MENTAL vs. Mathematical Universe Hypothesis, by Tegmark
MENTAL vs. Tegmark's Mathematical Universe Hypothesis
MENTAL vs. TEGMARK'S MATHEMATICAL UNIVERSE HYPOTHESIS
"The universe is a mathematical structure. There is only mathematics; that's all there is" (Max Tegmark).
"Everything that exists mathematically, exists physically. Physical existence is equivalent to mathematical existence" (Max Tegmark).
The Mathematical Universe Hypothesis
Swedish-American cosmologist Max Tegmark has developed a speculative, mathematically grounded "theory of everything" that goes beyond Platonism: the Mathematical Universe Hypothesis (MUH), which states:
Mathematics is the only thing that exists.
Mathematics, the mathematical world or cosmos, is the only thing that exists, for it is the deepest thing we can conceive of. Mathematics does not describe the universe, it is the universe.
The universe is a mathematical structure.
Our external physical reality is a mathematical structure. While traditional physics tries to describe reality through mathematics, MUH states that reality is mathematical, and more specifically, a mathematical structure. "Everything in our universe is purely mathematical, including YOU." Our universe, with its particular laws, is just one mathematical object in the cosmos of infinite mathematical structures. This mathematical object we are discovering little by little. We have not yet found the mathematical structure isomorphic to our world.
By the definition of mathematical structure, if our physical reality is isomorphic to a mathematical structure, then our physical reality is a mathematical structure. The physical world is an abstract mathematical structure, and the entities and relationships of that abstract mathematical structure describing physical reality must be completely abstract.
There are no physical objects.
Physical objects are just mathematical structures that we experience on a subjective level. This sensation is an illusion, in the same way that we experience the color red, because red is not a property of the object, but something conceived by our cognitive capacity. "A mathematical structure is our external physical reality, rather than being merely a description of it."
Sufficiently complex mathematical structures that include self-aware substructures (SAS), will subjectively perceive themselves as existing in a "real" physical world. We are an example of such self-aware substructures.
Mathematical independence.
Following Plato, he holds that this mathematical universe exists independently of us and in it there is neither space nor time. The entities of this universe are abstract and immutable. Mathematical structures have not been created, they simply exist, they are timeless. "We did not invent mathematical structures, we discovered them, and invented only the notation to describe them." The mathematical notation is unimportant. What is important are the relationships they describe.
Freedom.
In the mathematical universe there are no laws at all because it is a space of freedom.
Wigner's question.
Explain the question posed by Wigner: mathematics is so effective in describing reality because it is reality, because reality is a mathematical structure. The physical world is isomorphic to a mathematical structure. Mathematical existence and physical existence are equivalent.
Formulas.
Mathematical formulas or expressions constitute the manifested mathematical world. That is, the mathematical manifestations are not the physical worlds, but the mathematical expressions themselves.
Research.
Instead of exploring our universe, which is a particular mathematical structure, it is better to study all the structures of the mathematical cosmos, the supreme cosmos for one reason: because it is simpler.
Physical theories.
Our theories about the universe are not mathematics approximating physics, but mathematics approximating mathematics, universal mathematics reflected in particular mathematics.
External physical reality.
External physical reality is independent of us humans. Physical reality being independent of humans, it should be described by non-human entities, without human concepts. It should be completely abstract, without any preconceived meaning. The only properties of these entities should be the relationships between them. This is precisely where mathematics appears: in the relationships.
Valid mathematical structures.
Not all mathematical structures exist. There are only: 1) computable ones, i.e., those that can be constructed in a finite number of steps; 2) consistent ones, i.e., those free of contradiction; 3) decidable ones (in Gödel's sense), i.e., those that can be derived from a set of axioms, such as the ZF axioms of set theory.
Two perspectives: the frog and the bird.
There are two perspectives or ways of observing physical reality: that of the frog (or lower) and that of the bird (or higher):
For the bird everything is an eternal and immutable mathematical structure. The world sees it simply because it perceives the general. It is the vision of the mathematician, who only observes a mathematical structure where symmetry is never broken. This superior perspective is what occurs in Einstein's theory of relativity, where all events already exist within a mathematical structure called space-time.
The frog feels the passage of time. It sees the complex world because it perceives only the details, the particular, the superficial. At this level, the world is described by physical laws (laws of motion, law of gravitation, etc.). It is the vision of the observer within the mathematical structure, where symmetry is broken.
TOE (Theory of Everything).
A TOE is a complete mathematical structure of the physical world. The only properties of a mathematical structure are the timeless and immutable relationships between the elements of the structure. Since there is no time in mathematical structures, there are no "initial conditions" either. There is also no randomness, since this concept only makes sense in the context of external time. The universe is not a computer because there is nothing to compute, since the universe is a timeless mathematical structure. A TOE that included the beginning of creation would be meaningless or an incomplete description.
Physical-mathematical isomorphism.
"All structures that exist mathematically also exist physically." "Physical existence is equivalent to mathematical existence". It is the only postulate formulated in "Ultimate Ensemble Theory of Everything". Physical objects are not shadows or imperfect copies of ideal and timeless objects belonging to an inaccessible realm but are mathematical structures in themselves. This position is the opposite of traditional Platonism.
There is "mathematical democracy": physical and mathematical existence are equivalent. Therefore, all mathematical structures have the same ontological status.
Quantum physics.
There is no sense in the Copenhagen interpretation (the interpretation of the wave function as probability), which is why Einstein was right when he said "God does not play dice".
Boltzmann's paradox.
Boltzmann's paradox is clarified in the context of MUH. This paradox refers to the fact that, according to the second law of thermodynamics, the entropy (the degree of disorder) of the universe is always increasing, and yet we observe order in the universe because the universe is a timeless mathematical structure.
The implications of MUH
According to Tegmark, the MUH has important implications:
It predicts the existence of the multiverse (parallel universes). Paradoxically, "Describing a multiverse might be simpler than describing a single universe." "Without the multiverse, the explanation of reality would become very complicated and arbitrary." If there are many parallel universes, then we should find ourselves in a normal, typical one. If we observe something atypical or special in our universe, we could say that the MUH is not true.
It predicts that many mathematical regularities remain to be discovered to those already discovered, such as the standard model of particle physics.
MUH implies the "Theory of Everything" (TOE), the Holy Grail of theoretical physics: a complete description of reality. "The theory of everything shall be purely abstract and mathematical."
If MUH is true, it would unify physics and mathematics, and allow us to understand the essential unity of everything.
The multiverse
There are several theories or models of the multiverse. There are three that correspond to parallel universes that do not communicate with each other. Tegmark adds a fourth theory: the mathematical universe hypothesis. The 4 levels of the hierarchy go from least to most diverse:
The multiverse formed by different regions, and distant from each other, of our universe emerged from the Big Bang. They are temporarily cut off from each other because light has not had time to reach them. According to the theory of cosmological inflation, our universe is infinite.
Cosmological inflation refers to the hypothetical extraordinarily rapid growth (by a factor of the order of 1078 in 10−32 seconds) of an initial nucleus that formed the universe in the Big Bang. After the inflationary period, the universe continued to expand but at a slower rate. This theory was proposed by Alan Guth in 1980.
The multiverse formed by universes that are eternally incommunicado due to cosmological inflation of space. It can also be the multiverse arising from several Big Bangs, where each universe could have different types of physical laws, different dimensions and different types of particles. It may also be the multiverse associated with the fundamental equations of physics that have more than one solution.
The multiverse of the "many worlds", Hugh Everett's theory. When an observation is made at the quantum level, the universe splits into as many parallel versions of itself as there are possibilities. According to the Copenhagen interpretation, when an observation is made, the wave function "collapses". But in the many-worlds theory there is no such collapse, but rather the wave function branches and the entire universe is subdivided into as many universes as there are possibilities. All universes have exactly the same physical constants and laws and the same space-time structure, and exist in the abstract mathematical structure of "Hilbert space", with infinite spatial dimensions.
The mathematical multiverse. Our universe is just one mathematical structure in a cosmos full of mathematical structures. There are many parallel universes, but they are all mathematical objects. These universes may have the same physical laws as ours or have completely different ones.
The "Theory of Everything" (TOE)
According to Tegmark, a TOE must meet several conditions:
It must unify quantum theory and the theory of general relativity.
It must be a simple mathematical structure of great beauty. In general, a set is easier to generate (by an algorithm) than any one of its members. For example, the set of natural numbers can be generated with a trivial program. On the other hand, a many-digit random number can have enormous complexity.
The "algorithmic complexity" of a mathematical entity is called the smallest program that generates it. The "final set" of all mathematical structures must have the minimum algorithmic complexity.
It must be based on simple mathematical structures, such as integers. Just as it is said "All roads lead to Rome", all mathematical structures must lead to basic and simple structures.
It must explain or describe the lower vision (the frog's) from the higher vision (the bird's). We cannot verify TOE from the bird's vision, but we can falsify (in Popper's sense) TOE from the frog's observations.
The SASs
A SAS (a self-aware substructure) must meet at least 3 conditions:
Complexity. It must be compatible with Gödel's incompleteness theorem.
Predictability. It must be able to make inferences about its future insights.
Stability. There must be sufficient time to be able to make predictions.
SASs are the observers. We humans are an example of SASs. We are living "inside" a mathematical structure. We perceive ourselves as "local", stable, permanent, unique and isolated. And our perception is limited to what is useful, stable and permanent.
The mathematical structure that describes our world must be the most generic that is consistent with our observations. Our observations are the most generic that are consistent with our existence.
The frog's vision is the form that perceives a SAS within a mathematical structure. The bird's vision is that of the mathematician perceiving the mathematical structure. Different SASs can perceive different physical realities.
Self-awareness would merely be a side effect of advanced information processing.
SASs must exist in space-time:
Space must be of dimension n=3. According to Paul Ehrenfest [1917], in a space of n>3 neither atoms nor planetary orbits would be stable. And with n<3, there could be no gravitational force and organisms would have serious topological problems (e.g., two nerves could not cross each other).
Time must be of dimension m=1. Otherwise, certain physical constraints would be eliminated and physical laws would be different, including the possibility of reverse causation.
The concept of mathematical structure
A mathematical structure S is a set of abstract entities S1, S2, . .. with an abstract relations R1, R2, ..... For example, integers, real numbers, groups, etc.
An example of mathematical structure is the group formed by two elements (0 and 1) and 4 relations defined by an operation (denoted by "+") defined by the expressions 0+0=0, 0+1=1, 1+0=1, 1+1=0
This mathematical structure is purely abstract and can have different concrete meanings. For example, "0" can mean "even number", "1" "odd number", and "+" "arithmetic sum". Symbols are mere forms without intrinsic meaning. Relationships between elements are intrinsic properties. The notation is irrelevant; it is the abstract relationships that are important. For example, the number 4 can be referred to in different ways: "IV", "four", "four", etc.
Characteristics of mathematical structures:
A particular mathematical structure can be defined as an equivalence class of all possible descriptions or interpretations.
A mathematical structure is a description of itself, regardless of the notation used.
A mathematical structure is free of human description, of human baggage, so it asserts nothing.
By the definition of mathematical structure, if there is an isomorphism between one mathematical structure and another, both are the same structure.
A mathematical structure S is reducible if its entities can be divided into two sets T and U without relations between the entities of different sets. The sets T and U are level IV parallel universes. Most mathematical structures (the integers, Euclidean space, etc.) are irreducible. An example of a reducible structure is a set of elements, which have no relations between their elements.
Given two mathematical structures, it is always possible to define a structure that is the union of both, with their corresponding relations. Therefore, a finite number of parallel universes can be considered as belonging to a higher mathematical structure. From the frog's point of view, it is irrelevant whether or not the mathematical structure is part of a reducible structure. From the bird's point of view, what matters is the higher structure.
An automorphism of a mathematical structure S is defined as a permutation of the elements of S that preserve all their relations. It is easy to see that the set of all automorphisms of a structure S − Aut(S) form a group. Aut(S) can be considered as the group of symmetries of a structure, that is, the transformations that make the structure invariant. The laws of physics are invariant under Aut(S).
Remarks:
Not all imaginable universes exist, for there are things we can imagine that have no mathematical structure.
A mathematical structure is not a theory.
The level IV multiverse is not the set of all mathematical structures.
A natural language sentence involving relations is not a mathematical structure unless we either obviate all connotations or else add entities and relations that describe it exactly.
The External Reality Hypothesis (ERH)
The External Reality Hypothesis asserts that there exists a physical reality completely independent of us humans.
The ERH is not universally accepted. It is rejected by, among others:
Solipsists. Solipsism is a radical form of subjectivism according to which only the self exists or can be known.
Berkeley. The only reality is mental reality; there is no physical reality.
Physicists who accept the Copenhagen interpretation of quantum physics: the world ceases to exist when it is not observed.
The Computable Universe Hypothesis (CUH)
The computable universe hypothesis asks whether our external physical reality is a form of computer simulation, and asserts that:
The mathematical structure that is our physical reality is definable by computable functions.
Only computable mathematical structures exist physically, i.e. their results are obtained in a finite number of steps. Put from another perspective, only decidable mathematical structures exist physically, i.e., those that are derivable from a certain hypothetical finite set of primary axioms. This may explain why our physical laws seem to be so simple.
Relationships between entities in a mathematical structure are computations that do not involve physical time and do not evolve the universe, but only describe the universe and its history. For example, in the theory of relativity, time is just another variable, on the same level as the spatial variables.
It is possible to simulate our universe by means of a small computer program.
The HUC eliminates potential paradoxes related to the level IV multiverse. The complete level IV multiverse, i.e., the union of the infinite computable mathematical structures is not a computable structure, i.e., it is not a member of itself.
According to Tegmark, mathematical structures and computations are described by formal systems. But Tegmark recognizes that these 3 aspects −mathematical structures, computations and formal systems− are different aspects of a transcendent underlying structure "whose nature we do not fully understand."
Relations between MUH and other theories
Apart from being a form of radical Platonism, MUH has close connections or analogies with:
Rudy Rucker's MindScape concept [1982].
The MindScape is a Platonic world of ideas separate from the mental and the physical. Consciousness explores this world or "mindscape" that contains all possible thoughts. All humans share this space, just as we share the physical universe.
The "modal realism" of David Lewis.
Lewis argues that possible worlds are as real as our own world. All possible worlds have the same ontological level. There are an infinite number of possible worlds −ours, one of them−, all incommunicado from each other, without spatial, temporal or causal links.
The concept "pi in the sky" by John Barrow [2001].
Barrow uses the expression "pi in the sky" with a double meaning, because it is pronounced the same as "pie in the sky". The latter expression means an idea, project, plan, thought or dream that will probably never come true because it is unattainable.
For Barrow, mathematics is a collection of all the infinite possible models or patterns of the universe. Mathematics is infinite and everywhere. Barrow uses the phrase "pi in the sky" to explore the world of mathematics: its origin, what it is, whether it is something already existing that we discover or whether it is a human invention, and whether it can lead us to the ultimate meaning of the universe.
Robert Nozick's "fecundity (or fullness) principle" [1983].
The fruitfulness principle is an egalitarian or symmetrical principle. It asserts that all possibilities are realized or instantiated. No possibility has a privileged ontological status. All possibilities are given equal status. Our world is only one among infinite possible worlds, coexisting with all of them.
In turn, the principle of fecundity is a possible principle. Therefore, it is also realized or instantiated. It is an instance of itself. It is a principle that justifies itself.
The philosophical concept of "universal structural realism".
Structural realism is to accept as real the structural and mathematical content of our scientific theories. Only the structure of the world is knowable, not its nature. Structural realism is structural realism extended to the entire universe. MUH corresponds to the ontic version of this concept, i.e., what exists. The ERH corresponds to the epistemic version, i.e., what we can know.
The Mathematics - Matter - Mind Triad (MMM)
According to Roger Penrose [2006], there are relationships between mathematics, matter and mind, which he reflected in a triangular and circular diagram called "Penrose diagram" or "MMM diagram":
The 3 relationships are:
Mathematics arises from or is a product of the mind.
Matter can be explained in terms of mathematics.
Mind arises from matter.
This Penrose triangle has been challenged or nuanced by Tegmark and two other authors (Piet Hut and Mark Alford) in the article "On Math, Matter and Mind" [2006], who put forward 3 different views:
Tegmark's fundamentalist view
Rejects the "Mind → Math" arrow. Mathematics is not a product of the human mind because mathematics is independent of the human observer.
Defends the arrow "Mathematics → Matter" because the world is intrinsically mathematical, and furthermore mathematics and physical existence are equivalent.
Defends the arrow "Matter → Mind". Mind and consciousness emerge from matter, from certain complex physical systems that process information. Human mind emerges from matter as a self-aware mathematical substructure.
Alford's secular view
Defends the "Mind $rarr; Mathematical" arrow. Mathematics is an activity of the mind. It arises from the mind, not from an ideal or higher world independent of us. Reject the arrow "Mathematics → Matter." Mathematics is not the ultimate substance of the world. Reject the arrow "Matter → Mind." Mental processes are not material processes. Matter is less fundamental than mind. There is no duality or separation between mind and matter, but they are two aspects of the same reality.
The mystical vision of Hut
Questions or rejects the 3 arrows of Penrose's diagram as superficial. Argues that a deeper, unifying and transcendent vision is needed. Science has always advanced along the line of unification. For example, in physics it has discovered intrinsic connections between electricity and magnetism, between space and time, between matter and energy, and so on. In this sense, mathematics, mind and matter are not 3 independent concepts, and must be unified in the future. The arrows in the MMM diagram should be considered only as indicators of significant correlations, not as causal relationships. They are only like the shadows on the wall of Plato's cave.
MENTAL vs. Tegmark's Mathematical Universe Hypothesis
The idea that the universe is mathematical goes back to the Pythagoreans of ancient Greece. For Plato, mathematics exists on a higher or ideal plane of reality. Galileo said that "The book of nature is written in the language of mathematics". Eugene Wigner in his famous article of [1960] wondered about "the unreasonable effectiveness of mathematics in the natural sciences," which demanded an explanation. Tegmark goes further by identifying mathematical platonism with physical reality.
There are certain differences between Max Tegmark's mathematical universe hypothesis, and MENTAL:
Rationale.
Tegmark's position is speculative, with no concrete foundation. Instead, the foundation of MENTAL is the principle of descending causality and the primary archetypes or dimensions of consciousness.
According to Tekmark, there is an unknown underlying structure common to mathematical structures, computations and formal systems. This structure is precisely MENTAL, the language based on primary archetypes.
Applying the principle of descending causality, we can affirm that there exists the hierarchy MENTAL - Mathematical World - Physical World:
MENTAL manifests itself as mathematical structures, which are the simplest and most efficient structures of relations. In fact, mathematics studies the relationships between universal semantic primitives.
Some of these mathematical structures manifest themselves in the physical world.
The primary archetypes.
The deepest thing we can conceive of is not mathematics but the primary archetypes. From them derive (among other disciplines) mathematics and computer science, with their descriptive and operational characteristics, respectively. Reality, at the deep level, is abstract. Reality, the concrete, the superficial, is constituted by manifestations of the primary archetypes, which connect everything. Everything superficial is connected through the deep. True reality resides in the deep, in the abstract. The superficial reality is not the true reality.
Physical reality and mental reality share the same primary archetypes, which are abstract in nature and have only abstract relationships between them, relationships which are also primary archetypes.
Matter vs. mind.
Our theories of the universe are not mathematics approaching mathematics, but primary archetypes approaching primary archetypes, to recognize common archetypes. It is rather an identification or synchronization between the physical and mental levels.
Structures vs. expressions.
Tegmark talks about "mathematical structures". In MENTAL there are expressions.
Existence.
A mathematical object exists if we can construct or describe it at the formal level by means of the primary archetypes of MENTAL. There are two forms of existence of a mathematical object: the descriptive level (associated with the consciousness of the right hemisphere) and the operative level (associated with the consciousness of the left hemisphere). The descriptive level is more general, and the operative level is more particular. Irrational numbers are inaccessible from the operational point of view because they are incomputable in a finite number of steps. Only some of them are expressible at the descriptive level.
In MENTAL every expression is a computation because every expression is evaluated and produces a result, which may be the expression itself.
The mathematical universe is infinite. Our universe reflects a particular mathematical structure.
General-particular distinction.
In MENTAL we distinguish between general mathematical structure (containing some parameter) and manifestations of that general mathematical structure, which are particular mathematical structures.
Relations.
In MENTAL, the key lies in the relationships between the elements. Names are irrelevant. It is the relationships that are important. For example, the expression abc is the relationship between three elements defined by the sequence (a b c).
The universal meta-expression.
Ω, the set of all possible expressions, is not an expression. That is why it is said to be a meta-expression, so it does not include itself. The "MENTALverse" is the universe of all possible expressions of MENTAL. Therefore, the MENTALverse is Ω.
Degrees of freedom.
Degrees of freedom are the universal semantic primitives, the primary archetypes. But there are axioms that relate these primitives to each other. The exploration of all the possibilities of the mathematical world is the exploration of all the possibilities of MENTAL.
Platonism.
The mathematical universe is of the Platonic type, in which neither physical space nor time exists, but abstract space and time do exist. And, therefore, there is computational time, which manifests itself at the physical level as physical time, so the universe is computing at the deep level and manifesting it superficially as physical motions and processes. Physical processes can be considered computations.
Physical universe.
Our physical universe is a reflection or manifestation of the mathematical universe. Its laws and properties are mathematical, but it is not a mathematical object. The universe is, at its core, mathematical. But mathematics is not the ultimate level of reality. The ultimate level of reality resides in the primary archetypes. Only mathematical laws that are descriptive and operative are manifested at the physical level.
Invention and discovery.
Mathematics is both a discovery and an invention, for there are many ways or paradigms of approaching the mathematical world.
Wigner's question.
Mathematics is so effective in describing the physical world because this physical world is a manifestation of the mathematical one. Reality is not a mathematical structure, but a manifestation at the physical level of a mathematical structure. Physical world and mathematical world share the same primary archetypes.
Non-isomorphism.
There is no identification between the mathematical world and the physical world. The physical world is a manifestation of the mathematical world. Only some mathematical structures manifest at the physical level. Not all of them are manifested. Every physical structure has an underlying mathematical structure, but not every mathematical structure has physical reality. The unification of physics and mathematics is somewhat impossible because mathematics transcends the physical world.
The physical world and its laws are the sensible reference that allows us, through a process of abstraction, to transcend it and evoke or connect with the mathematical world. All possible mathematical structures already exist at the Platonic level, but only some of them manifest themselves at the mental or physical level.
The frog and the bird.
For the frog, the world is complex because he perceives it from the superficial and particular. For the bird, the world is simple, because it perceives it from the general and universal. MENTAL unites or connects the vision of the frog and the bird: the particular is a manifestation of the universal. All things are connected at a deep level, at a metaphysical level, where there is neither space nor time.
TOE.
MENTAL is a TOE because all things share the primary archetypes, including the physical world, the mental world, the mathematical world and the possible worlds. It is a TOE of great simplicity, and where each primitive has its corresponding opposite or dual.
Consciousness.
For Tegmark, consciousness arises from the complexity of mathematical substructures. It is just the opposite: consciousness resides in the primary archetypes that unite opposites or duals. It is impossible for a mathematical structure to be self-conscious. But every mathematical structure, whether simple or complex, reflects or manifests the archetypes of consciousness.
The infinite.
Nothing in the physical world corresponds to the concept of infinity. The presence of infinity in physical theories is dismissed because it is interpreted as an error or defect in the theory.
ERH.
ERH is false. External reality is not independent of humans because internal (mental) reality and external (physical) reality share the same primary archetypes.
The generalization of the MMM diagram
In spite of the above mentioned hierarchy MENTAL-MATH-PHYSICAL, they all share the same primary archetypes. Penrose's 3 arrows diagram is a diagram that reflects superficial relationships. The diagram we propose is a reflection of the principle of descending causality: it is radial, like Piet Hut's, but identifying the central element and generalizing the periphery:
At the center reside the primary archetypes, the archetypes of consciousness structured as the MENTAL language.
The ends of the spokes are the different manifestations of the primary archetypes structured as the different disciplines: mathematics, physics, psychology, philosophy, linguistics, etc.
This diagram − which we can call "MENTAL-centric"− is simpler and more generic than that of Penrose. And according to the principle of Occam's razor, it is more likely to be true.
Addenda
The Mandelbrot set
The paradigm of human-independent mathematical structure is the Mandelbrot set, introduced by Benoit Mandelbrot in 1980. It is produced by a very simple mathematical formula of recursive type. We are the explorers of the mathematical universe, which is independent of us, a universe in which we find the Platonic mathematical structure that is the Mandelbrot set, which can be considered a universe in itself. The complexity of the Mandelbrot set is only at the edge of the region of the set. According to Penrose −in his book "The Emperor's New Mind"− "The Mandelbrot set is not an invention of the human mind; it was a discovery. Like Mount Everest, the Mandelbrot set is right up there."
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