One of Leibniz's later works, The Monadology, contains an almost aphoristic condensation of Leibniz's lifetime of thought. Leibniz organizes his ideas around an idea he terms the monad (it is sometimes capitalized, i.e. Monad). In this post, we will be plundering Leibniz's ideas at will. The cosmology I am proposing here is a framework for physical reasoning - it is not a scientific method but it is intended to form a foundation on which a robust scientific method can be built.
In modern higher mathematics, the field of group theory has explored abstractions of algebra. Some of these abstract algebras are referred to as groups and the field takes its name from these groups. Group theory derives different kinds of algebras by individually relaxing the constraints on ordinary algebra, such as that an operation should be commutative or associative, and so on. In this way, group theory treats the properties of algebra almost like the properties of physical substances and categorizes the different kinds of mathematical structures that arise from combinations of these properties accordingly.
The property of closure, for example, holds that a set is closed under a given operation if applying that operation to elements in the set always yields another element of that same set. For example, the positive whole numbers are said to be closed under addition and multiplication because any two positive whole numbers can be added or multiplied, yielding another positive whole number. But the positive whole numbers are not closed under subtraction or division because these operations can yield negative or rational numbers. Closure is a general property of an algebra, however, and is therefore not restricted to the ordinary operations of addition, multiplication, and so on.
A common example of an abstract closure is the Rubik's cube. Twisting a face can be considered an operation on the cube. Each twist takes the cube from one state in its state-space to another state in its state-space. Thus, the set of all Rubik's cube states and the operation of twisting the faces form an algebraic closure. This property is important because if we are dealing with an algebraic closure, we do not have to worry about special cases, such as division-by-zero, for example. We can say that algebraic closures are very well-behaved mathematical objects, making them easy to reason about.
When we reason about the world, we alternate between treating the world as a scattered collection of unrelated particulars, on the one hand, and treating the world as a unitary, indivisible whole, on the other hand. In philosophy, the tension between these two ways of thinking about the world is termed the problem of the one and the many. Let us treat mathematical closures as a thinking tool and apply this tool to the world, per se. For example, when I mix two substances in a chemistry laboratory, whatever the result, it is again another substance. We can think of the world by analogy to a mathematical closure; we will just call it a closure, for short. The world-as-closures gives us a way to hold the tension between the one and the many without giving up logical consistency. The world is a duality. Viewed in one way, the world is one substance that is related to itself by many actions. Viewed in another, equally valid, way, the world is many substances that are related to each other by one action. But no matter which we look at the world, it is a closure.
In mathematics, the motivation for studying closures is that they are well-behaved - the motivation is, ultimately, aesthetic. In physical reasoning, however, this is not our primary motivation - we are constrained in our choice of aesthetic by the facts of the physical world itself. But as we already pointed out in the case of a chemistry experiment, the physical world does indeed behave like a closure, in the broadest sense.
Let's consider again the idea of digital physics, which we introduced in Part 14. One of the most common digital models used in digital physics is called the cellular automaton (CA). Easily the most widely studied CA is John Conway's Game of Life. It has been proved that the Game of Life is Turing universal, meaning, it is possible to implement a universal Turing machine, U, in a properly initialized Game of Life.
The video linked below shows an animation of a massive field of Game of Life that implements, on top of itself, another Game of Life. That is, the field is initialized to simulate the rules of the Game of Life, within the Game of Life itself.
This is a kind of fractal closure property. It is similar to software virtualization, where we say that we have virtualized the hardware environment by completely simulating it in software. This fractal closure property should not be confused with an attempt to solve what can be called "the substrate problem" in digital physics - if the world is made out of software, where is the hardware that it is running on?
The following video is fairly technical but provides an easy-to-follow introduction to the general activity that is occurring inside of a digital computer when it is computing, specifically, when it is adding.
The thesis of digital physics is this: what is happening, physically, when we do digital logic in a computer is the same as what is happening in the above simulation of Life-in-Life - we are merely observing the underlying rules of the physical world at a larger scale than their native scale. One of the motivating factors for reasoning about the physical world this way is that it clearly forms a closure - the world is made of information and evolves according to information transforms. We can think of the world as many individual pieces of information and a single transform. Or, equivalently, we can think of the world as a single piece of information (pattern, message) that is operated on by many transforms, giving rise to the many phenomena we observe.
I am not aware of a name for this principle in digital physics but I propose that it be named the projection principle. The projection principle arises from a principle in computation that can be called substrate-independence - I can compute with water pipes and valves, gears and levers, vacuum tubes and wires, or silicon transistors on an integrated circuit. The substrate is irrelevant. The essence of computation consists in the pattern that is moving across the susbstrate.
The problem immediately arises (as we have already seen in Part 10) that if the world is a computer, it is a quantum computer, not a classical digital computer. As Seth Lloyd explains it, "The universe is observationally indistinguishable from a giant quantum computer."[1] The quantum monad, then, is the result of combining the projection principle with Lloyd's quantum computation (QC) thesis - the Universe is indistinguishable from a giant quantum computer at every scale. Of course, the question immediately arises, "If we are in a giant quantum computer, how come we do not observe quantum effects at the macroscopic scale?" We will be addressing this question in upcoming posts.
The quantum monad is a stronger thesis than the QC thesis. Leibniz asserts, regarding the monad, that, "all simple substances or created monads ... [are], so to speak, incorporeal automatons." [2, §18] He contrasts this with corporeal bodies - "every organic body of a living being is a kind of divine machine or natural automaton" [2, §64] Between the corporeal and incorporeal, there is a duality that correlates well with the duality of quantum physics - that is, the duality between particles and waves. The key, here, is that Leibniz identifies monads as automata. An automaton - whether natural or artificial - is subject to complete description by a set of laws. Thus, the monad is an incorporeal entity that strictly obeys a finite set of laws.
This brings us full circle back to the category of logic as it is the expression of law itself. We began with the digital physics thesis, which derives from the projection principle. We then derived from the projection principle the idea of the quantum monad. The quantum monad, in turn, can be seen as nothing more than the strictest application of logic to phenomena that are directly observed as well as to phenomena that are only indirectly observed (inferential). Thus, the quantum monad is exactly equivalent to the evolution of quantum causality.
From the quantum monad, we may infer that you and I, and everything around us are all components of a massive, analog computation. This computation follows the projection principle and forms a physical closure - every combination of substance gives rise to substance of such a form that it again admits to re-combination by the set of combinations that were available originally. To refer to this computation as "a simulation" may be jarring, at first, but it actually fails to capture the true immensity of the implications. Not only might the Universe be stranger than we can suppose, we may be able to work out the strangeness of the Universe to a far greater degree than any of our ancestors had ever dared to suppose, simply by working out the logical implications of the universal prior in a quantum computation with iron rigor.
The PMM thought-experiment shows that, in a simulated world, there is no such thing as "weird" or "spooky". In fact, quantum mechanics is relatively boring in a simulated world. A simulated physics would contain para-consistent spacetimes. For example, when you walk through a door in one direction, it connects rooms A and B, but when you walk through it in the other direction, it connects rooms B and C. We can always make such a spacetime consistent by adding dimensions but that is beside the point - arbitrarily high dimensionality is the rule in computation, not the exception.
As we mentioned in an earlier post, physical properties can be understood as degenerate forms of the ideal set of all possible properties. We can imagine simulation-builders imposing physics-like properties because they are useful. Locality, solidity (mutual-exclusion), flatness, linearity, massiveness, continuity, and so on, are features that are useful for imposing inescapable resource-bounds. In short, any environment that is constructed in such a way as to impose scarcity upon operators in that environment will have to impose properties very much like those that are familiar to us from careful observation of physical materials. In turn, imposing scarcity upon operators in a virtual environment is crucial for any sort of game of incomplete information because privacy can only be guaranteed up to resource bounds in any environment where all events are public record.[3]
Arbitrary choice of a set of properties to impose upon operators in a virtual environment is almost certain to leave what hackers refer to as "attack surface" - logical holes in the security design that allow cheaters to take undue advantage of resources in the virtual environment. In MMOG's, this shows up as cheaters giving their characters unlimited resources such as health, weapons, ammunition, and so on, that is, granting themselves privileges that no fair participants have access to. Thus, truly robust virtual environments that support games of partial information must choose rule sets that have provable properties, which brings us right back to group theory because it utilizes abstractions that are easy to reason about, such as the closure property.
The Perl programming language has a package called Quantum::Superposition, originally authored by Damian Conway. This package enables a set of abstractions that are the discrete equivalent of the behavior of quantum superpositions in the continuous domain. The Prolog language implements a form of non-determinism (back-tracking search) that can be thought of as the discrete-choice equivalent of the quantum path integral in continuous path-space. These equivalences between digital programming paradigms and quantum systems are just a few examples of an important principle - if the "real" laws of the Universe are computational in nature, then geometric or spacetime-based rules are the wrong way to analyze the world.
While spacetime laws are truly fundamental, the reason they are fundamental is because they are mathematically natural, not the other way around. This explains the apparent paradox of Minkowski spacetime. Minkowski spacetime seems weird to us but it's mathematically natural. Thus, it is an efficient geometry relative to rectilinear, Euclidean space. Euclidean space, in turn, is suitable for human scale and, thus, it is how our brain constructs or interprets local material reality. When we examine questions of indirect phenomena, such as, "Where is the particle?" the true answer is that there is no "where" at all. There is only the particle's state. In the PMM, for example, we may examine the geometry of the VR Environment (the rendered video and audio), versus the physical geometry of the video and audio state within the computers that make up the Shared World State. While these are not completely un-correlated, whatever correlation exists is highly non-linear and a function of arbitrary resource limitations that fluctuate chaotically based on local conditions within the simulating computers themselves. In short, it doesn't matter where the state describing a particular polygon[4] within the simulated environment is "really" located within the geometry of the simulating computer's memory banks - these two geometries might as well be random with respect to one another.
Because we are visually-oriented beings and because the sense of sight is arguably the most exact of the five senses, we tend to attribute great importance to space and time. Space and time are, arguably, the two greatest organizing principles of modern scientific thought. But in a quantum monadic Universe, space and time are not the highest principle of organization. In fact, space and time sit low on the totem-pole. In the following image, I have arranged the organizing principles of the quantum monad - we will only be exploring part of the hierarchy in this post:
Next: Part 17, Cracks in the Standard Cosmology
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1. The Universe as Quantum Computer, Seth Lloyd (December 17, 2013)
2. The Monadology, Gottfried Leibniz
3. This is the entire basis of public-key cryptography.
4. In computer graphics, the basic unit of 3D rendering is the polygon - typically a 3-gon (triangle) sharing its edges with other polygons in such a way that they form a closed volume. The polygons are oriented in 3D space according to their relation with each other and their overall relation to the player's viewpoint, sometimes referred to as camera-coordinates.
5. Utilizing algorithmic information theory, we can objectively define what it means for one system to comprehend another.
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