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The Mathematical Universe
I explore physics implications of the External Reality Hypothesis (ERH) that
there exists an external physical reality completely independent of us humans.
I argue that with a sufficiently broad definition of mathematics, it implies
the Mathematical Universe Hypothesis (MUH) that our physical world is an
abstract mathematical structure. I discuss various implications of the ERH and
MUH, ranging from standard physics topics like symmetries, irreducible
representations, units, free parameters, randomness and initial conditions to
broader issues like consciousness, parallel universes and Godel incompleteness.
I hypothesize that only computable and decidable (in Godel's sense) structures
exist, which alleviates the cosmological measure problem and help explain why
our physical laws appear so simple. I also comment on the intimate relation
between mathematical structures, computations, simulations and physical
systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs;
more details at http://space.mit.edu/home/tegmark/toe.htm
Some comments on "The Mathematical Universe"
I discuss some problems related to extreme mathematical realism, focusing on
a recently proposed "shut-up-and-calculate" approach to physics
(arXiv:0704.0646, arXiv:0709.4024). I offer arguments for a moderate
alternative, the essence of which lies in the acceptance that mathematics is
(at least in part) a human construction, and discuss concrete consequences of
this--at first sight purely philosophical--difference in point of view.Comment: 11 page
The Relativity of Existence
Despite the success of modern physics in formulating mathematical theories
that can predict the outcome of experiments, we have made remarkably little
progress towards answering the most fundamental question of: why is there a
universe at all, as opposed to nothingness? In this paper, it is shown that
this seemingly mind-boggling question has a simple logical answer if we accept
that existence in the universe is nothing more than mathematical existence
relative to the axioms of our universe. This premise is not baseless; it is
shown here that there are indeed several independent strong logical arguments
for why we should believe that mathematical existence is the only kind of
existence. Moreover, it is shown that, under this premise, the answers to many
other puzzling questions about our universe come almost immediately. Among
these questions are: why is the universe apparently fine-tuned to be able to
support life? Why are the laws of physics so elegant? Why do we have three
dimensions of space and one of time, with approximate locality and causality at
macroscopic scales? How can the universe be non-local and non-causal at the
quantum scale? How can the laws of quantum mechanics rely on true randomness
How to create a universe
The purpose of this paper is (i) to expound the specification of a universe,
according to those parts of mathematical physics which have been experimentally
and observationally verified in our own universe; and (ii) to expound the
possible means of creating a universe in the laboratory
The Duality of the Universe
It is proposed that the physical universe is an instance of a mathematical
structure which possesses a dual structure, and that this dual structure is the
collection of all possible knowledge of the physical universe. In turn, the
physical universe is then the dual space of the latter
Quiescent cosmology and the final state of the universe
It has long been a primary objective of cosmology to understand the apparent
isotropy in our universe and to provide a mathematical formulation for its
evolution. A school of thought for its explanation is quiescent cosmology,
which already possesses a mathematical framework, namely the definition of an
Isotropic Singularity, but only for the initial state of the universe. A
complementary framework is necessary in order to also describe possible final
states of the universe. Our new definitions of an Anisotropic Future Endless
Universe and an Anisotropic Future Singularity, whose structure and properties
differ significantly from those of the Isotropic Singularity, offer a promising
realisation for this framework. The combination of the three definitions
together may then provide the first complete formalisation of the quiescent
cosmology concept.Comment: 7 pages, 3 figures, essay receiving honorable mention in the 2007
Gravity Research Foundation Essay award
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