‘Shut up and calculate!’ – is a very popular phrase in the quantum ...

Max Erik Tegmark is a Swedish-American cosmologist. Tegmark is a pr...

Metaphysical solipsism is the variety of idealism which asserts tha...

The Copenhagen interpretation was largely devised in the years 1925...

A theory of everything (ToE), final theory, ultimate theory, or mas...

Pythagoras interpreted the Pythagoras theorem and it's universal ma...

Galileo famously stated in Il Saggiatore in 1623: "Philosophy is wr...

The conclusion of "The unreasonable effectiveness of mathematics": ...

This reminds me of Alpha Go Zero, Google DeepMind's neural network ...

[This youtube video](https://www.youtube.com/watch?v=Qd28YfV5ZAY) w...

This is a big leap of faith. One could make the fairly un-controver...

In fact, this is already done on a number of [Fermat's library T-sh...

If the multiverse is a *purely* mathematical object, and if we have...

arXiv:0709.4024v1 [physics.pop-ph] 25 Sep 2007

Shut up and calculate

∗

Max Tegmark

Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139

I advocate an extreme “shut-up-and-calculate” approach to physics, where our external physical

reality is assumed to be purely mathematical. This brief essay motivates this “it’s all just equations”

assumption and discusses its implications.

What is the meaning of life, the universe and every-

thing? In the sci-ﬁ spoof The Hitchhiker’s Guide to the

Galaxy, the answer was found to be 42; the hardest part

turned out to be ﬁnding the real question. Indeed, al-

though our inquisitive ancestors undoubtedly asked such

big questions, their search for a “theory of everything”

evolved as their knowledge gre w . As the ancient Greeks

replaced myth-based explanations with mechanistic mod-

els of the solar s ystem, their emphasis shifted from asking

“why” to asking “how”.

Since then, the scop e of our questioning has dwindled

in some areas and mushroomed in others. Some ques-

tions were abandoned as naive or misguided, such as ex-

plaining the sizes of planetary orbits from ﬁrst principles,

which was popular during the Rena issance. The same

may happen to currently trendy pursuits like predict-

ing the amount of dark energy in the cosmos, if it turns

out that the amount in our neighbourhood is a histo rical

accident. Yet our ability to answer other questions has

surpassed earlier generations’ wildest expectations: New-

ton would have been amazed to know that we would one

day measure the age of our universe to an accuracy of 1

per cent, and comprehend the microworld well enough to

make an iPhone.

Mathematics has played a striking role in these suc-

cesses. The idea that our universe is in some s e nse math-

ematical goes back at le ast to the Pythagoreans of an-

cient Gree c e, and has s pawned centuries of discussion

among physicists and philosophers. In the 17th century,

Galileo fa mously stated that the universe is a “grand

book” written in the language of mathematics. More

recently, the physics Nobel laureate Eugene Wigner ar-

gued in the 196 0s that “the unre asonable eﬀectiveness of

mathematics in the natural sc ience s” demanded an ex-

planation.

Here, I will push this idea to its extreme and argue that

our universe is not just described by mathematics — it is

mathematics. While this hypothesis might sound rather

abstract and far-fetched, it makes startling predictions

about the structure of the universe that could be testable

by observations. It should also be useful in narrowing

down what an ultimate theo ry of everything can loo k

like.

∗

This is the “director’s cut” version of the September 15 2007 New

Scientist cover story. The “full strength” version is the much longer

article [1], which includes references.

The foundation of my argument is the a ssumption that

there exists an external physical reality indepe ndent of us

humans. T his is not too controversial: I would guess that

the majority of physicists favour this lo ng-standing idea,

though it is still debated. Metaphysical solipsists reject

it ﬂat out, and supporters of the so-called Copenhagen

interpre tation of quantum mechanics may reject it on the

grounds that there is no reality without observation (New

Scientist, 23 June, p 30). Assuming an external reality

exists, physics theories aim to describe how it works. O ur

most successful theo ries, such as general relativity and

quantum mechanics, describe o nly parts of this rea lity:

gravity, for instance, or the behaviour of subatomic parti-

cles. In contrast, the holy g rail of theoretical physics is a

theory of everything — a complete description of reality.

My personal quest for this theory begins with a n ex-

treme argument about what it is a llowed to look like. If

we as sume that reality exists independently o f humans,

then for a description to be complete, it must also be

well-deﬁned according to non-human entities — aliens or

supe rcomputers, say — that lack any understanding of

human concepts. Put diﬀerently, such a description must

be expressible in a form that is devoid of any human

baggage like “particle”, “observation” or other English

words.

In contrast, all physics theories that I have bee n

taught have two components: mathematical equations,

and words that explain how the equations are connected

to what we observe and intuitively understand. When

we derive the consequences o f a theory, we introduce new

concepts — protons, molecules, stars — because they are

convenient. It is important to remember, however, that

it is we humans w ho create these concepts; in principle,

everything c ould be c alculated without this baggage. For

example, a suﬃciently powerful supercomputer could cal-

culate how the state of the universe evolves over time

without interpre ting what is happening in human terms.

All of this raises the question: is it possible to ﬁnd a

description of externa l reality that involves no baggage?

If so, such a description of objects in this external re-

ality and the relations between them would have to be

completely abstract, forcing any words or symbols to be

mere labels with no preconceived meanings whatsoever.

Instead, the only properties of these entities would be

those embodied by the rela tions between them.

This is where mathematics comes in. To a modern lo-

gician, a mathematical structure is precisely this: a set of

abstract entities with relations between them. Take the

integers, for instance, or ge ometric objects like the do-

2

decahedron, a favourite of the Pythagorea ns. This is in

stark contrast to the way most of us ﬁrst perceive math-

ematics — either as a sadistic form of punishment, or as

a bag of tricks for manipulating numbers. Like physics,

mathematics has evolved to as k broader questions.

Modern mathematics is the formal study of structures

that can be deﬁned in a purely abstract way. Think of

mathematical symbols as mere labels without intrinsic

meaning. It doesn’t ma tter whether you write “two plus

two equals four”, “2 + 2 = 4” or “dos mas dos igual a

cuatro”. The notation used to denote the entities and the

relations is irrelevant; the only properties of integers are

those embodied by the relations between them. That is ,

we don’t invent mathematical structures — we discover

them, and invent only the notation for describing them.

So here is the crux of my arg ument. If you believe

in an external r e ality independent of humans, then you

must also believe in what I call the mathematical uni-

verse hypothesis: that our physical reality is a mathe-

matical structure. In other words, we all live in a g igantic

mathematical object — one that is more elaborate than a

dodecahedron, and probably also more complex than ob-

jects with intimidating names like Calabi-Yau manifolds,

tensor bundles and Hilbert spaces , which appea r in to-

day’s most advanced theories . Everything in our world

is purely mathematical — including you.

If that is true, then the theory of everything must be

purely abstract and mathematical. Although we do not

yet know what the theory would look like, particle physics

and cosmology have rea ched a point where all measure-

ments ever made can be explained, at least in principle,

with equations that ﬁt on a few pages and involve merely

32 unexplained numerical constants (Physical Review D,

vol 73, 023505). So it seems possible that the cor rect

theory of everything could even turn out to be simple

enough to describe with equations that ﬁt on a T-shirt.

Before discussing whether the mathematical universe

hypothesis is correct, however, there is a more urgent

question: what does it actually mean? To understand

this, it helps to distinguish between two ways of viewing

our external physical re ality. One is the outside overview

of a physicist studying its mathematical str ucture, like

a bird surveying a landsc ape from high above; the o ther

is the inside view of an obse rver living in the world de-

scribed by the structure, like a frog liv ing in the land-

scape s urveyed by the bird.

One issue in relating these two perspectives involves

time. A mathematical structure is by deﬁnition an ab-

stract, immutable entity ex isting outside of space and

time. If the history of our universe were a movie, the

structure would corres po nd not to a single frame but to

the entire DVD. So from the bird’s perspective, trajec-

tories of objects moving in four-dimensional space-time

resemble a ta ngle of spaghetti. Where the frog sees

something moving with constant velocity, the bird sees

a straight strand of uncooked spaghetti. Where the frog

sees the moon orbit the Earth, the bird sees two inter-

twined spa ghetti strands. To the frog, the world is de-

scribed by Newton’s laws of motion and gravitation. To

the bird, the world is the geometry of the pasta.

A further subtlety in relating the two perspectives in-

volves explaining how an observer could be purely math-

ematical. In this example, the frog itself must consist of

a thick bundle o f pa sta whose highly complex structure

corresponds to particles that store and process informa-

tion in a way that gives rise to the familiar sensation of

self-awareness.

Fine, so how do we test the mathematical universe

hypothesis? For a start, it predicts that further math-

ematical r egularities remain to be discovered in nature.

Ever since Galileo promulgated the idea of a mathemat-

ical cosmos, there has been a steady progression of dis-

coveries in that vein, including the standard mo del of

particle physics, which captures striking mathematical

order in the microcosm of elementary particles and the

macrocosm of the early universe.

That’s not all, however. The hypothesis also makes a

more dramatic prediction: the existence of parallel uni-

verses. Many types of “multiverse” have been proposed

over the years, and it is use ful to classify them into a four-

level hierarchy. The ﬁrst three levels corre spond to non-

communicating parallel worlds within the same math-

ematical structure: level I simply means distant regions

from which light has not yet had time to reach us; level II

covers regions that are fore ver unr eachable because of the

cosmologic al inﬂation of intervening space; and le vel III,

often called “many worlds”, involves non-communicating

parts of the so-called Hilbert space of quantum mechanics

into which the universe can “split” during certa in quan-

tum events. Level IV refers to parallel worlds in distinct

mathematical structures, which may have fundamentally

diﬀerent laws of physics.

Today’s best estimates suggest that we need a huge

amount o f information, perhaps a Googol (10

100

) bits,

to fully describ e our frog’s view of the observable uni-

verse, down to the positions of every star and g rain of

sand. Most physicists hope for a theory of everything

that is much s impler than this and can be speciﬁed in

few enough bits to ﬁt in a book, if not on a T-shirt.

The mathematical universe hypothesis implies that such

a s imple theory must predict a multiverse. Why? Be-

cause this theory is by deﬁnition a complete description

of reality: if it lacks enough bits to completely specify our

universe, then it must instead describe a ll possible com-

binations of stars, sand grains and such — so that the

extra bits that des c ribe our universe simply encode which

universe we are in, like a multiversal telephone number.

In this way, describing a multiverse can be simpler than

describing a single universe.

Pushed to its extreme, the mathematical universe hy-

pothesis implies the level-IV multiverse, which includes

all the other levels within it. If there is a par ticula r math-

ematical structure that is our universe, and its properties

correspond to our physical laws, then each mathemati-

cal structure with diﬀerent properties is its own universe

with diﬀerent laws. Indeed, the level-IV multiverse is

3

compulsory, since mathematical str uctures are not “cre-

ated” and don’t exist “somewhere” — they just exist.

Stephen Hawking o nce asked, “What is it that breathes

ﬁre into the equations and makes a universe for them to

describe?” In the case of the mathematical cosmos, there

is no ﬁre-breathing required, since the point is not that

a mathematical structure describes a universe, but that

it is a universe.

The existence of the level-IV multiverse also answers a

confounding questio n emphasised by the physicist John

Wheeler: even if we found equations that describe our

universe perfectly, then why these particular equations,

not others? The answer is that the other equations gov-

ern parallel universes, and that our universe has these

particular equations becaus e they are sta tistica lly likely,

given the distribution of mathematical structures that

can s upport observers like us.

It is crucia l to ask whether para llel universes are within

the purview of science, or are merely speculation. Paral-

lel universes are not a theory in themselves, but rather a

prediction made by 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. General re lativ-

ity, for instance, has successfully predicted many things

that we can observe, such as gravitational lensing, so we

also take seriously its predictions for things we cannot,

like the internal structure of black holes.

So here’s a testable prediction of the mathematical uni-

verse hypothesis: if we exist in many parallel universes,

then we sho uld expect to ﬁnd ourselves in a typical one.

Suppose we succeed in computing the probability distri-

bution for some quantity, say the dark energy density or

the dimensionality of space, mea sured by a typical ob-

server in the part of the multiverse where this quantity

is deﬁned. If we ﬁnd that this distribution makes the

value measured in our own universe highly atypical, it

would rule out the multiverse, and hence the mathemat-

ical universe hypothesis. Although we are still far from

understanding the requirements for life, we could start

testing the multiverse prediction by assessing how typical

our universe is as regards dark matter, dark energy and

neutrinos, because these substances aﬀect only better un-

derstood pr ocesses like galax y formation. This prediction

has passed the ﬁrst of such tests, bec ause the abundance

of these substances has been measur e d to be rather typi-

cal of what you might measure from a ra ndom galaxy in

a multiverse. However, more accur ate calculations and

measurements mig ht still rule out such a multiverse.

Ultimately, why should we believe the mathematical

universe hypothesis? Perhaps the most compelling ob-

jection is that it feels counter-intuitive and disturbing.

I personally dismiss this as a failure to a ppreciate Dar-

winian evolution. Evolution endowed us with intuition

only for those aspects of physics that had survival value

for our distant ancestors, such as the parabolic trajec-

tories of ﬂying rocks. Darwin’s theory thus makes the

testable prediction that whenever we look beyond the

human scale, our evolved intuition should break down.

We have repeatedly tested this prediction, and the re -

sults overwhelmingly support it: our intuition breaks

down at high speeds, where time slows down; on small

scales, where particles can be in two places at once; and

at high temperatures, where colliding particles change

identity. To me, an electron colliding with a positron and

turning into a Z-boson feels about as intuitive as two col-

liding car s turning into a cruise ship. The point is that if

we dismiss s e e mingly weird theories out of hand, we risk

dismissing the corre c t theory of e verything, whatever it

may turn out to be.

If the mathematica l universe hypothesis is true, then it

is grea t news for science, a llowing the p ossibility that an

elegant uniﬁcation of physics and mathematics will one

day allow us to understand reality more deeply than most

dreamed possible. Indeed, I think the mathematical cos-

mos with its multiverse is the best theor y of everything

that we could hope for, because it would mean that no

aspect of r e ality is oﬀ-limits from our scientiﬁc quest to

uncover reg ularities and make quantitative predictions.

However, it would also s hift the ultimate question

about the universe once again. We would abandon as

misguided the question of which particular mathemati-

cal equations describe all of r e ality, and instead ask how

to compute the frog’s view of the universe — o ur ob-

servations — fro m the bird’s view. That would deter-

mine whether we have uncovered the true structure of

our universe, and help us ﬁgur e out which corner of the

mathematical cosmos is our home.

[1] M Tegmark 2007, “The Mathematical Universe”, arXiv

0704.0646 [gr-qc], submitted to Foundations of Physics

‘Shut up and calculate!’ – is a very popular phrase in the quantum theory community to express the attitude of the typical working quantum physicist confronting people who express
doubts or anxiety over what quantum theory really means. The phrase originated in an article by the physicist David Mermin, which appeared in the journal Physics Today, in 1989. "If I were forced to sum up in one sentence what the Copenhagen interpretation says to me, it would be “Shut up and calculate!”" For more about the history of "Shut up and calculate", see: http://www.physicsandmore.net/resources/Shutupandcalculate.pdf
Galileo famously stated in Il Saggiatore in 1623: "Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth."
![Imgur](https://i.imgur.com/qOHs3kZ.png)
This reminds me of Alpha Go Zero, Google DeepMind's neural network that learns to become the best GO player in the world in 1.5 days from scratch, learning in an unsupervised manner. Similarly, perhaps supercomputers could learn in an unsupervised manner and calculate over time how the universe evolves. It could also develop concepts (or not) that are relatable to human terms.
Metaphysical solipsism is the variety of idealism which asserts that nothing exists externally to this one mind, and since this mind is the whole of reality then the "external world" was never anything more than an idea: https://en.wikipedia.org/wiki/Metaphysical_solipsism
This brings to my mind philosophical trends where the ontological claims about the world, and the epistemological methods that are valid for acquiring "knowledge" about the world(merely: correlated data under a certain network of interpretations) are in their sense or nature, relational.
It also brings forth several approaches to quantum mechanics- with the Relational Interpretation of Rovelli Carlo as the most intense- that are pointing out how an objectification of the observational action can't but fail on a conceptual level, hitherto leading to the minimalist approach( or apophatic) of the Copenhagen school(which is not so much on a disagreement with the above, but just admits it's inability to offer an ontological explanation).
It finally seems to me, that the name of the paper and the relational approaches( even QFT is based on the notion of interaction more than that of entities) is connected intimately with the modern understanding of correlation- be it QFT, Statistical Mechanics, Renormalization, Chaotic systems).
Might be worth mentioning his book The Mathematical Universe here too!
A theory of everything (ToE), final theory, ultimate theory, or master theory is a hypothetical single, all-encompassing, coherent theoretical framework of physics that fully explains and links together all physical aspects of the universe. Finding a ToE is one of the major unsolved problems in physics. Over the past few centuries, two theoretical frameworks have been developed that, as a whole, most closely resemble a ToE. These two theories upon which all modern physics rests are general relativity and quantum field theory. https://en.wikipedia.org/wiki/Theory_of_everything
In fact, this is already done on a number of [Fermat's library T-shirts](https://teespring.com/stores/fermats-library?aid=marketplace&tsmac=marketplace&tsmic=campaign) ;)
Max Erik Tegmark is a Swedish-American cosmologist. Tegmark is a professor at the Massachusetts Institute of Technology and the scientific director of the Foundational Questions Institute. He is also a co-founder of the Future of Life Institute, and has accepted donations from Elon Musk to investigate existential risk from advanced artificial intelligence. His early research focused on cosmology, while his current focus is on artificial intelligence. In August of 2017, he published the book "Life 3.0: Being Human in the Age of Artificial Intelligence".
The Copenhagen interpretation was largely devised in the years 1925 to 1927 by Niels Bohr and Werner Heisenberg. According to the Copenhagen interpretation, physical systems generally do not have definite properties prior to being measured, and quantum mechanics can only predict the probabilities that measurements will produce certain results. The act of measurement affects the system, causing the set of probabilities to reduce to only one of the possible values immediately after the measurement. This feature is known as wave function collapse.
There have been many objections to the Copenhagen interpretation over the years. These include: discontinuous jumps when there is an observation, the probabilistic element introduced upon observation, the subjectiveness of requiring an observer, the difficulty of defining a measuring device, and to the necessity of invoking classical physics to describe the "laboratory" in which the results are measured.
Alternatives to the Copenhagen interpretation include the many-worlds interpretation, the De Broglie–Bohm (pilot-wave) interpretation, and quantum decoherence theories. https://en.wikipedia.org/wiki/Copenhagen_interpretation
[This youtube video](https://www.youtube.com/watch?v=Qd28YfV5ZAY) with Neil deGrasse Tyson and Princeton theoretical physicist Paul Steinhardt explains some nice philosophical implications of one of the multiverse theories.
The conclusion of "The unreasonable effectiveness of mathematics": "Let me end on a more cheerful note. The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning."
Eugene Wigner was a was a Hungarian-American theoretical physicist, engineer and mathematician. He received half of the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles." To read the full document: http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html
This is a big leap of faith. One could make the fairly un-controversial statement that our physical reality *has* a mathematical structure. Prof. Tegmark says that reality *is* a mathematical structure, and one could interpret it a few different ways. But how does it necessarily follow that everything is *purely* mathematical? One would have to show matter, energy, the fundamental particles and quantum mechanics arising out of some axiomatic basis.
If the multiverse is a *purely* mathematical object, and if we have a theory of *everything*, it should occupy exactly 0 bits of information. Or perhaps a mysterious one bit : that the multiverse exists.
Pythagoras interpreted the Pythagoras theorem and it's universal mathematical truth as divine and it lead him and his followers to believe that numbers and mathematical relations could explain the universe. The Pythagorans believed that before the universe there existed infinity with no limits or structure and that God would have then created a limit upon all things so that infinity would be able to take on an actual size. The Pythagorans were a community of 300 that lived in Croton, southern Italy and lived by structured rules and standards. They worshipped numbers and believed in the divinity of mathematics. No one knows for sure if Pythagoras actually existed or if his theorem is attributable to him or people in the community of Pythagorans. Many believe he is a mythical figure. To read more about Pythagoras and Pythagorans: http://classicalwisdom.com/pythagoras-and-the-revolution-of-mathematics/