Chapter 1. What is information?
In 1948, Claude Shannon wrote in his famous article “A Mathematical Theory of Communication”:
“Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages.”
Shannon simplified the notion of information by stripping away meaning and semantics, to focus just on the message part, the raw data that is transmitted through a communication channel. Shannon showed that any message can be encoded as a string of bits, where each bit represents the answer to a yes-no question. In this way, information is the resolution of uncertainty, the level of surprise, the power to inform, and a measure of entropy. Shannon’s article caught on, communication theory became information theory, and the bit became the unit of information.
Information = bits
Shannon ignored the semantic aspects because they were irrelevant to the engineering problem of communication and storage. However, semantics are of course very much relevant to the concept of information. A string of bits is inert and meaningless without knowing the yes-no questions. In this article we will work with a broader definition and measure of information, one that does include the semantic aspects. We do this by including a description of the receiving machine. It is the receiving machine that asks the yes-no questions, so it is both the bits and the receiving machine together that define what the information is all about. We take language and machine for granted in everyday communication, understandably so, but if we were to communicate with aliens, then along with our first message we would somehow need to include a description of the machine that can read the message. This is of course what Carl Sagan and others had to think about when designing the Pioneer Plaques, and more so with the Voyager Golden Records. The point is, information always involves a machine. Without a machine there is no information.
Suppose, then, we include the machine in our definition of information.
Information* = bits + machine
What is the basic unit of a machine that processes bits? This can be a universal logic gate, like a NAND gate or a NOR gate, let’s just call it gate for short. The receiving machine is therefore a circuit of gates, a digital computer. Information and computation have the same components, so they must share the same phase space. Whatever can be expressed with information can also be expressed with computation, and vice versa. Information processing is therefore equivalent to computation. The point is, any piece of information can be represented as a string of bits and a circuit of gates. If we assume the circuit wiring to be properties of the gates, and a source of clock pulses as a given, then our basic unit of information is a tuple of bits and gates.
Information* = ( bits, gates )
John Wheeler famously said “it from bit”, and here we extend his idea by saying “it from bits and gates”. A bit can represent the outcome of a choice, but you still need something that does the actual choosing, and here we represent that choosing with a gate.
What does this computational model of information say about the information in our universe? Can we model our whole universe as a giant piece of information? Is the total amount of information conserved? What types of information are there?
It is of course unlikely that our quantum universe is actually a digital simulation, as quantum mechanics is beyond the practical reach of digital computation. Nevertheless, for the sake of argument, we can pretend our universe is a digital simulation. Bits and gates are easier to reason about than qubits and quantum logic gates, and the outcome of the reasoning should be the same. The point here is that simulating particle interactions require calculations, and here we represent those calculations as digital computations. Given enough bits, gates and clock pulses, we can simulate or approximate any physical phenomenon with arbitrary precision. I think most will agree that it is in theory possible to approximate our universe with digital simulation, albeit using astronomical numbers of bits and gates, such that the simulation produces the same phenomena as in our universe, like stars, planets, clouds and possibly even life.
Our universe can in theory be approximated by high-resolution digital simulation such that the simulation produces the same phenomena as we are familiar with in our universe.
In the next chapter we will carry out a thought experiment where we model the whole universe as a circuit of bits and gates. The thought experiment suggests that in any universe or simulation there exists two types of information: explicit information and implicit information.
Chapter 2. Explicit information
Our universe contains a massive but finite number of elementary particles, or waves. According to some estimates, the number of elementary particles in the observable universe is in the order of 10⁹⁷, including photons and neutrinos, but not dark matter or dark energy. Each particle is one of a fixed number of fundamental types, and we know of 31 of them, including the hypothesized graviton. Each particle has a finite set of states that describe properties such as mass, charge and spin. Each particle follows a common and fixed set of rules, the fundamental laws of physics.
A complete and non-redundant description of all these particles and rules is all that is required to account for everything that happens in the universe. Let’s call this description the explicit information of our universe. The reason for the word explicit here is because, at the bottom of it all, the information of particles must exist as something, something that explicitly exists. We do not know what it takes for something to exist, so instead we will just assign it a definition and give it a name, setting the limits of what we can know.
explicit information = a non-redundant description of the states of all elementary particles in our universe, plus a non-redundant description of the machine that repeatedly or continuously applies the laws of physics to these states
Let’s do a thought experiment. Imagine a massive digital circuit, ready to run a simulation of an entire universe. Suppose the particles in the simulated universe follow the same mathematical rules as in our universe, with a precision right down to Planck length and time, and initial conditions similar to that of our own universe. In our thought experiment this massive circuit represents the explicit information of the simulated universe. Let’s call this circuit the explicit circuit.
A brief technical description of the explicit circuit: The states of each particle are stored as bits in registers, representing properties such as mass, charge, spin, relative position and velocity. Each state register has an LED attached to its output, a light-emitting diode that is either on or off, 1 or 0, so that we can conveniently see its output value. The LEDs are arranged to form a massive matrix called the explicit matrix, representing the total state of the simulated universe. All state registers are connected to a common data bus, which in turn connects to a central circuit containing the logic of the fundamental laws of physics. When the circuit is running, the central circuit cyclically iterates all particles’ state registers to compute and update their next values, doing whatever buffering and sub-cycles are necessary. The whole explicit circuit is driven by a single clock, the explicit clock. The speed of the clock does not matter, as this does not affect the outcome of the calculations.
And now for the big moment in our thought experiment. We power up the explicit circuit and see the massive matrix of LEDs blinking away. The whole matrix of LEDs stretches farther than the eye can see. We walk along the matrix and look at some of the blinking LEDs for a while. Disappointingly, we see nothing interesting, it all seems quite random. Looking only at the bits of explicit information we see no interesting macroscopic phenomena like galaxies or planets. Of course, we cannot see these things because the information of macroscopic phenomena is not necessary in the non-redundant description of the simulation. On the other hand, we know there must be lots of interesting things happening in the simulation. After all, the simulated particles behave in the same way as the particles in our universe, so we can expect the eventual formation of atoms, molecules, stars, rocks, planets, rivers, clouds, and maybe even life. However, if we want to see what is happening in the simulation, then we would need to attach additional circuits of gates and LEDs. Similar to a graphics card and a screen, these additional circuits would read the bits of the explicit matrix, execute some rendering algorithm, and then display the result on its own matrix of LEDs. Given enough supply of additional gates and LEDs, we can potentially bring anything in the simulation into view.
We conclude that it costs computation to bring a simulation’s macroscopic phenomena into view. From the perspective of explicit information, the information of macroscopic phenomena simply does not exist, or rather, it does not need to exist. The point here is that the same logic should apply to our own universe. Given just the explicit information of our universe, it would require additional computation to bring its macroscopic phenomena into view. Assuming there is no additional computation going on, we conclude that the macroscopic phenomena in our universe do not explicitly exist. However, from within our universe, we as living creatures see views of macroscopic phenomena all the time, such as clouds, trees and chairs. The question is, what is doing the computation that makes these views possible? Obviously there is no extra computation going on. Instead, the transformations and computations that produce these views are done by our eyes and neurons, which ultimately consist of just elementary particles that follow the rules as described in explicit information. Explicit information accounts for everything that happens in our universe, by definition, so all other information must be “implicit information”, information that mathematically derives from explicit information. Therefore, the information of macroscopic phenomena, and all the information we see and experience, is implicit information, redundant information that exists only in a mathematical sense. If implicit information exists only in a mathematical sense, how come this information is so real to us?
Chapter 3. Implicit information
The non-redundant explicit information of our universe ultimately accounts for everything that happens inside it, but we as creatures living within the universe cannot see this explicit information directly. The information we do see and experience is what we can call implicit information. As the name suggests, implicit information is the information that does not need to be explicitly stated because implicit information is mathematically true, a derivation or transformation of the already existing explicit information. At any moment in time, implicit information is the snapshot set of all redundant information that holds true in our universe, i.e. everything that can be said about what is happening in our universe, from whatever location, scale or perspective. Obviously this is a huge set, much bigger than the explicit information from which it derives. What follows is a set of definitions that attempt to capture this massive mathematical structure.
implicit information = the set of all “instances of implicit information”
instance of implicit information = a “transformation” (or sequence of transformations) of explicit information
transformation = a machine, consisting of bits, gates and a clock with infinite frequency, which accepts explicit information (or an instance of implicit information) as input, and instantly produces an instance of implicit information as output
With infinitely many transformations and recursions, implicit information is an infinite tree-like structure, with explicit information at the root. When the bits of explicit information change, so do the bits of implicit information, instantly.
In the analogy of the thought experiment of the previous chapter, an instance of implicit information can be modeled as an implicit circuit. An implicit circuit is an external circuit that reads the bits of the explicit matrix, performs some transformation of these bits, and outputs its result on its own implicit matrix of LEDs. The implicit information of the simulated universe is the infinite tree of all possible implicit circuits and their implicit matrices. Among these implicit matrices there will be views of galaxies, stars, planets and clouds.
Furthermore, among the implicit matrices there will be descriptions of all the rules and laws that mathematically derive from explicit information. Let’s call these derived laws the implicit laws. The implicit laws are just as mathematically true as the laws of physics in explicit information, but the implicit laws are redundant because they are not necessary in the non-redundant description of our universe. The second law of thermodynamics is an example of an implicit law, implicitly true because our universe is a many-particle system that is continuously being updated by a common set of rules. In all such particle systems, whether a simulation or a universe, whether discrete or continuous, whether classical or quantum, there is the second law of thermodynamics in action, the statistical law that describes how concentrations of moving particles tend to spread out, and how all macroscopic phenomena gradually fade away until equilibrium. The explicit clock can tick forever, but implicit time will eventually stop. The second law of thermodynamics reveals itself in the simplest of simulations of bouncing particles, whereas this law is not explicitly stated in its code.
With all this in mind, let’s put it all together.
Whereas the explicit information of our universe somehow exists independently as its own thing, implicit information exists only in a mathematical sense. From the perspective of explicit information, only the non-redundant information of its elementary particles actually exist, and nothing else. However, from the perspective of implicit information, all instances of implicit information are just as mathematically true and real as the explicit information it derives from. For every macroscopic phenomenon we observe in our universe, whether a planet, a cloud or a chair, there mathematically exists an instance of implicit information that exactly represents that phenomenon. The space of implicit information is consistent with everything we observe, so why not say that our reality is that space? I hypothesize that our reality and existence is exactly this rich mathematical set of all instances of implicit information, the space where all mathematical transformations of explicit information actually exists.
Implicit Information Hypothesis
From a mathematical perspective, implicit information is as true and real as the explicit information it derives from. Our reality is that perspective, the mathematical space where all transformations of explicit information actually exists.
The space of implicit information, with all its mathematical transformations, explains the unreasonable effectiveness of mathematics in the natural sciences. It’s implicit information all the way down, all the way down to explicit information. Although our universe is nondeterministic at the particle level, for each observation or measurement there is a definitive outcome that is real, and all its implicit information at those moments are instantly real too. It is only in the mathematical space where implicit information is instantly calculated.
Chapter 4. Implicit machines
Ultimately, there is only one machine that actually exists, the explicit machine that calculates the progression of the universe’s particles’ states. Consequently, there is just one source of computation, from which we all tap. There is no extra computation going on in the universe when we turn on a laptop, or when we think harder. The amount of compute we can potentially extract and use in our implicit world can never exceed the amount of computations being done to compute the explicit information of our universe, and is further limited by Landauer’s principle.
At any moment in time, in the vast space of a universe’s implicit information, there may exist many instances of implicit information that describe a machine, an implicit machine. Many of these implicit machines will be short-lived, mere snapshots of coincidence. However, given the right conditions, driven by free energy and entropic forces (implicit energy and implicit forces), implicit machines can emerge as solutions for increasing entropy at a faster rate, which in turn encourages the formation of the solution, and so on. We can understand life as implicit machines that emerge in universes and simulations with laws of physics expressive enough for there to exist a design for a replicating survival-machine, and, in accordance with anthropic reasoning, we indeed live in such a universe. In our universe a solution for a sustained machine factory has been found in the form of a self-replicating cell. From a computational perspective, we can understand evolution as the apparent breadth-first search in entropic environments for machine-designs that survive, with the tree of life being the ongoing result of that search.
An implicit machine is a machine constructed from instances of implicit information. An implicit machine is itself an instance of implicit information. Living organisms are implicit machines that metabolize, replicate and survive in an entropic environment.
For each living organism there exists an instance of implicit information that contains the description of a machine that exactly matches what the living organism is and does. If the a particular instance of implicit information exactly describes a duck, then this instance of implicit information will exactly behave as a duck. Similarly, the full description of a living brain also exists as an implicit machine. A brain is an implicit machine, consisting of neurons that are themselves implicit machines. For any conscious thought a brain has, there exists a set of elementary particles in the brain that accounts for the thought, but it is only in its macroscopic implicit information where we can find the information that exactly is that thought. The information of a conscious thought is not explicitly present in the explicit information of our universe, just like the second law of thermodynamics is explicitly present in explicit information. Given just the explicit information of our universe, it would require transformations and computations to bring into view what a brain is thinking. Therefore, it is only in the mathematical space that the information of conscious thought exists, the same space where the information of clouds and chairs exist.
If we say we are conscious of the information in our brain, then this must mean that we are that information. In other words, we are instances of implicit information within our implicit machine. We can understand ourselves to be quite literally the mathematical definition of the machine in our brain. We are mathematics. Being the mathematical definition of the machine itself accounts for the subjective sense and belief that we have control because indeed that is what a machine does, by definition.
Consciousness is being an implicit machine that reads and interprets instances of implicit information. Consciousness is implicit information, just as implicit as the second law of thermodynamics, and just as real as clouds and chairs.
If consciousness is no more than an instance of implicit information, and if there is no bias towards any particular instance of implicit information being more real than any other, then surely this implies that all instances of implicit information are just as real as our consciousness. In the world around us there are stories happening everywhere, not just in brains. Consider for example the moving parts of a mechanical watch, or the swarm behavior in the murmuration of starlings, or the fluctuations in stock market prices, or even the digestive system in our own body. These are all macroscopic stories told by implicit machines, just like the stories in our brains, albeit very different in behavior and interestingness. Taking the argument to the extreme, the whole process of evolution, with its breadth-first search for survival machines, can hypothetically be described as an algorithm, a machine, and therefore that story must implicitly exist too, albeit operating at much longer time scales than our perception. We have of course no way of knowing what such machines experience because we are not those machines. We can only guess or imagine what qualia a burning flame experiences, a macroscopic phenomenon that appears to behave as if it wants to survive, wandering around in search for fuel. Of course there are no intelligent thoughts going on in a flame, that information is simply not there, not even in its implicit information, but the movement of a flame does convey some kind of behavior, and that behavior exists as instances of implicit information, so in this way there is some level of qualia going on, albeit minute when compared to what we experience. We can understand qualia as the first-person experience of being the implicit machine that reads and interprets implicit information. The sensation of qualia, like the color red or the flavor of sweetness, cannot be communicated in bits alone, it can only be experienced first-hand by an implicit machine.
Consciousness is an instance of implicit information, a macroscopic story told by many neurons and molecules, telling rich mathematical stories of behavior and qualia. We are quite literally mathematical machines that experience mathematical stories. Whether a living machine, a mechanical machine or an electronic computer, every machine in our world is an implicit machine, a machine constructed from instances of implicit information. Each machine’s story is as real as any other, although most stories are obviously not very interesting. We would not say that a desktop calculator is conscious, the information in a calculator is not as rich as the information in our brains, and there is no expression of fear or pain if we were to dismantle it. Calculators are not equiped with the survival and repair mechanisms for dealing with increasing entropy, so its information is just not very interesting, and in some ways just as boring as the bits and gates of explicit information. However, if we were to simulate a solar system on a giant supercomputer and life happens to emerge in that simulation in a similar way as it did in ours, then that life would be just as real and conscious as us. This is because a physical supercomputer is an implicit machine, constructed from instances of implicit information, which can all be traced down to the particles in explicit information, the same particles as we are made of.
The implicit information hypothesis says we are completely implicit, free choice is an illusion, while still holds true that we are the ones doing the actual choosing. Carl Sagan said “We are a way for the universe to know itself”. I expand on this by saying “The universe is a way for mathematics to know itself”. Without the existence of explicit information there is nothing for mathematics to act upon or reveal itself, and there would be no implicit information. Although we ultimately have explicit information to thank for all of this, our reality and behavior is in its implicit information, the mathematical space where all transformations of explicit information exists. This hypothesis and explanation, I believe, is a solution to David Chalmers’ hard problem of consciousness.