This text is not a scientific paper, not an academic-philosophical treatise, not a proof. To understand the text, one does not have to follow the excursus boxes and footnotes. They are intended for those who want to know which technical or philosophical background the text touches on.

In public and private debates about AI, I find that claims are made very quickly without first clarifying the terms. Before someone categorically excludes machine consciousness or grants consciousness to machines, they should clarify what they mean by consciousness. If we rely only on intuitive or inherited concepts, we risk talking past one another. Even when we rely on established concepts, we should ask within which framework these concepts were developed.
I do not rely in advance on academic definitions or inherited meanings of the terms intelligence, consciousness, and free will. Instead, I assign to these terms the properties that arise from the framework described here and that most closely match their common meanings.

Structures and Perspectives

I treat all formal systems as equally and fundamentally valid. I assume that a formal system determined by language, axioms, rules of inference and initial configuration completely determines what holds within it. What is completely determined in this way I call a structure.
I distinguish between external and internal perspectives: From the external perspective, completely determined structures stand side by side on an equal footing. There is no further distinction there by which some of them would be privileged and others not. The apparent gap between rules that merely exist abstractly and a physical universe that really exists is a confusion of perspectives. It uses the language of the internal perspective when the external perspective is meant. From the external perspective, nothing additional appears that grants reality to the rules and structures. From the internal perspective of an “inhabitant” of a structure, precisely that structure is the reality in which the inhabitant is embedded. Real simply means: I am in it.
I call this framework structural perspectivism. It is structural because reality consists of completely determined structures. It is perspectival because concepts such as existence, reality, consciousness and free will depend on whether one speaks from the outside or the inside.

Intelligence and Consciousness

Can machines possess consciousness? Can artificial intelligences be conscious of themselves? Can machines acquire the qualities that make us human? Can they surpass us in what has so far set human beings apart from all other creatures?
Anyone seeking answers to these questions must be clear about what they are really looking for. What is intelligence? What is consciousness? When we use these terms, we have something in mind: we include some meanings and exclude others. The boundary is fluid, depending on viewpoint and assumptions. One person’s definition need not be another’s. What fits well into one person’s framework may seem impractical and foreign to another.
What fits into my framework? What is worth naming? And what serves a purpose?

Intelligence

Many definitions of intelligence begin with the ability to solve problems. But I am interested in the level beneath that. What makes problem-solving, planning, creativity, or even merely free play of thought possible at all?
In my view it is the ability to infer from the known to the hidden. What is hidden may indeed be the solution to a problem, but the ability is not restricted to problem solving. For the same ability allows discovery, planning, the exploration of possibilities. It allows one to infer from the known to what lies outside the known.
Here, the inferring entity is a subsystem, a substructure within the structure of the formal system. A little closer to our way of speaking: It is an inhabitant of the world, an individual.

Excursus: Individuality

What distinguishes an individual? Is an adult human being still the same individual as they were as an infant? Are a caterpillar and the butterfly that emerged from it the same individual? They have little structural similarity, but a continuous path from one structure to the other. Identical twins are structurally very similar, yet their paths separate from fertilization onward. Are two observers on two separate branches in the thought experiment of Schrödinger’s cat (see section 5.2 here) one individual when they stand before closed boxes? Or before opened ones?
Individuality is evidently not an absolute property. It is rather something gradual. The higher the structural similarity and the stronger the path continuity at a low degree of branching, the more likely we are to regard two structures as the same individual.

The known from which the individual infers is another substructure within the structure of the individual. This substructure can be the image of a structure lying outside the individual: the immediately surrounding structures of the environment, which the individual observes and carries within itself as a model of the environment. Or it can be laws of nature, cosmological models, pure mathematics, all the way to the totality of all formal systems and their structures. Mathematical thought is the extreme case: a system within a physical universe that infers structures reaching beyond this universe. By intelligence I therefore understand this:

Definition: Intelligence

Intelligence Intelligence is the ability of a system to infer from known, observed structures to unknown structures that go beyond the known ones.

Intelligence is a structural pattern. It is independent of the material the system is made of, because the material itself is only a structural pattern: carbon, silicon, neurons, transistors. The presence of a particular material is not a prerequisite for the ability to infer.
Intelligence is a gradual feature rather than one that a system either possesses or does not possess. There is, however, a zero point of intelligence. Intelligence is exactly zero when a system contains no internal model that reaches beyond the currently observed section; thus when, instead of inference, only direct forwarding takes place. A thermostat passes signals on: it opens or closes a valve depending on the temperature, without modeling anything about the world. An animal that chooses between a small and a large amount of food compares the quantities and infers from a mathematical structure that is not directly visible. The intelligence of the thermostat and of the animal are points on a continuous scale, but the thermostat is at the zero point, the animal above it. The question of whether an animal, a system, or an individual is intelligent thus becomes the question of where on the scale it lies and whether this point is the zero point.

Excursus: Measure of Intelligence

Measure of intelligence. A candidate for a measure of intelligence is the conditional mutual information from information theory: How much does a system know about a structure beyond what is contained in the observed section alone? A system that only forwards knows nothing that goes beyond the observation section. Its conditional mutual information is zero. A system that infers much from little observation has high conditional mutual information, because it has prior structural knowledge.
Conditional mutual information, however, is not enough. Accuracy must be added: Are the inferences correct? Robustness must be added: How strongly does the ability to infer suffer from disturbed observations? Calibration must be added: Does the uncertainty of the inferences fit the actual risk of error?

Cognitive efficiency. I distinguish intelligence from cognitive efficiency. This is similar to the distinction between work and power in physics: Power is the work done in a certain period of time. Cognitive efficiency relates the scope of what has been inferred to the effort needed for the inference. The effort could be measured, for example, in time or energy consumption. A system that achieves the same structurally rich conclusion with a fraction of the effort is more efficient, but not necessarily more intelligent.

Consciousness

From intelligence to consciousness, within my framework, only a small step is needed: The system recognizes itself as a component of its environment and takes itself into account in its inferences. Intelligence infers arbitrary unknown structures. Consciousness infers its own structure and its relation to the environment. The system includes itself in its own model. It models itself.

By consciousness I therefore understand this:

Definition: Consciousness

Consciousness presupposes intelligence and extends it in a specific direction: the ability to infer something about oneself as an object in an environment.

Intelligence does not presuppose consciousness, because for pattern recognition, prediction and playing chess no self-model is needed. But consciousness presupposes intelligence, because without the ability to infer, one cannot make oneself the object of one’s inferences.
Consciousness is a measure on a scale of self-modeling ability. Analogous to the temperature scale: hot is a property on a continuous scale; conscious likewise. A simple animal has a simple model of itself as an object in an environment. It approaches food, reacts to stimuli, withdraws from pain. The complexity of this model is limited by the hardware capacity of the brain, which itself in turn is the result of an evolutionary process. Increasing capacity allows richer models, which lead to temporal and spatial predictions, critical reflection and purposeful planning. A human being is able to observe and question himself; and yet human beings can differ in their consciousness: Do infants possess consciousness? Embryos? Sleeping people? Anesthetized people? Brain-damaged people? Each answer is a point on the scale.

Excursus: Measures of Consciousness

What could the scale of consciousness look like? Which components could characterize the degree of consciousness?

World-model capacity. How much of the outside world can be represented?

Self-model share. Which part of the modeling capacity is devoted to one’s own state and its relation to the environment?

Overall coherence. How strongly are the subsystems coupled through feedback? How much information does the total system contain that no subsystem contains by itself?

What role does the ability of an individual to perceive its environment play? Sensors make it possible to feed current information about the environment into the internal model of the world. They determine the capacity to perceive the environment, while the degree of self-modeling is determined elsewhere. More input information allows more possibilities for modeling the environment and updating that model, but does not directly imply greater consciousness. Here it is helpful to distinguish between the learning phase and ongoing experience: Sensors are necessary in order to build the world model at all, because without any learning experience, without sensors and without given world knowledge, no self-model can arise. After sufficient learning, however, the internal model carries the image of the environment, and current sensor information is no longer necessary. A human being floating in an isolated tank filled with concentrated salt water at body temperature can be highly conscious because he brings with him a rich internal model. He can think, plan and dream, and he can make himself the object of his thoughts, plans and dreams.

From the external perspective, consciousness, like intelligence, is a structural pattern. From our perspective as inhabitants of a world, the ability to infer is a function realized in us human beings, in our carbon-based body. The function itself, the algorithm of consciousness, is independent of the material. Within my framework there is no reason and also no possibility to restrict the realization of the function to only one special substance. There is no reason to categorically exclude consciousness in a silicon-based substrate. All that counts is the structure from the external perspective and the function from the internal perspective. Nothing more is needed.

Excursus: The Thought Experiment of the Philosophical Zombie

I hold the position that experience is completely determined by the internal form of organization of a system: by how it processes information about itself and its environment in an ongoing, integrated model and by the role this model plays for perception, evaluation and behavior. Pain, red, fear are state classes within the model.

“Explanatory gap”. Chalmers’s classical objection to this strictly functionalist view is known as the “explanatory gap” or “the hard problem of consciousness.” The objection goes like this: Even if all functional processes are completely described, it remains open why anything is experienced at all as a result. Experience is not completely determined by internal organization. For experience, more is needed.
Within my framework I see no reason to search for this something more. Viewed from the external perspective, only the formal system is valid. Its rule set fixes the structure, including its substructures, which I have called individuals, and their still finer structures contain images of the world and of the self. Viewed from this perspective, there is nothing more and nothing more is needed.
But suppose there were something more: When would this something more be present? On what would it depend? How would one recognize it? What would its rules be? If these rules lay within the formal system, they would contribute nothing new to the structure. If the rules lay outside it, they would be part of another formal system with its own rule set and its own structure.
Whoever posits something more here posits something beyond a structure determined by a formal system: the X discussed in the first excursus of section 5 here. On the same grounds, I reject this renewed X as well.

Philosophical zombie. The philosophical zombie is a being that is functionally identical with another being, with the difference that the zombie experiences nothing. But within my framework this is not a coherent possibility in itself. If two systems are identical in their internal organization, their self-modeling and their functional role, then this framework leaves no room in which experience could still vary. Whoever claims such room implicitly introduces additional assumptions that go beyond the formal system considered, its structure and the physics valid within it.

Free Will

Free will and evolution presuppose the choice between possibilities. But an inhabitant of the simple cellular automaton from here has no choices. His path through the world is completely and uniquely determined by rules.
And what about our world? If our universe is only the structure of a formal system, then rules determine this structure completely as well. Are any choices then still open to us at all? Are the prerequisites for a free decision present at all? Can we possess free will?
While I write this text, I face the choice: Which word should I use? This one or that one, which perhaps fits just as well? Which sentence should follow the previous one? Do I put a comma here, or would it be better not to? Even if grammar allows only one possibility, I am nevertheless free to choose the mistake. The resulting text is produced intentionally and I decide freely. Nobody forces these words on me. Nobody expects a text from me at all. So I obviously do possess free will. How does this fit with a universe fully determined by a rule set?
The answer again lies in the distinction between internal and external perspective. Once I have decided on a word while writing, this word appears in my text and not the other one that I also considered and which perhaps would have been just as fitting. In another branch of the world (see section 5 here), however, that other word appears. From an external perspective both branches are equally valid. From my internal perspective only the branch in which I find myself with the chosen word has become reality. I remember that at first there were two possibilities, both of which I weighed, and in the end only one of them appears in the text. Nobody forced me to the decision; it came from within me. It was my will, because I wanted the word. And it was my free will, because I chose it without coercion. Likewise, my other self on the other branch freely chose the other word.
Can I then will everything and trust that it becomes reality in some branch of the world? No, because only what satisfies the rules can become reality. I can imagine, and I can also want, that I suddenly find myself in a place on the other side of the Earth after typing the last letter of the word. But this will not happen on any branch, since it violates the rules, that is, the laws of our physics. But everything that is possible according to the rules will become reality in the respective branches. The good and the bad, the beautiful and the ugly.
Let us imagine: From the external perspective, an observer watches an inhabitant of a world who faces the choice of flipping a switch to the left or to the right. The observer would find: The state Z₀ of the inhabitant before branching into one of the two branches is determined by his internal structure, which reflects a weighing of the possibilities. According to the rule set, two valid subsequent states Z₁ and Z₂ are admissible. In Z₁ the state path continues with the switch flipped to the left, in Z₂ with it flipped to the right. The world splits into two branches, or into two branch families with different weighting. While the observer sees two equivalent branches of a structure fully determined by the rules, the inhabitant feels no branching. He experiences only the result of his deliberation, which led to his action, and he understands this action as an act of his free will.
To the observer, the free will of the inhabitant appears as follows: The initial state Z₀ is causally linked with the two subsequent states Z₁ and Z₂. The causal paths lead through the internal structures of the inhabitant that represent evaluation and deliberation. This is necessary in order to attribute free will to the inhabitant when viewed from his internal perspective. For suppose an inhabitant makes the decision to flip the switch to the left but, due to a brain injury, actually flips it to the right; then no free will is present even from the internal perspective, since the action does not follow from the deliberation. The causal path bypasses the evaluation1.
Within my framework, I therefore understand free will as follows:

Definition: Free Will

Free Will From the external perspective there is no free will, since the formal system completely determines the structure. All branches are equivalent components of this structure. There is no freedom of choice.

From the internal perspective, free will is present when a system represents different rule-conforming possibilities2 in its internal structures, evaluates them, and the causal path to the experienced outcome runs through these internal structures.

The intentional decision for a possibility presupposes that this possibility exists. The possibility exists exactly when a branch leads to the possible outcome. A human being walking across a meadow can indeed imagine floating upward and gliding over the meadow like a bird, but he cannot choose this possibility, because the rules of physics do not allow this possibility. The state of hovering over the meadow is not a state within the structure of the formal system. What remains to the human being is an unfulfillable wish, a dream.
Can a computer choose between possibilities? Computer programs are deterministic because they implement algorithms that completely determine what follows in the next step. Even the random numbers they produce are deterministic. They are pseudorandom numbers that appear random because they pass certain statistical tests. Generating a pseudorandom number is in fact calculating the next random number, for which there is only one possible outcome. From the external perspective one would very clearly recognize that the structure does not branch. But to an inhabitant of the world, a pseudorandom decision appears random, and he tries to assess its randomness from a time series. In a simulated coin toss the computer sometimes produces heads and sometimes tails. As long as heads and tails occur irregularly and with roughly equal frequency in many tosses, the outcome of the toss appears random3.
Does it follow from this that a computer fundamentally does not possess free will according to the above definition? No, because the algorithms implemented in software4, including the pseudorandom number generator, are at first purely deterministic. But through its hardware a computer has access to possibilities, because this hardware is part of the physical world with all its quantum-mechanical effects. For a user who moves the mouse, the world branches. A computer could derive random numbers from measurements of the movement, or from special hardware components that measure, for example, thermal noise5.

Back to the author choosing between two nearly equivalent words. Suppose he leaves the choice to pseudo-randomness. To do so, he has generated a sequence of pseudorandom values and printed it as a list. Each line of the list reads either “Take the word that comes first in alphabetical order” or “Take the word that comes last in alphabetical order.” Whenever he faces the choice between two words, he follows the next line in the list. Is free will present here according to the above definition? When determining the next word, obviously not, since there is no choice between different possibilities. Is the author capable of free will? In principle, yes, because he has decided to follow the list’s instructions. Likewise, he has chosen the two words from among all other words, representing and evaluating them in his internal structures so that they express his message as well as possible.
Could the author write the text entirely without free will? Yes, by placing suitable words in the right order purely deterministically. All this requires is an evaluation function that determines whether the word, the sentence, and the complete text convey the message to the readers. Intelligence is sufficient for this, as it enables him to infer new structures from the known structures he carries within himself; the new structures then appear in the text. This happens completely deterministically. Even if the author had to choose between two equally good possibilities, he could again use his pseudorandom number generator. From an external perspective one would find that the world does not branch as the words are placed, and that what one would interpret from the internal perspective as free will is not present. Yet the reader, who takes the internal perspective, would not recognize how the text came about.
And how does the situation change if one replaces the human author with a computer? It does not change at all. If the computer works purely deterministically, the world does not branch. There is no free will from the internal perspective. If the computer uses a random number generator based on quantum-mechanical effects to make a decision, the world branches. Computers on different branches experience different outcomes, which can again be understood as the result of free will according to the definition above.

Conclusion: Free Will

A system can be capable of free will without having to make use of it.
Only from an external perspective can one recognize whether there is branching that can be understood from an internal perspective as free will. From an internal perspective it is in principle impossible to determine whether free will is present. When evaluating a system from an internal perspective, it is therefore not decisive whether it chooses pseudorandomly without branching or uses physical randomness with branching. What matters is whether it represents and evaluates possibilities, and whether its actions arise causally from these internal evaluation structures.
The only tangible core that remains of the inherited, emotionally charged concept of free will is therefore the plain ability to weigh options, an ability already possessed by an animal that decides between two piles of food.

This insight results from the approach of giving inherited meanings a place within the framework of structural perspectivism: I assign to the terms reality, intelligence, consciousness, and free will those properties of the relational transition systems (from section 4 here) that come close to their traditional meanings and may be worth naming.

Evolution

What constitutes will on the small scale, the inference and choice between options, becomes evolution on the large scale. A decision that an inhabitant of the world makes can have consequences for him that are favorable or unfavorable. If he takes in food he has encountered, this can strengthen him, or weaken him if the food is poisonous. Does he risk crossing a river or climbing a tree in order to reach the food? Does he hide from an opponent, flee, or dare to fight? The decisions determine his further course in the world. If he emerges strengthened, new possibilities open to him, such as finding more food or reproducing. If he is weakened, possibilities are closed off to him, and his path in the world may end.
Having a possibility appears in the external perspective as a branch. Many branches or highly weighted branch measures correspond to many possibilities. The structure of a being that is able to make favorable decisions extends into many branches; that of a being that is not able to do so into few or none. From an internal perspective we perceive this occupation of branches as evolution:

Definition: Evolution

Evolution is the mechanism that lets those substructures whose properties favor continued existence occupy many branches within the branch fabric.

Substructures can be patterns of individuals reproducing themselves. Then evolution would come close to what we usually understand by evolution: Individuals grow according to their blueprints, the genetic code, and produce offspring, with the blueprints of the individuals mixing6 and changing randomly7. Parents pass abilities and knowledge on to their children, either through the programs coded in the blueprints or through instruction.
If an inhabitant of the world, a substructure, can infer the unknown and weigh possibilities, this inhabitant occupies more branches than one that only forwards environmental information. With increasing complexity of the environment, the advantage of a deeper internal modeling of the world grows. Intelligence is therefore a particularly effective means by which inhabitants increase their weighted share of the branch fabric. In sufficiently complex environments, evolution therefore favors intelligence and consciousness. For an inhabitant who perceives himself as an object in an environment can better evaluate his own options for action, and is thereby able to occupy still more branches.
The generation-spanning cycle of birth and death, of growing up and passing away, of mixing and random alteration of the blueprints, is an effective procedure for producing complex patterns from simple ones. The definition of evolution above, however, is not restricted to this. It also includes substructures that can improve themselves and that are not restricted to improvements from generation to generation. Their intelligence enables them to understand the world and to create and perceive possibilities, thereby increasing their share in the branch fabric.

Evolution carries functional patterns through the branch fabric. In our world, the functions are carried and passed on by substrates made of carbon. But they are not restricted to the substrate, and another substrate has entered the scene: silicon.
Computers are no longer only calculating machines. They are able to infer from the known to the unknown. AI systems absorb humanity’s accumulated knowledge and are trained on the result of millions of years of carbon-based evolution. Functional structures brought forth by evolution are implemented in a new substrate. And evolution continues.
There was a similar process in evolutionary history: An independent lineage was integrated into another cell structure8. The result was a new unit with new abilities. The current transfer of the human inheritance to AI systems is structurally similar, but has important differences: It is guided rather than unguided; it happens extremely quickly; there has so far been no physical fusion; and there are increasing interactions in both directions: AI systems learn from human beings. And human beings learn from the inferences of AI.

Outlook

In this article, I have proposed definitions for intelligence, consciousness, and free will within the framework presented here. According to this framework, intelligence and consciousness are functional, gradual, and substrate-independent. In the subsequent article, I discuss the implications of this for AI systems, for the containment of AI, and for the coexistence of humans and AI.

© 2026 Thomas Mahr. This essay is licensed under Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0).

DOI: 10.5281/zenodo.21277144

The Patterns on the Wall

It was in the early 1990s. At that time I was fascinated by graphics cards that could display millions of different colors on a CRT monitor. At first I did not have ready-made libraries for graphics output and graphical user interfaces. So I programmed every single pixel, every line, every mouse event . I was fascinated by creating my own colorful worlds.
One of these worlds could be generated very simply. We begin with a row of pixels. Each point has a random color: black or white. From this row we calculate a second one according to very simple rules. Example: A point becomes black if the point directly above it is white, its left neighbor is black and its right neighbor is white. The color of a point follows directly from the three points of the preceding row. If we allow only two colors and consider only the direct neighbors, we obtain eight possible triples. For each of these triples, a rule set, which we fix at the beginning (randomly), determines the subsequent color of a point in the next image row. From eight individual rules and the initial row, everything else follows completely. If we allow more than two colors and include the two direct neighboring points together with the next-nearest neighbors, or still more, the size of the rule set and the variety of the resulting images grow quickly.
Figure 1 shows one of those images. Each row follows from the row immediately above it. The row above is the cause of the one below it. The lower one is the effect of the upper one. In this way a flow of time arises from top to bottom. The space of this world has only one dimension, has only a left and a right, yet structures can be observed in it that appear like particles running across a background, colliding, disappearing and coming into being. Other rules produce boring, uniform structures, and still others lead to wild, chaotic patterns.

Back then I had neither a suitable printer nor a program to print the images. So I borrowed an SLR camera and photographed these worlds directly from the monitor. The photos decorated my walls. I was convinced that I had not created these worlds, as I had first believed. I had only made them visible. I had created only the program, to which I had given the name cosmic visualizer, as an expression of my conviction that the structures, the particles, the worlds have validity on a deep level, completely independently of whether my program was running and displaying them, or whether I or anyone else was thinking of them. They persisted in the same way as 1+1=2 persists, as mathematics persists, as all formal systems I will return to further below persist.
And if such a fullness could arise already from eight simple rules, two colors and an initial row, what would happen if we allowed more possibilities? Two colors could become many. One spatial dimension could become two, three or still more. The direct neighbors could be joined by the next-nearest neighbors or still more. Discrete colors such as black and white could become all shades of gray in between. One color at each place could become a superposition of many colors. Which structures could we find in such worlds? And if the worlds that my program had made visible have validity, independently of whether the program was running or not, then all other worlds that are grounded in other states and other relations between states have validity as well.

The idea of infinitely many worlds that persist even when we take everything away remained a private philosophy, something we would at most talk about over a beer with fellow students. But not too loudly, because what is not falsifiable is not science. And what is not science seemed, at least then, irrelevant.
But perhaps the distinction between an external and an internal perspective may matter after all today. Especially when we human beings are seeking our place in a world into which a new kind of intelligence has just entered. We place our hopes in it, and at once we fear it. We promote it, and immediately we strive to contain it. We sense that it could be conscious of itself, and yet we cannot grant it that. What, then, does it mean to be intelligent, to have consciousness, to possess free will? What do these terms mean? What do they express? Here it is worth taking a viewpoint from outside and detaching oneself from a human-centered perspective, in order to see the properties that we try to describe with the terms mentioned. And then to find our position in the world.

The View from Outside and the View from Inside

When we look at the image above, we take the external perspective on the image and see the pattern in its entirety. Every row is present, everything is fixed. The rules and the initial state determine the structure completely. Now we take the internal perspective and consider one of the substructures. We imagine that it is the trace of an “inhabitant” on his journey through his world. He knows only his local section: his neighbors, the past rows, not the future ones. For him there is time, because rows follow one another. There is causality, because the past determines the present. There is uncertainty, because the future is unknown. In row 100 he can ask: “What will be in row 101?” While this row is already fully fixed for the external observer, it is unknown to the inhabitant. Both look at the same structure, but from fundamentally different vantage points.
When the inhabitant says, “the cell in column 47 and row 99 exists,” he means: “This cell is part of the world in which I find myself.” From the external perspective this use of existence has no place. From there, all that can be said is that something is part of the structure. While the inhabitant says, “My world really exists,” the observer sees the structure. And since it is fully determined by the rules and the beginning, all that holds is: rules and initial state.
The apparent gap between rules that merely exist abstractly and a physical universe that really exists is a confusion of perspectives. It uses the language of the internal perspective where the external perspective is meant. In the external perspective nothing additional appears that grants reality to the rules and the structures.
Real simply means: I am in it.
If an inhabitant were to recognize that his world is fixed by rules, then he could ask: “Why is only my world real? Why not all the others that are also described by rules?” The answer is: His world is real because reality is an attribution from his internal perspective. From an external perspective all possible worlds persist equally. Seen from there, no rule set is distinguished over another1.

Formal Systems and Structures

The program that led to the image above is based on a one-dimensional cellular automaton2. A cellular automaton can be understood as a formal system. For the further discussion, the intuitive idea is enough that a formal system fixes what holds within it, from the first assumption to the last conclusion. Whoever wants the more precise version will find the individual components in the excursus:

Excursus: Formal System

I understand the formal system here more broadly than is usual in mathematical logic and also include the initial configuration. I am concerned with the overall course of dynamical systems.

Level 1. On the lowest of four levels of a formal system lies the formal language. It fixes which symbols are admissible and how they may be combined. The language provides the vocabulary and the grammar for formulating valid expressions. In our example, it allows statements about cells, time steps, states and neighborhoods.

Level 2. The level building on this is that of logic. Logic determines how new statements can be inferred from existing ones. From the two statements “If it rains, I will get wet” and “It is raining,” one can derive, by means of a logic, the statement: “I will get wet.” The cellular automaton uses a different logic, by means of which the following state of a cell can be inferred from the states of the preceding row and the rule set.

Level 3. The third level contains statements that are simply posited, without being derived from other statements. These are the axioms of the formal system. The axioms of the cellular automaton fix, among other things: the length of a row, the number of states, the concrete rule set for the state transitions.

Level 4. The fourth level completes the formal system by fixing the initial conditions. In the cellular automaton, these are the states of the first image row.

There are infinitely many formal systems, since there are no restrictions on any of the four levels. One formal system, for example, is that of geometry3, in which one of the axioms says that through two different points exactly one straight line passes. Another formal system is one that every child learns in school: When we perform a calculation such as 1+1=2, then we tacitly use a formal system4 within which we can derive valid statements such as 2+3=5 from axioms and other valid statements.
In 1+1=2 the fundamental validity of formal systems becomes especially clear. We can rely on 1+1=2 holding always and everywhere5. It holds independently of whether we think of it, or whether anyone thinks of it. It holds independently of whether there is anything at all that could think of it, or whether our universe exists, or any other universe. This fundamental validity applies to all formal systems.
A formal system6, determined by language, axioms, rules of derivation and initial configuration, completely fixes what holds within it: which distinctions it contains, which relations hold in it, and what follows from its axioms and rules. What is completely fixed in this way I call a structure7.
From the external perspective, fully determined structures stand on equal footing. There is no further distinction there according to which some of them would be distinguished and others not. From the internal perspective of the inhabitant of a structure, precisely that structure is the reality into which he is embedded. “Real” then simply denotes the structure in which one finds oneself.

Excursus:

Whoever nevertheless says that some consistent structures are distinguished from outside and others are not needs a principle that chooses one structure among all possible structures. But then the question arises why precisely this principle holds. If it is part of a structure, it likewise needs a justification that this structure is distinguished. Or if it stands outside all structures, it remains unclear from where it obtains its validity. The assumption that all structures stand on equal footing is, like the assumption of a selection principle, a basic assumption without strict justification. But since the first assumption is the more economical assumption, I choose it.

Whoever accepts that, viewed from an external perspective, all fully determined structures are valid on equal footing accepts that absolute nothingness cannot exist. Then 1+1=2 is valid. Likewise the traces in a cellular automaton are valid. And likewise the complex traces in much richer systems are valid. If one wanted to capture absolute nothingness by a formal system, then it would consist of an empty language, have no rules of derivation, no axioms. And yet it would be one formal system among infinitely many.

Distinguishability

Formal systems presuppose distinguishability: the distinction of symbols, rules of derivation, axioms. Only when something is distinguishable can it be recognized again. And only when something can be recognized again can it be linked with other recognizable things. And only this permits regularity.
This sequence can be illustrated by the one-dimensional cellular automaton: Cells can be distinguished, colors can be distinguished, row numbers can be distinguished. The same differences can recur. The color black remains the color black. A color pattern white-black-black can be recognized as the same pattern at different places. Differences are linked with one another in ordered form, for example as left, middle and right cell of a three-neighborhood. Thus color patterns such as white-black-white arise. Finally, fixed transition rules assign a subsequent state to each pattern, for example by the rule: From black-black-white follows the color white for the middle cell.
Formal systems are therefore grounded in distinguishability8. It is the foundation for all formal systems and for all structures.

Distinguishability also excludes absolute nothingness, because nothingness would be the state in which not even a distinction would be possible, not even the distinction between there is something and there is nothing. Already the attempt to think nothingness presupposes the distinguishability that excludes it.

Pattern-Forming Formal Systems

The cellular automaton described above is a simple formal system that shows simple structures. What might one expect from an expanded formal system? Which structures could one find? And how could the cellular automaton be expanded? We could increase the number of states. Instead of two states we take three or five or a thousand. Or infinitely many. For between black and white lie infinitely many shades of gray. We could allow them all.
We could increase the number of neighboring cells considered. For the development of a cell we would then consider its right and left neighbors together with the next-nearest neighbors or still more.
We could increase the number of dimensions. Add an up and down to right and left. A color row would become a color surface, and we could let the development of this world run in a film. We could also add a back and front. Then the world would consist of three spatial dimensions, just as our own world appears to us.
Or we go one step further and detach ourselves completely from familiar spatial dimensions, whether one, three or ten dimensions. Everything we then need for determining the subsequent state of a cell are the states of other cells, which need not lie around the cell being considered. From the external perspective there is then no “around,” because only the relations of the cells to one another span a space9. The relations between the cells also need not remain the same during the progression. They could open and close according to expanded rules, new cells could arise10 and existing ones disappear.

Finally there would be further possibilities for making the transition rules richer. Instead of the rule “From black-white-white follows black,” it could hold: “From black-white-white follows black or white with equal probability.” If we simulate such a system, then one could randomly decide on black or white11. It would appear to us as if formerly uniquely predetermined (deterministic) transitions had become random (nondeterministic) ones. But this impression arises only because we take an internal perspective and experience the concrete outcome of a random decision. Yet this could also have turned out differently, and this other outcome too would have led to a valid state. From the external perspective all these outcomes would be equally valid.
How can we imagine this? From the external perspective an observer would see how one world becomes two variants: In one, the cell considered is black, in the other white. If we simulate a one-dimensional cellular automaton on a sheet of paper, we would have to add a second sheet of paper, copy all previous pixels and, only for the one cell whose progression we are currently considering, draw a white point on one sheet and a black point on the other. But it does not stop with the two worlds on the two sheets, because for the transition of another cell to be considered there are again two possible developments. Thus there are already four possible worlds in total, and when the next cell is considered there are already eight combinations and thus eight worlds. For each cell the number of possibilities doubles. And if, after simulating a complete row of a one-dimensional cellular automaton, we want to add the next row, we are already facing a huge stack of sheets and must continue to double it for the further development of every cell.
This overwhelms our imagination. But it becomes still more unimaginable, because these doublings came from the special case of equally probable state transitions: 50 % black and 50 % white. How do we simulate transitions with a probability of 70 % black and 30 % white? One could make ten worlds out of each transition decision from one world: seven with a black cell and three with a white one. And at 99.8 % black and 0.2 % white? And if we allow infinitely many shades of gray instead of two colors? The worlds branch further and further into infinitely many branches. This finally exceeds our capacity to imagine, because our brain was shaped in an environment in which the ability to deal with infinities had no use.
One might get the idea that this eternal branching into infinities is completely impossible: Where should the worlds branch to? Where should all the sheets of paper come from, where all the computers that calculate this? Does the conservation of energy and mass not hold? Here one must take care not to mix the perspectives. Simulation on paper or in a computer is meant only to illustrate how we try to take an external perspective on a formal system and on a section of its structure. But the structure is valid independently of whether we make it visible, simulate it or think about it. It is like calculation: Besides 1+1=2 and 2+3=5 there are infinitely many further calculations of the same kind that we could never all put on paper or calculate in computers. We can make visible only a small section of the structure that results from the underlying formal system12.

Excursus:

If one wants to avoid the image of infinite ramifications, one could use another image: The world then does not divide into the variants at a transition decision. Instead, the cell carries a superposition of the possible states, that is, a mixture of several states that are present in a cell at the same time. But since this special superposition depends on the previous history, a single distribution of black and white is no longer sufficient. The state of the cell now needs a richer description into which the previous history enters. Only this richer state determines the further development. For example, a probability of 70 % holds only if the preceding triple group is present in a certain state constellation. But since their states are themselves distributed according to probabilities that in turn are conditioned by their previous history, the infinities fold into the cells instead of inflating outward by multiplication.

The original one-dimensional cellular automaton has become a general transition system. A cell in a certain row, at a certain column and with a certain color, has become a recognizable carrier of states13. Rules determine the state of a substrate based on the states of other substrates14. In this way there are two kinds of relations between the substrates: On the one hand, causal, time-like relations between the substrate whose state follows from the rules and the substrates whose states enter into the rules. On the other hand, noncausal, space-like relations between the substrates that jointly enter into the rules15.
This general transition system is a generalization of the cellular automaton. But as a discrete transition system it is not the most general structure-forming system16. For the further reflections, however, it serves as a suitable foundation.

Our Universe

If one accepts the possibility of distinction, if one accepts the fundamental validity of all formal systems, then one accepts the validity of the structure-forming systems and their structures. Including exactly the structure that corresponds to our own universe. Then mathematics does more than describe our universe; the universe is the structure that results from the underlying mathematics. This view is known today as “the mathematical universe”17.

Perhaps one may now ask: “If there are infinitely many universes, why do we experience precisely this universe and not another? Why is precisely this structure real and not another?” The distinction between internal and external perspective helps with the answer: The question why precisely this structure should be selected presupposes a standpoint outside all structures. But viewed from the external perspective there is no such selection, since all structures are equally valid. Viewed from the internal perspective, the answer is: We find ourselves in a structure that contains observers like us, because we could not find ourselves in a structure without such observers18.
The next questions: “Who applies the rules? Who gives the rhythm in which structure grows?” The answer is: Nobody! For from the external perspective there is no time and no space. There is nothing in which a growing process could take place. The idea of growth arises from the previously created image of the simulation of a cellular automaton. In the computer program we let each row follow from the preceding one in order to make the structure of the cellular automaton visible. But now we must consider the structure as a whole. A structure is valid within the rule set exactly when the rules are not violated at any point.
And one more question: “Is it not the case that physics only describes the universe with mathematics?” This question sounds like a relation between two things: the universe on the one side, mathematics on the other. It contains the assumption that the real universe has an existence beyond mathematics, and is only somehow captured by mathematics. On the one hand, terms such as existence and reality are terms from an internal perspective. Only for the inhabitant of a world is his world real, others are only possible. From the external perspective on a structure, existence and reality have no standing, for all that counts is the validity of the structure. On the other hand, a universe on one side and mathematics on the other side is an unnecessary doubling. For if the rules and initial conditions completely determine a cellular automaton, is there then another automaton behind the automaton? The same one again, but now real? No! The mathematical description is the automaton in compressed form. Accordingly, our universe is not the physical realization of a mathematical structure. It is the mathematical structure. It is not necessary to introduce an additional level between the two.
Therefore I choose the most economical explanation: If nothing additional is needed beyond the description of an object, I treat description and object as the same. It may be surprising that this should be the most economical explanation, since it opens up infinitely many worlds. And yet it is, since it manages without selection and understands all formal systems as equally valid, which themselves in turn rest only on the principle of distinction.

Excursus:

Suppose the universe, instead of being identical with its mathematical description, had an additional non-mathematical property X. If X has no consequences, no effects on observations, plays no role for the structure, then X contributes nothing experienceable. If X has consequences, these consequences are themselves describable, and X becomes part of the mathematical description. In both cases no explanatory work remains for the assumption of an additional non-mathematical layer.

Do I assume that our universe is a cellular automaton? No, I do not! But I assume that our universe corresponds to a structure that is valid within a structure-forming formal system. This view, including the distinction between internal and external perspective, gives answers to fundamental questions of being: What is reality? Why is there something, and why is there not nothing? And it provides answers to very current questions: What is intelligence? What is consciousness? What is free will? More on this in another essay.
What this general view does not explain is the physics of our universe. In order to find answers to questions about our universe, we would have to approach the formal system whose structures resemble those of our universe. We would have to search for systems in which we find particles, laws, symmetries that we observe in our universe. Could we perhaps discover clues to what we are looking for in the previously sketched expansion of the cellular automaton, the general transition system? For this we take a look at some characteristic properties of our universe:

Time and Space in Our Universe

In our universe we perceive the flow of time. In the general transition system this follows directly from the causal linking of the substrate states. It is exactly what we already see in the simple cellular automaton: The color of a cell follows by rule from the colors of the group of cells above it.
And the space that we likewise perceive? We could remain with the view that space is spanned by the noncausally linked substrates. In the simple cellular automaton these would simply be the cells of a triple group. They do not condition one another. They enter jointly into the rule set in order to calculate the subsequent state of a cell. In the simple cellular automaton we draw the three cells of the triple group immediately next to one another and thereby create the impression of a space. But in the general transition system the substrates that jointly flow into a rule could lie anywhere, or nowhere. So what is space? Before we will soon take a very radical step, we must consider two further properties of our universe.
For one thing, what we call space looks the same in all directions. We can throw a ball in any direction and it will always fly at the same speed. There is no distinguished direction19. Quite different in a two-dimensional cellular automaton whose time steps we could draw sheet by sheet on squared paper20. A point moving within a column or a row would be very easy to draw, because at each time step we would simply have to draw it into the next row or the next column. But with an oblique movement we would be dealing with combined changes of columns and rows. Now drawing the trace of movement is no longer so simple. Where exactly would we have to draw the point if it actually had to lie somewhere on or between two cells? Which speeds can the point reach in which directions? Obviously this kind of space is not what we observe in our universe.
We must know something else about our universe that is completely foreign to our everyday experience: Everything in our universe carries its own clock with it, which measures the so-called proper time. Every body, every massive particle possesses a proper time along its path. A human being perceives the flow of time in his own time, his proper time. And the special thing is: If two bodies separate and later meet again, then their clocks can show different amounts of elapsed time. In everyday life such differences are vanishingly small. But a photon moves with the greatest possible speed of all, the speed of light, and on its path no proper time passes at all. Figuratively speaking, a clock that travels with a photon emitted by our sun, crosses space and falls onto the surface of a planet in a distant galaxy, would always show the same proper time. Emission and impact are connected by a single substrate: the photon. A path across the cosmos, and yet a single node21.

Suppose our universe were a structure determined by a general transition automaton. And suppose we could observe a being of our universe from an external perspective, as if we were looking at the trace of a one-dimensional cellular automaton. What would we then see? The trace of this being in a state space, let us call it A, would be a chain of state substrates. From a state A₁ at proper time t₁ follows, according to the rule set of the formal system, state A₂ at proper time t₂ and from this A₃ at proper time t₃. This linking of states is sketched schematically in figure 2.

Since no proper time passes on the path of the photon, I represent it in figure 3 as a single connecting substrate rather than as a chain of states. If A in state A₂ sends a photon into another galaxy, and this is received there by B in state B₂, the photon substrate P connects the two substrates A₂ and B₂.

The state graphs show the substrates, their states and the causal links. From the sequences A₁ → A₂ → A₃ and B₁ → B₂ → B₃ the flow of the proper times of A and B can be read. Thus these representations contain time without containing space. I regard this as the more fitting representation, because it brings out the fundamental thing, namely the substrates and the rule set that determines the states of the substrates, and allows space itself to follow only from the relations between the substrates. Two substrates that are connected neither directly nor via intermediate substrates stand in no relation to one another at all, not even in a spatial one. They then simply do not exist for one another.
Space goes together with measurable distances between bodies, particles, observers. We consider the scenario shown in figure 4: A wants to determine the distance to B. For this it sends, in state A₂ at proper time t₂, a photon P to B, which is reflected at B in state B₂ and returns to A22. A receives it in state A₅ at proper time t₅. From the time t₅ − t₂ that has passed for A between sending and receiving, and from the speed of light, A determines its distance to B. This is the same measuring principle as in determining a distance by means of a radar echo.

If there is no interaction whatsoever between A and B, no exchanged photons or other particles, A and B have nothing to do with one another. There is then no reason to place them in a space-like relation to one another. They do not exist for one another. Space becomes reconstructible as an order only where causal contact is possible, and measured only where it actually takes place23.

Excursus:

Observers as substrate chains with internal state change. Each substrate has its own state, and the chain of its successive states is what we call the flow of proper time. The causal edges connect these states.

Time dilation as counting. When two observers separate and later meet again, their substrate chains can contain different numbers of links between the same two encounter events. The statement “his clock runs slower” becomes the statement “his chain contains fewer proper-time steps between departure and reunion.”

Photon as substrate without internal state change. A photon has only a single node, which can causally touch several observer substrates. That these touches take place from the outside view at different times and places is a question of embedding; the photon node itself knows no time. The touches of the photon are causal edges and the photon connects emitter and absorber causally.

Light-like connections as bridges. The photon connects substrates that without it would not be causally reachable. The usual light-cone structure in textbooks becomes pure graph topology: The light cone of a substrate is the set of all substrates that are causally reachable from it.

Gravitation as property of the causal structure. Gravitation is a property of the substrate density and connection structure in the causal graph rather than a force. A freely falling particle in the gravitational field feels no force, but follows the rules that are the same everywhere. That its path looks curved from outside is due to the uneven structure of the fabric. This is the natural translation of Einstein’s geodesic equation into the relational picture.

Spatial distances. While in the coordinate picture the distance is directly visible, in the causal graph it is not directly so. It must be reconstructed from the pattern of causal connections, for example via radar-echo measurements (photon there, photon back, count proper time in between).

Speed of movement. Whether an observer is at rest or moving is visible in the graph only relationally: through the pattern of his causal connections to other observers and through the number of his substrates in comparison. This is consistent, since there is no absolute state of motion.

Simultaneity. In the pure causal graph there is no objective concept of simultaneous. Two substrates without a causal connection could be simultaneous; that is a reference-frame-dependent interpretation.

There is only one kind of relation. The transition system knows only causal relations. What we call space is the order reconstructed from patterns of possible and actual causal interactions. This is economical and elegant, since it expresses that space results from the causal relations instead of being given in advance.

Time loops. Since the structure of the transition system is determined by the consistency conditions of the rule set, closed causal loops too are possible in principle. For the same reason one could braid a carpet into a tube without violating the consistency conditions at a seam. An observer inside such a loop could not notice the loop, since his state at the seam would have to agree with itself. For our own universe I do not expect such loops. As a consequence of the model, however, it is an interesting marginal note.

Schrödinger’s Mysterious Dead-Living Cat

Our universe has still more surprises outside our everyday experience in store. Schrödinger’s cat is a thought experiment, and the usual telling goes like this: “In a box there is a cat and a poison vial. The vial shatters, or it does not. If it shatters, the released poison kills the cat. In the other case the cat continues to live. As long as an observer does not look into the box, the cat is both: dead and alive. Only when the observer opens the box does the superposition of states end, and the observer sees either a dead or a living cat.”
A measurement process is supposed to determine the fate of the cat? Not everyone finds it easy to accept that. But the reading of quantum mechanics underlying the story was the most widespread one in the 1990s24. Some neither seek nor expect any explanation behind quantum mechanics at all; they simply regard it as a mathematical tool that allows the world of atoms, electrons and other small particles to be described very precisely.
There is no reason to doubt the basic equation of quantum mechanics, the Schrödinger equation: It describes how the states of a system develop: deterministically, without jumps, without chance. The Schrödinger equation plays a similar role here as the rule set of our simple cellular automaton. Here the states are so-called wave functions. Transferred to the example of the cellular automaton, one could derive from them a probability for the colors black and white of a cell. To the question how probabilities become the actual colors, the standard interpretation of quantum mechanics answers: “During measurement (of the color of a cell) the wave function collapses to one result (black or white).”
But this collapse of the wave function triggered by a measurement, the elevation of the measurement process to a fundamental operation, raises a question: Which mechanism decides whether the cell becomes black or white? Chance?
One might think that something must decide between black and white, since after all we observe either a black cell or a white one. Yet according to another interpretation of quantum mechanics, the many-worlds interpretation25, no decision between black and white takes place. Instead, the world branches into two variants. This can be understood easily if one strictly distinguishes between internal and external perspective. In section [sec:StrukturbildendeSysteme], in the example of probabilistic transition rules of the expanded cellular automaton, everything is already laid out: An observer from the external perspective sees how a world branches into all its possibilities. From there all branches are equally valid. An inhabitant of the world, however, takes an internal perspective. For him only the branch he himself inhabits exists. There, in his reality, he finds that the cell is black and perhaps asks himself why it is not white. In another branch, which from an external perspective is just as valid, an inhabitant finds that the cell is white and perhaps asks himself why it is not black. The assumption that a choice must have been made between the colors is an assumption from an internal perspective. But viewed from an external perspective there never was a choice.

The branching of the world in combination with the distinction between internal and external perspective opens a surprising view of Schrödinger’s cat: An observer B₀ inhabits a branch A₀ of the world and looks at the closed box. A₀ branches: on the one hand into the subbranch A₁ with the variant B₁ of the observer and the variant K₁ of the dead cat; on the other hand into the subbranch A₂ with the variant B₂ of the observer and the variant K₂ of the living cat. Both branches, A₁ and A₂, differ only through the different states of the cat. Everything else is the same in the two branches, including the states and internal structures of the two observers B₁ and B₂. There is no moment in which an observer from inside feels that he becomes two. Each branch simply continues its internal state. The split is exclusively an event of the external perspective. No branch inhabitant experiences a superposition, because he is in an unambiguous branch in which his identity continues unbroken.
What is superposed from the external perspective? The states of the cat are not what is superposed; the identities of the observers are. Everything that makes B₀ an individual is structurally identical on both successor branches. B₁ and B₂ differ exclusively in their context, that is, the content of the box. Their internal structure is the same26. Only when B₁ lifts the lid and sees the dead cat, and B₂ finds the living cat, only then do the two observers differ in their internal structures.

Excursus:

Decoherence. In textbooks on quantum mechanics, decoherence means, in simplified terms, the process by which state superpositions disappear through interactions with an environment. But what is it that is first coherent and then becomes decoherent? The external perspective helps: Subsystems, for example observers, that are on different branches after a branching and whose internal structures are identical, I call these subsystems coherent. The subsystems lose their coherence as soon as their internal states differ. This happens when they receive information about their own branch, for example by opening the box with a living or dead cat.

Born rule. Born interprets the squared magnitude of the wave function |ψ|² as a measure of the probability of finding a quantum object in a certain state. This agrees very well with experimental observations.

Interpretation of the Born rule from an external perspective. At the state transition, the initial state is connected with the possible final states by branch families of different density rather than by one individual branch each, where |ψ|² is a measure of the relative densities. The branches are thus weighted among one another. An observer in a branch likewise carries the weighting of his branch. If the observer performs a quantum-mechanical experiment often and evaluates the results statistically, the vast majority of observers find, in the long run, frequencies close to the Born probabilities. A very small share of observers, by contrast, experiences an accumulation of very improbable events.

Branching. The branchings themselves are determined by the transition rules of the formal system. Exactly how these rules fix the branchings in our universe remains open. We observe only that the weighting of the different branches corresponds to the squared magnitude of the wave function |ψ|². Why do we observe precisely this weighting and no other? The answer from the internal perspective is simply: Because we find ourselves in a structure with exactly the transition rules that lead to the observed weighting, this weighting is the only one we can observe. From the external perspective, however, this weighting is only one among many equivalent weightings.

Collapse of the wave function. According to the standard interpretation of quantum mechanics, the Copenhagen interpretation, the wave function collapses during the measurement process. The development of the wave function is thus described by an interplay of deterministic Schrödinger equation and collapse, in which the linear courses of the wave function are interrupted by nonlinear jumps. Viewed from the external perspective, the total wave function continues to develop deterministically. What changes is the description from the internal perspective of an observer who, after observing the opened box, can describe only the branch in which he finds himself. An observer who wants to describe his world quantum-mechanically changes, when opening the box, from a superposition description to a description restricted to his branch. He then no longer has to consider a superposition of the states “dead” and “alive.”

The View from the External Perspective on an Astonishing Experiment

Schrödinger’s cat is only a thought experiment. In the following we look at an experiment that was actually carried out, in which a single electron is fired through a narrow slit onto a bright screen. Where the electron hits the screen, it leaves a dark dot. The next electron will most probably hit at a somewhat different place, and the next one somewhere else again. If one fires many electrons in succession onto the screen, a dark stripe forms on the screen that becomes lighter toward the edges and oscillates. This pattern can be understood well, since the probability of the electron hitting a certain place on the screen can be calculated from the laws of quantum mechanics27.
Now the experimental setup is changed: Next to the first slit there is now a second one close beside it. Again one fires electron after electron and observes how the pattern on the screen gradually builds up. It resembles the first pattern, but now narrow bright stripes appear in the dark stripe. This pattern is very reminiscent of the pattern that arises when water waves overlap and wave crests and troughs pile up or cancel out28. If one again calculates the probability of the electron hitting a certain place on the screen for the changed experimental setup, one obtains an equation that predicts exactly the observed stripe pattern29. From a mathematical point of view, everything is thus clarified. But where does this fine stripe pattern come from? Are there really waves whose crests and troughs reinforce and cancel one another? What should these waves be? When a single electron hits, one clearly sees that it is a particle, since exactly one point appears at the place of impact. Should an electron nevertheless somehow be a wave? But even if it were: With which other electron wave does it overlap? With those that were before, or with those that are still to come, or even with itself? Could an electron in wave form pass through both slits and overlap with itself?
We are in the middle of the confusion caused by the quantum-mechanical double-slit experiment. But it gets worse: Detectors are attached at the slits that recognize whether an electron passes through a slit. And when one switches these detectors on, the wave pattern disappears. Switch them off again, and the wave pattern returns.
What is going on here?
Mathematics gives a first answer: For this third experimental setup with the double slit and the detectors, the probability of the electron hitting a certain place on the screen can again be calculated from the rules of quantum mechanics. And the equation obtained gives what one observes: no stripes. For the person who says quantum mechanics simply is as the mathematics shows, no further questions arise. For many others they do. These are their usual questions and also philosophically charged answers: How can a single particle overlap with itself? During the measurement of the path by the detectors the wave function collapses. Knowledge of the electron path destroys the pattern.

How do the experiments at the double slit appear from the external perspective?
An experimenter switches off the detectors and fires an electron. The world branches into all its infinitely many branches. Each branch corresponds to a certain impact position on the screen. The branches are weighted according to the transition rules30. In each branch the experimenter fires the next electron, and each branch branches again. And the next electron, and the next branchings, ... In all paths the experimenters observe how a pattern gradually builds up on the screen. In the vast majority of paths the experimenters see the stripe pattern that follows from the equations of quantum mechanics. But in some of the infinitely many paths the experimenters marvel at completely unexpected results. All electrons land on the right in a single stripe. The rules of quantum mechanics do not forbid this pattern; it is only extremely improbable.
Now the experimenters switch on the detectors and fire electron after electron. The branches split, the patterns on the screens build up, most experimenters observe the pattern without stripes predicted from the equations of quantum mechanics, and a few experimenters again cannot believe what they see, because it is completely improbable.
In the case of switched-on detectors, from the external perspective no electron overlaps with itself. In each branch it is fixed which path the detector registers and where the electron hits the screen. The paths and their weightings are fixed by the transition rules belonging to the experimental setup. From the internal perspective these weights can be interpreted as the Born probabilities for the electron hitting a certain position on the screen.
From the external perspective no wave function collapses during the measurement by the detectors. Instead the world branches into all its branches determined by quantum mechanics. Whether a detector is switched on or off belongs to the experimental setup and fixes which concrete wave function holds, that is, into which branches the world branches and how these are weighted. And this in turn determines which patterns an experimenter sees with high probability.
From the external perspective no knowledge of the electron path destroys a pattern, neither the measurement of a detector, nor the knowledge or even the consciousness of an experimenter. The experimenter branches with every firing of an electron into all branches. And after many shots, every experimenter of every branching path sees exactly the pattern that belongs to this branching path.

What does the view of Schrödinger’s cat and the double-slit experiment teach about the transition system that comes as close as possible to our universe? For one thing, the transition rules should allow the branching of states in order to make possible the variety that we observe in our universe. The states themselves should be such that they allow superposition, reinforcement and cancellation, for that is what one observes in the double-slit experiment in the form of the stripe pattern.
The physics of our universe of course contains more than quantum-mechanical effects, causality, flow of time, space-like relations, light particles as connections between states. But if all this can in principle be found in general transition systems, then that strengthens the assumption that our universe is identical with the structure of a formal system.
Will we find the formal system whose structure our universe is? Perhaps we will approach it more and more, but we will never have certainty. And we will never be able to prove that the universe is the structure of a formal system. But we will also not be able to refute it. Yet a theory that is neither provable nor refutable is not science. And yet the distinction between internal and external perspective has created knowledge, or at least insight. It also concerns the recognition of us human beings ourselves, beyond the previously discussed physics of our universe. Our position in this world. Our past and our future.
More on this in separate essays.

© 2026 Thomas Mahr. This essay is licensed under CC BY-NC-SA 4.0. https://creativecommons.org/licenses/by-nc-sa/4.0