieved. Theories are simply more probable or less probable than the alternatives on the market. The idea that experimental scientists struggle through their entire life simply to prove theorists wrong is a very strange one, although I definitely know some experimentalists who chase theories like lions chase gazelles. To a Bayesian, the right criterion is not falsifiability but testability, the ability of the theory to be rendered more or less probable using further data. Nevertheless, scientific theories generally do have untestable components. Any theory has its interpretation, which is the untestable baggage that we need to supply to make it comprehensible to us. But whatever can be tested can be scientific.

Popper’s work on the philosophical ideas that ultimately led to falsificationism began in Vienna, but the approach subsequently gained enormous popularity in western Europe. The American Thomas Kuhn later took up the anti-inductivist baton in his book The Structure of Scientific Revolutions. Kuhn is undoubtedly a first-rate historian of science and this book contains many perceptive analyses of episodes in the development of physics. His view of scientific progress is cyclic. It begins with a mass of confused observations and controversial theories, moves into a quiescent phase when one theory has triumphed over the others, and lapses into chaos again when the further testing exposes anomalies in the favoured theory. Kuhn adopted the word paradigm to describe the model that rules during the middle stage,

The history of science is littered with examples of this process, which is why so many scientists find Kuhn’s account in good accord with their experience. But there is a problem when attempts are made to fuse this historical observation into a philosophy based on anti-inductivism. Kuhn claims that we “have to relinquish the notion that changes of paradigm carry scientists ..closer and closer to the truth.” Einstein’s theory of relativity provides a closer fit to a wider range of observations than Newtonian mechanics, but in Kuhn’s view this success counts for nothing.

Paul Feyerabend has extended this anti-inductivist streak to its logical (though irrational) extreme. His approach has been dubbed “epistemological anarchism”, and it is clear that he believed that all theories are equally wrong. He is on record as stating that normal science is a fairytale, and that equal time and resources should be spent on “astrology, acupuncture and witchcraft”. He also categorised science alongside “religion, prostitution, and so on”. His thesis is basically that science is just one of many possible internally consistent views of the world, and that the choice between which of these views to adopt can only be made on socio-political grounds.

Feyerabend’s views could only have flourished in a society deeply disillusioned with science. Of course, many bad things have been done in science’s name, and many social institutions are deeply flawed. One can’t expect anything operated by people to run perfectly. It’s also quite reasonable to argue on ethical grounds which bits of science should be funded and which should not. But the bottom line is that science does have a firm methodological basis which distinguishes it from pseudo-science, the occult and new age silliness. Science is distinguished from other belief-systems by its rigorous application of inductive reasoning and its willingness to subject itself to experimental test. Not all science is done properly, of course, and bad science is as bad as anything.

The Bayesian interpretation of probability leads to a philosophy of science which is essentially epistemological rather than ontological. Probabilities are not “out there” in external reality, but in our minds, representing our imperfect knowledge and understanding. Scientific theories are not absolute truths. Our knowledge of reality is never certain, but we are able to reason consistently about which of our theories provides the best available description of what is known at any given time. If that description fails when more data are gathered, we move on, introducing new elements or abandoning the theory for an alternative. This process could go on forever. There may never be a final theory. But although the game might have no end, at least we know the rules….


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Get thee behind me, Plato

Posted in The Universe and Stuff with tags , , , , , , , , , , on September 4, 2010 by telescoper

The blogosphere, even the tiny little bit of it that I know anything about, has a habit of summoning up strange coincidences between things so, following EM Forster’s maxim “only connect”, I thought I’d spend a lazy saturday lunchtime trying to draw a couple of them together.

A few days ago I posted what was intended to be a fun little item about the wave-particle duality in quantum mechanics. Basically, what I was trying to say is that there’s no real problem about thinking of an electron as behaving sometimes like a wave and sometimes like a particle because, in reality (whatever that is), it is neither. “Particle” and “wave” are useful abstractions but they are not in an exact one-to-one correspondence with natural phenomena.

Before going on I should point out that the vast majority of physicists are well away of the distinction between, say,  the “theoretical” electron and whatever the “real thing” is. We physicists tend to live in theory space rather than in the real world, so we tend to teach physics by developing the formal mathematical properties of the “electron” (or “electric field”) or whatever, and working out what experimental consequences these entail in certain situations. Generally speaking, the theory works so well in practice that we often talk about the theoretical electron that exists in the realm of mathematics and the electron-in-itself as if they are one and the same thing. As long as this is just a pragmatic shorthand, it’s fine. However, I think we need to be careful to keep this sort of language under control. Pushing theoretical ideas out into the ontological domain is a dangerous game. Physics – especially quantum physics – is best understood as a branch of epistemology. What is known? is safer ground than what is there?

Anyway, my  little  piece sparked a number of interesting comments on Reddit, including a thread that went along the lines “of course an electron is neither a particle nor a wave,  it’s actually  a spin-1/2 projective representation of the Lorentz Group on a Hilbert space”. That description, involving more sophisticated mathematical concepts than those involved in bog-standard quantum mechanics, undoubtedly provides a more complete account of natural phenomena associated with the electrons and electrical fields, but I’ll stick to my guns and maintain that it still introduces a deep confusion to assert that the electron “is” something mathematical, whether that’s a “spin-1/2 projective representation” or a complex function or anything else.  That’s saying something physical is a mathematical. Both entities have some sort of existence, of course, but not the same sort, and the one cannot “be” the other. “Certain aspects of an electron’s behaviour can be described by certain mathematical structures” is as I’m  prepared to go.

Pushing deeper than quantum mechanics, into the realm of quantum field theory, there was the following contribution:

The electron field is a quantum field as described in quantum field theories. A quantum field covers all space time and in each point the quantum field is in some state, it could be the ground state or it could be an excitation above the ground state. The excitations of the electron field are the so-called electrons. The mathematical object that describes the electron field possesses, amongst others, certain properties that deal with transformations of the space-time coordinates. If, when performing a transformation of the space-time coordinates, the mathematical object changes in such a way that is compatible with the physics of the quantum field, then one says that the mathematical object of the field (also called field) is represented by a spin 1/2 (in the electron case) representation of a certain group of transformations (the Poincaré group, in this example).I understand your quibbling, it seems natural to think that “spin 1/2″ is a property of the mathematical tool to describe something, not the something itself. If you press on with that distinction however, you should be utterly puzzled of why physics should follow, step by step, the path led by mathematics.

For example, one speaks about the ¨invariance under the local action of the group SU(3)” as a fundamental property of the fields that feel the nuclear strong force. This has two implications, the mathematical object that represents quarks must have 3 ¨strong¨ degrees of freedom (the so-called color) and there must be 32-1 = 8 carriers of the force (the gluons) because the group of transformations in a SU(N) group has N2-1 generators. And this is precisely what is observed.

So an extremely abstract mathematical principle correctly accounts for the dynamics of an inmensely large quantity of phenomena. Why does then physics follow the derivations of mathematics if its true nature is somewhat different?

No doubt this line of reasoning is why so many theoretical physicists seem to adopt a view of the world that regards mathematical theories as being, as it were,  “built into” nature rather than being things we humans invented to describe nature. This is a form of Platonic realism.

I’m no expert on matters philosophical, but I’d say that I find this stance very difficult to understand, although I am prepared to go part of the way. I used to work in a Mathematics department many years ago and one of the questions that came up at coffee time occasionally was “Is mathematics invented or discovered?”. In my experience, pure mathematicians always answered “discovered” while others (especially astronomers, said “invented”). For what it’s worth, I think mathematics is a bit of both. Of course we can invent mathematical objects, endow them with certain attributes and proscribe rules for manipulating them and combining them with other entities. However, once invented anything that is worked out from them is “discovered”. In fact, one could argue that all mathematical theorems etc arising within such a system are simply tautological expressions of the rules you started with.

Of course physicists use mathematics to construct models that describe natural phenomena. Here the process is different from mathematical discovery as what we’re trying to do is work out which, if any, of the possible theories is actually the one that accounts best for whatever empirical data we have. While it’s true that this programme requires us to accept that there are natural phenomena that can be described in mathematical terms, I do not accept that it requires us to accept that nature “is” mathematical. It requires that there be some sort of law governing some  of aspects of nature’s behaviour but not that such laws account for everything.

Of course, mathematical ideas have been extremely successful in helping physicists build new physical descriptions of reality. On the other hand, however, there is a great deal of mathematical formalism that is is not useful in this way.  Physicists have had to select those mathematical object that we can use to represent natural phenomena, like selecting words from a dictionary. The fact that we can assemble a sentence using words from the Oxford English Dictionary that conveys some information about something we see doesn’t not mean that what we see “is” English. A whole load of grammatically correct sentences can be constructed that don’t make any sense in terms of observable reality, just as there is a great deal of mathematics that is internally self-consistent but makes no contact with physics.

Moreover, to the person whose quote I commented on above, I’d agree that the properties of the SU(3) gauge group have indeed accounted for many phenomena associated with the strong interaction, which is why the standard model of particle physics contains 8 gluons and quarks carrying a three-fold colour charge as described by quantum chromodynamics. Leaving aside the fact that QCD is such a terribly difficult theory to work with – in practice it involves  nightmarish lattice calculations on a scale to make even the most diehard enthusiast cringe –  what I would ask is whether this  description in any case sufficient for us to assert that it describes “true nature”?  Many physicists will no doubt disagree with me, but I don’t think so. It’s a map, not the territory.

So why am I boring you all with this rambling dissertation? Well, it  brings me to my other post – about Stephen Hawking’s comments about God. I don’t want to go over that issue again – frankly, I was bored with it before I’d finished writing my own blog post  – but it does relate to the bee that I often find in my bonnet about the tendency of many modern theoretical physicists to assign the wrong category of existence to their mathematical ideas. The prime example that springs to my mind is the multiverse. I can tolerate  certain versions of the multiverse idea, in fact. What I can’t swallow, however is the identification of the possible landscape of string theory vacua – essentially a huge set of possible solutions of a complicated set of mathematical equations – with a realised set of “parallel universes”. That particular ontological step just seems absurd to me.

I’m just about done, but one more thing I’d say to finish with is concerns the (admittedly overused) metaphor of maps and territories. Maps are undoubtedly useful in helping us find our way around, but we have to remember that there are always things that aren’t on the map at all. If we rely too heavily on one, we might miss something of great interest that the cartographer didn’t think important. Likewise if we fool ourselves into thinking our descriptions of nature are so complete that they “are” all that nature is, then we might miss the road to a better understanding.


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