Our Mathematical Universe *
Quest for the Ultimate Nature of Reality
Author: Max Tegmark
A book I recommended to my friends as I was reading it, which prompted the beginning of “Read Between the Minds” book club.
Here are some bullet point notes.
Four Levels of Parallel Universes
Level I - unobservably distant regions of space
Level II - other post-inflationary regions
Level III - elsewhere in quantum Hilbert space
Level IV - other mathematical structures
Mainstream Physics
Einstein’s theory of general relativity allows for the possibility that space goes on forever. It also allows the option where space is finite without having an end by being “curved”.
About 14 billion years ago, everything we can now observe was hotter than the core of the Sun, expanding so fast that it doubled its size in under a second. We don’t know what happened earlier.
Our Universe spent a few minutes being a giant fusion reactor, converting hydrogen into helium and other light elements, until the cosmic expansion diluted and cooled our Universe enough to stop the fusion. This turned about 25% of hydrogen into helium.
After about 400,000 years of expansion and dilution, the hydrogen-helium plasma cooled into transparent gas. We see this transition as a distant plasma wall with a faint glow called the cosmic microwave background, triggering two Nobel Prizes.
Over the next billions of years, gravity transformed our Universe from uniform into clumpy, amplifying the tiny density fluctuations that we see in the cosmic microwave background into stars, planets, galaxies, etc.
The cosmic expansion predicts that distant galaxies should be receding from us, which is in agreement with what we actually observe.
The entire history of our Universe is accurately described by simple physical laws that let us predict the future from the past, and the past from the future.
These physical laws that govern the history of our Universe are all cast in terms of mathematical equations.
Einstein’s gravity theory is one of the most mathematically beautiful theories, explaining gravity as a manifestation of geometry. It shows that the more mass space contains, the more curved space gets.
Einstein’s theory also inferred the total amount of mass in our Universe. Atoms were found to make up only 4% of this mass, with 96% being unexplained.
This missing mass is ghostly - invisible and able to pass through regular matter undetected. Its gravitational effects suggest two separate substances of opposite character.
Dark matter, which clusters, dilutes as it expands, attracts, and helps galaxy formation.
Dark energy, which doesn’t cluster, doesn’t dliute, repels, and sabotages galaxy formation.
There are serious problems with the earliest stages of Friedmann’s Big Bang model. Inflation theory solves them all, and explains the mechanism that caused the Big Bang.
Inflation explains why space is so flat, and why on average, our distance Universe looks the same in all universe - the 0.002% fluctuations we observe are quantum fluctuations stretched by inflation from microscopic to macroscopic scales, then amplified by gravity into today’s galaxies and cosmic structures.
Inflation even explains cosmic accelerations (which triggered a 2011 Nobel Prize), and says that our Universe grew much like a human baby: an accelerating growth phase, in which the size doubled at regular intervals, followed by a more leisurely decelerating growth phase.
Inflation created our Hot Big Bang, and inflation’s early stages are better thought of as a Cold Little Swoosh, because it was neither hot nor big nor much of a bang.
What we call our Big Bang wasn’t the beginning, but the end (of inflation in our part of space) - and inflation typically continues forever in other places.
Inflation generally predicts that our space isn’t just huge, but infinite, with initial conditions generated randomly by quantum fluctuations.
Controversial Multiverse Stuff
Parallel universes are not a theory, but a prediction of certain theories
Eternal inflation predicts that our Universe is just one of infinitely many universes in a Level I multiverse where everything that can happen does happen somewhere.
Inflation is the leading theory for our cosmic origins because it’s passed observational tests, and parallel universes seem to be a non-optional implication.
Inflation converts potentiality into reality: if a mathematical equation governing uniform space have multiple solutions, then eternal inflation will create infinite regions of space instantiating each of those solutions - this is the Level II multiverse.
Many physical laws and ocostants that are unchanged across a Level I multiverse may vary across a Level II multiverse.
This could explain why many constants in our own Universe are so fine-tuned for life.
This would also give many numbers we’ve measured in physics a new number: they’re not telling us something fundamental about reality, but merely something about our location in it.
Critique of parallel universes has largely shifted from “this makes no sense and I hate it” to “I hate it”.
Everything seems to be made of particles
These particles are purely mathematical objects in the sense that their only intrinsic properties are mathematical properties.
These particles don’t obey the classical laws of physics.
Mathematically, the state of these particles can’t be described by numbers, but a wavefunction, describing the extent to which they are in different places.
This gives them properties of both traditional particles and of waves (superposition).
Particles aren’t allowed to be in only one place (the Heisenberg uncertainty principle), which prevents atoms from collapsing.
The future behaviour of particles is described by the Schrodinger equation, which shows that microscopic superpositions can get amplified into macroscopic superpositions. The textbook formulation postulates that the wavefunction sometimes collapses, violating the Schrodinger equation and introducing fundamental randomness into nature.
The textbook formation of quantum mechanics is either incomplete or inconsistent.
In the mathematically simplest quantum theory, there’s something more fundamental than our 3D space and the particles within it: the wavefunction and the infinite-dimensional place called Hilbert space where it lives.
In this theory, particles can be created and destroyed, and can be in several places at once, but there is, was, and always will be only one wavefunction, moving through Hilbert space as determined by the Schrodinger equation.
This mathematically simplest quantum theory, predicts the existence of parallel universes where you live out countless variations of your life.
It also implies that quantum randomness is an illusion, caused by quantum cloning of you.
There’s nothing quantum about apparent randomness, which happens even if you’re classically cloned.
This mathematically simplest quantum theory also predicts a censorship effect called decoherence, which hides most such weirdness from us, mimicking wave function collapse.
Decoherence happens constantly in your brain, debunking suggestions about quantum consciousness.'
The quantum multiverse is unified with the spatial multiverse, so that a wavefunction for a system describes its infinite copies throughout space, and quantum uncertainty reflects your ignorance about which particular copy you’re observing.
If we live in an infinite uniform space, it doesn’t matter whether the wavefunction ultimately collapses - all of Everett’s many worlds are indistinguishable, and collapse doesn’t prevent all quantum outcomes from actually happening.
The wavefunction and Hilbert space, which constitute arguably the most fundamental physical reality, are purely mathematical objects.
In the internal reality of your mind, the only information you have about the external reality is the small sample transmitted through your senses.
This new information is distorted in many ways, and arguably tells you as much about how your senses and brain work as it tells you about external reality.
The mathematical description of external reality that theoretical physics has uncovered appears very different from the way we perceive this external reality.
Midway between internal and external reality lies the “consensus reality”, the shared description of the physical world that all self-aware observers agree on.
This ultimately splits the “ultimate question of life, the universe, and everything” into two parts that can be tackled separately:
The challenge for physical sciences is deriving the consensus reality from the external reality
The challenge for the cognitive sciences is to derive the internal reality from the consensus reality
Extremely Controversial Multiverse Stuff
Since antiquity, people have puzzled over why our physical world can be so accurately described by mathematics.
Ever since, physicists have kept discovering more patterns in nature that are describable by mathematical equations.
The fabric of our physical reality contain dozens of pure numbers, from which all measured constants can in principle be calculated.
Some key physical entities such as empty space, elementary particles, and the wavefunction appear to be purely mathematical in the sense that their only intrinsic properties are mathematical properties.
The External Reality Hypothesis (ERH) - that there exists an external physical reality completely independent of us humans - is accepted by most but not all physicists.
With a sufficiently broad definition of mathematics, the ERH implies the Mathematical Universe Hypothesis (MUH), that our physical world is a mathematical structure.
This means that our physical world not only is described by mathematics, but that it is mathematical, making us self-aware parts of a giant mathematical object.
A mathematical structure is an abstract set of entities with relations between them. The entities have no baggage - no properties except these relations.
It is crucial not to conflate the language of mathematics (which we invent) with the structure of mathematics (which we discover).
The MUH solves the infamous infinite regress problem where the properties of nature can only be explained from the properties of its parts, which require further explanation ad infinitum: the properties of nature stem not from the properties of its ultimate building blocks (which have no properties at all), but from the relations between these building blocks.
Mathematical structures are eternal and unchanging: they don’t exist in space and time - rather, space and time exist in (some of) them.
The MUH implies that the flow of time is an illusion, as is change.
The MUH implies that it’s not only spacetime that is mathematical, but also everything therein, including the particles we’re made of. Mathematically, this seems to correspond to “fields”: numbers at each point in spacetime that encode what’s there.
The MUH implies that you’re a self-aware substructure that is part of the mathematical structure.
In Einstein’s theory of gravity, you’re a remarkably complex braid-like structure in spacetime, whose intricate pattern corresponds to information processing and self-awareness.
In quantum mechanics, your braid pattern branches like a tree.
The movie-like subjective reality that you’re perceiving right now exists only in your head, as part of your brain’s reality model, and it includes not merely edited highlights of here and now, but also a selection of prerecorded distant and past events, giving the illusion that time flows.
You’re self-aware rather than just aware because your brain’s reality model includes a model of yourself and your relation to the outside world: your perceptions of a subjective vantage point you call “I” are qualia, just as your subjective perceptions of “red” and “sweet” are.
The theory that our external physical reality is perfectly described by a mathematical structure while still not being one is 100% unscientific.
You should expect your current observer moment to be a typical one amount all observer moments that feel like you.
Such reasoning leads to controversial conclusions regarding the end of humanity, the stability of our Universe, the validity of cosmological inflation, and whether you’re a disembodied brain or simulation.
It also leads to the measure problem, a serious scientific crisis that calls into question the ability of physics to measure anything at all.
The MUH implies that mathematical existence equals physical existence.
This means that all structures that exist mathematically exist physically as well, forming the Level IV multiverse.
Exploring the Level IV multiverse doesn’t require rockets or telescopes, simply computers and ideas.
The simplest mathematical structures can be listed by a computer algorithmically, each one with its own unique number.
Mathematical structures, formal systems, and computations are closely related, suggesting that they’re all aspects of the same transcendent structure whose nature we still haven’t understood.
The MUH is compatible with Godel’s incompleteness theorem - the mathematical structure we inhabit can be inconsistent even if we can’t prove it.
The Computable Universe hypothesis (CUH) that the mathematical structure that is our external physical reality is defined by computable functions would make everything provably consistent.
The Finite Universe hypothesis (CUH) that our external physical reality is a finite mathematical structure implies the CUH and eliminates all concerns about reality being undefined.
The CUH/FUH may help to solve the measure problem and explain why our Universe is so simple.
The MUH implies that there are no undefined initial conditions - initial conditions tell us nothing about physical reality, merely about our address in the universe.
The MUH implies that there is no fundamental randomness: randomness is simply the way cloning feels subjectively.
The MUH implies that most of the complexity we observe is an illution, existing only in the eye of the beholder, being merely information about our address in the multiverse.
A collection of things can be simpler to describe than the sum of its parts. Our multiverse is simpler than our Universe, in the sense that it can be described with less information, and the Level IV multiverse is simplest of all, requiring almost no information to describe.
We probably don’t live in a simulation.
THe MUH is in principle testable and falsifiable.
Back to “Normally” Controversial Multiverse Stuff
On the largest and smallest scales, the mathematical fabric of reality becomes evident, while it remains easy to miss on the intermediate scales that humans are usually aware of.
If the ultimate fabric of reality really is mathematical, then everything is in principle understandable to us, and we’ll be limited only by our own imagination.
Although the Level IV multiverse is eternal, our particular Universe might end in a:
Big Chill
Big Crunch
Big Rip
Big Snap
Death Bubbles
Evidence suggests that there’s no other life-form as advances as us in our entire Universe. This is based on three assumptions:
Interstellar colonization is physicall possible and can be easily accomplished if a civilization as advanced as ours has a million years to develop the required technology.
There are billions of habitable planets in our galaxy, many of which formed not only millions, but billions of years before Earth.
A non-negligible fraction of civilizations that can colonize space would choose to do so.
From a cosmic perspective, the future potential of life in our Universe is vastly greater than anything we’ve seen so far.
Yet, humans devote only meager attention and resources to existential risks that threaten life as we know it, including accidental nuclear war and unfriendly artificial intelligence.
Although it’s easy to feel insignificant in our vast cosmos, the entire future of life in our Universe may arguably be decided on our planet in our lifetime!
