Why does anything exist?

Why does anything exist? Why is there something rather than nothing? Wouldn’t nothing have been so much easier?

This question has awed and mystified people throughout time.

The first question which we have a right to ask will be, “Why is there something rather than nothing?”

Gottfried Wilhelm Leibniz in “The Principles of Nature and Grace, Based on Reason” (1714)

Not how the world is, is the mystical, but that it is.

Ludwig Wittgenstein in “Treatise on Logic and Philosophy” (1921)

No question is more sublime than why there is a Universe: why there is something rather than nothing.

Derek Parfit in “Why Anything? Why This?” (2008)

Martin Heidegger called this question the “fundamental question of metaphysics.” But it might as well be the fundamental question for any being — our existence poses a mystery that demands an answer.

Every society in every time has wrestled with this dilemma. It’s our most enduring question. For we all seek to know: why we are here?

Lacking an answer, we are like a ship adrift. Our ignorance on this question makes us like an amnesiac who awakens in a dark and strange place — knowing neither where we are, nor how we got here.

Some say without an answer to this question, we can’t know anything:

It is possible to think that one cannot answer any question if one cannot answer the question of why there is something rather than nothing. How can we know why something is (or should be) a certain way if we don’t know why there is anything at all? Surely this is the first philosophical question that has to be answered.

Robert Nozick in “Philosophical Explanations” (1981)

With an answer to this question we could orientate ourselves. We would know our place in reality, and understand the reason behind it all. An answer to this question would tell us not only why we exist, but also what else exists, both within the universe we see and beyond.

But can this question even be answered?

Some have suggested the answer is unknowable.

Who knows truly? Who here will declare whence it arose, whence this creation? The gods are subsequent to the creation of this. Who, then, knows whence it has come into being? Whence this creation has come into being; whether it was made or not; he in the highest heaven is its surveyor. Surely he knows, or perhaps he knows not.

The Hymn of Creation in “Rig Veda” (c. 1500 B.C.)

For most of history, the question remained beyond the possibility of being answered. But we live in a most-exciting point in time: one where this question has fallen to the progress of human knowledge.

In the past decades, results from physics, cosmology, mathematics, and computer science, have coordinated at last to solve this timeless question. We can now say, with some confidence, why we exist.

The answer we have is more than an idle philosophical speculation — it can be observationally tested and thereby be confirmed or falsified.

So far, observations are in agreement with this answer.

Let us retrace humanity’s steps in finding this answer, and see what this answer reveals about the nature of reality and our place in it.

Two Paths to Existence

One reason we find “Why does anything exist?” so difficult is that there are only two possible answers — both are repugnant to our intuition as each contradicts our commonsense understanding of the world.

Given something exists, either:

1. Something emerged from nothing, or
2. There are self-existent things

The idea that something came out of nothing is contrary to reason. How can nothingness do, nevermind create, anything?

The idea that there exist self-existent things, is contrary to experience. Everything we know appears to have a preceding cause. How could anything create itself, or exist without some creative act?

And yet, that one of these answers must be right seems inescapable. There’s no other way to reach “something exists” without either starting with something at the beginning, or starting with nothing and having something emerge from nothing.

If we seek an answer to this question we have to be willing to accept an idea contrary to our commonsense understanding of the world.

But which of these paths leads to the correct answer?

Something from Nothing?

The first of the two answers is that something emerged from nothing. But how is this possible? Does it even make sense logically?

For at least 2,500 years, humans have debated whether anything can come from nothing. The Greek philosopher Parmenides made the earliest recorded argument that “nothing comes from nothing.”

I will not permit thee to say or to think that [being] came from not-being; for it is impossible to think or to say that not-being is. What would then have stirred it into activity that [being] should arise from not-being later rather than earlier? So it is necessary that being either is absolutely or is not.

Parmenides in “The Way of the Truth” (c. 475 B.C.)

To decide whether existence emerging from nothingness is even logically possible, we need a precise definition of nothing. For instance, by ‘nothing’ do we mean no things, or do we mean absolute nothingness: no laws, structures, properties, or principles?

Defining Nothing

It might have been true that nothing ever existed: no living beings, no stars, no atoms, not even space or time. When we think about this possibility it can seem astonishing that anything exists.

Derek Parfit in “Why Anything? Why This?” (2008)

What is nothing? It seems like a straightforward question. Just keep removing things until there is nothing left.

Start with the universe as it is. Wipe away all the matter and energy. Take away all the quantum fields of the vacuum, and any virtual particles popping in and out of existence. And voilà: nothingness.

But wait, there’s still space. It still has dimensionality, and curvature. There is still time and physical law, even if there are no particles or fields left to be governed by them. Let us delete those too.

Let’s erase the volume of space, erase time, and erase physical law.

When we say out of nothingness we do not mean out of the vacuum of physics. The vacuum of physics is loaded with geometrical structure and vacuum fluctuations and virtual pairs of particles. The Universe is already in existence when we have such a vacuum.

No, when we speak of nothingness we mean nothingness: neither structure, nor law, nor plan.

John Archibald Wheeler in “Law Without Law” (1983)

What are we left with? If we eliminate all the dimensions of space and time, we’re left with a zero-dimensional, changeless point.

But a point is still a thing. Can we delete that too?

Kinds of Nothing

So long as we operate from a theory of geometry, we can’t define nothingness as anything less than a space of zero-dimensionality.

This leaves us with a point.

If we want to eliminate the point, we need to define nothingness not as a space of zero dimensionality, but as something non-geometric. For this, we must define nothingness in terms of some other theory.

But any theory we might choose has its own notion of nothing. In other words, nothingness is theory-dependent.

There is an unlimited number of possible theoretical systems. Does this mean there are also unlimited conceptions of nothing?

Nothing is simple. Not even Nothing.

Bruno Marchal

Might there be a true nothing — one with no laws, principles, nor any theory behind it? Or might every conception of nothing require a theory of things in order to declare that there are none of them?

Rules for Nothing

Wheeler called for absolute nothingness: “neither structure, nor law, nor plan.” But is this kind of absolute nothing achievable?

For instance, the law of identity holds that for any A, “A = A“. Without such a rule, there would be nothing to ensure that nothing stayed nothing, and didn’t later become equal to something.

For nothingness to persist, the rules of logic must apply. Further, if nothingness is the state where “zero things exist”, then the rules of arithmetic must also hold to ensure that “0 = 0” rather than “0 = 1”.

For there to remain no things requires some minimum set of laws. There might be no things as such, but the idea of no laws seems incompatible with there being and remaining no things.

In the beginning, there was only truth, logic and their relation. No possible reality can do without them.

C. W. Rietdijk in “Four-dimensional reality continued” (2018)

If there were no logic, what logic or reason ensures that nothing comes from nothing? If there were no laws, what law or principle would prohibit the spontaneous emergence of a universe?

The Trouble with Nothing

Can we define nothing in a way that suppresses all forms of existence?

That is, to not only have no things, but an absolute nothingness — a nothingness of no objects, neither abstract nor concrete, no properties, no laws, no principles, and no information content?

Or is this a fool’s errand? One that leads to a logical inconsistency and thus an impossibility? Might nothingness be, in some sense, unstable?

If absolute nothingness can be shown to be an impossible dream, it will advance us on our path to discover the reason for existence. It might even reveal some self-existent or necessarily-existent thing.

Properties of Nothing

Any time we delete something from reality, we leave something else in its place. When we deleted matter, we created a vacuum. When we eliminated light, we created darkness. When we removed heat we created cold. When we deleted space, we created a point.

The idea of nothingness has not one jot more meaning than a square circle. The absence of one thing always being the presence of another — which we prefer to leave aside because it is not the thing that interests us or the thing we were expecting — suppression is never anything more than substitution, a two-sided operation which we agree to look at from one side only: so that the idea of the absolution of everything is self-destructive, inconceivable; it is a pseudo-idea, a mirage conjured by our own imagination.

Henri Bergson in “The Two Sources of Morality and Religion” (1935)

If every deletion is a substitution for something else, then a pure nothing, devoid of any properties whatever, is impossible.

So while we might succeed in removing all material things from reality, we could not remove all properties from reality.

The existence of properties appears inescapable.

Nothingness, of any kind will always have some description and properties, even when it’s just a cold, dark, empty, vacuum.

But how far can we go in eliminating properties?

For instance, if we define nothingness as the empty set from set theory, what properties would remain? Temperature has no meaning for a set.

Would any properties remain for such a nothing?

Properties of Zero

Every conception and definition of nothing contains at its heart: zero. For any conception of a thing, nothing will always be zero of them.

The vacuum: zero energy. Geometry: zero dimensionality. The empty set: zero elements. Arithmetic: zero magnitude. Information theory: zero bits.

If zero is a universal property of nothing, we must ask: what are the properties of zero? What does zero bring to the table of reality?

Zero has many properties. It’s even, it’s the additive identity, it’s the only number that’s neither positive nor negative. It’s the number of elements in the empty set and the number of even primes greater than two.

In fact, zero has more properties than we could list if we recruited all the atoms in the observable universe to serve as paper and ink. This effort is doomed because zero’s properties are infinite in number.

Zero’s factors couldn’t be listed, as zero has infinitely many of them. Every number evenly divides zero and hence is one of zero’s factors.

Aside from zero’s factors, we could list infinite trivial properties of zero: zero is “the difference between 1 and 1” and it’s “the difference between 2 and 2” and it’s “the difference between 3 and 3” and so on.

But there are also an infinite number of non-trivial properties of zero. Some are even beyond the understanding of today’s mathematicians. As an example, mathematicians have for centuries wondered:

are there even numbers >2 that aren’t the sum of two primes?

This question is known as Goldbach’s conjecture after Christian Goldbach who posed it in 1742. Nearly three centuries later, it remains unsolved.

Between 2000 and 2002, a $1,000,000 prize was offered to anyone who could answer this question. All this money to settle a question about a property of zero. To decide: is zero the number of exceptions to Goldbach’s rule?

We now see why “Nothing is simple. Not even nothing.” All definitions of nothing include the concept of zero. Far from being simple, zero is an object of unlimited complexity.

An Explosion of Entities

Can zero exist in isolation — completely alone from other numbers? Or do relationships between numbers make them inseparable?

Zero’s properties reference other numbers. And each of these numbers carries its own set of properties and relations to the other numbers.

Are the properties of one any less real than the properties of zero? Perhaps in a reality having no things, ‘one’ is meaningless.

In a reality containing nothing, there are no things as such — at least no material things. But in such a nothing, there is an abstract thing: zero.

Zero reflects the number of material things to count. But how many abstract things are there to count? There is at least one. The one number that exists to define the number of material things is zero.

But if we have one number and it is one thing to count, now another number exists: one. We then have zero and one together as the only numbers. But now we have two numbers. Now two exists…

This is how numbers are defined in set theory. Within set theory, each number is formed as the set of all previous sets. The process starts with the empty set (represented by { } or ∅) which contains zero things.

• 0 = { } = ∅
• 1 = { 0 } = {∅}
• 2 = { 0, 1 } = { ∅, {∅} }
• 3 = { 0, 1, 2 } = { ∅, {∅} , {∅, {∅}} }
• 4 = { 0, 1, 2, 3 } = { ∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}} }

It seems once a single abstract number is admitted, each next number comes to life as the count of the abstract numbers that preceded it. Is there any way to stop the proliferation of infinite abstract entities?

If zero exists by virtue of there being zero things to count, then on that basis, shouldn’t every number have the same right to exist by virtue of being the number of preceding numbers there are to count?

The existence of any number, in virtue of its properties, entails the existence of all the others i.e. a system of mathematics couldn’t exist bereft only of the number, say, 42; and the existence of any number, in virtue of the full set of its properties/structural relationships, entails the existence of every other number.

David Pearce in “Why Does Anything Exist?” (1995)

Set theory and building up numbers from the empty set are modern ideas — they appeared around the turn of the 20^th century. Yet the idea of numbers giving rise to themselves goes back much farther.

The Tao gives birth to One.
One gives birth to Two.
Two gives birth to Three.
Three gives birth to all things.

Laozi in chapter 42 of “Tao Te Ching” (c. 600 B.C.)

A True Nothing

Whenever we specify or define nothing, we invoke theories and concepts which, in turn, lead to properties and abstract entities.

But what if we forgo even specifying nothing? Might this be a path to achieve absolute nothingness? A true nothing, having:

• No things, no objects
• No definitions, no properties
• No abstract entities, no concepts
• No sets, no numbers
• No set theory, no mathematics
• No specifications, No information

Avoiding all this we have no theories of any kind. We are left with a plain and simple, pure, unadulterated nothing at all.

But again this leads to trouble. There’s a problem with this kind of nothing — a nothing of no information is identical to everything.

We note that the collection of all possible descriptions has zero complexity, or information content. This is a consequence of algorithmic information theory, the fundamental theory of computer science. There is a mathematical equivalence between the Everything, as represented by this collection of all possible descriptions and Nothing, a state of no information.

Russell Standish in “Theory of Nothing” (2006)

At first this sounds counter-intuitive, if not outright wrong. Yet this consequence is something we intuitively understand in other contexts.

Let’s review three such cases: Unsculpted Marble, an Unsent E-mail, and the Library of Babel. Each demonstrates an equivalence between the nothing of no specification, and the everything of all possibilities.

Unsculpted Marble

Before marked by a sculptor’s chisel, a block of marble contains every figure — or at least every figure fitting the dimensions of the block.

Michelangelo’s Pietà was in the block before he uncovered it. It was there with all the other figures. To bring forth the Pietà alone, required the addition of information. Michelangelo had to uniquely specify the Pietà from among the set of all possibilities.

There is a beautiful angel in that block of marble, and I am going to find it? All I have to do is to knock off the outside pieces of marble, and be very careful not to cut into the angel with my chisel. In a month or so you will see how beautiful it is.

George F. Pentecost in “The Angel in the Marble” (1883)

This specification requires adding information to the block, by way of chisel marks. It is only in the absence of this information — in the absence of any chisel marks — that all possible figures remain.

In this sense, information is subtractive rather than additive. When information specifies, it eliminates from the preexisting infinite set of possibilities. Absent such information, all possibilities remain.

An Unsent E-mail

You are at your desk, awaiting an important e-mail from your boss.

Before this message arrives you know nothing about the contents of this e-mail — you are in a state of having no information.

But there is one thing you know before the e-mail arrives: the e-mail will be one message from among the infinite set of possible e-mails.

Only after the e-mail arrives in your inbox do you learn which from among the infinite set of messages the boss chose to send you.

But consider the case where instead of sending a single e-mail, the boss sent you every possible e-mail. Would you be able to learn anything from these infinite messages about what your boss wants?

The lack of specification in the infinite set of messages is equal to the lack of specification that existed prior to receiving anything. Both states are equivalently unspecified. Therefore, both represent states of complete ignorance and a state of having zero information.

Having every message is as informative as having no message.

The Library of Babel

One of the best illustrations of the uselessness of all information comes from Jorge Luis Borges’s concept of a ‘Total Library’, described in his short story The Library of Babel.

This library is described as follows:

The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries, with vast air shafts between, surrounded by very low railings. From any of the hexagons one can see, interminably, the upper and lower floors. […]

There are five shelves for each of the hexagon’s walls; each shelf contains thirty-five books of uniform format; each book is of four hundred and ten pages; each page, of forty lines, each line, of some eighty letters which are black in color. […]

This thinker observed that all the books, no matter how diverse they might be, are made up of the same elements: the space, the period, the comma, the twenty-two letters of the alphabet. He also alleged a fact which travelers have confirmed: In the vast Library there are no two identical books. From these two incontrovertible premises he deduced that the Library is total and that its shelves register all the possible combinations of the twenty-odd orthographical symbols.

Jorge Luis Borges in “The Library of Babel” (1941)

From the provided information, we can calculate the number of books in this library. This total library contains every possible 410-page book, representing every possible arrangement of 25 characters.

Each page, with 40 lines and 80 characters, contains 3,200 characters. Each book, with 410 pages, contains 410 \times \text{3,200} = \text{1,312,000} characters. With an alphabet of 25 characters, this gives 25^{\text{1,312,000}} possible books.

This number is 25 multiplied by itself over a million times. To put its magnitude in context, the number of atoms in the observable universe is only 25^{57} or 25 multiplied by itself 57 times.

This library is a great treasure. For in this library we can find every book, article, poem, and novel ever written, or that could be written.

We’ll find descriptions of every scientific theory: from Newton’s Principia, to Einstein’s Relativity, to the presently unknown theory of quantum gravity. We’ll find blueprints to world-changing technologies not yet invented based on principles not yet discovered.

This library possesses the greatest works of literature: the complete works of Shakespeare, Dickens, and Tolstoy. It also has every work yet to be written: the completed Game of Thrones series, as well as the unfinished works of Tolkien, Hemingway, and Twain.

The library has the untold histories of every civilization, including civilizations now lost to time. It has the contents of every scroll burned in the fire of Alexandria. The library has biographies of every person who’s ever lived — and even biographies of those yet to be born.

What could be more valuable than this boundless trove of information, with its complete knowledge, its answers to every mystery, and its articulated solutions to every problem?

This is where the equivalence between all information and no information rears its ugly head. It renders the library worthless.

There are issues with this library. To start, for every valid theory, technology, history, and biography in the library, there are countless others that are subtly wrong, inaccurate, or utterly bogus.

Worse, finding any book with more than a few grammatically sensible words is next to impossible. Most books are pure gibberish (or babble) — indistinguishable from random sequences of characters.

Perhaps all hope is not lost. Since this library contains every possible book, surely this library contains books that serve as indexes to find all the other meaningful and sensible books in the library.

But this dream is also impossible.

Given the number of books, it’s impossible to uniquely reference any other book with a descriptor shorter than the length of the book. Thus it takes all 410 pages to reference a specific book in this library.

What if we organize the books somehow, such as by sorting them in alphabetical order? Then finding any particular book would be easy.

This too suffers from a pathological breakdown.

While this makes it easy to find any particular book, the difficulty shifts from finding the book to deciding which book we want to find.

This is a consequence of the library having every possible book. As one seeks a book of interest, one is faced with 25 choices: to choose which of the 25 characters is next in the content of the book we seek.

During the search, the seeker must choose each next letter, and must do this for all 1,312,000 characters in the book. Thus, finding a book in this library is as difficult as writing the book in the first place!

In a way, we already have access to this library — as we are already free to put down any sequence of characters we want, and thus “find” a book that is already present somewhere in this total library.

Thus, this library provides no new knowledge or information. Its set of all books is as helpful to us as if it had no books. And so a total library offers nothing. It’s equivalent to having no information at all.

You can explore this frustrating enigma of the Library of Babel. Jonathan Basile created an online version at libraryofbabel.info.

Everything From Nothing?

Information theory reveals the equivalence between the totality of all information and the nothingness of zero information. Both lack any specification. Both are completely uninformative. Both contain within them the complete and infinite set of every possibility.

We’ve seen this equivalence first-hand. We saw it in the unsculpted block of marble, in the unsent e-mail, and in the Library of Babel.

So is a nothing of no specification, a nothing or an everything?

Less Information, More Reality

How much information is in the Library of Babel?

To determine this, we need only consider what is the shortest description that can generate the content of the library.

For instance:

A library containing one of each possible 410-page book with 3,200 characters per page and a fixed alphabet of 25 characters.

The preceding description for the library is 125 characters long. There could be shorter descriptions, but this sets an upper-bound for the information content of the Library of Babel.

It takes next to no information to describe the vast Library of Babel. Paradoxically, there’s more information in a single page from a single book in the Library, than in the entire library itself!

How could this be? How can there be less information in the library as a whole than there is in a single book or page from the library?

This is a consequence of algorithmic information theory, which includes the science of data compression. It reveals that it is simpler (in terms of needing a shorter description) to generate every book in the library than it is to generate only a single book, or a single page of a book in the library.

A shorter, less specific, and more general description casts a wider net:

The description “All possible books” needs fewer characters than the description of the Library of Babel, but it defines a much larger set of books, in fact, it defines an infinite set of books, of all possible lengths and character sets. The Library of Babel, though vast, was still finite.

Might the same apply to our universe and reality?

To describe one universe like ours requires a vast amount of information. It requires specifying not only the physical laws, but also the position, direction, and speed of every particle in the universe.

This is estimated to require on the order of 10^{90} bits.

Yet to specify every possible universe of our kind — a multiverse of every possible arrangement of particles ruled by our laws of physics — needs much less information.

Such a multiverse requires only the information to define the physical laws, particle types, fundamental forces and constants of nature.

This can be done in just a few pages of equations.

Describing our specific universe is like describing a specific book from the Library of Babel. It needs more information than the library itself.

In theories such as the string theory landscape, the constants of nature are not specified by the theory, leading to an even greater multiverse consisting of every possible universe having every set of possible values for the constants of nature (e.g. different values for things like the electron mass and the strength of electromagnetism).

There are reasons to suspect this, or something like it is true. For one, it explains why laws of physics and constants of nature appear fine-tuned for the emergence of life. (See: “Was the universe made for life?“)

This description of a “string theory landscape” needs less information. It might save a page by not having to include the 30 some odd constants of nature. And yet, it describes a vastly larger multiverse.

What happens when the length of reality’s description goes to zero?

This would leave the equations themselves unspecified — implying an even greater multiverse. This multiverse includes universes not just of every arrangement of matter, nor universes of every set of constants, but universes ruled by every kind of physical equations.

If all possible string vacua, space-time geometries, masses of elementary particles and interaction strengths, and (by)laws of physics are realized, then all possible descriptions are satisfied. This is equivalent to zero information.

David Pearce in “Why Does Anything Exist?” (1995)

Thus, to specify all possible physical laws, all possible physical constants, for all possible universes, needs no information at all.

Might we inhabit such a nothing?

This is the thesis of Russell Standish’s 2006 book Theory of Nothing.

Standish believes our universe, with its seemingly vast quantity of information, is something like a book in the Library of Babel. We would then be denizens of nothing, occupying a place within a total reality which altogether amounts to zero information.

Such a reality — one of zero information — is the simplest state of existence. It’s simpler than an empty vacuum or a geometrical point, as these both need a non-zero amount of information to describe.

Necessary Existence

We’ve attempted but frustratingly failed to define a true nothing.

When we tried to specify a nothing, whether as a vacuum, a point, or an empty set, we inevitably invoke properties, abstract entities, the number zero and the infinitude of numbers and their relationships.

Furthermore, this specification is not an absolute nothing as it requires reality to have a nonzero amount of information to specify it.

Alternatively, if we attempt a nothing of zero information and zero specification we get a total reality containing all possibility. Neither approach succeeds in bringing about absolute nothingness.

Moreover, these approaches rely upon and assume the validity of logical principles and consistency. No reality, not even a nothing, appears possible without laws and principles of logic.

And so the goal of the philosopher’s nothing, the “neither structure, nor law, nor plan” kind of true nothing at all, seems an impossible dream.

The nothings we attempt break down and lead to somethings.

• With no structure, there are zero structures, this introduces zero and with it the structure of all numbers and their interrelations.
• With no law, there are no restrictions on what can or cannot exist nor any law to prevent things spontaneously popping into existence.
• With no plan, there is no information, which is equivalent to a totality.

The motto reads:

“Omnibus ex nihil ducendis sufficit unum”
For producing everything out of nothing, one principal is enough.

Gottfried Wilhelm Leibniz in “letter to Duke” (1679)

If a true and absolute nothing is impossible or unstable, does this mean there must be self-creating or self-existent things? Can a thing exist out of logical necessity, because its absence is impossible?

What might the nature of such things be?

A Self-Existent Thing?

If something did not emerge out of nothing, then there’s only one other possibility: that there is something that has always existed.

In other words, nothingness is not the default state of reality.

It is extraordinary that there should exist anything at all. Surely the most natural state of affairs is simply nothing: no universe, no God, nothing. But there is something.

Richard Swinburne in “Is There a God?” (1996)

Given that something exists, it either came from nothing or else something has existed from the beginning. The existence of this thing is somehow necessary. It existed without any preceding cause.

This, we also find contrary to intuition. It’s strange because everything we are familiar with can trace its existence to some earlier cause:

Manufactured things are made by people, or by machines that were made by people. Life comes from other life. Things not created by humans or other life, like rivers and mountains are created by natural forces acting on matter. It seems to defy reason for a thing to exist without a cause. And yet, we know the universe exists.

The universe either came from some preceding cause, or else the universe has always existed, is self-existent, or self-creating.

There is no third option. If the universe is not the end of this causal chain, then something else is. Therefore we must accept some things are self-creating (come out of nothing) or are self-existent.

Let’s call such a thing causeless.

Existing without Cause

Take anything that exists: the chair you’re sitting in, your conscious thoughts, the Eiffel tower. For the purposes of the reasoning, it doesn’t matter what thing we start with.

Given that this thing exists, there are two possibilities: either that thing was caused or it was not caused. If a thing has no cause, then it is causeless. Otherwise, the thing has a cause and its existence is owed to some other thing. If we follow the chain of causality back towards an ultimate root cause, there are three possibilities:

1. First Cause: The chain of causality comes to an end in a ‘first cause‘
2. Infinite Regression: The chain of causality continues forever
3. Causal Loop: The chain of causality forms a closed cycle, or a loop

These represent all possibilities. The trace either ends (a first cause) or it continues forever. If it continues forever it forms an infinite chain that’s either open (an infinite regression) or closed (a causal loop).

In all three cases we find something that has always existed: either the first cause, the infinite chain itself, or the causal loop itself. This thing, which has always existed, we can describe as causeless.

First Cause

If when tracing back through the series of causes we happen upon something causeless, then our existence results from a first cause.

Leading cosmological theories, such as the big bang and cosmic inflation posit that the universe is not infinitely old, but rather underwent an abrupt event where it came into existence.

That our universe has a point that may be marked as a beginning leaves open the possibility that there is a preceding cause for our universe. Another possibility is that the universe is its own cause, emerging as a random quantum fluctuation allowed by laws of physics.

Many religions speak of the first cause as a divine act of creation. In such a case God would be the first cause. Yet some other non-theistic object could as well be responsible for our existence.

If the universe is not eternal, we should look for some reason for the sudden appearance of the universe: to explain how it could arise by itself, be self-existent, or be the product of some prior cause.

Infinite Regression

If our universe has an eternal history, or if it belongs to a reality having an eternal history, then we exist due to an infinite regression.

A number of scientific theories propose that our universe is eternal.

Prior to wide acceptance of the big bang, the steady-state model was popular. It proposed that the universe is eternally expanding with new matter perpetually created to fill the void in the newly made space.

Since the acceptance of the big bang, various new models suppose that the big bang is itself part of an eternal succession of big bangs.

Roger Penrose’s conformal cyclic cosmology supposes that the heat death of our universe could appear as a new big bang in the next ‘aeon’.

Lee Smolin proposed cosmological natural selection wherein a new universe spawns every time a black hole forms. Accordingly, if the laws mutate, he suggests that universes might even ‘evolve’ towards having laws that maximize the production of black holes.

Sean Carroll notes that the equations of quantum mechanics, unlike those of general relativity, permit physicists to calculate eternally into the past or future. With a theory of quantum gravity, we could in principle predict backwards to times preceding the big bang.

The Schrödinger equation has an immediate, profound consequence: almost all quantum states evolve eternally toward both the past and the future. Unlike classical models such as spacetime in general relativity, which can hit singularities beyond which evolution cannot be extended, quantum evolution is very simple. […] If this setup describes the real world, there is no beginning nor end to time.

Sean Carroll in “Why Is There Something, Rather Than Nothing?” (2018)

If an infinite regression is true, there is no ultimate cause. However, we might still look for an ultimate explanation for the chain of causes.

Causal Loop

It might be that our existence is part of an infinite series, but one that repeats forever. If true, we are stuck in a never ending causal loop.

In 1922, Alexander Friedmann applied Einstein’s equations of general relativity to the universe as a whole. He found that for certain values of the density of the universe and the cosmological constant, the universe will expand for a period of time, slow down, and eventually recollapse.

In his 1923 book, The World as Space and Time, Friedmann speculates that the collapse (or Big Crunch) could rebound (in a Big Bounce), causing a new Big Bang. The process could repeat forever.

The idea of a cyclic cosmology has appealed to many scientists, including Georges Lemaître, Richard Tolman, George Gamow, William Bonnor, Herman Zanstra and Robert Dicke, among others.

We can now ask ourselves two important questions: why was our universe in such a highly compressed state, and why did it start expanding? The simplest and mathematically most consistent, way of answering these questions would be to say that the Big Squeeze which took place in the early history of our universe was the result of a collapse which took place at a still earlier era, and that the present expansion is simply an “elastic” rebound which started as soon as the maximum permissible squeezing density was reached.

George Gamow in “The Creation of the Universe” (1952)

Cyclical cosmologies can be found in many religions. For example, there is the concept of the Wheel of Time in the Dharmic religions.

The most elegant and sublime of these is a representation of the creation of the universe at the beginning of each cosmic cycle, a motif known as the cosmic dance of Shiva.

The god, called in this manifestation Nataraja, the Dance King, has four hands. In the upper right hand is a drum whose sound is the sound of creation. In the upper left hand is a tongue of flame, a reminder that the universe, now newly created, will billions of years from now be utterly destroyed.

Carl Sagan in “Cosmos” (1980)

But cyclic models, lacking observational evidence and theoretical support, remained on the periphery of cosmology.

In 1998, observations revealed the expansion of the universe was not slowing, but accelerating. This seems to rule out a future collapse.

The driver of this acceleration, dark energy, remains little understood. If it is constant, the expansion will continue forever. But in some theories, it varies with time and so a later collapse may be possible.

Cyclic models have seen a revival. In 2001, Justin Khoury, Burt Ovrut, Paul Steinhard and Neil Turok proposed the ekpyrotic universe.

This idea marries string theory and cosmology to give a model where periodic brane collisions trigger cycles of Big Bangs and Big Crunches.

If our universe is part of a causal loop, no beginning or end is identifiable. But what got it started? Did one of the succession of states spring forth out of nothing, or might the loop have always existed?

The Nature of Uncaused Things

Given that reality exists, we know there must be an entity that is causeless. What is it about causeless entities that makes them existent?

If a first cause, how did it bring itself into existence? If an infinite regression or causal loop, how did it come into being?

Might it exist out of logical necessity? Or is it a result of chance? Or might it exist simply because it can exist, and nothing forbids it?

Tracing causes backwards can tell us where the previous state came from, but it won’t answer where the chain or loop itself came from.

Some believe that, if all events were caused by earlier events, everything would be explained. That, however, is not so. Even an infinite series of events cannot explain itself. We could ask why this series occurred, rather than some other series, or no series.

Derek Parfit in “Why Anything? Why This?” (2008)

What we are looking for is not a cause, but a reason — an explanation.

For in the cases of the loops or infinite regression, we can always find an earlier cause, but may never reach a satisfactory reason.

For the question to be properly, fully answered, we need a sufficient reason that has no need of any further reason—a ‘Because’ that doesn’t throw up a further ‘Why?’—and this must lie outside the series of contingent things, and must be found in a substance which is the cause of the entire series. It must be something that exists necessarily, carrying the reason for its existence within itself; only that can give us a sufficient reason at which we can stop, having no further Why?-question taking us from this being to something else.

Gottfried Wilhelm Leibniz in “The Principles of Nature and Grace, Based on Reason” (1714)

If we seek a final ‘Because’ that puts an end to any further ‘Whys’, we must find something that we can show must exist. Not only must this thing exist, but we must also show how this thing can account for the reality we experience — only then will we have succeeded in our quest.

Candidates for Self-Existence

Throughout history, philosophers, scientists and religions have suggested candidates for self-existence.

These causeless entities generally fall into one of seven categories:

1. Logic
2. Truth
3. Numbers
4. Possibility
5. The Universe
6. A Higher Plane
7. Consciousness

Let’s review each candidate and its merits for self-existence. Afterwards, we will consider whether that entity could further serve as an ultimate explanation: a self-existent starting point from which the rest of reality emerges as a direct consequence of that thing.

Logic

Some suppose rational principles, like the laws of logic, are self-existent. Unlike physical laws, logical laws have an air of inevitability to them.

These are laws such as:

• The law of identity (things are identical to themselves, e.g. A = A)
• The law of the excluded middle (statements are either true or not true)
• The law of noncontradiction (no statement is both true and false).

These are laws that seem inevitable, and necessary in any reality, as it’s hard to imagine any reality where logical laws would not hold.

If logical laws apply in all universes and all possible realities, they represent universal laws, applying everywhere and to everything.

If we can say laws of physics exist because all matter in our universe adheres to physical laws, then could we say laws of logic exist, because all things in all possible realities adhere to these logical laws?

If so, then laws of logic are self-existent. They are necessary even in a reality of no things, as logical laws ensure “nothing = nothing”.

If I ask myself why bodies or minds exist rather than nothing, I find no answer; but that a logical principle, such as A = A, should have the power of creating itself, triumphing over the nought throughout eternity, seems to be natural.

Henri Bergson in “Creative Evolution” (1907)

This idea, that logical law and rational principles have eternally existed predates modern philosophers. It’s a cornerstone belief in Taoism.

There was something formless and perfect before the universe was born. It is serene. Empty. Solitary. Unchanging. Infinite. Eternally present. It is the mother of the universe.

For lack of a better name, I call it the Tao.

Laozi in chapter 25 of “Tao Te Ching” (c. 600 B.C.)

Tao translates as “the way”, “principles”, and “natural order”. A similar sentiment is expressed in Christianity. The Gospel of John begins:

In the beginning was the Word, and the Word was with God, and the Word was God.

Gospel of John, chapter 1 verse 1 (100 A.D.)

The term Word is a translation of verbum in Latin, which is a translation of logos (λόγος) in Greek. Logos has a deep and rich meaning. Aside from “word” logos also means, “reason”, “principles”, and “rational law”.

Logos is the root from which we get the word logic. It is also the origin of the suffix -logy (as in biology, geology and psychology). Where it means the principles, explanation, and story thereof.

If however he be admitted to exist apart from Matter in virtue of his character as a principle and a rational law [logos], God will be bodiless, the Creative Power bodiless.

Plotinus in “The Enneads VI.1.26” (270 A.D.)

In Chinese bibles, Logos has been translated as Tao. In this way, Both Taoist and Christian ideas suppose that the Tao/Logos (order, reason, principles, logic, rational law) exists prior to the material universe.

Truth

Some believe that truth is causeless.

There seems to be some essential difference between “zero is even” and “zero is odd” — only one of them is true. Did anything make it so?

When did this statement become true? Did it require a human mind to conceive of it as being true, or has it always been true? Might this property of truth have an independent and necessary existence?

If logical laws apply universally, then any well-formed statement is either true or false. The law of noncontradiction says a statement can’t be both true and false. The law of excluded middle says a statement must be either true or false — there is no middle ground.

Thus, if logical laws apply to everything, they apply to all statements, forcing on them the objective property of being either ‘true’ or ‘false’. As Derek Parfit said, “Some truth is logically necessary when its denial leads to a contradiction.”

Accordingly, the truth that “zero is even” would exist before humans proved it. It would be true before it was first spoken. Presumably, it would be true absent a universe of things, for even in the case zero things exist, it remains true that “an even number of things exist.”

When we imagine how things would have been if nothing had ever existed, what we should imagine away are such things as living beings, stars and atoms. There would still have been various truths, such as the truth that there were no stars or atoms, or that 9 is divisible by 3. We can ask why these things would have been true. And such questions may have answers. Thus we can explain why, even if nothing had ever existed, 9 would still have been divisible by 3. There is no conceivable alternative.

Derek Parfit in “Why Anything? Why This?” (2008)

Ultimately, nothing is responsible for creating this truth. Truth exists out of its own necessity. It has always existed and could never not exist.

The idea of the primacy of truth is very old. It can be found in many religions, some of which draw an equivalence between God and Truth.

In the 3,000 year old religion of Zoroastrianism, it is said that Asha (meaning truth and order) is the divine law behind all things.

Iran, as India, presents us with a term which has had to signify first of all ‘true statement’; that this statement, because it was true, had to correspond to an objective, material reality; and that, as the discourse did, this reality must embrace all things; and, finally that one recognized in it a great cosmic principle since all things happen according to it.

Jacques Duchesne-Guillemin in “Heraclitus and Iran” (1963)

In the book of Psalms 31:5, God is called the “God of truth.” In the Qur’an, Al-Ḥaqq (meaning The Truth), is one of the 99 Names of God.

Similar ideas are found in Dharmic religions.

The Mūl Mantar (or root mantra), is the most important verse of the Sikh religion. It begins: “There is one creator, whose name is truth” and is described as timeless, beyond birth or death, and self-existent.

In the Brahma Saṁhitā, a Hindu prayer book, the primeval Lord Govinda is described as the “indivisible, infinite, limitless, truth.”

If it is possible for the human tongue to give the fullest description of God, I have come to the conclusion that God is Truth.

Mahatma Gandhi in “All Men Are Brothers” (1953)

Numbers

Some speculate that numbers, or their relationships, are self-existent.

If truth has an independent existence, this truth includes the infinite truths describing all true relationships between the numbers.

These include arithmetical statements, such as:

• 2 is even
• 7 is prime
• 1 is greater than 0
• 2 + 2 = 4
• n × 0 = 0
• the square root of 9 = 3

Truths concerning the numbers are boundless.

Might this infinite truth, provide a scaffolding and structure to all the numbers? And if there is nothing more to numbers than their properties and relations, then might numbers — in some sense — really exist?

It’s been said, “math is the science we could still do if we woke up tomorrow and there was no universe.” The idea that math holds some claim to reality is known as mathematical realism, or Platonism. It’s believed by many, if not most, mathematicians.

It is an idea that many mathematicians are comfortable with. In this scheme, the truths that mathematicians seek are, in a clear sense, already “there”, and mathematical research can be compared with archaeology; the mathematicians’ job is to seek out these truths as a task of discovery rather than one of invention.

Roger Penrose in “The Big Questions: What is reality?” (2006)

But can number relations have any reality in the absence of things?

If zero things exist, it would have to be true that “0 not equal 1”, and also that “0 not equal 2”, and true that zero not equal any other number.

So even with no things, an infinite number of arithmetical relations are needed to avoid contradiction and preserve a nothing of zero things.

If all things were absent, would Two And Two Make Four be a non-reality, remaining like that until at least four things had come to exist? Presumably the answer must be No.

John A. Leslie and Robert Lawrence Kuhn in “The Mystery of Existence” (2013)

This idea that numbers have an independent existence is ancient. It can be traced to some of the earliest records of human thought. It was taught by ancient philosophers and is found in the oldest religious texts.

Taoism, for instance, sets the existence of numbers as prior to things.

The Tao gives birth to One.
One gives birth to Two.
Two gives birth to Three.
Three gives birth to all things.

Laozi in chapter 42 of “Tao Te Ching” (c. 600 B.C.)

The Greek mathematician Pythagoras taught “All things are number.“

Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through studying it they came to believe that its principles are the principles of everything.

Aristotle in “Metaphysics 1.985b” (c. 350 BC)

Pythagoras was the first to propose that the motions of the planets are governed by mathematical equations, which he called the harmony of the spheres. When Newton discovered his law of universal gravitation some 2,000 years later, he credited Pythagoras for the discovery.

Across times, mathematicians have described a seemingly divine connection between mathematics and reality:

Geometry, which before the origin of things was coeternal with the divine mind and is God himself (for what could there be in God which would not be God himself?), supplied God with patterns for the creation of the world, and passed over to Man along with the image of God.

Johannes Kepler in “The Harmony of the World” (1619)

From these considerations it is now wonderfully evident how a certain divine mathematics or metaphysical mechanics is employed in the very origination of things.

Gottfried Wilhelm Leibniz in “On the Ultimate Origination of Things” (1697)

To all of us who hold the Christian belief that God is truth, anything that is true is a fact about God, and mathematics is a branch of theology. […]

An old Greek, a French child, and a self-taught Indian, each finds for himself the same theory of geometrical conics. The simplest and therefore the most scientific way of describing this, is that they have discovered, not created, a geometry that exists by itself eternally, the same for all, the same for teacher as for taught, the same for man as for God. The truth that is the same for man as for God is pure mathematics.

Hilda P. Hudson in “Mathematics and Eternity” (1925)

Possibility

Some speculate that simply not being impossible, is sufficient for being actual. If true, then every possible object, structure, and entity exists.

What then is impossible?

At a minimum, we can say self-contradictory things. For example: square circles, married bachelors, triangles with five sides, and so on. We might also include things proven to not exist: odd numbers evenly divisible by two, a largest prime number, a sixth platonic solid.

If consistency and provability are the requirements for possibility, then possible existence is mathematical existence. As David Hilbert said, “Mathematical existence is merely freedom from contradiction.”

The idea that all possible things exist has enjoyed many names:

In 1936, Arthur Lovejoy dubbed it the principle of plenitude. In 1981, Robert Nozick named it the principle of fecundity. David Lewis, in 1986, developed it as a theory he called modal realism. In Max Tegmark’s 1998 model of multiverses, he called it the mathematical universe hypothesis. Most recently, in 2008, Derek Parfit coined the all worlds hypothesis.

If all possible objects are actual, then our universe is just one such possible structure among an infinite, and total, set of all possible structures. Anything that could happen, happens somewhere.

There are so many other worlds, in fact, that absolutely every way that a world could possibly be is a way that some world is. And as with worlds, so it is with parts of worlds. There are ever so many ways that a part of a world could be; and so many and so varied are the other worlds that absolutely every way that a part of a world could possibly be is a way that some part of some world is.

David Lewis in “On the Plurality of Worlds” (1986)

If the universe is inherently mathematical, then why was only one of the many mathematical structures singled out to describe a universe? A fundamental asymmetry appears to be built into the heart of reality.

As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe.

Max Tegmark in “Parallel Universes” (2003)

The idea that possibility is sufficient for actuality is not new.

Arthur Lovejoy, who wrote about the history of this idea, traced it to 360 B.C. beginning with Plato’s theory of forms. Plato hypothesized a realm containing all possible forms (eternal, perfect, idealizations).

We find this idea expressed in a variety of ways throughout history.

The One is all things and not a single one of them. […] It is because there is nothing in it that all things come from it: in order that being may exist, the One is not being, but the generator of being.

Plotinus in “The Enneads V.2.1” (270 A.D.)

But to explain more distinctly how from eternal or essential metaphysical truths there arise temporal, contingent or physical truths, we must first observe that, from the very fact that there exists something rather than nothing, it follows that in possible things, or in possibility or essence itself, there is a certain need of existence, or so to speak, a claim to exist, in a word, that essence of itself tends to existence.

Gottfried Wilhelm Leibniz in “On the Ultimate Origination of Things” (1697)

Others have linked God’s infinite nature to an infinite creation.

From God’s supreme power, or infinite nature, an infinite number of things—that is, all things have necessarily flowed forth in an infinite number of ways, or always flow from the same necessity; in the same way as from the nature of a triangle it follows from eternity and for eternity, that its three interior angles are equal to two right angles.

Baruch Spinoza in “Ethics” (1677)

Know thou of a truth that the worlds of God are countless in their number, and infinite in their range. None can reckon or comprehend them except God, the All-Knowing, the All-Wise.

Baháʼu’lláh in “Tablet to Vafá” (circa 1885)

It makes sense that an infinitely creative deity would create other universes, not just our own. […] For the theist, the existence of multiple universes would simply support the view that creation reflects the infinite creativity of the creator.

Robin A. Collins in “Spiritual Information” (2005)

The Universe

Some say that the universe, or the physical law that enabled it to come into existence, has always existed and so is self-existent.

The reasoning is simple. If we know at least one thing is causeless, why not just presume this causeless thing is the universe itself?

I should say that the universe is just there, and that’s all.

Bertrand Russell in “Russell-Copleston debate” (1948)

Perhaps there is no reason. It simply is, and has no explanation.

Given the universe exists, we know the universe is possible. Perhaps it exists because it is possible, and nothing forbade it from existing.

But there are other tracks to follow. Perhaps we can demonstrate that the universe is self-creating. Or that it exists due to some higher law.

Modern cosmology made progress along these directions.

The theory of cosmic inflation uses general relativity to explain how a tiny quantum fluctuation can inflate into the huge universe we now see.

Inflation is radically at odds with the old dictum of Democritus and Lucretius, “Nothing can be created from nothing.” If inflation is right, everything can be created from nothing, or at least from very little. If inflation is right, the universe can properly be called the ultimate free lunch.

Alan Guth in “Inflation and the New Era of High-Precision Cosmology” (2002)

According to the laws of quantum mechanics, the quantum fluctuation that seeded our universe appeared because it was possible, emerging out of nothing but the physical laws themselves.

Is there any bound to how small the initial universe could be? […] To my surprise, I found that the tunneling probability did not vanish as the initial size approached zero. I also noticed that my calculations were greatly simplified when I allowed the initial radius of the universe to vanish. This was really crazy: what I had was a mathematical description of a universe tunneling from a zero size — from nothing! […]

And yet, the state of “nothing” cannot be identified with absolute nothingness. The tunneling is described by the laws of quantum mechanics, and thus “nothing” should be subjected to these laws. The laws of physics must have existed, even though there was no universe.

Alexander Vilenkin in “Many Worlds in One” (2006)

General relativity and quantum mechanics are the two cornerstone theories of modern physics. From them alone we can explain a self-emerging universe. Quantum mechanics shows how possible fluctuations spontaneously pop into existence. General relativity explains how such a fluctuation could expand exponentially to reach an unfathomable size. (See: “What caused the big bang?”)

But we must wonder, why these laws? What, if anything, is special about them? Who or what anointed these equations with existence?

What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?

Stephen Hawking in “A Brief History of Time” (1988)

The idea that the universe is uncreated, or exists due to some laws, predates the successes of modern physics and cosmology.

The ancient Greeks and Romans believed that the material of the universe has always existed, since nothing comes from nothing.

The first principle is that nothing can be created from the non-existent: for otherwise anything would be formed from anything without the need of seed.

Epicurus in “Letter to Herodotus” (c. 300 B.C.)

This matter was originally in a state of disorder (or kháos).

Before the ocean and the earth appeared — before the skies had overspread them all — the face of Nature in a vast expanse was naught but Chaos uniformly waste.

Ovid in “Metamorphoses” (8 A.D.)

It was not until a divine craftsman imposed mathematical order on this chaos that the ordered universe — the kosmos — appeared.

In religions with past-eternal cosmologies, the universe is believed to be causeless. Jainism explicitly says the universe was not created.

The doctrine that the world was created is ill advised and should be rejected. If God created the world, where was he before the creation? If you say he was transcendent then and needed no support, where is he now? How could God have made this world without any raw material? If you say that he made this first, and then the world, you are faced with an endless regression. If you declare that this raw material arose naturally you fall into another fallacy, For the whole universe might thus have been its own creator, and have arisen quite naturally.

Jinasena in “Mahapurana” (898 A.D.)

A Higher Plane

Some suppose our universe exists on account of a higher plane and that this higher plane, rather than the universe, is self-existent.

There are many conceptions of what this higher plane of reality is.

Some describe this plane as a cause of being, be it God, a creator, divine will, a first cause, or an unmoved mover. Others describe it as a source of being, the Mind of God, The One, or the Tao. Still others describe it as a ground of being, The Absolute, The All, or what Hindus call Brahman.

Not all theories of higher planes of existence need be supernatural. There are also naturalistic descriptions of higher realities.

In multiverse theories, a higher reality contains our universe among others. In brane cosmology, our universe is caused by collisions in a literal ‘higher dimension’. In the simulation hypothesis, our universe is the result of computations occurring in a more fundamental reality. (See: “Are we living in a computer simulation?”)

Though these theories deal with phenomena that are beyond the nature of our universe, and hence supernatural, evidence is accumulating for some of these higher realms.

Every experiment that brings better credence to inflationary theory brings us much closer to hints that the multiverse is real.

Andrei Linde in interview (2014)

Various theories imply that various types of parallel universes exist, so that by modus ponens, if we take any of these theories seriously, we’re forced to take seriously also some parallel universes. […] Parallel universes aren’t a theory, but predictions of certain theories.

Max Tegmark in “Are Parallel Universes Unscientific Nonsense?” (2014)

The idea of a preexistent cause, source, or ground of being, one that’s external to and beyond our universe, is as old as religion itself.

By means of the Higher Knowledge the wise behold everywhere Brahman, which otherwise cannot be seen or seized, which has no root or attributes, no eyes or ears, no hands or feet; which is eternal and omnipresent, all-pervading and extremely subtle; which is imperishable and the source of all beings.

Mundaka Upanishad I.6 (c. 800 B.C.)

In the beginning God created the heavens and the earth.

Genesis 1:1 (c. 600 B.C.)

Consciousness

Some posit that consciousness is self-existent. If true, consciousness could be the cause of a universe that exists only in appearance.

The idea seems strange, but we must admit all knowledge of existence comes to us through experiences that exist in our conscious minds.

This fact hasn’t escaped the attention of scientists.

It is difficult for the matter-of-fact physicist to accept the view that the substratum of everything is of mental character. But no one can deny that mind is the first and most direct thing in our experience, and all else is remote inference.

Arthur Eddington in “The Nature of the Physical World” (1927)

I regard consciousness as fundamental. I regard matter as derivative from consciousness. We cannot get behind consciousness. Everything that we talk about, everything that we regard as existing, postulates consciousness.

Max Planck in “Interviews with Great Scientists” (1931)

The relation between mind and matter perplexes scientists to this day.

It leads to philosophical conundrums like brains in a vat, Boltzmann brains, and the simulation argument. All of which suppose that perceived reality is an illusion — a byproduct of a deluded mind.

It’s also led physicists to propose theories where conscious minds play a fundamental role in shaping reality as we see it. Physics, after all, is fundamentally about experiences. Physics is the science of predicting future observations from prior observations.

In 1970, Heinz-Dieter Zeh proposed the many-minds interpretation of quantum mechanics, which proposes that differentiation of an infinity of observer mind states explains quantum phenomena.

A many minds theory, like a many worlds theory, supposes that, associated with a sentient being at any given time, there is a multiplicity of distinct conscious points of view. But a many minds theory holds that it is these conscious points of view or ‘minds,’ rather than ‘worlds’, that are to be conceived as literally dividing or differentiating over time.

Michael Lockwood in “‘Many Minds’. Interpretations of Quantum Mechanics” (1995)

The mysterious link between consciousness and reality inspired John Wheeler’s idea of a participatory universe. As Martin Redfern described, “Many don’t agree with John Wheeler, but if he’s right then we and presumably other conscious observers throughout the universe, are the creators — or at least the minds that make the universe manifest.”

The idea that consciousness precedes the material world has a rich history. It is found across philosophies and religious traditions. Where physical reality is seen as a dream or construct of a mind or soul.

For it is the same thing that can be thought and that can be.

Parmenides in “fragment 3” (c. 475 B.C.)

A few millennia later, the philosopher George Berkeley echoed Parmenides, concluding that “to be is to be perceived.”

It is indeed widely believed that all perceptible objects — houses, mountains, rivers, and so on — really exist independently of being perceived by the understanding. But however widely and confidently this belief may be held, anyone who has the courage to challenge it will — if I’m not mistaken — see that it involves an obvious contradiction. For what are houses, mountains, rivers etc. but things we perceive by sense?

George Berkeley in “The Principles of Human Knowledge” (1710)

Hindus believe the universal mind, or world soul Atman, became the universe. Accordingly, the universe is not real, but the dream of a God under the spell of māyā — a temporary ignorance of the true reality.

Buddhists believe that the mind underlies and forms everything.

All the phenomena of existence have mind as their precursor, mind as their supreme leader, and of mind are they made.

Gautama Buddha in “The Dhammapada” (c. 500 B.C.)

The Taoist philosopher Zhuang Zhou said the world is a dream.

While he is dreaming he does not know it is a dream, and in his dream he may even try to interpret a dream. Only after he wakes does he know it was a dream. And someday there will be a great awakening when we know that this is all a great dream.

Zhuang Zhou in “Zhuangzi” (c. 300 B.C.)

Reviewing Answers

We’ve considered seven proposals for self-existent things:

1. Logic
2. Truth
3. Numbers
4. Possibility
5. The Universe
6. A Higher Plane
7. Consciousness

Yet so far, none of these is satisfactory as an ultimate explanation. None stands out as a final Because that doesn’t throw up a further Why.

Abstract Entities: Logic, Truth, Numbers

First, we have abstract entities: logic, truth, and numbers. But though these things are plausibly causeless, how could they cause anything?

These things are eternal and unchanging, not to mention abstract. How can they cause anything like the huge dynamic universe we see?

So the cause of the universe must (at least causally prior to the universe’s existence) transcend space and time and therefore cannot be physical or material. But there are only two kinds of things that could fall under such a description: either an abstract object (like a number) or else a mind (a soul, a self). But abstract objects don’t stand in causal relations. This is part of what it means to be abstract. The number 7, for example, doesn’t cause anything.

William Lane Craig in “Reasonable Faith” (1994)

Possibility: Mathematical Consistency

What about all possibility? If all possible things exist, then our universe would be counted among those possible things.

But why should possible things be actual? As J. J. C. Smart remarked, “That anything should exist at all does seem to me a matter for the deepest awe.” Existence is what we seek to explain.

And there is another issue: why is our universe so simple and ordered compared to all else that exists in the space of all possibility?

Tegmark’s proposal, however, faces a formidable problem. The number of mathematical structures increases with increasing complexity, suggesting that “typical” structures should be horrendously large and cumbersome. This seems to be in conflict with the simplicity and beauty of the theories describing our world.

Alexander Vilenkin in “Many Worlds in One” (2006)

The Physical: The Universe, Physical Law

If the universe alone exists, it explains exactly what we see.

But there would be lingering questions. Why does consciousness exist? Are abstract entities real? And perhaps the biggest mystery of all: “Why should this universe, or its laws, be the only real ones?

As Lee Smolin asked, “Why do these laws, and not others, hold in our universe?” Does the existence of laws require some higher principle?

Although science may solve the problem of how the universe began, it can not answer the question: Why does the universe bother to exist? Maybe only God can answer that.

Stephen Hawking in interview (1988)

Higher Planes: God, Multiverse, Simulation

We might appeal to a higher cause to explain the universe we see.

But as J. J. C. Smart reminds us, “If we postulate God in addition to the created universe we increase the complexity of our hypothesis. We have all the complexity of the universe itself, and we have in addition the at least equal complexity of God.”

This seems true for any higher principle.

For example, if we presume our universe is the result of a simulation in a higher reality, what’s responsible for that higher reality?

Whatever our final theory of physics, we will be left facing an irreducible mystery. For perhaps there could have been nothing at all. Not even empty space, but just absolutely nothing […] If you believe God is the creator, well, why is God that way? The religious person is left with a mystery which is no less than the mystery with which science leaves us.

Steven Weinberg in “Closer to Truth: Cosmos, Consciousness, God” (2008, 2009)

The Mental: Mind, Soul, Consciousness

If consciousness is causeless, it could explain why perceptions exist.

But if reality is only a dream or illusion, why do our perceptions appear to follow along with a universe adhering to physical laws?

If it’s all an illusion, what’s the source of this illusion?

Even if everything in this universe were an illusion, there would still have to be something outside this universe that generates the illusion.

John A. Leslie and Robert Lawrence Kuhn in “The Mystery of Existence” (2013)

A Causeless Cause

What we seek, and have so far have failed to identify, is a causeless cause.

This is, something that not only has a plausibly self-existent and causeless nature, but also plausibly accounts for the reality we see.

We’ve found things that appear to be causeless: logic, truth, and numbers — but these things also appear incapable of being a cause.

Conversely, we’ve found things that could be a cause: the universe, a higher plane, and consciousness — but they don’t seem causeless.

Then there is possibility, for which we have reason to question whether it is causeless and whether it causes what we see.

We find an almost inverse relation: The more plausibly something is causeless, the less plausible it seems to be the cause for what we see.

A causeless cause would provide us with a complete explanation.

It would explain both itself and the properties of observed reality. It would describe the relation between the mental and material. It would tell us why the universe exists and why it has simple, ordered laws.

To progress we need to find the connecting glue — the missing piece of the puzzle that shows either how a causeless thing accounts for the reality we see, or alternatively, why the reality we see is causeless.

Three Modes of Existence

In reviewing the seven categories of possibly causeless things, we encountered three modes of existence. Loosely speaking they are:

• Mathematical Existence
• Material Existence
• Mental Existence

Mathematical existence includes: abstract entities, logic, truth, numbers, math, properties, forms, equations, relations, possibility, structures, laws, and principles. This mode might include religious concepts of divine law/will/order (Tao/Logos), the infinite indivisible truth (Asha/Govinda), and divine mathematics.

Material existence includes: matter, energy, the vacuum, spacetime, physical law, the universe, the multiverse, particles, forces, fields, and physical systems. This mode might include what religions refer to as creation, kosmos, the material plane, and māyā/illusion.

Mental existence includes: mind, consciousness, observations, perceptions, ideas, and dreams. This mode might include religious concepts of the mind of god, world soul, Atman, and souls or spirits.

My viewpoint allows for three different kinds of reality: the physical, the mental and the Platonic-mathematical, with something (as yet) profoundly mysterious in the relations between the three.

Roger Penrose in “The Big Questions: What is reality?” (2006)

Math, Matter, Mind

Of the three modes of existence, does any stand out as being more fundamental than any of the others? What is their relation?

If one of these modes of existence can be shown as primary, while the others are derivative, then we might close in on a causeless cause.

A common view of physicists is that matter produces mind, and mind produces math. But even among physicists, this view isn’t universal.

The triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. Non-physicists should be wary of any claim that modern physics leads us to any particular resolution of this circularity, since even the sample of three theoretical physicists writing this paper hold three divergent views.

Piet Hut, Mark Alford, and Max Tegmark in “On Math, Matter, and Mind” (2006)

What is the reality of these modes of existence? Are all on equal footing? Or is one more fundamental while the others are derivative?

Materialism: Matter is Primary

Materialism is the view that matter is fundamental.

It assumes mental states are the byproduct of particular material arrangements (e.g. brains) and that mathematical objects, if they exist at all outside of minds, have no bearing on the material world.

Materialism is a popular, if not conventional, view among physicists.

Materialism can explain why our perceptions follow the patterns of physical law, but it has difficulty explaining why matter gives rise to mental states — this is the so-called hard-problem of consciousness.

Materialism also hits an explanatory dead-end trying to answer why matter exists and why it follows simple physical laws.

If eager to know the world’s structure, ask the scientists. Science, however, seems unable to answer some key questions concerning the structure. For start, why is the structure an orderly one? Why do events so often develop in fairly simple and familiar ways, leading us to talk of causal laws? […]

Then there is what can seem the biggest question of all. Science investigates the world’s structure, but why is there anything at all to be structured? Why is there a cosmos, not a blank? Why is there something rather than nothing? Science cannot answer this.

John Leslie in “A Cosmos Existing through Ethical Necessity” (2009)

Idealism: Mind is Primary

Idealism is the view that mind is fundamental.

It assumes mental states are the basis of reality, and that the matter that seems to exist, exists only as thoughts and perceptions in minds.

Idealism is expressed by Eastern religions, theologians, and mystics. But increasingly, physicists recognize they can’t so easily do away with the observer. It seems the observer plays a necessary, if not fundamental, role in any description of reality.

Consciousness cannot be accounted for in physical terms. For consciousness is absolutely fundamental. It cannot be accounted for in terms of anything else.

Erwin Schrödinger in interview (1931)

But idealism doesn’t answer everything. It doesn’t explain why minds are bound up with the patterns of matter in a material world.

We find that our perceptions obey some laws, which can be most conveniently formulated if we assume that there is some underlying reality beyond our perceptions. This model of a material world obeying laws of physics is so successful that soon we forget about our starting point and say that matter is the only reality, and perceptions are nothing but a useful tool for the description of matter. This assumption is almost as natural (and maybe as false) as our previous assumption that space is only a mathematical tool for the description of matter. We are substituting reality of our feelings by the successfully working theory of an independently existing material world. And the theory is so successful that we almost never think about its possible limitations.

Andrei Linde in “Inflation, Quantum Cosmology, and the Anthropic Principle” (2002)

Platonism: Math is Primary

Platonism is the idea that math is fundamental.

It assumes abstract objects are the most real, and that everything we see and perceive is somehow derivative from this higher existence.

Platonism is popular among philosophers and mathematicians, whose job is to study the objective properties of abstract things.

If mathematical objects form the basis of reality, it might explain why the material world is so mathematical in its form.

In a famous 1959 lecture, physicist Eugine P. Wigner argued that “the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious.” Conversely, mathematical structures have an eerily real feel to them. They satisfy a central criterion of objective existence: they are the same no matter who studies them. A theorem is true regardless of whether it is proved by a human, a computer or an intelligent dolphin. Contemplative alien civilizations would find the same mathematical structures as we have. Accordingly mathematicians commonly say that they discover mathematical structures rather than create them.

Max Tegmark in “Parallel Universes” (2003)

Where Platonism falls short is in explaining how abstract objects lead to material or mental existence. According to Leibniz, the difficulty is explaining, “how from eternal or essential metaphysical truths there arise temporal, contingent or physical truths.”

What Came First?

For each of the three modes of existence, there is an ancient school of thought holding that mode of existence as most fundamental.

The Mathematical: Plato believed that abstract entities were the most real and that the material world was derivative.

The Material: Plato’s foremost student Aristotle, disagreed, saying material substances were more real than abstract forms.

The Mental: Several centuries later, Plotinus argued that mind was more real than the material reality it perceives.

Today’s scientists, mathematicians, and philosophers seem no closer to an answer on whether math, matter, or mind came first.

• Does mind give rise to math, or does math give rise to mind?
• Does matter give rise to mind, or does mind give rise to matter?
• Does math give rise to matter, or does matter give rise to math?

To unravel the mystery of existence requires that we understand the relationship between these modes of existence. Only then do we have any hope of identifying an ultimate explanation: a causeless cause.

To address the nature of reality we need to understand its connection to consciousness and mathematics.

Roger Penrose in “The Big Questions: What is reality?” (2006)

Are They One?

Various thinkers have suspected the three modes of existence to be connected and perhaps are all aspects of one ultimate reality.

Mind and Matter as One?

Modern physical experiments have revealed something inseparable between the mind and the observed physical reality.

As we penetrate into matter, nature does not show us any isolated ‘basic building blocks’, but rather appears as a complicated web of relations between the various parts of the whole. These relations always include the observer in an essential way. The human observer constitutes the final link in the chain of observational processes, and the properties of any atomic object can only be understood in terms of interaction with the observer. This means that the classical ideal of an objective description of nature is no longer valid. The Cartesian partition between the I and the world, between the observer and the observed, cannot be made when dealing with atomic matter. In atomic physics, we can never speak about nature without, at the same time, speaking about ourselves.

Fritjof Capra in “The Tao of Physics” (1975)

Aren’t we mistaken in making this separation between ‘the universe’ and ‘life and mind’? Shouldn’t we seek ways to think of them as one?

John Archibald Wheeler quoted in “Trespassing on Einstein’s Lawn” (2014)

Math and Matter as One?

Likewise, mathematicians and scientists cannot help but notice a mysterious link connecting mathematics and the physical world.

There exists, unless I am mistaken, an entire world consisting of the totality of mathematical truths, which is accessible to us only through our intelligence, just as there exists the world of physical realities; each one is independent of us, both of them divinely created and appear different only because of the weakness of our mind; but, for a more powerful intelligence, they are one and the same thing, whose synthesis is partially revealed in that marvelous correspondence between abstract mathematics on the one hand and astronomy and all branches of physics on the other.

Charles Hermite in “Eloges Académiques et Discours” (translation p. 323) (1912)

Maybe the relationships are all that exist. Maybe the world is made of math. At first that sounded nuts, but when I thought about it I had to wonder, what exactly is the other option? That the world is made of “things”? What the hell is a “thing”? It was one of those concepts that fold under the slightest interrogation. Look closely at any object and you find it’s an amalgamation of particles. But look closely at the particles and you find that they are irreducible representations of the Poincaré symmetry group―whatever that meant. The point is, particles, at bottom, look a lot like math.

Amanda Gefter in “Trespassing on Einstein’s Lawn” (2014)

All as One?

If matter and mind are two aspects of one reality, and if math and matter are likewise two aspects of one reality, then all three must be connected — all would be reflections of one underlying reality.

So how do the elements of the trinity fit together: the “phenomenological” world, the “physical” world, and the “mathematical” world? On the unargued assumption that the principle underlying ultimate reality is radically simple, it will here be conjectured that these three realms are one-and-the-same under different descriptions.

David Pearce in “Why Does Anything Exist?” (1995)

A Path to Reality

For millennia, philosophers have debated the relation between math, matter, and mind. For millennia, they’ve sought a causeless cause.

Despite this, philosophy has not yielded any definitive answers.

Perhaps science can shed new light on this question. Science allows us to test and decide among competing theories. Science provides opportunities to discover the missing piece of the puzzle and explain how and why a causeless thing gives rise to the reality we see.

As it happens, discoveries in the field of mathematics in the 20^th century found this missing puzzle piece. We now know a viable link between “eternal or essential metaphysical truths” and “temporal, contingent or physical truths.” We can explain how reality can emerge from self-existent, causeless truth concerning numbers and their relations.

But without hard science and observational evidence to back it up, how can we ever know if this explanation is right? How can we ever escape from the morass of inconclusive philosophy?

Fortunately, discoveries in the fields of physics and cosmology — also occurring in the 20^th century — provide exactly this support. We not only have found a plausible path to reality, we have evidence for it.

20^th Century Mathematics

Many consider the field of mathematics to be mostly uneventful — unchanged since Euclid defined the laws of geometry 2,300 years ago.

But at the turn of the 20^th century, the field of mathematics was in a state of crisis. The field was shaken to its foundation. Math was broken, and it had to be rebuilt from scratch. During this reformation, monumental discoveries shocked and dismayed mathematicians.

In the first half of the 20^th century, logicians and mathematicians discovered a provably self-existent thing. In the second half of the 20^th century, they showed how — under certain assumptions — this self-existent thing could account for the reality we see.

Might this thing be our causeless cause?

Let’s see what mathematicians found, and how they came to find it.

The Foundational Crisis

At the turn of the 20^th century, math was in trouble. It was undergoing what came to be called the foundational crisis of mathematics.

At the time, set theory had come to serve as the foundation of mathematics. All mathematical proofs ultimately relied on it.

But in 1899, Ernst Zermelo noticed this set theory had a fatal flaw. Zermelo told other math professors at the University of Göttingen about it, including David Hilbert, but Zermelo didn’t publish it.

In 1901, Bertrand Russell also noticed this flaw. But Russell didn’t stay quiet. He wrote a letter in 1902 to Gottlob Frege, just as his second volume on set theory was going off to the publisher.

Frege had spent decades laying the foundation of set theory. It was his life’s work. But one letter, showing one flaw, brought it all down.

Russell showed Frege’s set theory allows two contradictory statements to both be proved. This flaw is known as Russell’s paradox.

One flaw might not sound so bad, but in math it is fatal. For if in math, just one falsehood can be proved, then any falsehood can be proved.

This is known as the principle of explosion.

For example, assume mathematics had a flaw that allowed you to prove that “2 + 2 = 5”. You could use this false proof to prove anything. You could prove that the $1 in your bank account equals $1,000,000.

\def\arraystretch{1.5}
   \begin{array}{l:l}
   2 + 2 = 5                      & \text{(given)} \\ \hline
   0 = 1                            & \text{(subtract 4 from both sides)} \\ \hline
   0 = \text{999,999}     & \text{(multiply both sides by 999,999)} \\ \hline
   1 = \text{1,000,000}  & \text{(add 1 to both sides)} \\ 
\end{array}

If math allowed proofs of false statements, then contracts, commerce — even society as we know it — couldn’t function. This was the state of mathematics in 1900 — it’s no wonder it was considered a crisis.

Math was broken. It had to be fixed. It needed a rallying cry.

A Call to Action

In 1900, mathematicians from around the world gathered in Paris for the International Congress of Mathematicians.

David Hilbert, considered the greatest mathematician of his time, was invited to speak. He used the opportunity to present what he considered to be the 23 most significant open problems in mathematics.

The second of Hilbert’s problems called for a proof that the foundational rules of mathematics were free of contradictions.

This would, once and for all, put math on a solid foundation. Never again would mathematicians need worry that a new contradiction might one day surface and torpedo the whole of mathematics.

New Foundations

The collapse of Frege’s set theory and Hilbert’s call for a provably solid foundation for math served as an inspiration.

Under Hilbert’s direction, Zermelo, began work on fixing set theory.

Similarly, Bertrand Russell began work with his supervisor, Alfred North Whitehead, on a solution. Their aim was to lay a new foundation for mathematics based on a precise logic, and produce a set theory rid of paradoxes and contradictions.

It was a massive undertaking that took over a decade. It culminated in the three volume tome Principia Mathematica, published in 1910, 1912, and 1913. It was so detailed, that it famously required several hundred pages to work up to the point where it proved ‘1 + 1 = 2’.

Owing to its complexity and unique notation, Principia Mathematica never gained much popularity with mathematicians.

It also had a competitor.

By 1908, Zermelo developed a new set theory, consisting of just eight rules. And in 1921, it was further improved by Abraham Fraenkel. Their combined result is called Zermelo-Fraenkel set theory. It became the default foundation of mathematics, and remains so to this day.

Hilbert’s Program

Although no one had discovered contradictions in either Russell’s or Zermelo’s new foundational systems, no one had been able to prove they were free of contradictions either.

Mathematics still rested on a foundation of uncertain stability.

This led Hilbert, in 1921, to push for finding a mathematical theory that was provably consistent. And not only did he want this theory to be provably consistent, he wanted it to be provably complete.

A complete system of mathematics means any true statement can be proven within that theory. There would never be a need to add to this complete theory, as it would cover everything that mathematicians might think up in the future. It would be a final theory and the last theory any mathematician would ever need.

It was the mathematician’s equivalent of a theory of everything — where all of mathematics is derivable from one rock-solid foundation.

The effort to find this theory became known as Hilbert’s Program.

It was a noble goal. But less than a decade after launching his program, Hilbert’s dream of a final theory was shattered.

In 1930, at a conference in Königsberg, Hilbert remained confident in the eventual success of his program, proclaiming: “Wir müssen wissen. Wir werden wissen.” — “We must know. We will know.”

Gödel’s Incompleteness Theorems

Unknown to Hilbert, his dream had already been crushed. The day before, at the very same conference, the 24-year-old Kurt Gödel presented his PhD thesis. It proved Hilbert’s dream is impossible.

At the conference Gödel presented his First Incompleteness Theorem. It showed that in any finite mathematical foundation, there will be true statements that can’t be proved in that theory.

Thus Hilbert’s dream of completeness is impossible.

The most comprehensive current formal systems are the system of Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian axiom-system of set theory on the other hand. These two systems are so far developed that you can formalize in them all proof methods that are currently in use in mathematics, i.e. you can reduce these proof methods to a few axioms and deduction rules. Therefore, the conclusion seems plausible that these deduction rules are sufficient to decide all mathematical questions expressible in those systems. We will show that this is not true.

Kurt Gödel in “On formally undecidable propositions of Principia Mathematica and related systems I” (1931)

Gödel’s first incompleteness theorem showed there could never be a final theory that would serve mathematicians for all time.

Gödel wasn’t finished.

Shortly thereafter, he published his Second Incompleteness Theorem. This proved that no consistent theory of mathematics can ever prove itself to be consistent. The 2^nd of Hilbert’s 23 problems was impossible.

This explained the failure of Zermelo in proving the consistency of his set theory. It was actually a good sign that he was unable to — had he been able to prove it consistent, it would imply that it was not.

So now, not only was completeness impossible, but it was also impossible for a theory to prove its own consistency.

This was a double whammy to Hilbert. Hilbert lived another 12 years but he never publicly acknowledged Gödel’s result. Privately, he was crushed — he didn’t want mathematics to be this way.

But others greatly admired Gödel and his achievement.

When Harvard gave Gödel an honorary degree, he was introduced as “The discoverer of the most significant mathematical truth in the century.” Some called Gödel “the greatest logician since Aristotle.” Edward Nelson called Aristotle “the greatest logician before Gödel.”

John von Neumann said, “Gödel is absolutely irreplaceable; he is the only mathematician alive about whom I would dare make this statement.”

Undecidability

In 1673, Leibniz invented and later built the first digital calculator. He declared, “It is beneath the dignity of excellent men to waste their time in calculation when any peasant could do the work just as accurately with the aid of a machine.”

After he built the device, Leibniz began to wonder about the limits of what machines can calculate: was it possible to build a machine that could answer any mathematical question?

Several centuries later, David Hilbert together with Wilhelm Ackermann refined Leibniz’s question. At a conference in Berlin in 1928, they defined the Entscheidungsproblem (or decision problem).

The decision problem asks: is it possible to build a machine that can decide whether or not any mathematical question can be proved in some mathematical system?

Gödel showed that not every true statement was provable, but was there a way to decide whether or not a statement was provable?

It was an important question.

Such a method would be most useful to mathematicians. It would tell them when they ought to give up, and thereby save them from wasting their lives searching for proofs that don’t exist.

Alonzo Church got the first result on the Entscheidungsproblem. He defined a programming language, and proved certain questions about it are undecidable.

It follows that the Entscheidungsproblem is unsolvable in the case of any system of symbolic logic which is [consistent] in the sense of Gödel.

Alonzo Church in “An Unsolvable Problem of Elementary Number Theory” (1935)

The next year, Church’s student, Alan Turing, published another example of an undecidable problem — the halting problem.

Gödel has shown that there are propositions U such that neither U nor [not U] is provable. […] On the other hand, I shall show that there is no general method which tells whether a given formula U is provable.

Alan Turing in “On Computable Numbers, with an Application to the Entscheidungsproblem” (1936)

It was in this paper that Turing introduced the concept of a general purpose programmable computer, birthing the digital age.

Hilbert never got the answers he hoped for. We can’t prove the consistency of a mathematical foundation. We can’t prove everything that is true, and given undecidability, we can’t even be sure whether a statement has a proof or not.

And yet, despite not getting the answers he hoped for, Hilbert knew the right questions to ask. The answers produced great discoveries.

I’d like to make the outrageous claim, that has a little bit of truth, that actually all of this that’s happening now with the computer taking over the world, the digitalization of our society, of information in human society, you could say in a way is the result of a philosophical question that was raised by David Hilbert at the beginning of the century.

Gregory Chaitin in “A Century of Controversy Over the Foundations of Mathematics” (2000)

Hilbert’s 10^th Problem

Of Hilbert’s 23 problems, his 10^th problem asked for a general method to solve Diophantine equations. These are equations that allow only whole numbers, (no decimals or fractions), which are named after Diophantus who studied them in the 3^rd century.

Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

David Hilbert in “Mathematical problems” (1902)

Deceptively simple Diophantine equations were often notoriously difficult. A famous example is the Diophantine equation:

a^n = b^n + c^n

This equation is easy when n = 1, or when n = 2. Millennia ago Pythagoras proved there were infinite solutions when n = 2.

And yet, no one had found even one solution for n \ge 3. No one knew of a cube number (a^{3}) that was the sum of two other cube numbers.

In 1673, Pierre de Fermat wrote in his notes that he had a proof that there were no solutions when n \ge 3. But no one had ever found it, nor was anyone able to rediscover a proof.

The missing proof became known as Fermat’s last theorem.

The problem went unsolved for 321 years. Until in 1994, after seven years of work, Andrew Wiles completed a 129-page proof that no whole number solutions exist when n \ge 3.

If mathematicians had a procedure to solve Diophantine equations, Andrew Wiles wouldn’t have had to spend seven years working on this problem. Instead, he could program a computer to follow the procedure and the computer would crank out a solution.

In 1970, Hilbert’s 10^th problem was solved.

Solving it required 21 years of work by four mathematicians: Martin Davis, Julia Robinson, Hilary Putnam and Yuri Matijasevič.

Their proof, called the MRDP-Theorem (after their initials), gave a negative result. They proved there is no general procedure for solving Diophantine equations — and they proved it in a shocking way.

They showed an equivalence between solutions to Diophantine equations and what is computable. In other words, for any imaginable computer program, there is a Diophantine equation whose solutions equal all the outputs of that computer program.

This was so surprising that many mathematicians had difficulty believing it. It meant there is a Diophantine equation that picks chess moves like Deep Blue, and there’s a Diophantine equation that does your taxes like TurboTax, and there’s yet another Diophantine equation that does spellchecking like Microsoft Word.

For anything a computer can compute, there’s a Diophantine equation that gives the exact same answers.

But despite how surprising their result was, it was true.

And this is why there can be no general method for solving Diophantine equations: because the question of whether or not a program finishes (Turing’s halting problem) is equivalent to asking whether or not some Diophantine equation has solutions.

Since the halting problem is not generally solvable, the equivalence between Diophantine equations and computers meant Diophantine equations weren’t generally solvable either.

Yet again, what Hilbert asked for couldn’t be provided.

Hilbert’s questions probed at the heart of consistency, provability, decidability, and computability. They didn’t lead where he expected, but they did reveal deep truths about the nature of mathematics.

Universal Equations

In 1978, the mathematician James P. Jones went a step further. Just as it is possible to make a computer program that runs all other computer programs, it is also possible to make a Diophantine equation that includes all other Diophantine equations.

Matijasevič’s Theorem implies also the existence of particular undecidable diophantine equations. In fact there must exist universal diophantine equations, polynomial analogues of the universal Turing machine.

James P. Jones in “Undecidable Diophantine Equations” (1980)

Such Diophantine equations are general purpose computers: plug in the ‘program’ as one of the variables to the equation, and the solutions to the equation will be the outputs of that program.

Jones provided an example of such an equation. It is complex, but the truths concerning this single equation include all truths concerning the executions and outputs of all computer programs.

\text{In order that } x \in W_{v} \text{ it is necessary and sufficient that the} \\ \text{following equation has a solution in positive integers:}
\\ \text{}
 \\ 
(v - (((zuy)^2 + u)^2 + y))^2 + (elg^2 + \alpha - (b - xy)q^2)^2 + \\
(q - b^{5^{60}})^2 + (\lambda + q^4 - (1 +\lambda b^5))^2 + \\
(\theta + 2z - b^5)^2 + (l - (u + t \theta))^2 + (e - (y + m \theta))^2 + \\
(n - q^{16})^2 + (r - ([g + eq^3 + lq^5 + (2(e - z \lambda)(1 + xb^5 + g)^4 + \\
\lambda b^5 + \lambda b^5 q^4)q^4][n^2 - n]+ [q^3 -bl + l + \theta \lambda q^3 + \\
(b^5 - 2)q^5][n^2 - 1]))^2 +  (p - 2ws^2 r^2 n^2)^2 + \\
(p^2k^2 - k^2 + 1 - \tau^2)^2 + (4(c - ksn^2)^2 + \eta - k^2)^2 + \\
(k - (r + 1 + hp - h))^2 + (a - (wn^2 + 1)rsn^2)^2 + \\
(c - (2r + 1 + \varphi))^2 + (d - (bw + ca -2c + 4\alpha \gamma - 5 \gamma))^2 + \\
(d^2 - ((a^2 -1)c^2 + 1))^2 + (f^2 - ((a^2 - 1)i^2 c^4 + 1))^2 + \\
((d + of)^2 - (((a + f^2(d^2 - a))^2 - 1)(2r + 1 + jc)^2 + 1))^2 = 0

As [v] varies through the positive integers, the [equation] defines every recursively enumerable set. This is, to our mind, the attraction of the universal equations. At once, [this equation defines] primes, Fibonacci numbers, Lucas numbers, perfect numbers, theorems of ZF, or indeed theorems of any other axiomatizable theory.

James P. Jones in “Three Universal Representations of Recursively Enumerable Sets” (1978)

We might consider such universal equations as ‘God Equations‘ — equations whose solutions contain and include all the others.

In his 1987 book Algorithmic Information Theory, Gregory Chaitin describes one such equation: the “Exponential Diophantine Equation Computer.” It has 20,000 variables and is two hundred pages long.

This equation perfectly replicates the behavior of the LISP programming language. He describes the equation as follows:

If the LISP expression k has no value, then this equation will have no solution. If the LISP expression k has a value, then this equation will have exactly one solution. In this unique solution, n = the value of the expression k.

Gregory Chaitin in “META MATH! The Quest for Omega” (2004)

Chaitin showed that even modern day computers and programming languages have counterparts in the form of Diophantine equations.

Universal Diophantine equations are remarkable. They exist in pure arithmetic. The arithmetical relations they encode represent every program that can be computed along with all of their outputs.

Among these solutions we can find the valid proofs of every theorem in every mathematical system, every way of playing every computer game that has or will ever be invented, and simulations of every galaxy in the observable universe down to the atomic level.

Jones’s discovery of universal Diophantine equations inspired him to quote chapter 11, verse 7 of the Bhagavad Gita: “Whatever you wish can be seen all at once right here. This universal form can show you all that you now desire. Everything is here completely.”

Given that such equations include everything computable, including all physical laws and systems as well as simulations of any observer’s mind and brain — could these equations be the glue connecting eternal mathematical truths with contingent physical truths?

The Universal Dovetailer

In 1991, Bruno Marchal wrote a program he called the Universal Dovetailer — a program that generates and runs all programs.

In order to run every program without getting stuck on a program that never ends, the Universal Dovetailer interleaves, or dovetails, on the processing, doing a little bit of work on each program at a time.

The program is simple. The full program is quite short, consisting of about 300 lines of LISP code. Its pseudocode is even simpler:

for k from 0 to ∞:
    for j from 0 to k:
        for i from 0 to j:
            // Compute k steps of program i on input j

This program sequentially generates every program and runs it for every input. The longer the Universal Dovetailer runs, the more programs it generates and the more steps of each program it performs. If allowed to run forever, it runs every program there is.

This program, like universal Diophantine equations, contains all.

While studying the consequences of the existence of all computations, Marchal made an incredible discovery: what he describes as the “many-histories interpretation of elementary arithmetic.”

The discovery served as the basis of his 1998 PhD thesis, Computability, Physics and Cognition. This paper explains how we can explain the appearance of a multiverse given two assumptions:

1. All computations exist
2. Computation supports cognition

We will explain that once we adopt the computationalist hypothesis, which is a form of mechanist assumption, we have to derive from it how our belief in the physical laws can emerge from only arithmetic and classical computer science.

Bruno Marchal in “The computationalist reformulation of the mind-body problem” (2013)

Given there exist universal Diophantine equations, all computations exist as a consequence of arithmetical truth concerning them.

While there is no physical realization of the perpetual execution of the Universal Dovetailer, its complete execution exists in number theory as a consequence of arithmetical truth. There are, for instance, Diophantine equations whose solutions exactly equal all the sequentially generated states reached by the Universal Dovetailer.

So if we accept the self-existent truth of ‘2 + 2 = 4’, we must also accept truths concerning universal Diophantine equations — truths that concern all computational histories and all simulated realities.

To be sure, the existence of the UD is a logical consequence of elementary arithmetic with Church’s thesis or Turing’s thesis.

Bruno Marchal in discussion list (2019)

It therefore becomes a purely mathematical question to prove whether some Diophantine equation contains in its solutions a computational state equivalent to some person’s physical brain state.

We would then exist for the same reason that ‘2 + 2 = 4’ — as an inevitable consequence of mathematical truth. The question “Why is there anything at all?” is reduced to “Why does 2 + 2 = 4?”

A Story of Creation

We have arrived at a plausible story of creation.

We can now connect the causeless abstract entities: logic, truth, and numbers, with a viable cause for our perceptions of a physical reality.

Why does anything exist?

Because necessity requires logical laws; logical laws imply incontrovertible truth; such truth includes mathematical truth; mathematical truth defines numbers; numbers possess number relations; number relations imply equations; equations define computable relations; computable relations define all computations; all computations include algorithmically generated observers; and these observers experience apparent physical realities.

Ancient Anticipations

This account of how eternal mathematical truths could give rise to contingent physical truths depended on recent discoveries.

It required a deep understanding of modern ideas: universal equations, computers, computation, virtual reality, and simulation.

Only a century ago, we didn’t even have words for these concepts. Despite this, a few ancient thinkers gave theories for existence that are eerily similar to this modern creation story.

They postulated something primal and simple that gave rise to the numbers, and from numbers arose beings, consciousness, and matter.

2,600 years ago, Laozi wrote that numbers proceed from The Tao and that from numbers that all things are born:

The Tao gives birth to One.
One gives birth to Two.
Two gives birth to Three.
Three gives birth to all things.

Laozi in chapter 42 of “Tao Te Ching” (c. 600 B.C.)

Diogenes Laërtius was a biographer of eminent philosophers. The following is his account of 2,500-year-old Pythaogrean beliefs:

That the monad (one) was the beginning of everything. From the monad proceeds an indefinite duad (two), which is subordinate to the monad as to its cause. That from the monad and the indefinite duad proceed numbers. And from numbers signs. And from these last, lines of which plane figures consist. And from plane figures are derived solid bodies. And from solid bodies sensible bodies.

Diogenes Laërtius in “The Lives and Opinions of Eminent Philosophers” (c. 225 A.D.)

1,750 years ago, Plotinus developed neoplatonism — a rich theory concerning the relations between various levels of being.

Wikipedia describes Plotinus’s chain of being as a series of emanations: “The first emanation is Nous (Divine Mind, Logos, Order, Thought, Reason) […] From Nous proceeds the World Soul. […] From the world soul proceeds individual human souls, and finally, matter, at the lowest level of being and thus the least perfected level of the cosmos.”

The One is not a Being but the generator of Being. [V.2.1]

The greatest, later than the [One], must be the [Intellectual-Principle], and it must be the second of all existence. [V.1.6] For what emanates from the Intellectual-Principle is a Reason-Principle, a Logos. [III.2.2]

And as soon as there is differentiation, number exists. [V.1.5]
Thus Number, the primal and true, is Principle and source of actuality to the Beings. [VI.6.15]

[The Soul’s] substantial-existence comes from the Intellectual-Principle. [V.1.3] The Soul, itself a Divine Thought and possessing the Divine Thoughts, or Ideas, of all things, contains all things concentred within it. [III.6.18]

This gives the degree in which the kosmos is ensouled, not by a soul belonging to it, but by one present to it; it is mastered not master; not possessor but possessed. [IV.3.9] This one universe is all bound together in shared experience. [IV.4.32] So matter is actually a phantasm. [II.5.5]

Plotinus in “The Enneads” (270 A.D.)

1,570 years ago, Proclus wrote that mathematical existence occupies a middle ground. He said mathematical being sits between the simple reality that’s grounded in itself and the things that move about in matter.

Mathematical being necessarily belongs neither among the first nor among the last and least simple of the kinds of being, but occupies the middle ground between the partless realities—simple, incomposite, and indivisible—and divisible things characterized by every variety of composition and differentiation. The unchangeable, stable, and incontrovertible character of the propositions about it shows that it is superior to the kinds of things that move about in matter. But the discursiveness of [mathematical] procedure, in dealing with its subjects as extended, and its setting up of different prior principles for different objects—these give to mathematical being a rank below that indivisible nature that is completely grounded in itself.

Proclus in “A Commentary on the First Book of Euclid’s Elements” (c. 450 A.D.)

The Causeless Cause Found?

Could this be the answer? Could things be so simple?

In order for this explanation of existence to be correct, mathematical truth must be causeless. Mathematical existence must depend on neither human minds, nor on physical or material things.

In addition, mathematical truth must be something capable of generating observers — observers who consciously perceive their environment and which they consider as existing physically.

Ideally, this causeless cause will illuminate the relation between the mental and material, and explain why the universe obeys simple laws.

Can the theory achieve this?

Is it Causeless?

For mathematical truth to serve as a causeless cause, it must exist causelessly. Math must depend on neither minds nor matter.

Independent of Minds

Do numbers and their properties exist beyond the minds of mathematicians and their scribblings on blackboards?

Had Hilbert’s program succeeded, and given a mathematical theory capable of proving all true statements, then arguably, mathematics might only be that which follows from this theory. Math would then be an invention of the human mind.

But the failure of Hilbert’s program, and Gödel’s proof of the impossibility for any finite theory to define all mathematical truth, meant that mathematical truth is infinite and beyond description, and therefore cannot be a product of human minds.

[The existence of] absolutely undecidable mathematical propositions, seems to disprove the view that mathematics is only our own creation; for the creator necessarily knows all properties of his creatures, because they can’t have any others except those he has given to them. So this alternative seems to imply that mathematical objects and facts (or at least something in them) exist objectively and independently of our mental acts and decisions, that is to say, [it seems to imply] some form or other of Platonism or ‘realism’ as to the mathematical objects.

Kurt Gödel in “Some basic theorems on the foundations of mathematics and their implications p. 311″ (1951)

(See: “Is math invented or discovered?“)

Independent of Matter

The infinite nature of mathematical truth also implies an independence from matter. Our observable universe has an information capacity of 10^{120} bits. This number is large, but finite.

Nowhere in physics, is there room to store, represent, or hold the infinite true statements of mathematics. If there are infinite primes, infinite factors of zero, infinite digits of \pi, they don’t exist physically.

If these infinite properties don’t and can’t depend on physical processes operating within a material universe, it follows that mathematical properties must exist independently of matter.

It is our firm belief that the Pythagorean theorem needs not be created, nor the fact that the circumference of a circle is 3.14… times the diameter; the laws of nature and the collection of truths, values and their interrelations are primordial and have always existed.

C. W. Rietdijk in “Four-dimensional reality continued” (2018)

Is it the Cause?

For this story to work, abstract objects: truth, numbers, equations, and so on, must play a causal role in generating reality and perceptions.

The default position of philosophers has been that abstract objects have no effects — they cause and do nothing. But we must admit that this has always been an assumption, it’s never been proven.

Although philosophers deny that abstract objects can have causal effects on concrete objects (abstract objects are often defined as causally inert), their potential, say as a collective, to be an explanatory source of ultimate reality cannot be logically excluded.

John A. Leslie and Robert Lawrence Kuhn in “The Mystery of Existence” (2013)

Recent advances in mathematics give us pause. The discovery that all computations exist as a consequence of mathematical truth, makes us wonder whether abstract mathematics is really so ineffectual.

But can mind or matter really be created by math?

The Cause of Minds

Consciousness remains one of humanity’s last great mysteries. While science has not settled the question of what consciousness is, it has progressed by developing a testable theory of consciousness.

In the 1600s, thinkers such as René Descartes and Thomas Hobbes advanced the idea of mechanism — the theory that our brains and bodies are machines that operate according to mechanical rules.

In 1936, the discovery of universal machines (or computers) led to the Church-Turing thesis, which says the behavior of any finite machine can be perfectly replicated by an appropriately programmed computer.

This is their special power. It is what makes computers so useful.

Some were quick to recognize the implications of the Church-Turing thesis for theories of minds, brains, and consciousness.

The two fathers of computing, Alan Turing and John von Neumann, noticed parallels between computers and the mind. In 1948, Turing wrote the first chess playing program and in his 1950 paper Computing Machinery and Intelligence, Turing asked, “Can machines think?”

The last work of John von Neumann was a lecture series: The Computer and the Brain, published posthumously in 1958. In it von Neumann explains that it is not that the brain acts like a computer, but that computers are so varied in what they can do, that they can be set up to imitate any machine — presumably even the human brain.

The important result of Turing’s is that in this way the first [universal] machine can be caused to imitate the behavior of any other machine.

John von Neumann in “The Computer and the Brain” (1958)

In the 1960s and 1970s, philosophers of mind including Hilary Putnam and his student Jerry Fodor developed what they called functionalism. In its digital form, functionalism is known as the Computational Theory of Mind (or computationalism). This is the idea that function, or computation, is the foundation of consciousness.

The computational theory of mind remains as the most popular theory for consciousness among scientists and philosophers.

Computationalism, or digital mechanism, or simply mechanism, is [a] hypothesis in the cognitive science according to which we can be emulated by a computer without changing our private subjective feeling.

Bruno Marchal in “The computationalist reformulation of the mind-body problem” (2013)

If the computational theory of mind is true, then mathematics can explain where observers come from. Observers would be found among the infinite computational histories within arithmetical truth. (See: “What is consciousness?” and “Can a machine be conscious?“)

Recent discoveries in physics lend support to computationalism.

In 1981, Jacob Bekenstein discovered a physical limit now known as the Bekenstein bound. This bound says that a physical system of finite mass and volume can contain, at most, a finite amount of information. This applies to any finite physical system: a brain, the Earth, the Solar System, our galaxy, or the observable universe.

Given that the observable universe has a finite mass and volume, it follows by the Bekenstein bound that it has a finite description.

Given that it has a finite description, it follows by the Church-Turing thesis that the evolution of the observable universe is something that is perfectly replicated by a certain computer program.

This program contains a version of you, me, the Earth and everyone and everything present in our universe. Our shared histories and memories would be identical. But the question remains: are these computational doppelgängers conscious like we are?

If we inspected the contents of this computer program, we would find analogues of all the objects of our own universe. We would find the same books, articles, and movies. Among these we would even find many works on the mysterious nature of consciousness.

If these purely computational versions of us are not conscious, what drives them to write and read books about consciousness?

If, on the other hand, they are just as conscious as we are, then the idea of a separately existing physical reality becomes redundant. In that case, for all we know, we are these computational versions!

We would then exist as pure computations. We would inhabit the computational histories of simulated realities that exist only as a consequence of mathematical truth concerning universal equations.

Every imaginable computation is realized in arithmetic as true relations about these universal equations. This includes the computations that describe you, your environment, and even the evolving state of your brain as it processes this very sentence.

If computationalism is right, this is who we are.

We’ll explore the fascinating relations between computation, mathematics, physics and mind, and explore a crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, making us self-aware parts of a giant mathematical object.

Max Tegmark in “Our Mathematical Universe” (2014)

The Cause of Matter

Can mathematical truth, with its inherent infinite collection of computational histories, explain matter, physical laws and universes?

How can abstract things, like truth, numbers, computations, give rise to concrete things like chairs, bricks, and houses?

What’s the difference between abstract existence vs. concrete existence?

Some say the difference is only a matter of perspective. To a being who inhabits an abstract object, (be it an abstract mathematical object or abstractly existing computation), it seems concrete to them.

This equivalence between physical and mathematical existence means that if a mathematical structure contains a [self-aware substructure], it will perceive itself as existing in a physically real world, just as we do.

Max Tegmark in “The Mathematical Universe” (2007)

The relative aspect of concrete existence, is explicit in Markus Müller’s definition of physical existence:

Given two objects A and B, we say that they [physically exist] for each other if and only if, under certain auxiliary conditions, modifying the state of A will affect the state of B, and vice versa.

Markus Müller in “Could the physical world be emergent instead of fundamental, and why should we ask?” (2017)

Whenever a conscious observer experiences or interacts with another object, that object appears concrete to that observer — even if, from another point of view, both that observer and object seem abstract.

Of the modes of existence, this understanding implies mind over matter.

Math produces an infinity of conscious minds, and the perceptions of these minds include experiences of material realities.

Computationalism, together with the mathematical existence of all computations, leads to a causal reversal between mind and matter.

What results is not a primitive matter with consciousness emerging from its organisation but the reverse: consciousness is now the more primitive and matter, or rather the appearance of material organisation, emerges from all the possible experiences of all the possible consciousnesses.

Bruno Marchal in “The Amoeba’s Secret” (2014)

Matter is then, as Plotinus supposed, a phantasm.

Is This Testable?

This is a big pill to swallow. Are we to take as serious the idea that we live inside an equation? And this equation somehow produces all computations by virtue of its solutions? And that the whole physical universe is just some kind of shared hallucination?

Extraordinary claims require extraordinary evidence.

Unless there is a way to test and either confirm or falsify this theory, we are not operating in the realm of science, but fantasy.

Fortunately, there is a way to test this theory.

Due to the fact that not all programs appear with equal frequency, a particular bias should appear in the resulting computational histories.

We can then check for this bias by comparing our observations of the character of physical law and the properties of our universe against the predictions made by the theory.

If our observations match the predictions, we gain evidence in support of the theory. If they don’t match, we rule the theory out.

This is how all theories are tested.

Algorithmic Information Theory

The reason not all programs occur with equal frequency is due to a consequence of algorithmic information theory (or AIT).

This field was developed by Ray Solomonoff, Andrey Kolmogorov and Gregory Chaitin, starting in the 1960s.

Chaitin says, “AIT is the result of putting Shannon’s information theory and Turing’s computability theory into a cocktail shaker and shaking vigorously. The basic idea is to measure the complexity of an object by the size in bits of the smallest program for computing it.”

Across the infinite programs executed by universal equations, some programs exhibit identical behavior. This is because the program’s code may instruct it to read only a fraction of its total available code.

Programs that complete are naturally self-delimiting. They define their own length by virtue of reading only a finite number of bits.

Above we see two equivalent programs highlighted in blue, both of these programs are defined by the same 9-bit prefix.

Given that this program length is 9 bits, we can calculate that this program should appear once every 2^{9} (or 512) bit strings. Self-delimited 10-bit programs would be half as common, appearing once every 2^{10} (or 1,024) programs. Conversely, 8-bit programs are twice as common as 9-bit ones.

We can use this consequence of algorithmic information theory to make several predictions about the character of physical law.

The main point is that the derivation is constructive, and it provides the technical means to derive physics from arithmetic, and this will make the computationalist hypothesis empirically testable, and thus scientific in the Popperian analysis of science.

Bruno Marchal in “The computationalist reformulation of the mind-body problem” (2013)

Confirming Evidence

Could such a bold theory be true?

For now, let’s neither accept nor reject this theory. To do either before weighing the evidence would be premature.

So let us not believe anything and maintain an open mind. For the time, we will only play with the idea and see where it leads.

As with any theory, the only path forward is to see what this theory predicts and then to compare the predictions with our observations.

If we find it leads in a fruitful direction, by making predictions we can confirm and by not making predictions we can refute, then we will have cause to tentatively accept this theory.

Predictions of the Theory

Does the reality we see fit predictions of a reality generated by the infinite computations inherent to causeless arithmetical truth?

For that matter, what are the predictions?

At first blush, it seems impossible to get any useful predictions from a theory that includes all computations and all observations. For if they all exist, any observation is compatible with the theory. As Victor Stenger noted, “Theories that explain everything explain nothing.”

Fortunately, there is a catch: not all observations are equally likely.

If our conscious states result from the existence of all computations, then they are subject to the rules of algorithmic information theory.

This enables us to make testable predictions, and thereby tie it back to hard science, observation, and measurement.

Some of the predictions of this theory provide clues to otherwise unsolvable questions in physics and cosmogeny. These predictions offer answers to such fundamental mysteries as:

• Why the universe obeys simple, mathematical, life-friendly laws
  • Why empiricism (i.e. experimental reproducibility) works
  • Why Occam’s Razor works
  • Why the laws appear fine-tuned for life
• Why the laws are quantum mechanical
  • Why uncertainty and randomness exist in physics
  • Why infinite descriptions are needed to explain any occurrence
• Why observation and information are fundamental in physics
• Why the universe has time and a beginning (e.g., the big bang)

These results are the work of pioneers in the theory, who include Bruno Marchal, Max Tegmark, Russell Standish, and Markus Müller.

Using the tools of computer science, math, information theory, and algorithmic information theory, they revealed how these traits of the universe result from our mind states being computationally-generated.

The appearance of a universe, or even universes, must be explained by the geometry of possible computations.

Bruno Marchal in “The Amoeba’s Secret” (2014)

Let’s review the evidence for this most speculative of theories, which is presently at the forefront of mathematics and physics.

Why Laws?

We take for granted that our universe obeys laws. But why should it?

What’s the source of these laws? Why are they so simple? Why aren’t they ever violated? Why these laws and not others?

All these questions are mysteries left unaddressed by science.

In the orthodox view, the laws of physics are floating in an explanatory void. Ironically, the essence of the scientific method is rationality and logic: we suppose that things are the way they are for a reason. Yet when it comes to the laws of physics themselves, well, we are asked to accept that they exist “reasonlessly”.

Paul Davies in “The flexi-laws of physics” (2007)

With the equations, when they are not too complicated, we can predict phenomena. But in truth, the equation doesn’t explain anything. It compresses, certainly, in a very ingenious way, the description of the physical world, but it does not explain the nature of bodies nor why these bodies obey laws, nor from where these laws come.

Bruno Marchal in “The Amoeba’s Secret” (2014)

That laws are never violated, on its face, seems highly improbable. For in the space of possibility, for each way there is for the universe to obey the laws, there are infinite ways it might deviate from them.

For each law-governed world, there are countless variants that would fail in different ways to be wholly law-governed.

Derek Parfit in “Why Anything? Why This?” (2008)

Why the laws hold is unknown to science. And yet, this feature of reality is the very basis that allows us to do science.

A lawful universe is the basis of empiricism. It is why we can repeat experiments, and make predictions about the future based on past observations. But why does this work, and why should it work?

Marchal explains the emergence of laws as a consequence of the computational reality. He says the laws are the ‘consistent extensions’ of programs that produce the observer’s mind state.

Arithmetic contains or executes all computations. Your first person is distributed on all computations going through your current first person state. To make any prediction on the future of your possible inputs, you need to take all the computations into account, and the laws of physics is what is invariant in all consistent extensions.

Bruno Marchal in discussion list (2019)

Müller goes further, and gives a mathematical proof that shows why, given algorithmic information theory, observers will, with high probability, observe a ‘persistence of regularities‘ (i.e. laws).

That is, computable regularities that were holding in the past tend to persist in the future. […]

Intuitively, highly compressible histories are those that contain regularities which can be used to generate shorter descriptions.

Markus Müller in “Law without law: from observer states to physics via algorithmic information theory” (2020)

Because most programs are simple, and simple programs tend to keep doing what they have been doing, this gives the appearance of a fixed set of laws that holds into the future as the program unfolds.

So in a sense, the laws of physics are the rules of the programs that instantiate us, as seen by those of us inside those programs.

Why the Laws are Mathematical

It has long been recognized that mathematics is “unreasonably effective” in describing the physical laws. In 1623, Galileo wrote, “[The universe] is written in the language of mathematics.”

This connection between math and physics still puzzles scientists.

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 in “The Unreasonable Effectiveness of Mathematics in the Natural Sciences (1960)

But why should physics be so mathematical? Tegmark offers a simple explanation: because physical theories result from our perceptions of what are ultimately mathematical structures.

The various approximations that constitute our current physics theories are successful because simple mathematical structures can provide good approximations of how a [self-aware substructure] will perceive more complex mathematical structures. In other words, our successful theories are not mathematics approximating physics, but mathematics approximating mathematics!

Max Tegmark in “Is ‘the theory of everything’ merely the ultimate ensemble theory?” (1998)

Why the Laws are Simple

In the 2^nd century, Ptolemy wrote, “We consider it a good principle to explain the phenomena by the simplest hypothesis possible.”

This rule of thumb is called the law of parsimony or Occam’s razor.

It is the idea that in science, the simplest answer that fits the facts is usually right. Occam’s razor is no doubt a useful and effective rule, but until recently no one understood why it works.

Deep truths of nature can be expressed by short formulas, like F = ma and E = mc². Physical equations rarely involve more than a few terms, rather than dozens or hundreds.

Physicists are awestruck by this simplicity. Einstein remarked, “The eternal mystery of the world is its comprehensibility.”

Given there are far more ways for these formulas to be more complex, it’s especially odd that they should be so simple.

Compared with simple laws, there is a far greater range of complicated laws. […] We would have some reason to believe that there are at least two partial Selectors: being law-governed and having simple laws.

Derek Parfit in “Why Anything? Why This?” (2008)

But the lesson is that at present the idea that the ultimate laws are as simple as possible is a hope, not something suggested by the evidence. Moreover, the prospect still faces the challenge of explanatory regression, as one would [be] left to explain why the underlying laws should be so simple.

Sean Carroll in “Why Is There Something, Rather Than Nothing?” (2018)

The mystery of simple comprehensible laws can now be answered.

We have found a selector that preferentially selects universes with simple laws. Algorithmic information theory tells us that for each bit saved in a program’s description, its occurrences double.

This adds up fast. A program that’s 30 bits shorter, say 120-bits vs. 150-bits occurs 2^{30} — or over 1 billion — times more often.

Ray Solomonoff, the father of algorithmic information theory, was the first to draw a connection between AIT and Occam’s razor.

On a direct intuitive level, the high a priori probability assigned to a sequence with a short description corresponds to one possible interpretation of “Occam’s Razor.”

Ray Solomonoff in “A Formal Theory of Inductive Inference” (1964)

When Müller applied algorithmic information theory to observer states, he found that it led to the prediction of simple physical laws.

Observers will, with high probability, see an external world that is governed by simple, computable, probabilistic laws.

Markus Müller in “Law without law: from observer states to physics via algorithmic information theory” (2020)

Why the Laws are Life-Friendly

One of the most surprising discoveries in physics of the past 50 years was the discovery that the laws of physics and constants of nature appear specially selected to allow complexity and life to arise.

Wheeler wrote, “A life-giving factor lies at the centre of the whole machinery and design of the world.”

That the constants of nature, the strengths of the forces, the particle masses, etc., are just right to permit complex structures to arise is mysterious. Why are the laws this way? Why are they life friendly?

Physicists ask, why does the universe appear fine-tuned?

As we look out into the universe and identify the many accidents of physics and astronomy that have worked together to our benefit, it almost seems as if the universe must in some sense have known we were coming.

Freeman Dyson in “Energy in the Universe” (1971)

The fine tunings, how fine-tuned are they? Most of them are 1% sort of things. In other words, if things are 1% different, everything gets bad. And the physicist could say maybe those are just luck. On the other hand, this cosmological constant is tuned to one part in 10^{120} — a hundred and twenty decimal places. Nobody thinks that’s accidental. That is not a reasonable idea — that something is tuned to 120 decimal places just by accident. That’s the most extreme example of fine-tuning.

Leonard Susskind in “What We Still Don’t Know: Are We Real?” (2004)

The first step in explaining fine-tuning is to recognize that for any universe to be perceived, requires that it be populated with conscious observers. This reasoning is known as the anthropic principle.

The next step is to explain why any universe exists that supports conscious observers. Typical answers are that the universe was either designed or it is just one among a vast set of mostly dead universes.

We imagine our universe to be unique, but it is one of an immense number—perhaps an infinite number—of equally valid, equally independent, equally isolated universes. There will be life in some, and not in others.

Carl Sagan in “Pale Blue Dot” (1994)

The existence of infinite computational histories, guarantees that some will be of a type that can support life. Moreover, algorithmic information theory tells us the resulting physics should be maximally simple while respecting the constraint of being life-friendly.

In this paper I show why, in an ensemble theory of the universe, we should be inhabiting one of the elements of that ensemble with least information content that satisfies the anthropic principle. This explains the effectiveness of aesthetic principles such as Occam’s razor in predicting usefulness of scientific theories.

Russell Standish in “Why Occam’s Razor” (2004)

And indeed, this is what we find when we examine our physics:

A very interesting question to me is: is the universe more complicated than it needs to be to have us here? In other words, is there anything in the universe which is just here to amuse physicists?

It’s happened again and again that there was something which seemed like it was just a frivolity like that, where later we’ve realized that in fact, “No, if it weren’t for that little thing we wouldn’t be here.”

I’m not convinced actually that we have anything in this universe which is completely unnecessary to life.

Max Tegmark in “What We Still Don’t Know: Why Are We Here” (2004)

(See: “Is the universe fine-tuned?“)

Why Quantum Mechanics?

Quantum mechanics is a cornerstone theory of modern physics. It’s among the most thoroughly tested of all theories in science, and it’s given us the most accurate predictions in all of physics.

But quantum mechanics is incredibly strange.

It suggests the existence of many (infinite) histories (i.e. many-worlds or many-minds). Observation or measurement appears to cause the infinite set of possibilities to “collapse” to just one of the possibilities, and the selected result is absolutely unpredictable.

Quantum mechanics includes apparent absurdities, like unobserved cats being simultaneously alive and dead, non-local faster-than-light influences, and unlimited computation underlying physical reality.

I have never been able to let go of questions like: How come existence? How come the quantum?

John Archibald Wheeler in “Geons, Black Holes, and Quantum Foam” (1998)

Of the mysteries in physics, “How come the quantum?“, ranks highly:

Niels Bohr said, “Those who are not shocked when they first come across quantum theory cannot possibly have understood it.” Werner Heisenberg admits, “I repeated to myself again and again the question: Can nature possibly be so absurd as it seemed to us in these atomic experiments?” And Richard Feynman said, “I think I can safely say that nobody understands quantum mechanics.”

Wheeler thought if an ultimate theory could explain quantum mechanics, it would be a sure sign the theory was on the right track.

The most important test is whether it gives anything like quantum mechanics. If it does, we have a go-ahead sign, if not, we have to revise our thinking.

John Archibald Wheeler quoted in “Trespassing on Einstein’s Lawn” (2014)

Marchal’s 1998 thesis, Computability, Physics and Cognition, gave the first hints that features of quantum mechanics, such as indeterminism, the many parallel histories, the non-clonability of matter, and quantum logic, could be explained as a consequence of computationalism.

As in quantum mechanics, computationalism highlights a strong indeterminism, as well as a form of non-locality. […] Computationalism entails the existence of a phenomenology of many-worlds or parallel states.

Bruno Marchal translated from “Computability, Physics and Cognition” (1998)

Marchal writes, “The quantum empirical clues happen to be serious hints that the physical emerges from an internally defined statistics on the numbers dreams or computations seen from inside.”

Standish went further. In a 2004 paper and in his 2006 book, he showed one could derive the basic rules, or postulates, of quantum mechanics, including the Schrödinger equation, purely from basic assumptions about observation within an infinite set of possibilities.

The explanation of quantum mechanics as describing the process of observation within a plenitude of possibilities is for me the pinnacle of achievement of the paradigm discussed in this book. I can now say that I understand quantum mechanics. […]

So when I say I understand quantum mechanics, I mean that I know that the first three postulates are directly consequences of us being observers. Quantum mechanics is simply a theory of observation!

Russell Standish in “Theory of Nothing” (2006)

Irreducible Randomness

One of the strangest features of quantum mechanics is the presence of irreducible randomness that creates absolute unpredictability.

Compounding this strangeness is the fact that the equations of quantum mechanics are entirely deterministic. And yet, when a measurement is made, it seems the universe momentarily stops following these equations to randomly select one possibility to make real, from among the many possibilities present in the equations.

This was a pill too hard for Einstein to swallow. He declared, “God doesn’t play dice with the world.” And in the end he never accepted it.

In the single-electron double-slit experiment, an electron is put into a superposition — where the electron exists in multiple locations at once.

Then it’s location is measured.

But when we measure the electron’s location, it will appear in only one location, seemingly at random. Before measurement, it’s impossible, even in theory, to predict where the electron will be.

If we inhabit a computational reality why do we see any randomness or unpredictability? Computations are perfectly predictable. Might this observation of randomness give us cause to doubt or rule out our being in a computational reality?

The opposite is true. The existence of an infinite computational reality explains why we encounter absolute unpredictability.

If only one computational history existed, observing randomness would be cause to dismiss the theory. But here there are infinite computational histories. Some of these histories will be similar to each other, some, so similar as to be almost indistinguishable.

Since there are infinite computational histories, each observer’s mind state can be found within infinite parallel computational histories.

In a 1988 conference and in a 1991 paper, Mechanism and Personal Identity, Marchal explained how the appearance of randomness emerges from multiple instantiations of a single observer’s mind.

He calls the phenomenon first-person indeterminacy.

To predict the first person observable outcome of any physical experiment, you have to assume that your current computational state will not be obtained in some other part of the universe, or the multiverse, with different output for your experience.

Bruno Marchal in “The computationalist reformulation of the mind-body problem” (2013)

In summary: no brain that belongs to multiple distinct universes, or computational histories, can ever be sure what it will see next.

It is impossible for any observer to deduce with certainty on the basis of her observations and memory which world she is a part of. That is, there are always many different worlds for which being contained in them is compatible with everything she knows, but which imply different predictions for future observations.

Markus Müller in “Could the physical world be emergent instead of fundamental, and why should we ask?” (2017)

So even in a fully deterministic reality, the existence of infinite histories makes the appearance of randomness inevitable.

The physicist shining a photon at a piece of glass is in an infinity of histories where the photon will reflect, and is in an infinity of histories where the photon will pass through. The physicist can’t tell which until after the experiment is performed and she learns the result.

Ultimately, randomness stems from our inability to self-locate within the infinite sea of indistinguishable computational histories.

Tegmark notes how randomness appears in deterministic processes:

It gradually hit me that this illusion of randomness business really wasn’t specific to quantum mechanics at all. Suppose that some future technology allows you to be cloned while you’re sleeping, and that your two copies are placed in rooms numbered 0 and 1. When they wake up, they’ll both feel that the room number they read is completely unpredictable and random.

Max Tegmark in “Our Mathematical Universe” (2014)

Einstein is vindicated. God doesn’t play dice with the world. But perhaps, not even God can predict what universe you will find yourself in once you perform a measurement that splits yourself.

(See: “Does everything that can happen, actually happen?“)

Infinite Complexity

In 1948, Richard Feynman developed the path integral formulation, which provided a new way to understand quantum mechanics.

Feynman showed that you get the same results quantum mechanics predicts by taking into account and adding up every one of the infinite combinations of possible particle paths and interactions.

It was bizarre, but it worked. And this new formulation provided key insights that helped develop quantum electrodynamics (or QED). In 1965, Feynman, together with Sin-Itiro Tomonaga and Julian Schwinger shared the 1965 Nobel Prize in physics for developing QED.

But why adding up all of these infinite possibilities gave the right answers presented a great puzzle, which bothered Feynman.

It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do?

Richard Feynman in “The Character of Physical Law” (1965)

The appearance of infinite happenings, infinite computations and infinite logical operations underlying physical reality is mysterious.

Perhaps the simplest answer for why reality appears this way is, “It appears this way because that is the way reality is.”

If infinite computational histories form the foundation of reality, then infinities in physics might just be a reflection of this reality.

In short, within each universe all observable quantities are discrete, but the multiverse as a whole is a continuum. When the equations of quantum theory describe a continuous but not-directly-observable transition between two values of a discrete quantity, what they are telling us is that the transition does not take place entirely within one universe. So perhaps the price of continuous motion is not an infinity of consecutive actions, but an infinity of concurrent actions taking place across the multiverse.

David Deutsch in “The Discrete and the Continuous” (2001)

Matter is only what seems to emerge at infinity from a first person plural point of view (defined by sharing the computations which are infinitely multiplied in the [Universal Dovetailer’s] work) when persons look at themselves and their environment below their substitution level. The non-cloning results from the fact that such a matter emerges only from an infinity of distinct computations.

Bruno Marchal in “The computationalist reformulation of the mind-body problem” (2013)

Quantum Computers

Richard Feynman and David Deutsch are the two fathers of the quantum computer. Feynman proposed their possibility in 1982 and in 1985, Deutsch described how to build one.

These computers exploit the unlimited complexity inherent in quantum mechanics to build computers of incredible power.

How quantum computers do what they do is puzzling.

Each qubit added to a quantum computer doubles its power. A quantum computer with 300 qubits can simultaneously process 2^{300} states. This number of states exceeds the 2^{265} atoms in the observable universe.

How could a tabletop device process more states than there are atoms? How could it solve problems that no conventional computer could solve in the lifetime of the universe, even if all matter and energy in the observable universe were recruited for that purpose?

Some found the abilities of these computers so incredible, they concluded quantum computers simply weren’t possible. After all, where exactly would all that computation be occurring?

Deutsch and Tegmark offer some answers.

Since the Universe as we see it lacks the computational resources to do the calculations, where are they being done? It can only be in other universes. Quantum computers share information with huge numbers of versions of themselves throughout the multiverse.

David Deutsch in “Taming the Multiverse” (2001)

Given engineering challenges, for decades quantum computers remained only theoretical. Today, quantum computers are a reality.

In 2019, engineers at Google reported that their 53-qubit quantum computer solved in 200 seconds a problem that would take the world’s most powerful supercomputer 10,000 years.

Today, anyone can sign up for free to program and use IBM’s quantum computers over the internet.

What makes quantum computers difficult to build, is that to work, they must be completely isolated from the environment — such that they are not measured by anyone or anything until it finishes its work.

By isolating the quantum computer from the environment, observers temporarily make their existence compatible with all the possible states the quantum computer might simultaneously be in.

The parallel computations performed by quantum computers can then be explained by the work of parallel computational histories.

If current efforts to build quantum computers succeed, they will provide further evidence for [the quantum multiverse], as they would, in essence, be exploiting the parallelism of the [quantum] multiverse for parallel computation.

Max Tegmark in “Parallel Universes” (2003)

(See: “How do quantum computers work?“)

Why Time?

The universe, our lives, and even our thoughts are inextricably linked with the march of time. Few things are as familiar to us as time, and yet time remains little understood. (See: “What is time?“)

2,500 years ago, Heraclitus recognized change to be the only constant in life, saying, “All entities move and nothing remains still.” But it doesn’t seem logically necessary for a universe to have time.

Mathematical structures are eternal and unchanging: they don’t exist in space and time—rather, space and time exist in (some of) them. If cosmic history were a movie, then the mathematical structure would be the entire DVD.

Max Tegmark in “Our Mathematical Universe” (2014)

Why should our universe have a property like time?

All computers process information in an ordered sequence of steps. This ordering defines a notion of time that exists for any computation.

A Turing machine requires time to separate the sequence of states it occupies as it performs a computation.

Russell Standish in “Why Occam’s Razor” (2004)

Müller further showed that with algorithmic information theory, we can predict the appearance of a universe that evolves in time.

Our theory predicts that observers should indeed expect to see two facts which are features of our physics as we know it: first, the fact that the observer seems to be part of an external world that evolves in time (a ‘universe’), and second, that this external world seems to have had an absolute beginning in the past (the ‘Big Bang’).

Markus Müller in “Could the physical world be emergent instead of fundamental, and why should we ask?” (2017)

Assuming we are part of an unfolding computation, then we should expect to find ourselves in a universe with time.

A Beginning in Time

Current evidence suggests our universe has a beginning.

But why should it?

Until the middle of the 20^th century, most scientists believed the universe was infinitely old, without a beginning. They considered theories of an abrupt creation event to be inelegant.

Accordingly, scientists resisted the idea of a beginning until overwhelming evidence came out in its favor. It wasn’t until we could actually see the afterglow of the big bang, in the form of microwaves, that scientists were convinced the universe began a finite time ago.

We call this point the beginning because in tracing the history of the universe backwards, we hit a point where predicting earlier states breaks down and further backwards tracing becomes impossible.

The physics either stops providing sensible answers, or we run into an explosion of possibilities and can’t tell which of them is real.

The theory of cosmic inflation, gives an account of what caused the hot, dense, early phase of the universe. (See: “What caused the big bang?“)

But inflation makes further backwards prediction (or retrodiction) impossible. It wipes its footprints with a set of infinite prehistories.

Since our own pocket universe would be equally likely to lie anywhere on the infinite tree of universes produced by eternal inflation, we would expect to find ourselves arbitrarily far from the beginning. The infinite inflating network would presumably approach some kind of a steady state, losing all memory of how it started, so the statistical predictions for our universe would be determined by the properties of this steady state configuration, independent of hypotheses about the ultimate beginning.

Alan Guth in “Eternal Inflation: Implications” (2013)

Müller shows that algorithmic information theory predicts most observers will find themselves in a universe with simple initial conditions and an absolute beginning in time.

He explains this reasoning for a hypothetical observer named Abby:

If she continues computing backwards to retrodict earlier and earlier states of her universe, she will typically find simpler and more “compact” states, with measures of entropy or algorithmic complexity decreasing — simply because she is looking at earlier and earlier stages of an unfolding computation.

At some point, Abby will necessarily arrive at the state that corresponds to the initial state of the graph machine’s computation, where simplicity and compactness are maximal. At this point, two cases are possible: either Abby’s method of computing backwards will cease to work; or Abby will retrodict a fictitious sequence of ‘states before the initial state’, typically with increasing complexity backwards in time.

Markus Müller in “Law without law: from observer states to physics via algorithmic information theory” (2018)

This mirrors what cosmic inflation does for our universe.

In an alternate history where humans developed algorithmic information theory before microwave telescopes, we might have predicted a beginning of the universe before telescopic evidence came in.

Information as Fundamental

Physicists are increasingly recognizing that information plays a fundamental role in physics.

Scientists have long understood that matter and energy can be neither created nor destroyed. They are, in all interactions, conserved.

But only recently have physicists realized the same is true for information. Physical information can neither be copied nor deleted. There is an equivalent law for the conservation of information.

This discovery stemmed from the black hole information paradox.

According to general relativity, dropping something into a black hole destroys its information, like an ultimate furnace. But according to quantum mechanics, information can’t be destroyed. At best, a black hole can only rearrange information, like an ultimate shredder.

In 1981, this paradox sparked the “black hole war” — waged by two camps of physicists. After decades of debate, the black hole war settled in favor of quantum mechanics.

Information can’t be destroyed, not even by a black hole. Physicists now understand a kind of mass-energy-information equivalence.

There is also an equivalence between entropy in thermodynamics and entropy in information theory. And constants of nature are closely linked to the ultimate physical limits of computational speed, efficiency and storage density. (See: “How good can technology get?“)

Why is the link between physics and information so tight?

Wheeler dedicated his life to the pursuit of fundamental questions. Ultimately, he reached the conclusion that everything is information.

It from bit symbolizes the idea that every item of the physical world has at bottom — a very deep bottom, in most instances — an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin.

John Archibald Wheeler in “Information, Physics, Quantum: The Search for Links” (1989)

Now I am in the grip of a new vision, that Everything is Information. The more I have pondered the mystery of the quantum and our strange ability to comprehend this world in which we live, the more I see possible fundamental roles for logic and information as the bedrock of physical theory.

John Archibald Wheeler in “Geons, Black Holes, and Quantum Foam” (1998)

Why is information fundamental? The answer is easy if reality is computational. Information lies at the heart of computation.

In the end, all that computers do is process information. So to say, “computation is the foundation of reality,” is another way of saying, “information processing is the foundation of reality.”

The burgeoning field of computer science has shifted our view of the physical world from that of a collection of interacting material particles to one of a seething network of information.

Paul Davies in “The flexi-laws of physics” (2007)

What we can learn from these reconstructions is that a few simple and intuitive constraints on encoding and processing of information will automatically lead to (aspects of) the Hilbert space formalism of quantum theory.

Markus Müller in “Law without law: from observer states to physics via algorithmic information theory” (2019)

Observation as Fundamental

Observation also appears to have a fundamental role in reality.

The universe and the observer exist as a pair. […] The moment you say that the universe exists without any observers, I cannot make any sense out of that. You need an observer who looks at the universe. In the absence of observers, our universe is dead.

Andre Linde in “Does the Universe Exist if We’re Not Looking?” (2002)

Quantum mechanics revealed that observation somehow forces reality to choose from among many possibilities.

More recently, physicists have speculated that the observer’s power to force reality’s hand applies not only to the here and now, but perhaps all the way back to the beginning of the universe.

We are participators in bringing into being not only the near and here but the far away and long ago. We are in this sense, participators in bringing about something of the universe in the distant past.

John Archibald Wheeler in “The anthropic universe” (2006)

The top down approach we have described leads to a profoundly different view of cosmology, and the relation between cause and effect. Top down cosmology is a framework in which one essentially traces the histories backwards, from a spacelike surface at the present time. The no boundary histories of the universe thus depend on what is being observed, contrary to the usual idea that the universe has a unique, observer independent history. In some sense no boundary initial conditions represent a sum over all possible initial states.

Stephen Hawking and Thomas Hertog in “Populating the landscape: A top-down approach” (2006)

The observer might even, in some sense, choose the laws of physics.

It is an attempt to explain the Goldilocks factor by appealing to cosmic self-consistency: the bio-friendly universe explains life even as life explains the bio-friendly universe. […] Cosmic bio-friendliness is therefore the result of a sort of quantum post-selection effect extended to the very laws of physics themselves.

Paul Davies in “The flexi-laws of physics” (2007)

Observation and its relation to observed reality is an enigma.

Wheeler believed the relation between them was our best clue to finding an answer to why there is something rather than nothing.

« Omnibus ex nihil ducendis sufficit unum », Leibniz told us; for producing everything out of nothing one principle is enough. Of all principles that might meet this requirement of Leibniz nothing stands out more strikingly in this era of the quantum than the necessity to draw a line between the observer-participator and the system under view. […] The necessity for that line of separation is the most mysterious feature of the quantum. We take that demarcation as being, if not the central principle, the clue to the central principle in constructing out of nothing everything.

John Archibald Wheeler in “Quantum Theory and Measurement” (1983)

In the view that all computational histories exist, observation does play a role in selecting both histories and physical laws.

It is a tautology that observers only find themselves in computational histories capable of producing their observations.

Since every imaginable program exists, implementing every imaginable set of laws, then in a very real sense, the observer does force reality to select both the laws and history they observe.

To derive the effective laws of physics, one needs to do statistics over the ensemble of identical observers. This involves performing summations over the multiverse, but these summations are with a constraint that says that some given observer is present.

Saibal Mitra in discussion list (2018)

It’s curious that Buddhist thinkers reached similar conclusions about observers well ahead of modern physicists.

The Buddhist does not believe in an independent or separately existing external world, into whose dynamic forces he could insert himself. The external world and his inner world are for him only two sides of the same fabric, in which the threads of all forces and of all events, of all forms of consciousness and their objects, are woven into an inseparable net of endless, mutually conditioned relations.

Anagarika Govinda in “Foundations of Tibetan Mysticism” (1969)

Reviewing the Evidence

We have found evidence in support of this theory. The existence of infinite computational histories predicts many features of reality.

It predicts a universe of inviolable, but simple, mathematical, and life-friendly laws. It predicts a multiverse of parallel histories, infinite computational complexity, and a fundamental unpredictability, as we find in quantum mechanics.

The theory predicts a universe that evolves in time, has simple initial conditions, and a point that we can’t retrodict beyond: a beginning. Further, it predicts information and observation are fundamental.

So far, all of these predictions are confirmed by current physical and cosmological observations. For the first time in history, humanity has an answer to why we exist that is backed by physical evidence.

Conclusions

Given the observational evidence, we have reason to suspect that this theory, or something close to it, is correct.

It implies: We live within the total set of all computations.

Moreover, we have traced the existence of this set to something that’s a strong candidate for having necessary existence: self-existent truths concerning numbers and their relations.

One option, following Leibniz and others, is that we reach a level at which further explanation is not required, because something is necessarily true.

Sean Carroll in “Why Is There Something, Rather Than Nothing?” (2018)

This truth not only seems causeless, but because from it we can deduce much of physics, it is also a candidate for being the cause.

The supreme task of the physicist is the discovery of the most general elementary laws from which the world-picture can be deduced logically.

Max Planck in “Where is science going?” (1932)

Under this theory, the most general laws from which we can deduce the world-picture become the laws of arithmetic. Thus, arithmetic, as a theory of arithmetical truth, becomes a theory of everything.

This brings a whole new meaning to Leopold Kronecker’s edict: “God made the integers, all else is the work of man.”

This is why, with Church’s thesis, and the quantum confirmation of mechanism, intuitive arithmetic, a.k.a. number theory and its intensional variants, may well be the simplest and richest “theory of everything” that we can have at our disposal.

Bruno Marchal translated from “Computability, Physics and Cognition” (1998)

This theory — arithmetic — has been under our noses the whole time!

Behind it all is surely an idea so simple, so beautiful, so compelling, that when—in a decade, a century, or a millennium—we grasp it, we will all say to each other, how could it have been otherwise? How could we have been so stupid for so long?

John Archibald Wheeler in “How Come the Quantum?” (1986)

The Journey Here

It’s been a long road to reach the point where humanity can scientifically address the question: “Why does anything exist?”

Humans have walked the earth for some 500,000 years. But only in the last 1% of that time, or the past 5,000 years, have we had writing. Only in the last 0.1% of that time, or the past 500 years, have we had the scientific method. And only in the past 0.01% of that time, or the past 50 years, has humanity known about universal equations.

To get an answer to our question, required that humans discover numbers, equations, computation, and wrestle with topics at the foundation of mathematics, including consistency, completeness, and decidability. In the end, this led to our discovery of universal equations that define all computation.

To find evidence linking this computational reality to physics, humans had to discover the expanding universe and gather evidence of the big bang. We also had to probe the smallest scales and through careful study of particles, discover the quantum nature of reality.

A century ago, we had none of this understanding.

A Strange Answer

We can’t help but notice how strange this answer is.

But perhaps we should have expected this. Would we expect that the final answer to the greatest mystery of the cosmos would be ordinary?

Now, my own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose.

J. B. S. Haldane in “Possible Worlds and Other Essays” (1927)

Whatever may be the truth about the universe, it is bound to be astonishing.

Bertrand Russell

We will first understand how simple the universe is when we recognize how strange it is.

John Archibald Wheeler in “Geons, Black Holes, and Quantum Foam” (1998)

Tegmark cautions against rejecting theories just for being weird. And admits he would be disappointed if the answer weren’t a bit weird.

It’s very important for us physicists to not dismiss ideas just because they are weird, because if we did we would have already dismissed atoms, black holes, and all sorts of other marvelous things. And actually, you know when you ask a basic question about the nature of reality, you know, don’t you expect an answer which is a bit weird? I think anything but weird would be a big letdown.

Max Tegmark in “What We Still Don’t Know: Are We Real?” (2004)

A Triumph of Human Reason

I believe when the history of science is written, then what’s being discovered about our universe in the last decade or two will be one of the most exciting chapters.

Martin Rees in “What We Still Don’t Know: Are We Real?” (2004)

We now have viable answers to great questions of existence:

• Leibniz’s question: Why is there something rather than nothing?
• Einstein’s question: Why is the universe so comprehensible?
• Wigner’s question: Why is the universe so mathematical?
• Wheeler’s question: How come the quantum?
• Smolin’s question: Why these laws and not others?
• Feynman’s question: Why does infinite logic underlie physics?
• Hawking’s question: What breathes fire into the equations?

It required us to assume math, rather than matter, is fundamental. Given the evidence supporting this view, we might consider the 2,400-year-old debate between Plato and Aristotle as settled.

If we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason — for then we should know the mind of God.

Stephen Hawking in “A Brief History of Time” (1988)

Hawking believed if we could discover what breathes fire into the equations, then we should know the mind of God.

But do we? By postulating infinite, eternal, mathematical truth as the ultimate explanation, and the cause and source of reality, have we succeeded in explaining God, or have we explained God away?

Open Questions

While this theory provides answers to many questions, it does not answer everything, and much additional work is required.

Room for God

This theory provides a purely natural and rational account for why anything exists. Is there any room for God in this picture?

We now have a view of reality where everything emerges from absolute truth. This infinite truth embodies all knowledge. Being a container of all knowledge, as well as all minds and things, can we compare this infinite set of truth to an omniscient mind?

This truth is infinite and incomprehensible, eternal and indestructible, without a beginning or end. It is uncreated and self-existent. It is transcendent, immaterial, immanent, and indivisible. It’s the reason and cause behind all things. It serves as the creator, source, and ground of being, supporting us and the material universe.

Does this infinite truth or ‘omniscient mind‘ lead to the existence of God? Might it even be God? It’s not a simple question.

But knowing why anything exists, leaves us in a better position to answer questions about what exists and what doesn’t.

(See: “Does God Exist?“)

Deriving Physical Law

How much of physical law can we derive from the assumption of all computations together with the requirement of life-friendliness?

Can we predict things like types of particles and forces, or the dimensionality of spacetime? Might we even be able to predict values of constants like particle masses and force strengths?

What really interests me is whether God could have created the world any differently; in other words, whether the requirement of logical simplicity admits a margin of freedom.

Albert Einstein

It remains to be seen how much of physical law is universal (applying to all observers in all computational histories), and how much is geographical (depending on which histories an observer belongs to.

As a theoretical physicist, I would like to see us able to make precise predictions, not vague statements that certain constants have to be in a range that is more or less favorable to life. I hope that string theory really will provide a basis for a final theory and that this theory will turn out to have enough predictive power to be able to prescribe values for all the constants of nature including the cosmological constant. We shall see.

Steven Weinberg in “Dreams of a Final Theory” (1992)

But this hope, of deriving every aspect of physics, is waning.

Max Tegmark recounts, “as recently as 1997, the famous string theorist Ed Witten told me that he thought string theory would one day predict how many times lighter an electron is than a proton. Yet when I last saw him at Andrei Linde’s sixtieth birthday party [in 2008], he confessed after some wine that he’d given up on ever predicting all the constants of nature.”

Implications

If all computations exist, and if those computations explain our observed reality, it leads to many surprising implications.

The Universe is a Dream

The theory lends support to the ancient idea, expressed by Taoist, Greek, and Christian philosophers, and a tenet of Hindu and Buddhist belief: that the material universe is a kind of dream or illusion.

It implies that the material and physical are byproducts of mind.

Collective karmic impressions, accumulated individually, are at the origin of the creation of a world. […] The outside world appears as a result of the acts of sentient beings who use this world. […] The “creator of the world,” basically is the mind.

The 14^th Dalai Lama in “Beyond Dogma” (1994)

For the things which one thinks are most real, are the least real.

Plotinus in “The Enneads V.5.11” (270 A.D.)

Only recently have modern scientists begun to embrace this view, with a few even doubting the “realness” of physical existence.

Neils Bohr said, “Everything we call real is made of things that cannot be regarded as real.” In an interview, Marvin Minsky admitted, “We don’t know that we exist because […] maybe we’re just what a program would do if the computer were turned on and it’s not even running.”

We Live in a Simulation

The simulation hypothesis and simulation argument raise the question of whether or not we inhabit a vast computer simulation.

If we exist as a consequence of mathematical truth, the simulation hypothesis is made true by default. For we would then find ourselves living within the infinite set of computationally generated histories.

This blurs the distinction between virtual reality and real reality.

It remains an open question: is anyone in control of the simulation we happen to be in? (See: “Are we living in a computer simulation?”)

Our Place in Reality

With an answer to why anything exists, we can orientate ourselves in reality. We now understand our position and place in it.

Mathematical truth implies the existence of all computations. The existence of all computations implies the existence of all observers. The existence of all observers leads to a quantum mechanical reality populated with all possibilities and ruled by simple laws.

So what exists? Almost everything.

Reality becomes so big and so comprehensive that it includes everything and everyone that can be. Every thought that can be had and every experience. Every story and scenario plays out, eventually and somewhere. Actually, they all recur an infinite number of times.

Indeed, in this view, reality is so large that it guarantees the existence of an afterlife. See (“Is there life after death?“)

Confession: if I love [this theory], it is because it entails the existence of many things not “physically present”, notably those incredible deep universal dreamers which keep [losing] themselves in an incredible labyrinth of partially sharable dreams, meeting ladders and ladders of surprises, self-multiplying and self-fusing, and which are partially terrestrial and partially divine creatures.

Bruno Marchal in discussion list (2011)

Reasoned study of the mysteries of existence has brought us to a coherent theory of why there is something rather than nothing.

The best evidence suggests our universe is one among an infinite number of possible realms, with the full extent of reality being unbounded. The source of this reality is logical necessity, via infinite mathematical truths which are independent of any material universe.

We can count ourselves among the first generation of humans able to reason logically, with the support of observational evidence, to arrive at answers for why our universe has the laws it does, why we are here, and why there is something rather than nothing.

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