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- Jun 14, 2018
The Mathematical Universe Hypothesis is a theory of everything constructed by Max Tegmark which asserts that first an foremost, that our physical universe is an entity existing wholly independently of human existence and experience.
http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
https://arxiv.org/pdf/gr-qc/9704009.pdf
https://arxiv.org/pdf/0704.0646.pdf
Much of what is written below comes directly from this, edited and reformatted for forum viewing.
PART I: INTRO TO MUH
From the assertion that our physical universe is an entity existing wholly independently of human existence and experience, we can assert that for a complete description, a theory of everything, of the universe to exist, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”.
The physical theories we use to describe our world can be crudely put into a sort of family tree, where each might, at least in principle, be derivable from more fundamental ones above it, of which they are only approximations thereof. All of them have two components: mathematical equations and “baggage”, words that explain how they are connected to what we humans observe and intuitively understand.
Quantum mechanics as usually presented in textbooks has both components: some equations as well as three fundamental postulates written out in plain English. At each level in the hierarchy of theories, new concepts (e.g. , protons, atoms, cells, organisms, cultures) are introduce d because they are convenient, capturing the essence of what is going on without recourse to the more fundamental theory above it. It is important to remember, however, that it is we humans who introduce these concepts and the words for them: in principle, everything could have been derived from the fundamental theory at the top of the tree, although such an extreme reductionist approach appears useless in practice.
Crudely speaking, the ratio of equations to baggage decreases as we move down the tree, dropping near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top are highly mathematical, and physicists are still struggling to articulate the concepts, if any, in terms of which we can understand them. The MUH implies that the TOE indicated by the question mark at the top is purely mathematical, with no baggage whatsoever.
However, could it ever be possible to give a description of the external reality involving no baggage? If so, our description of entities in the external reality and relations between them would have to be completely abstract, forcing any words or other symbols used to denote them to be mere labels with no preconceived meanings whatsoever. A mathematical structure is precisely this: abstract entities with relations between them.
PART II: COSMOLOGICAL IMPLICATIONS OF MUH
Parallel universes are now all the rage, cropping up in books, movies and even jokes: “You passed your exam in many parallel universes — but not this one.” However, they are as controversial as they are popular, and it is important to ask whether they are within the purview of science, or merely silly speculation. They are also a source of confusion, since many forget to distinguish between different types of parallel universes that have been proposed, whereas [15, 16] argued that the various proposals form a natural four-level hierarchy of multiverses (Figure IV C) allowing progressively greater diversity:
• Level I: A generic prediction of cosmological inflation is an infinite “ergodic” space, which contains Hubble volumes realizing all initial conditions — including an identical copy of you about 10^10^29 m away.
• Level II: Given the mathematical structure corresponding to the fundamental laws of physics that physicists one day hope to capture with equations on a T-shirt, different regions of space can exhibit different effective laws of physics (physical constants, dimensionality, particle content, etc.) corresponding to different local minima in a landscape of possibilities.
• Level III: In unitary quantum mechanics [29, 30], other branches of the wave function add nothing qualitatively new, which is ironic given that this level has historically been the most controversial.
• Level IV: Other mathematical structures give different fundamental equations of physics for that T-shirt.
The key question is therefore not whether there is a multiverse (since Level I is the rather uncontroversial cosmological standard model), but rather how many levels it has.
The issue of evidence and whether this is science or philosophy has been discussed at length in the recent literature. The key point to remember is that parallel universes are not a theory, but a prediction of certain theories. For a theory to be falsifiable, we need not be able to observe and test all its predictions, merely at least one of them. Consider the following analogy:
General Relativity — Black hole interiors
Inflation — Level I parallel universes
Inflation+landscape — Level II parallel universes
Unitary quantum mechanics — Level III parallel universes
MUH — Level IV parallel universes
Because Einstein’s theory of General Relativity has successfully predicted many things that we can observe, we also take seriously its predictions for things we cannot observe, e.g., that space continues inside black hole event horizons and that (contrary to early misconceptions) nothing funny happens right at the horizon. Likewise, successful predictions of the theories of cosmological inflation and unitary quantum mechanics have made some scientists take more seriously their other predictions, including various types of parallel universes.
If the TOE at the top of Figure 1 exists and is one day discovered, then an embarrassing question remains, as emphasized by John Archibald Wheeler: Why these particular equations, not others? Could there really be a fundamental, unexplained ontological asymmetry built into the very heart of reality, splitting mathematical structures into two classes, those with and without physical existence? After all, a mathematical structure is not “created” and doesn’t exist “somewhere”. It just exists.
As a way out of this philosophical conundrum, I have suggested that complete mathematical democracy holds: that mathematical existence and physical existence are equivalent, so that all mathematical structures have the same ontological status. This can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato’s realm of ideas, the Mindscape of Rucker, exist “out there” in a physical sense, casting the so-called modal realism theory of David Lewis in mathematical terms akin to what Barrow refers to as “π in the sky”. If this theory is correct, then since it has no free parameters, all properties of all parallel universes (including the subjective perceptions of self-aware substructures in them) could in principle be derived by an infinitely intelligent mathematician.
In the context of the MUH, the existence of the Level IV multiverse is not optional. As was discussed in detail in Section II D, the MUH says that a mathematical structure is our external physical reality, rather than being merely a description thereof. This equivalence between physical and mathematical existence means that if a mathematical structure contains a SAS (self-aware substructure), it will perceive itself as existing in a physically real world, just as we do (albeit generically a world with different properties from ours). Stephen Hawking famously asked “what is it that breathes fire into the equations and makes a universe for them to describe?”. In the context of the MUH, there is thus no breathing required, since the point is not that a mathematical structure describes a universe, but that it is a universe.
In summary, there are three key points to take away
from our discussion above:
1. The ERH implies that a “theory of everything” has no baggage.
2. Something that has a baggage-free description is precisely a mathematical structure.
Taken together, this implies the Mathematical Universe Hypothesis formulated on the first page of this article, i.e., that the external physical reality described by the TOE is a mathematical structure
3. From this all mathematical structures are physical realities, and vice versa.
What do you people think of this hypothesis?
http://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
https://arxiv.org/pdf/gr-qc/9704009.pdf
https://arxiv.org/pdf/0704.0646.pdf
Much of what is written below comes directly from this, edited and reformatted for forum viewing.
PART I: INTRO TO MUH
From the assertion that our physical universe is an entity existing wholly independently of human existence and experience, we can assert that for a complete description, a theory of everything, of the universe to exist, it must be well-defined also according to non-human sentient entities (say aliens or future supercomputers) that lack the common understanding of concepts that we humans have evolved, e.g., “particle”, “observation” or indeed any other English words. Put differently, such a description must be expressible in a form that is devoid of human “baggage”.
The physical theories we use to describe our world can be crudely put into a sort of family tree, where each might, at least in principle, be derivable from more fundamental ones above it, of which they are only approximations thereof. All of them have two components: mathematical equations and “baggage”, words that explain how they are connected to what we humans observe and intuitively understand.
Quantum mechanics as usually presented in textbooks has both components: some equations as well as three fundamental postulates written out in plain English. At each level in the hierarchy of theories, new concepts (e.g. , protons, atoms, cells, organisms, cultures) are introduce d because they are convenient, capturing the essence of what is going on without recourse to the more fundamental theory above it. It is important to remember, however, that it is we humans who introduce these concepts and the words for them: in principle, everything could have been derived from the fundamental theory at the top of the tree, although such an extreme reductionist approach appears useless in practice.
Crudely speaking, the ratio of equations to baggage decreases as we move down the tree, dropping near zero for highly applied fields such as medicine and sociology. In contrast, theories near the top are highly mathematical, and physicists are still struggling to articulate the concepts, if any, in terms of which we can understand them. The MUH implies that the TOE indicated by the question mark at the top is purely mathematical, with no baggage whatsoever.
However, could it ever be possible to give a description of the external reality involving no baggage? If so, our description of entities in the external reality and relations between them would have to be completely abstract, forcing any words or other symbols used to denote them to be mere labels with no preconceived meanings whatsoever. A mathematical structure is precisely this: abstract entities with relations between them.
PART II: COSMOLOGICAL IMPLICATIONS OF MUH
Parallel universes are now all the rage, cropping up in books, movies and even jokes: “You passed your exam in many parallel universes — but not this one.” However, they are as controversial as they are popular, and it is important to ask whether they are within the purview of science, or merely silly speculation. They are also a source of confusion, since many forget to distinguish between different types of parallel universes that have been proposed, whereas [15, 16] argued that the various proposals form a natural four-level hierarchy of multiverses (Figure IV C) allowing progressively greater diversity:
• Level I: A generic prediction of cosmological inflation is an infinite “ergodic” space, which contains Hubble volumes realizing all initial conditions — including an identical copy of you about 10^10^29 m away.
• Level II: Given the mathematical structure corresponding to the fundamental laws of physics that physicists one day hope to capture with equations on a T-shirt, different regions of space can exhibit different effective laws of physics (physical constants, dimensionality, particle content, etc.) corresponding to different local minima in a landscape of possibilities.
• Level III: In unitary quantum mechanics [29, 30], other branches of the wave function add nothing qualitatively new, which is ironic given that this level has historically been the most controversial.
• Level IV: Other mathematical structures give different fundamental equations of physics for that T-shirt.
The key question is therefore not whether there is a multiverse (since Level I is the rather uncontroversial cosmological standard model), but rather how many levels it has.
The issue of evidence and whether this is science or philosophy has been discussed at length in the recent literature. The key point to remember is that parallel universes are not a theory, but a prediction of certain theories. For a theory to be falsifiable, we need not be able to observe and test all its predictions, merely at least one of them. Consider the following analogy:
General Relativity — Black hole interiors
Inflation — Level I parallel universes
Inflation+landscape — Level II parallel universes
Unitary quantum mechanics — Level III parallel universes
MUH — Level IV parallel universes
Because Einstein’s theory of General Relativity has successfully predicted many things that we can observe, we also take seriously its predictions for things we cannot observe, e.g., that space continues inside black hole event horizons and that (contrary to early misconceptions) nothing funny happens right at the horizon. Likewise, successful predictions of the theories of cosmological inflation and unitary quantum mechanics have made some scientists take more seriously their other predictions, including various types of parallel universes.
If the TOE at the top of Figure 1 exists and is one day discovered, then an embarrassing question remains, as emphasized by John Archibald Wheeler: Why these particular equations, not others? Could there really be a fundamental, unexplained ontological asymmetry built into the very heart of reality, splitting mathematical structures into two classes, those with and without physical existence? After all, a mathematical structure is not “created” and doesn’t exist “somewhere”. It just exists.
As a way out of this philosophical conundrum, I have suggested that complete mathematical democracy holds: that mathematical existence and physical existence are equivalent, so that all mathematical structures have the same ontological status. This can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato’s realm of ideas, the Mindscape of Rucker, exist “out there” in a physical sense, casting the so-called modal realism theory of David Lewis in mathematical terms akin to what Barrow refers to as “π in the sky”. If this theory is correct, then since it has no free parameters, all properties of all parallel universes (including the subjective perceptions of self-aware substructures in them) could in principle be derived by an infinitely intelligent mathematician.
In the context of the MUH, the existence of the Level IV multiverse is not optional. As was discussed in detail in Section II D, the MUH says that a mathematical structure is our external physical reality, rather than being merely a description thereof. This equivalence between physical and mathematical existence means that if a mathematical structure contains a SAS (self-aware substructure), it will perceive itself as existing in a physically real world, just as we do (albeit generically a world with different properties from ours). Stephen Hawking famously asked “what is it that breathes fire into the equations and makes a universe for them to describe?”. In the context of the MUH, there is thus no breathing required, since the point is not that a mathematical structure describes a universe, but that it is a universe.
In summary, there are three key points to take away
from our discussion above:
1. The ERH implies that a “theory of everything” has no baggage.
2. Something that has a baggage-free description is precisely a mathematical structure.
Taken together, this implies the Mathematical Universe Hypothesis formulated on the first page of this article, i.e., that the external physical reality described by the TOE is a mathematical structure
3. From this all mathematical structures are physical realities, and vice versa.
What do you people think of this hypothesis?
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