Groundless Emergent Multiverse (2.0)

Bridging the gap between non-duality and science

This post is also available as pdf and in German.

Introduction

Asking the old question, “Why does anything exist, rather than nothing?” one already makes several assumptions: a difference between existence and non-existence, and some causality that leads to existence. The answer, however, is found beyond these categories—outside of causality and time, neither existing nor non-existing. In this post, I try to point to an answer so simple that it transcends all concepts and language.

When someone asks about the nature of reality, they usually expect it to be possible to find a definite answer, some fundamental layer or substance—a ground to rest on. No such thing is necessary. Groundlessness and not-knowing are sufficient as a starting point to explain how and why anything exists. Even more than that, the belief in and need for some grounding stand in the way of realizing this. Fully letting go of the need for closure leads to the freedom to see things as they are. The first part of this post will point to this understanding. The second part will give an intuition for how the multiverse and all physical laws can emerge out of uncertainty.

The following summary will likely only make sense in hindsight. It’s a very abstract understanding you have to acquire through contemplating these questions on your own.

Without constraints, all coherent universes can exist. No matter what universe one thinks of as the “true” one, it does not exclude all other possible universes. Ultimately, the biggest possible reality contains all possible universes. In this way, however, any distinction made would favor one universe over the others. Ultimate reality, therefore, can only be without properties. When we see every universe as a collection of distinctions, then there is structure in how they are related. This way, they are not separate but form one whole, branching out into infinite possibilities. It is, by logic, impossible to think of any more fundamental explanation because this one transcends and includes all others.

To put it another way: We cannot say anything about reality without a perspective, a frame of reference, an observation, or a subjective experience. All we can talk about is our perspective, our experience. There is nothing we could know about ultimate reality but our subjective observation.
The only way that parts come together into a whole, or that a whole is separated into parts, is by constructing boundaries. The combination of boundaries, weighted by confidence, constitutes a perspective. Since everything is made of parts, and every part is a limited view of a whole, all that exists is collections of boundaries with some confidence in their existence. Every thing that exists is a perspective. Every perspective is a limited and distorted view of reality.
We have to conclude that ultimate reality in itself is devoid of any properties. Reality, rather, is the superposition of all possible perspectives. Without properties, it is indeterminate, neither truly existing nor not existing. Empty of inherent existence, it is completely transparent. Each possible perspective on reality is a view of all other perspectives. They are interdependent, with no substance, no ground to be found anywhere.

There is nothing one can say about reality other than one’s perspective on it. Ultimate reality neither exists nor does not exist but is fully described as the combination of all possible perspectives on itself.

The accepted scientific worldview posits that our world is purely physical. Some physicists propose that physics consists entirely of mathematical structures. Set theory can explain all mathematical structures and can itself be constructed from the empty set—a collection without elements. If we accept this, then it isn’t too far a leap to conceive of everything that exists as variations of a reality that can be derived from the empty set. This does not mean that “nothing exists,” but that no thing exists independently. Everything is structure only, with no substance. Everything exists dependent on everything else.

Eventually, we will be able to tell a complete story to explain everything from the groundless nature all the way to your subjective experience in this moment. This post is only a rough outline for the first part of the story.

Deconstructing

What is Fundamental?

Every theory is based on assumptions (or axioms). These are statements that cannot be proven from within the theory but are accepted or assumed to be true.

If we keep questioning these assumptions and require that they be justified and explained, then we—simply by the act of questioning—cannot be left with any unexplained assumptions. What remains cannot be questioned further, nor can it be justified or explained further. Such a theory would be groundless, though it’s questionable whether the term “theory” should still apply.

As soon as we can explain some phenomena in terms of other phenomena, a hierarchy is implied, denoting some objects as more fundamental (and therefore more “real”) than those they constitute. Your body is made of organs, which are made of cells, molecules, atoms, subatomic particles, and the elementary particles of the standard model, which are actually just fields—and that’s how far we’ve gotten so far. Fields are mathematical descriptions. The discussion then is: Does the math just describe something more fundamental, or is it math?

Let’s start with the realization that, apparently, everything can be explained—usually in terms of smaller things. At least, that’s the experience scientific investigation has made so far. Whenever a fundamental, indivisible substance or mechanism was proposed, we later found an explanation and smaller parts. Should we update this all the way, and if so, what would that look like? There are a few possible answers:

• Dogmatic - There is a fundamental building block or substance. We just haven’t found it yet.

• Regressive - It’s turtles all the way down. There is no end to deconstructing.

• Circular - At some point, we end up where we began.

These are known as the Münchhausen trilemma because, according to common logic, we can’t pull ourselves out of the mud by our own hair, like Baron Münchhausen did. We have to stand on some ground. However, this is like saying the earth under your feet has to be placed on some immovable bedrock layer because everything else would be absurd. We know that reality is absurd under this logic, as the Earth floats in space, supporting its structure by counteracting forces balancing each other out. Therefore, let’s add a fourth option:

• Groundless - Everything exists because symmetric counterparts cancel out the need to justify their existence.

Infinite regress and circularity are usually dismissed as absurd, which leaves dogmatism and groundlessness. Before we take a look at the groundless, let’s see why the dogmatic argument doesn’t make sense either.

The idea of a fundamental layer of reality is an assumption that has to be filled with an answer to what that layer should be. Yet, by the very nature of assuming something as fundamental, one doesn’t and can’t explain it. The process might often be the reverse: We can’t explain X, so it must be fundamental. Many great minds fall in love with the things they can’t explain. They awe in the mystery, and instead of deconstructing it, they build upon that one idea, obscuring it even further.

To tackle the question of fundamental building blocks, one has to be able to let go of every idea and worldview. For everything that is called “fundamental,” we can ask: What is it made of? Why does it have the properties it has? Why this particular thing and not something else? Is it possible to define it? If so, can you deconstruct the definition? Is it possible to define other fundamental objects? Even questioning the assumption of the basic opposition between existent and non-existent. What does it mean for anything to exist?

If the thing in question is truly fundamental, then it must be possible to answer these questions from within itself, with no outside reference. Yet, all these questions could be rephrased as “Why is this theory true and not another theory?” It cannot be answered from within that theory because the theory cannot make statements outside its own assumptions. Even if a theory were consistent and complete, with a limited view, we could not know if another theory exists that is also consistent and complete, nor could we decide between them. That’s assuming an ideal case, but Kurt Gödel proved that every system has to be inconsistent or incomplete. So we don’t even get that. Within the hypothesis of “there has to be something fundamental,” we can never find a satisfying answer.

It takes some courage to abandon all assumptions and beliefs. But when we do so, what remains is sufficient as a starting point.

With this, there is no fundamental layer of reality. When considering all possible observations of reality, there is no content left other than observations. Observations, in turn, only describe their relationship to all other observations. Everything is empty of inherent existence and full of dependent existence. Without a fundamental layer, the reductionist approach breaks down. No layer of reality is more real than any other. There is no ontological hierarchy, no up or down.

Sooner or later, we come to realize that perhaps the most fundamental, and most fundamentally important, fact about any experience is that it depends on the way of looking. That is to say, it is empty. Other than what we can perceive through different ways of looking, there is no ‘objective reality’ existing independently; and there is no way of looking that reveals some ‘objective reality.’

— Rob Burbea, Seeing That Frees: Meditations on Emptiness and Dependent Arising

Foundations of Mathematics

All mathematical structures can be described as sets. Set theory is, therefore, one way to talk about the foundations of mathematics. However, this statement is slightly circular since, with the advent of set theory, edge cases have been discovered that certain variants of set theory cannot address. According to standard (ZF) set theory, recursive sets don’t exist. The need for this restriction arose due to Gödel’s incompleteness theorems and Russell’s paradox, which will be discussed in the following sections.

So, what are sets?

A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit.

— Felix Hausdorff

This description is especially useful here because it does not affirm sets as things on their own. It’s by merely thinking of a plurality as a unit that it is a set. To be precise, thinking is not required—just by the potential of drawing a boundary and grouping together, there is a set. For example, the integers from 0 to 5 form a set {0, 1, 2, 3, 4, 5}, and so do all songs in my playlist.

To formalize set theory and avoid paradoxes, axioms have been formulated. The most widely accepted axiomatic system is the Zermelo–Fraenkel set theory (ZF, with 8 axioms, or ZFC with the addition of the axiom of choice). The selection of these axioms allows for the construction of all permitted sets, starting only with the empty set. These axioms still present assumptions, chosen for functionality. This can be seen in the debate over whether to include the axiom of choice. Other axioms are also challenged, particularly the axiom of foundation, which excludes sets that contain themselves as members (or any kind of infinite regress). By this axiom, x = {x} is not a valid set. This restriction allows axiomatic set theory to avoid paradoxes but comes at the cost of limiting what it can talk about.

Set theory allows us to start from an empty set {}—a collection without objects—and define all mathematical objects. Yet, there is no single way to do it. The natural numbers can be alternatively expressed as:

{} = Ø
{{}} = {Ø} = 1
{{}{{}}} = {Ø, 1} = 2
{{}{{}}{{}{{}}}} = {Ø, 1, 2} = 3

or:

{} = Ø
{{}} = {Ø} = 1
{{{}}} = {1} = 2
{{{{}}}} = {2} = 3

or several other ways. What gives identity to the natural numbers is not the sets that define them but their relationships. Sets can be constructed in a way that exhibits those relationships. This implies that the sets themselves are a language needed to talk about the structure; they are not the structure itself. Consequently, other ways exist to discuss the foundations of mathematics that are not based on sets. Most prominently, category theory describes all structures in their relations to all other structures.

The core insight here is that every mathematical structure can be constructed from simpler structures and that ultimately, there is no smallest or initial element (urelement), no substance needed. However, what is left unexplained is the set of axioms.

Gödel’s Incompleteness

The first incompleteness theorem states that in any consistent formal system F within which a certain amount of arithmetic can be carried out, there are statements of the language of F which can neither be proved nor disproved in F. According to the second incompleteness theorem, such a formal system cannot prove that the system itself is consistent (assuming it is indeed consistent).

— Stanford Encyclopedia of Philosophy

This is a negative result in the sense that it shows us what cannot exist. Related theorems have been proven in other areas, such as computational complexity. It represents a hard limit on reality and on what we can know. It says that reality cannot be a formal system that is both complete and consistent. Of these three requirements, at least one does not hold.

Truth in a mathematical system is what can be derived from that system. Gödel shows that there are statements that are true for that system but cannot be derived from it. This, however, does not mean that they are universally true, but rather that no formal system can prove all true statements because it assumes statements to be true which it cannot prove. Since no formal system can give us the perfect answer for what is true, truth is always dependent on the system used. What is true or not depends on the assumptions that have been made. No single axiomatic system can describe reality perfectly.

What if we simply accept this result and realize that truth and reality are entirely dependent on our assumptions? The way we see reality is shaped by the glasses we put on. Without a formal system and no axioms (no glasses), reality may be consistent and complete, but we cannot make any statements about it—maybe not even that it is consistent and complete. Reality without constraints has no properties.

Any formal system, and therefore any theory, contains assumptions that lack justification outside that theory. What is “true” in one theory may be “false” in another. The concept of true or false itself is therefore dependent on assumptions. Without assumptions, even the distinction between true and false falls away.

With Gödel’s proof, we must accept that no theory can describe ultimate reality. We, however, consider theories that describe our experience better as more true. Math can be so abstract that it has no representation in our physical world. So much so that some physicists deny the existence of mathematical concepts that can’t exist in our physical universe, like infinities. Related is the idea that computation is fundamental to reality, and not math. If we accept, however, that the purpose of math is not to describe only our universe but every possible one, then the perspective changes. Infinities may not exist inside our local universe, but our universe exists within the infinite.

Conversely, this has a secondary effect. We can no longer discriminate between different theories of math by how well they describe our universe. Instead, there are different models that describe different universes; none of them is fundamentally more true than the others. The model of math one uses then just depends on the particular problem one is working on.

Set-Theoretic Multiverse

Related to Gödel’s incompleteness is the discovery that there are statements that cannot be decided to be true or false from within a system. A prime example of this is the continuum hypothesis (CH). George Cantor observed that he could not put the infinite set of natural numbers in a one-to-one correspondence with the infinite set of real numbers. This showed that there are at least two sizes (cardinalities) of infinity. The continuum hypothesis asks whether there is a cardinality of infinity in between those of the natural and real numbers.

It turns out that this question cannot be decided in ZF. Assuming it to be true gives one branch of set theory; assuming it to be false gives another, incompatible branch. Following this result, many similar statements have been found that simply are undecidable.

The CH and others could be added as new axioms to ZF, but deciding this would always carry some sense of arbitrariness. On the other hand, the already accepted axioms can also be called into question. Any axiom that is introduced both constrains the possible math that can be explored and provides a foundation upon which math can build.

Joel David Hamkins takes the position that we should simply accept and explore this set-theoretic multiverse. While he does not relate it to the physical multiverse, I think there is the same principle at work.

Russell’s Paradox

According to the unrestricted comprehension principle, for any sufficiently well-defined property, there is the set of all and only the objects that have that property. Let R be the set of all sets that are not members of themselves. (This set is sometimes called “the Russell set”.) If R is not a member of itself, then its definition entails that it is a member of itself; yet, if it is a member of itself, then it is not a member of itself, since it is the set of all sets that are not members of themselves. The resulting contradiction is Russell’s paradox.

— Wikipedia on Russell’s paradox

The “unrestricted comprehension principle” is what I previously described as: “just by the potential of drawing a boundary and grouping together, there is a set.” This is what was later called “naive” set theory, to distinguish it from the axiomatic approach. What Russell pointed out is that if this principle holds true, then one could create a set of all sets that do not contain themselves—but this leads to the above paradox.

In ZF set theory, this paradox is avoided by only permitting sets that are not members of themselves (the axiom of foundation).

Let’s consider a thought experiment from the opposite perspective. Suppose we only permit sets that do contain themselves. In this system, x = {x, y} is permitted, while x = {y} is not. By this logic, the “set of all sets that are not members of themselves” has no elements—i.e., it would be the empty set.

We can also adopt a meta-perspective that includes both approaches. From this view, the Russell set is both empty and contains every other set. An apparent paradox, but we can make sense of it if we reconsider what the empty set actually is.

We said that sets can be thought of as collections of objects. Then, what is a collection without objects? For this, we start with absolutely nothing and then create a set.

{}

A boundary drawn around nothing. At the very beginning, we haven’t constructed any other sets yet. So there is really nothing inside or outside the empty set.

Let’s be clear about this: absolutely nothing means no space, no universe, no math, no observer, no set—there is nothing that this boundary could separate. The boundary itself is not a thing; it’s not even the assertion that nothing exists. It’s a separation with nothing to separate. “Inside” and “outside” the set are both nothing. There is no difference between inside and outside. What the set indicates is: There is nothing, and within that nothing, there is still nothing. The empty set is within nothing and contains nothing. This means the empty set is recursive.

With no axioms in place that prevent an empty, recursive set from existing, the set is already present without a cause. It creates itself out of unconstrained potentiality.

We have to be careful here. The empty set is not nothing; it already is something—a boundary, a differentiation. While everything could create itself out of unconstrained potentiality, the empty set is the simplest form of existence. It is just the assertion that it exists.

When the empty set is recursive, then we can iterate over it, inserting the empty set into every instance of the empty set.

{}
{{}}
{{}{{}}}
{{}{{}}{{}{{}}}}

These iterations are the same as the definition of the natural numbers from above. Each of those sets contains the empty set and is itself an instance of the empty set.

We don’t need to stop there; we can see every set as a particular, constrained perspective of the empty set. Conversely, every set we can describe does contain the empty set. Because this goes two ways, every set described this way does contain itself. The empty set contains all sets and is empty nonetheless. In this view, the set containing all sets, the Russell Set, and the empty set are all the same (V = R = {}). It’s a single fractal containing all mathematical structures. Russell’s paradox is no longer paradoxical. We achieved this by redefining differences between sets as differences in perspective. The multitude of all sets has become one emptiness with multiple ways to view this emptiness.

Note that along the way, we already made at least one subtle assumption: that there is a binary contrast between existence and non-existence. But that need not be. We can see each pair of curly brackets (let’s call them “dualities”) on a continuum from non-existence to existence or from uncertainty to certainty. To switch from one set to another is then just a continuous change in the degree of certainty.

Nested parenthesis with degrees of certainty representing a continuous transition from not existing to existing.

Russell’s paradox is paradoxical because it makes conflicting assumptions. The standard solution is to only consider those cases that are not paradoxical—that is, to limit the view (via the axiom of foundation). Another way out is to find, recognize, and drop the hidden assumption. All paradoxes point to a conflict in assumptions and can always be resolved by letting go of those assumptions. The paradox is a property of the view one takes, not of reality itself.

Viewless View

Superposition of Views

Russell’s paradox was resolved in the previous thought experiment because we took a superposition of all possible sets. We can drop into any particular set with our perspective but are not confined to it. This process can be described for two options as one original state 0 which contains, but is undecided about, L and R, where either excludes the other. When we constrain our view to (take the perspective of) L being true, then R is false, and vice versa. This gives us four possible ways of viewing the situation:

• L is true

• R is true

• L & R: both are true

• 0: neither is true

This fourfold view is called catuṣkoṭi or tetralemma. It is affirmation, negation, both, or neither. This contrasts with mainstream Western philosophy, where the law of excluded middle (either true or false, with nothing in between) is foundational to logic. However, this law of excluded middle is also just an assumption that has to be justified. To think of a justification, you have to think outside of it, taking all possible options into account.

Nāgārjuna, a circa 200 CE Buddhist philosopher, adds a fifth option, which is a viewless view, or openness to views—the middle way. It does not cling to any of the options as the only right one but sees all truth claims as insubstantial. The point of this kind of logic is explicitly not to establish a fixed dogma of how things really are. Nāgārjuna, like Gödel, uses logic to expose the limits and vacuity of logic.

Neither from itself nor from another,
Nor from both,
Nor without a cause,
Does anything whatever, anywhere arise.

Chapter 1, verse 1
Nāgārjuna’s Mūlamadhyamakakārikā, translation by Jay L. Garfield

Take the following example: Is a line a collection of points, or are points sections on a line?

When you try to construct a line out of points, you would have to add indefinitely many points. Taking the view that points exist, you might conclude that lines are indefinitely far away and therefore impossible.

When you, on the other hand, take a line and cut it in pieces, you will never reach a point, since a point has infinite precision. Taking the view that lines exist, you might conclude that points are indefinitely far away and therefore impossible.

So which is it? Or is there a third option? One solution is to conclude that points and lines are not reality itself but descriptions of reality. Abstractions that only exist because you are taking a certain view of reality. Neither points nor lines exist. Another solution is in the objection that, when I think of points or lines, the concept does exist by me thinking it. Both lines and points exist. But then “neither” and “both” are also views. So according to them, both and neither “both” and “neither” are true. The actual problem is in thinking that reality is any particular way and that you could know it. Letting go of the need to assert truth, you will be able to rest in uncertainty.

As established above, any mathematical object can be described as nested sets—ultimately made up of, and presenting a perspective on, the empty set. That is, we can say that every object is empty. The information to describe or locate that object or perspective is equivalent to that object and not reducible. That is, it exists entirely on its own. A third view is that the object in isolation could not be said to have any properties; it is only in relation to other objects that it can be said to exhibit properties—its existence is entirely other-dependent. Assuming any of these views as the ultimate truth would exclude the others, but we have no objective way of deciding between them. The best we can do is to not decide but to use them only provisionally. Even the realization that all phenomena are empty of inherent existence is only a provisional truth.

“Empty” should not be asserted.
“Nonempty” should not be asserted.
Neither both nor neither should be asserted.
They are only used nominally.

Chapter 22, verse 11

Many Worlds

It doesn’t stop there. In quantum logic, all combinations of amplitudes for L and R are possible. There are not only four but an infinite number of possible values. Yet, when the state is measured, only one of these options is realized. The probability of measuring either outcome is given by the square of the amplitudes (the Born rule). This leads to the question: What is an “observation” in quantum mechanics?

All possible states of a QBit can be represented as points on the Bloch sphere. (source)

The double-slit experiment introduced a problem into physics that sparked many interpretations but no consensus. Before measurement, the particle behaves like a wave of probability, interacting with itself. After measurement, it seems as if a particle was observed at a single location, determined by the previous probability. So, what is a “measurement”? Does observation influence reality? Where and when does “observing” happen, and who or what is observing? The question becomes even more pressing when we add the problem of non-locality through entanglement into the picture. Luckily, the many-worlds interpretation allows us to make sense of this without the need to invoke a new fundamentally unexplained process or entity.

When Hugh Everett first proposed the solution, he called it “Relative State Formulation of Quantum Mechanics” or later “The Theory of The Universal Wave Function.” The term “Many Worlds” came with a later reinterpretation and, while catchy, sometimes leads to confusion. There is no need to assume that new worlds pop into existence each time we measure. On the contrary, Everett’s interpretation is very conservative, as it does not propose any entities apart from the wave function it tries to explain. The wave function does not change with measurement, but our view of it.

Then where does the measurement come into the picture? All possible outcomes are already present in the wave function before measurement. It’s a superposition of possible worlds. To measure means to interact with the wave function. This interaction constrains what worlds are accessible from the observer’s perspective. It introduces more certainty about any particular perspective, but without anything to choose the outcome, any possible perspective is realized. While several other interpretations of QM require hidden information or randomness to be fundamental, in MW, randomness is a perceptual artifact. The chain of events only appears random from the point of view of an observer.

When measuring a particle at some position, we constrained our view of the wave function to the part where the particle is more likely in this position. Due to the uncertainty principle, we never know the position exactly, but still have a probability distribution. Because two probability distributions interact (the observer and the observed), the resulting distribution will have fewer degrees of freedom.
The wave-particle duality is the duality of less and more certainty. A particle does not exist as a separate thing unless observed. The location of a particle is not a feature of “objective” reality, but one of the subjective perspective on absolute reality.

In other words, every world in many worlds is a perspective on reality, and every perspective is a probability distribution. Like the empty recursive set, the universal wave function already contains all possible perspectives. What appears to us as taking a measurement is only a perspective “already” existing within the multiverse.

Free from, but Not Without

This understanding is itself not a view. It is rather that, by understanding the nature of all views, one no longer mistakes them for reality.

As an analogy, imagine you are in a train, looking out the window. You can take yourself as a frame of reference and see the landscape as passing by. Alternatively, you take the landscape as a reference and realize you are traveling through it. You could play this game further and find other frames of reference, or you could realize that all frames are a construction, a belief—something you do. When you stop doing it entirely, then all phenomena are moving relative to each other. From there, a new question can arise: What does not change while changing points of view? Logically, that which all views have in common will not change. What all views have in common is that they are of the nature of being views, of being conditioned. This nature is unconditioned. Yet, you cannot take an unconditioned view on reality. You can only become aware of the conditioned nature of your view.

To observe commonalities is always an abstraction. Since every perspective is unique, one could say that no two things are the same. What allows you to perceive forms and name things is the ability to abstract by ignoring information. Yet, without abstraction, you would not be able to make sense of the world. All physical systems you may describe have only boundaries from the perspective of an observer, drawing the boundary and abstracting details away. Without boundaries, all differences fall away. Without imposing differences, there would be nothing to perceive. Form exists through observation; observation exists through form. Views and their content co-arise dependent on each other.

Reconstructing

We established that no single conceptual system can consistently explain why anything exists. Even nothingness is an incoherent concept, since “nothing” would have to be defined as the absence of something, but this would give properties to “nothing”—turning it into something. Therefore, something, some universe, has to exist. But of all the possible universes that could exist, there seems to be no way to prefer one over the other. To decide between them, we would have to find rules outside all possible universes that constrain which universe is actually realized. There is also no way to choose the meta-rules. There is no reason to prefer any possible universe over another. Therefore, we have to assume that all universes exist or don’t exist equally. Yet, some universes share properties and definitions; their relation results in a higher-level structure. The superposition of all universes is undecided but has an internal structure. This is our starting point. By not knowing, we can extrapolate.

Since no universe is preferred over another, any possible universe can be used as a frame of reference. This frame of reference is a perspective on reality. Every perspective can be defined only by its differences to other perspectives. Through differences, there are relations; with relations, there is structure. We’ll see that this structure gives rise to (the local impression of) time, causality, space, locally broken symmetries, and all the rest. The breaking of symmetry is what gives rise to information. Following this structure from most symmetric to less symmetric, one finds that structures branch, merge, or reach dead ends. This diversification and selection provide a universal evolution that selects for stable structures. Life, intelligence, self-awareness, and cooperation arise through selection for stability. The universe we observe is a region within the structure of reality that allows for this level of complex stability to emerge. With nothing that reality ultimately consists of, neither real nor not real, with the biggest and smallest being the same, there is no fundamental layer to reality. It is truly a groundless emergent multiverse.

Absolute

Symmetry and Symmetry Breaking

By symmetry, we mean the existence of different viewpoints from which the system appears the same. It is only slightly overstating the case to say that physics is the study of symmetry.

— Philip Warren Anderson, More Is Different

Under the mathematical notion of symmetry or invariance, liquid water is more symmetric than a snowflake. Something is invariant when it stays the same under transformation. You can mirror and rotate a snowflake, and it will look the same, but you can mirror or rotate water in many more ways, plus translate it in any direction. Pure symmetry is pure homogeneity, zero information. It is invariant under every transformation. It always stays the same. Symmetry, in a way, is the requirement that there is no change by outside cause. It’s empty of substance, not a thing in itself.

Without differentiation, everything being equivalent, there is pure symmetry. Also, not negating anything, it carries the potential for all things. Pure symmetry includes everything and its counterpart. The reverse view is also possible. Starting with everything, we see that every thing has symmetries or is symmetric with other things. The superposition of all things, then, is pure symmetry. In a way, there is only one symmetry, which is the equivalence of all things. When we observe something, it is only because we have a limited view and ignore some of its parts. Every experience, observation, perspective, or phenomenon is a limited view of symmetry. What makes something exist is its limited view; it is completely defined by its asymmetries, by what it ignores, by its relations to everything else.

One could say that no thing is fundamental to reality or that every thing is fundamental to itself. Both views would be equally valid but incomplete when taken alone.

Some analogies may help to illustrate this idea:

A butterfly and half a butterfly representing reduction in symmetry. (source)

• A butterfly has mirror symmetry. When we constrain our view to one side, the symmetry disappears.

• A flat plane has many symmetries. When we pick out a region (e.g., a square), that region has fewer symmetries (for a square, it’s the dihedral group D4), but all of those are contained in the flat plane. All imaginable flat regions exist in that plane.

• An infinite space filled with a superposition of all possible waves will be uniform. When we look at a finite region in that space, only wavelengths that are an integer fraction are stable within that region (the particle-in-a-box model).

Every way to conceptualize still includes thinking and, therefore, assumptions. An observer, any thought, or perception implies asymmetries; as long as they exist, there cannot be pure symmetry. If we let go of all assumptions, then there is no more thinking and no more experience. It is, therefore, impossible to observe, conceptualize, or experience pure symmetry.

Symmetry and superposition are related. To not observe parts is a superposition of perspectives and appears more symmetric. To observe a limited part is to take a measurement and locate one’s perspective within the superposition of worlds.

Frames of Reference

We have established that in the absence of all differentiation, there is complete uncertainty about how the world is, which is equivalent to pure symmetry. When taking a perspective or frame of reference that ignores some symmetries, the resulting view is less symmetric but contains more information. Constraints, information, and certainty are complementary. In this way, a perspective is completely defined by its asymmetries or, equivalently, by its remaining symmetries. This notion of “perspectives” can be thought of as equivalent to the fuzzy, recursive, perspectival sets discussed earlier. There is no one to take a perspective, just like there is no one needed to define a set. They exist just by being possible. The perspective is its own observation.
Any perspective contains and is a superposition of all more constrained versions of itself. Pure symmetry, or ultimate reality, is the superposition of all possible perspectives.

For any perspective, one can define a related perspective that adds or removes an asymmetry. In this way, all perspectives are related to each other, which in turn implies that perspectives can also be defined by their relations alone. The resulting structure of relations can be imagined as a network or fractal containing all possible mathematical objects, similar to category theory.

Every perspective is empty of any inherent substance. Remember that it is simply a limited perspective on symmetry, which is devoid of any properties. Insofar, every perspective contains pure symmetry, which in turn contains all other perspectives. Everything is mutually dependent on every other thing. Each perspective contains, and therefore knows about, the constraints that make it up. It contains every less constrained perspective all the way to pure symmetry, which again contains everything. Like the jewels in Indra’s net, every thing reflects everything else.

Three seemingly different interpretations of quantum mechanics can be thought of as different ways of looking at the same structure:

• QBism is taking a subjective view. Any observed world is a set of beliefs that have to be consistent.

• Relational QM is taking a relative view. Any phenomenon is only actualized in the relation between the observer and the observed. The relations between subjective experiences.

• Many Worlds is taking an absolute view: The probability of any observation is given by the wave function. Every world is defined by the state of its wave function. The superposition captures all possible relations.

In the same manner, one could systematize the language of perspectives by merging an axiom-free version of set theory (bottom-up approach) and an equivalently groundless category theory (top-down approach).

Conservation of Quantities

From the above logic follows that no thing can ever come into or go out of existence without respecting symmetry. For a closed system, this means that every quantity must remain constant. The conservation laws we observe in nature are naturally explained by assuming symmetry. Emmy Noether demonstrated how continuous symmetry corresponds to the conservation of momentum and energy in physics. For something to be accelerated in one direction, there must be an equal change in momentum in the opposite direction. For a thing to spin in one way, there must be an equal angular momentum in the opposite direction somewhere else. To observe a thing moving, spinning, or having any property at all is the result of a limited perspective that excludes the counterparts of those asymmetries.

Furthermore, none of the observed quantities has any inherent existence other than the asymmetries given by the constrained perspective. Every quantity expresses a relation and is never absolute.

Asymmetries form relations. An electron and a positron in the same place would annihilate, but placed apart, they attract each other. Because they tend back toward a more symmetric configuration, they have potential energy. The energy of a system is a measure of its internal asymmetry. perspectives are information; information is energy. The total energy of the multiverse is zero.

Subjective

The Arrow of Time

The way it is presented above, it might seem that pure symmetry is more fundamental. This is only one way of looking at it. On the other hand, it is possible to see every perspective as fundamental to itself. They, in turn, are related to each other. Pure symmetry, then, is only special as a fixed point in the structure of relations between perspectives. This emergent direction, pointing to more symmetry, less information, less entropy, is what we experience as the arrow of time.

From the statement that all quantities must be preserved and cancel out, there are some notable exceptions. Most of all, entropy always increases with time. Entropy is not a quantity that arises from the breaking of symmetry but a measure of how many ways symmetry is broken. Or, conversely, taking any perspective, it is a measure of how it could be otherwise.

For any perspective with a finite number of asymmetries, it is straightforward to deconstruct those and arrive at the undefined. On the other hand, any perspective can be further constrained in an infinite number of ways. Visualizing it as a tree of possibilities, it becomes apparent that for every branching, a path can be traced to the root, but no single preferred path forward exists. From each point, it seems like a coherent, linear past exists, yet the future is unknown. “Time,” in this image, is not flowing but a perceptual feature that arises by the way all possible perspectives are related to each other. It has an arrow pointing from the past to now but is not flowing.

Take the metaphor of a bug with bad memory sitting on the branch of a tree. Looking back, it can trace a path all the way back across the branches to the trunk and to the base of the tree. It may think, “This must be how I got here.” Looking ahead, it will only see branching with no clear path. It may think, “Where I will go is uncertain.” Then it flies off and lands on another spot on the tree, repeating the same thoughts.

The concept of “time” is a confused one since it combines several related phenomena: the arrow of time, causation, entropy, spacetime, and perceptual time. We get the first three from the above network. The time in spacetime might be just the impression of causal time overlaid on 4D space. Perceptual time is more complicated and not relevant to this discussion.

Note that the image of a tree is just one very limited way of conceptualizing this structure. It can also be the network of all (fuzzy, perspectival) sets, a hyperbolic and infinite-dimensional space, a directed acyclic graph, an empty recursive hypergraph, the structured space of all computations, etc.

This explanation of time is connected to most paradoxes. As we have seen earlier, a paradox can be resolved by taking a perspective that does not decide whether a proposition is true or false. Any fixed view on the proposition is a more constrained perspective and therefore a step in time. Classical logic and math can be caught up in paradox because they assume absolute simultaneity. Hence, the liar paradox “This sentence is a lie.” can be resolved by jumping in and relating the logical cycle indefinitely or by refusing to assert a truth value—breaking out of the cycle.

If the arrow of time is an emergent feature, then what about spacetime? Matter bends spacetime; the experience of spacetime depends on one’s frame of reference. Perspectives are information; information is energy, and energy is matter. How this works is still a big blank for me, but it suggests that spacetime also is an emergent feature of the relation between perspectives.

Uncertainty and Probability

The breaking of symmetry is a discrete event—or so it seems at first glance. Imagine an idealized ball on the top of an idealized hill. When it tilts to any side, it is already, irredeemably, on its way down that route. Any divergence from symmetry, no matter how small, would break that symmetry. What does that say about perspectives and time? Are they also discrete, or do they form a continuum?

There is another way to look at it when we let uncertainty back into the picture. The ball on the hill is, through inherent uncertainty, in a superposition of going down any path. To pick out a particular direction as a view amounts to continuously increasing the amplitude of probability of that view. It may seem like we only moved the problem, and now we need to ask if the asymmetry in the probability distributions arises discretely or not. But this makes the error of assuming time outside of time. It is resolved when we realize that symmetry already contains all asymmetries. One might visualize it as a landscape of all possible probability distributions continuously transforming into each other. Discrete and continuous views are complementary. They form a duality where neither is more true, and which is more useful depends on the context.

Above, we observed that because of symmetry, any quantity in a closed system must be conserved.
What is not conserved is every quantity that changes when we redefine the boundaries of the observed system. Through our perception of time, we order perspectives that take a more constrained view, from past to future. In the view of flowing time, each new perspective is more constrained. It excludes more symmetry and includes more information. The number of states the universe could have to explain our observation increases. Therefore, entropy increases with time.

Probability and information, in this way, are directly linked. Since time arises out of uncertainty, every moment in time has some inherent uncertainty. The end of uncertainty would also be the end of time. This would mean 100% certainty of something and therefore infinite information. If any point in space would contain infinite information, then it would collapse into a singularity where time ends.

Time, in this way, is a result of the uncertainty in the perspective. That means, because we experience time, we can conclude that our observed universe is not a singular, infinitely precise mathematical object.

With relations between perspectives and their direction, we get a natural notion of causality. From the perspective of every perspective, the preceding, less constrained perspective caused it. Since this is embedded within a larger network, causality is local and can travel only locally. Or rather, locality is defined by causation. For each unit distance a causal event travels, an equal unit of time has to pass. This is the maximum speed by which any two perspectives can influence each other—the speed of causality and light. This, in turn, means that out of time, we get the prerequisite for space.

The standard model of particle physics has proven to be a robust model of our world. For the most part, it is a descriptive theory. It does not explain what fields are or where they come from. Various interpretations exist, and some, like string theory, attempt to explain them in terms of more fundamental elements. The standard model describes these fields as a broken version of the gauge symmetries U(1)×SU(2)×SU(3).

With the idea of perspectives established, we can accept that the symmetries that describe particles are all there is to them. What we call a field is the symmetry underlying a particle. The wave function describes the degree of certainty of the symmetry being broken in a location. Particles are excitations in this symmetry. The probability of observing a particle is a measure of its degree of existence from the point of view of the observer. The world we observe is fluctuating somewhere between existence and non-existence. We cannot be sure about what exists, but we can be sure about our degree of uncertainty. Our own lack of knowledge about the world is the only thing we can know for sure.

Objective

What is Objective Reality?

The following might sound like I am confusing objective science with subjective experience. I am aware that one is not sufficient to extrapolate about the other. The point I’m trying to make is that it is not possible to separate the two. Your subjective experience is a perspective of reference and your only access to reality. Your set of beliefs about the world can be seen as a probability distribution. Inspecting one’s phenomenology with this in mind allows us to reason about how perspectives are structured.

As discussed earlier, everything that exists can be seen as a perspective, and each perspective is a fuzzy collection of asymmetries. You know—more than anything else—that your experiences in this moment exist. Therefore, it is a perspective and can be described as a fuzzy collection of asymmetries. It is a particular mathematical structure and exists in relation to other structures. Note that the reverse is not given—not every perspective is an experience.

Your experience of the start of this sentence is closely related (causally and by similarity) to your experience of reading its end. In any moment, you can only access memories of a previous moment, i.e., a residual of the causal influence. Other people can reproduce the same observation. You reading this text means we share information and therefore have connections to similar perspectives. Because we share similar experiences from the past (same physics, planet, language), we are able to communicate with each other. These assumptions allow you to form predictions about how I see the world. However, the only information I communicate to you right now are these words. In order to fully know my experience in this moment, you would have to be me in this moment. Almost all details of the actual experience are lost.

If your body and brain are made up of smaller parts, then how do they come together to form a mind with a unified subjective experience? How can anything be a “thing” if it is compounded and ultimately without substance? Perspectives are not objects in themselves but structure. They are the boundary that binds your experience into a whole (i.e. the question of mereology), yet they contain no information except the boundary itself. Perspectives do contain a view on other perspectives. This is apparent in the collection of real numbers. The set of real numbers is a perspective, but so is every number contained within it. The set of all positive integers contains an infinite number of perspectives and is itself contained in the perspective of real numbers.

Every concept is an abstraction. It ignores details about reality in order to match patterns. In this, the concept is not in the thing observed but in the observation. This means that concepts are also constrained views. Any concept may be included in my experiences, but usually does not capture the entirety of it. With practice, it is possible to modulate how much attention one gives to the concept—not thinking about it at all, or with a very unified mind. That is, moving further or closer towards that perspective, increasing the degree of certainty. With steady attention and letting go of distractions, one can go to the extreme and unify one’s mind around a concept to such a degree that no other thoughts arise and most sense experiences fade out. But it does not take a lot of concentration to see this. With every moment, sensations and thoughts fade in and out of the mind; it just takes some sensual clarity to notice. A unified mind will be better able to model a particular mathematical object than a distracted one. Thinking about a mathematical object, you not only invoke a label but model it in your mind. Your beliefs about the world in every moment are a probability distribution, hence a perspective.

One can implement a triangle in the physical world using sticks. Whatever happens inside the brain and body can, in the same way, implement a particular perspective. For example, the thought of a triangle. The collection of thoughts and sensations in turn also forms a perspective—an experience. This means that the mind is also part of a larger perspective of the physical world around it. Matter in the configuration of a triangle is a triangle insofar as it functions as one. The thought of a triangle is a triangle insofar as it functions as one. Combined perspectives form a new one to the degree that they function like that new perspective. This is an ontological claim, not just practical.

In order to model our environment, we give up part of our individuation, that is, reducing the degree of certainty of one particular perspective. As above, unifying the mind around a concept increases the degree of certainty of that concept and the absence of other concepts. Less certainty about any particular structure allows for more structures to be represented. To combine several concepts into one perspective requires that any particular concept has to have a reduced degree of certainty in the resulting perspective. The case that all certainty is reduced to zero is pure symmetry and therefore includes every possible structure equally.

This is similar to how, in a Bose-Einstein Condensate, atoms lose their individual positions and connect into one greater whole. Note that this process is scale-invariant. If you pick out any subset of your experience, this again is a perspective, more certain, less connected. When perspectives are connected (include each other with a certain probability) in a coherent way, then the overall perspective they form represents their consensus.

When you look for your face right now, you won’t be able to see it. Try to point at it, and you will see that there is nothing you are pointing at. Richard Lang says, “This is the only place in the world where I can point at nothing.” But to think that you have no face seems absurd. Objectively, you have a face.
Then, what is this objective reality that conflicts so strikingly with your unfiltered experience? Who or what determines what is objectively real when you don’t even trust your most direct sensory data? Another person would tell you that you have a face, but for them, the situation is the same—they don’t see their own face. The introduction of a third or fourth person won’t help but only push the problem further. It’s impossible to find a neutral observer for whom reality presents itself as it objectively is. Then how and why do we talk about an objective reality, independent of our subjective experience? The answer is that there is no single objective reality. The objective reality we can talk about is the consensus of several subjective realities, of perspectives connected in a coherent way.

If we relate perspectives, with subjective experience, does a stone or the number 5 have “experience” too? This is where the language breaks down, and we have to be careful not to equate experience (as I use it here), having a world model, self-awareness, and whatever “consciousness” means to you. In a future post, I will address the question of consciousness, the differences between perspective, experience, consciousness, and self-awareness, and show how they can be explained by combining existing theories.

Stability and Evolution

The above argument does not explain why we live in 3+1 dimensions, why we only observe the quantum fields we have found so far, or all of the other fine-tuning required that allows for entities to exist that can ask these questions. We find ourselves in a small subsection of all possible worlds. We could refer to the anthropic principle, but that begs the question of how it is possible to go from the general to this specific world. This means, for each apparent fine-tuned parameter, we need a generator that allows for a spectrum which includes the desired value. Yet this is not enough. A continuous range of values would not explain why we perceive exactly three spatial dimensions. In mapping out generators, we have to look for two features: the range from most to least symmetric, with the most symmetric being the most general and therefore most likely (e.g., spacetime being flat) and islands of stability. Stability in this context means invariant under added constraints (e.g., being subject to entropy).

This definition of stability is very similar to the definition of energy as invariance under time translation. This matches our intuition. A cup on a table is stable and has potential energy. When it tips over the edge, it loses its potential energy until it reaches a new stable state. But it is important to note the two different meanings of “time” are used here. Stability is broader in that it is not only a stable state but adaptively stable and resisting outside influence. Adaptive stability is always contextual within a given environment (set of constraints). Karl Friston’s free energy principle provides a precise formulation for what it means for a thing to persist.

Perspectives are a way to talk about mathematical structures. This means that perspectives can include all kinds of structures and therefore all kinds of properties we may need to account for. Perspectives that are a more constrained version of another perspective inherit properties of the parent. It may happen that some constraints are not possible, and other combinations of constraints turn out to lead to the same perspective. This scheme is a very simplistic version of diversification and selection. It allows for a kind of universal evolution (literally universal).

As an analogy, think of the real number line—first as a continuous line without numbers and without origin. Some ways to pick out numbers from that line are special in that they are invariant under operations. Recursively add 0 to itself, and you still have 0. Recursively multiply 1 by itself, and you still have 1. Similarly, pi and Euler’s number are defined by invariance and symmetry. The number 0 has more symmetries than the number 42 and, therefore, is more general. We can only deal with numbers that are stable in some way. The vast majority of all numbers cannot be defined in finite terms. They have no symmetry that would allow us to specify them, i.e., they are unstable and, therefore, unknowable and not actualized in our world.

As far as we know, there is no restriction for what kinds of particles to exist. Yet only a small number turns out to be stable and make up our everyday matter and forces. These, in turn, allow for a theoretically large number of different elements to exist. Most of these elements would be so massive that they never got created in the first place, and in most cases are so unstable that they decay quickly. An even smaller set of the stable elements generate the complexity of organic chemistry, which in turn allows for DNA and life.

The question then is, when all symmetries are possible, why do we only observe the broken symmetry of U(1)×SU(2)×SU(3)? What makes this one set special? Is it constrained by the dimensions of space? While I can only speculate at this point, an understanding of universal evolution points to the hypothesis that this is the case because these symmetries are very generic and stable in some way.

A different view on physics is required here. Physics not as laws of nature but as the results of universal evolution. There are no laws written down anywhere for particles to follow. Nature has no laws. It exists on its own accord, interacting with itself. Out of these interactions arise patterns, some more stable than others. Stable patterns can be described by emergent laws because they present a consensus of the smaller patterns that make them up. Stable patterns proliferate. Some patterns gain a higher level of stability by dynamically reacting to outside forces. Some manage to reproduce. Life is a higher level of the same process—dynamically preserved patterns. But the process of evolution is not limited to life. It functions before life and continues to function beyond life, in intelligence and higher levels of cooperation. Notably, competition between patterns results from a narrow boundary of self-identification. This limits the scale on which patterns can be stable. As an urgent example, if we humans don’t start to cooperate on a larger scale, we won’t be able to reverse climate change, thereby risking the destruction of civilization instead of seeding the universe with life.

When simpler perspectives include more complex ones, this means that for us to inhabit and observe a particular physics, the emergent laws must be very general (close to pure symmetry), particularly stable, a necessary ingredient for intelligent life, or—most likely—the combination of these. While everything that is possible also exists, not everything is possible, and not all possibilities are equally likely. One growing branch of physics and math might be the task of mapping out the landscape of all possible worlds (like the string theory landscape). Algorithmic information theory, impossibility results, and the study of symmetry from a groundless perspective can be important tools for this task.

Life happens at the edge of chaos—the fine line between pure order and pure randomness. Because it is constituted of interacting parts that find consensus, life can seek out and remain at this middle ground. The edge of chaos is a dynamic resting place from which many paths are available. Not being caught up in any particular path allows life to react quickly and in various ways to disturbances. Being on the edge of a phase shift allows for fast reaction. Because our universe is able to support life, it too is close to the edge of chaos—complex enough for life to emerge but still orderly enough for patterns to be stable and reproduce.

Conclusion

When we come into this world, we try to make sense of the things within our experience. Focused on this task, we forget to make sense of what experience itself is. When you seek out the edge of your visual field and the limits of your other senses, the things you don’t know and can’t imagine—then everything contained between those edges is you. As we look deeper into reality, we realize more and more that our frame of reference is one of many, that we are not the center of the universe, and that we are not separate from what we observe. We are what we observe. There is nothing outside experience that would “have” that experience. There is no preferred frame of reference and no unique, true way to model reality. It is entirely dependent on perspectives and cannot be conceptualized without taking a limited view. The universe is indeterminate regarding its own existence. It neither is nor is not—and therefore is in constant flux. The way our universe presents itself as one with time is because it is indeterminate regarding existence.

Like a polished gem, transparent and with many facets, you can see the whole from every perspective.

This insight is not entirely new. Similar ideas have emerged in various forms. In Taoism, the concept of Dao. In Buddhism, Pratītyasamutpāda (dependent arising) and Śūnyatā (emptiness). Mystics from nearly all traditions are pointing this way.

Modern thinkers have recontextualized these ideas by bridging Eastern philosophy with Western thought and scientific inquiry. For example Nishida Kitarō’s basho and Rob Burbea’s ways of looking. Similarly, in philosophy and science, theories like Ontic Structural Realism, David Pearce’s Zero Ontology, and Max Tegmark’s mathematical universe hypothesis.

There are still many open questions, missing pieces of the puzzle, but there are also many people already working on those, approaching them from different angles. I wish that in the future, we will be able to bring all of those threads together and weave a beautiful picture of reality.

When I say that everything is an aspect of something undefinable, I don’t deny that anything exists. Instead, I’m pointing to a false duality of existence and non-existence. I’m not talking about some form of nihilism, solipsism, or anything that denies a real world existing “out there.” It also is not realism, idealism, physicalism, functionalism, panpsychism, not even hiveism or whatever ism. It is that which avoids any fixed views—the middle way.

When you read all of this, the most likely reaction will be to see it as an interesting perspective that goes too far. You then go on with life and maybe even forget about this. But if you are still searching and keep asking questions, then you will rediscover insights. You might even see them as your own discovery—after all, they are. Since those insights are about seeing through your beliefs and not just information to absorb, they cannot be transmitted through words, and you will have to rediscover them in order to understand. The risk then is to identify with your insights, build a world model around them, and get stuck. If you keep questioning your world model, keep uncovering your hidden assumptions, and keep deconstructing them, then you will end up back here.

There is nothing to believe here. Quite the opposite, part of understanding is to hold your beliefs lightly, to not objectify them as real or not. This is just a guide pointing to a different way of understanding. I can only scratch the surface using words. It goes way deeper than that. Don’t accept it unless you’ve understood it. There is also no end to understanding. Don’t get stuck.