Cats are curious and fluffty.
They’re also made of cells that are made of molecules that are made of atoms that are
made of particles that are made of quantum fields.
But quantum fields are neither curious nor fluffy.
They have no subjective qualities and have questionable physicality.
They seem to be completely describable by only numbers, and their behavior precisely
defined by equations.
In a sense, the quantum world is made of math.
So does that mean cats are made of math too?
If you believe the Mathematical Universe Hypothesis then yes.
And so are you.
In his essay “The Unreasonable Effectiveness of Mathematics”, the physicist Eugine Wigner
said that “the enormous usefulness of mathematics in the natural sciences is something bordering
on the mysterious”.
This statement was inspired by the observation that so many aspects of the physical world
seem to be describable and predictable by mathematical equations to incredible precision.
The idea that the universe is in some sense mathematical is thousands of years old.
Pythagoras taught that all things are made of numbers.
Plato had some notions too, which I’ll come back to.
But it was the program of reductionism of modern physics that really inspired Wigner’s
statement.
One by one, the seemingly disparate theories in physics revealed themselves to be special
cases of more elegant, underlying mathematical laws until we were left with only two equations
describing all of nature: the Lagrangian of the standard model of particle physics and
Einstein's field equations of general relativity.
All the incredible complexity we see in the universe — from biology to human culture
— appears to ultimately emerge from matter, space and time as numerical quantities obeying
these equations.
So if we can describe baseline reality in purely mathematical terms, can we take this
one step further?
What if we say that the universe isn’t just described by mathematics, but in a more fundamental
way it actually is mathematics?
This is the proposal of MIT cosmologist Max Tegmark.
He calls it the mathematical universe hypothesis.
It states that: our external physical reality is a mathematical structure.
Tegmark makes the case for his hypothesis by first defining the concept of baggage.
Any theory in physics always comes with two components: a set of equations, and a bunch
of sentences explaining exactly how these equations relate to our human intuitions and
what we can observe via experiment.
For example, quantum mechanics is usually presented in textbooks as a handful of equations
along with a list of postulates relating to the equations, written in plain english - baggage,
as Tegmark calls it.
I have to say that his choice of the word “baggage” may be a hint of Tegmark’s
bias towards the primacy of math over anything non-math.
Tegmark points out a pattern in the math-to-baggage ratio in theories of nature.
First, let’s arrange the world into a hierarchy of emergence.
For example, you might say that human culture emerges from individual minds, which emerge
from the neural dynamics of brains, which emerge from the actions of cells, which emerge
from molecules, which emerge from atoms and quantum fields.
In terms of scientific fields, that list corresponds to sociology then psychology then neuroscience
then biology then chemistry then physics.
And maybe then you have a presumptive theory of everything at the very bottom.
In principle, each theory in one of these fields could be derived from theories below
it.
Tegmark points out that theories at the top are mostly regular human language, while theories
at the bottom are mostly math.
A sociology textbook will be filled with paragraphs of text, while a string theory text book is
wall-to-wall group theory and algebraic topology.
If you’re willing to extrapolate this trend, then maybe at the very bottom of the hierarchy,
our coveted theory of everything, there need be no baggage at all, simply a set of elegant
equations.
If we define baggage as the stuff that encodes the relation between the equations and our
human perspective, then if you remove humans you remove the baggage.
So now we invoke the external reality hypothesis: the assumption that there’s an external
physical reality that exists independently of us; and that humans and our baggage-heavy
minds are emergent from this baggage-free baseline reality.
In Tegmark’s view, that leaves only math for our description of that reality.
If there’s nothing else to, say, an electron but a set of numbers describing its position,
momentum and charge, can we say that the electron simply is that set of numbers?
The mathematical universe hypothesis hinges on the notion that mathematical existence
equals physical existence.
This is the most controversial part of the idea, but the most disturbing part is the
implication of this statement.
If mathematical existence implies physical existence, then, according to Tegmark, anything
that exists mathematically also exists physically.
More precisely, any self-consistent mathematical structure that can be written down can be
said to mathematically exist, and so it therefore also physically exists in the same way that
our universe does.
By the way, in this context a mathematical structure is a set of mathematical objects
that obey certain axioms.
And mathematical objects are number-like entities such as the integers, the real numbers, vectors,
etc. and the things that relate these entities like operators, functions, and so on.
Every independent, self-consistent such structure manifests as a universe out there somewhere.
So what started off as a sort of intuitive claim about physics has resulted in a claim
about an infinity of parallel universes—what Tegmark calls the Level 4 multiverse.
It’s a much bigger multiverse than the other kinds—from the level 1 multiverse you get
from the infinite space generated by eternal inflation, to the level 2 multiverse where
the fundamental constants change over that space, to level 3 where the universe splits
into parallel tracks of the quantum multiverse.
Level 4 potentially encompasses all of these, potentially enabling a realization of each
of the lower levels for every consistent set of mathematical laws.
OK, let’s reign this in a bit.
Before I go believing in a new type of multiverse, I have just one or two questions.
What does it even mean for the universe to be “made of math”, really?
What types of mathy universes exist out there?
Can this idea tell us anything about why do we live in a universe with our particular
set of equations?
And is this whole idea even remotely plausible and testable?
First up - what does “made of math” mean?
How can an equation or a number have a type of fundamental existence?
The idea is akin to Plato’s Theory of Forms - the idea that there’s a realm of idealized,
archetypal versions of the things we find in the real world.
It’s a sort of concept-space, where exist the quintessential notions of things - the
fundamental cat, the essence of beauty, the perfect sphere, etc.
The forms in the material world are just crude approximations of these Platonic ideals.
In mathematical platonism, mathematical entities - numbers, geometric forms, functions, operators,
etc. exist independently of the material world.
But Tegmark takes this one step further and suggests that the abstract existence of math
is all there is.
There is an ongoing and unsettled debate over whether math has a fundamental existence “out there” and is so discovered
by humans, or rather is it invented by us.
For now let’s just assume the former.
But then we still have to ask how a bunch of abstract math can end up looking like a universe.
To generate a universe from equations, those equations have to be implemented.
Numbers need to be plugged into the cosmic calculator or whatever.
That suggests you need a computational substrate to implement the math.
But in Tegmark’s story, that’s not the case.
He defines his Platonic mathematical structures to include not just the raw mathematical objects
like numbers and operators, but also every possible instantiation of them.
Actual implementation isn’t required - the possibility of implementation alone is enough
to grant a mathematical structure its full physical realization.
In this view, our own universe is just one of many possible mathematical structures.
We don’t know its full nature, but it incorporates the known mathematical structures of general
relativity and quantum field theory.
We can imagine an enormous - perhaps infinite number of different mathematical structures.
We could change the number of dimensions, the topology of the spacetime, the degrees
of freedom of the quantum fields, in general - we could change the laws of physics.
But completely different mathematical structures are possible too.
For example, the set of integers with the rules of arithmetic is a mathematical structure.
According to the Mathematical Universe Hypothesis, all of these have a physical existence.
Cool.
So does this mean I could whip out my portal gun and visit any universe that I can imagine?
Not at all.
Saying that all mathematical structures physically exist is not the same as saying that all imaginable
universes exist.
Tegmark’s condition is that only self-consistent mathematical structures exist.
Within a mathematical structure it’s possible to make theorems - these are statements in
the language of that structure that are provably true.
According to the mathematician David Hilbert, a consistent mathematical structure is one
in which it’s not possible to simultaneously prove and disprove a given theorem in that
structure.
Tegmark proposes that all mathematical structures that do not have internal contradictions have
a corresponding physical manifestation.
In order to explain why we see the particular universe that we do, we need to invoke the
anthropic principle in its weak version.
We’re naturally only going to see a mathematical structure in which observers can emerge.
Our structure has the internal complexity and fine tuning of its details so that self-aware
mathematical structures arise within the larger structure.
That’s us, by the way.
We observe the structure from the inside, when you play time in the right direction.
But there’s also an outside view, where the whole structure, including all points
in time, just exists.
This is related to the idea of block time in relativity, in which the past and future
have an eternal existence when viewed from outside reality.
OK, that all sounds totally reasonable and intuitive, right?
Hm.
Let’s look at some possible objections.
Several of the great thinkers of modern physics felt very strongly that mathematics is NOT
a fundamental aspect of external reality.
Rather, they believed that our mathematical descriptions of nature were at best models
of our experience of reality. These sorts of sentiments have been expressed by
Arthur Eddington, James Jean, Erwin Schrodinger, Neils Bohr, and Werner
Heisenberg, among others.
Jean described the mathematical methods of physics as being unable to put us in contact
with ultimate reality.
Instead they lead to a shadow-world of symbols, as Eddington put it.
And there’s actually some good evidence that math is really just a limited human construction
rather than a perfect, cosmic language.
For a long time it was thought that all statements in a good mathematical structure had to be
unambiguously either true or false, and that any truth value could be rigorously proven.
But in the 1930s, the mathematician Kurt Gödel rocked the foundations of math with his famous
incompleteness theorems.
There’s way too much to unpack this in detail, but for our purposes, it says that in any
mathematical framework that contains at least basic arithmetic, it’s possible to write
down mathematical statements that are neither true nor false - they cannot be proved right
or wrong using the rules of that mathematical framework.
We say such statements are undecidable.
So if the mathematical structure that describes our world contains equations that are neither
true nor false, then there are laws of physics that are neither true nor false.
This violates one the the basic tenets of the mathematical universe hypothesis - that
its mathematical structures must be internally consistent.
It also gives us some doubt that a Platonic world of perfect math is even a meaningful
concept.
The incompleteness theorems seem problematic, but perhaps we can sidestep the issue by saying
that the only mathematical structures that exist are the ones that are fully decidable.
One way to do that is to demand that the structure only contain computable functions.
A computation is the abstract concept of a program that takes in some numerical value,
does some operations on it using a computational model like a Turing machine, and spits out
some other numerical value.
If a function can be computed then presumably it’s decidable.
However not all computations are immune to undecidability.
The halting problem of Alonzo Church and Alan Turing states that it’s not even possible
to determine whether any given function will be computable in a finite number steps, and
this is related to Godel undecideability.
To satisfy the self-consistency requirement of the Mathematical Universe Hypothesis, we
now need to require that, to qualify for physical existence, a mathematical structure only includes
functions that can be executed as computations that require finite steps.
With all this in mind, Tegmark posits an updated Mathematical Universe Hypothesis called the
Computational Universe Hypothesis. It imbues physical existence to all computable mathematical
structures.
But note that he’s still not requiring that the computations actually be run—rather,
they exist by virtue of being in-principle computable.
Tegmark is certainly not the first to imagine the universe as a computation by the way—although
his hypothesis would encompass all possible such universes.
Fun side note: this idea has a weird consequence for the simulation hypothesis: the idea that
we ourselves might be living in a simulation.
If all simulatable universes exist already, do we need to posit a simulator for our own
universe?
OK, so is the Mathematical Universe Hypothesis plausible?
Well, the fact that Tegmark has to perform some serious intellectual contortions to get
around Gödel undecidability is a point against the simple elegance of the basic hypothesis.
The notion of Platonic existence of mathematics is also fraught, as I’ve mentioned.
On the other hand, Platonic existence may not be any more weird than the idea of physical
reality popping into existence without prior cause.
The mystery of the primal cause is often expressed with the question “Why is there something
rather than nothing?”
That framing has an implied assumption that “nothing” is the default state of reality.
If you flip that, you might instead say that existence is the default state, and that everything
that is possible exists.
This is the Principle of Fecundity, or Principle of Plenitude.
Tegmark’s hypothesis refines what “possible” means in those contexts.
And finally, is this idea even testable?
Not currently.
The only obvious test is via anthropic reasoning.
We should expect to find ourselves in the most typical mathematical structure capable
of supporting our existence, but until we know what structures are possible this is
impossible to evaluate.
So, are we made out of math?
Well to find out the answer, the mathematical universe hypothesis is going to need to be developed
to the point that it makes testable predictions.
But perhaps then it’ll be able to tell us why, among all of the possible level 4 multiverses,
we find ourselves in this particular mathematical spacetime.