Not only is the Universe stranger than we think, it is stranger than we can think. [Heisenberg]
One of the flaws in current thinking about the nature of the physical environment in which we live is a product of the determination not to fall into the trap of unreasonable speculation. This has led to great advances in our understanding of the physical environment that we are able to observe directly. But yet, as the reach of our investigation extends beyond the abilities of our senses it has become abundantly clear that the fundamentals of the physical environment are very alien in respect of our ability to understand it. One of these fundamentals is space and time, which are inherent to our perception of the external environment [Kant]. It has become increasingly clear that the space and time as it is presented to us by perception is very different from the space-time of the external world in which we live.
Much of what we know about space and time derives from the idea that space and time are not absolutes as they appear perceptually, but that different observers should be treated as having their own frames of reference within space and time. Investigating the relationships necessary to translate between different frames of reference has led to considerable advances in determining many of the fundamentals of our physical environment. What makes it possible to link different frames of reference is a constant speed of light as observed by all frames of reference. This insight has led to considerable advances in the understanding of the mathematical relationships that exist within the structures of our physical environment. A frame of reference in the external environment can appear unintuitive but it is very common in the realm of computation. Postscript is a language used to code two-dimensional visual representation which is very reliant on building visual information within local frames of reference. Operating systems are also highly reliant on local frames with respect to memory, where the memory of individual processes are managed so that the local memory space appears to the local process as if it was global.
One of the most important implications of relative frames of reference is that it can be shown that space and time are not seperate entities but must exist together as a single four-dimensional manifold. While a unified space-time is implicit in the mathematical relationships that have been established there is no standard way to view (or interpret) this four-dimensional space. Indeed, the aim to understand space-time is considered by many to be counterproductive; the view being that there is little to be gained beyond the mathematical relationships. This in effect places it outside the bounds of our understanding, and more importantly limits our ability to reason about the nature of the physical world not as it appears to us but how it really is. Moreover, the focus on the mathematics hides artefacts of our perception in the guise of the observer. Implicit within the idea of local frames of reference is an observer for whom the frame of reference is local. The idea of the world as it is being dependant on an observer also extends to the empirical investigation of our physical environment at very small scales where matter appears to be both particle and wave. It is suggested that observation causes the wave function of what is possible to collapse into the predictable particles we are able to observe. But yet it is almost self-evident that the idea of an observer has no place in a theory of the physical environment. It is simply an artefact of our role as perceptual agents attempting to understand. The formalization of the observer is a placeholder for our lack of understanding or our inability to understand. The error in reasoning is supposing that this placeholder has a physical reality. The imperative to focus on the mathematics to the exclusion of a broader understanding is a short-cut in the attempt to understand something which is perhaps ouside the boundary of what we are able to understand. The observer is in effect a humunculous placed into the unknown darkness to see what cannot be seen. Humunculi are the essenence of what we imagine ourselves to be which are then used to explain the unknown. They are very common in explanations of perception where rather than understanding the underlying computational mechanics of perception it is described as a kind of internal perceiver. While it may well be that our physical environment is too complex for us to understand, we know it is not populated by the cultural decendants of mythology. We are able to understand that there are limits to our ability to understand and while it is reasonable to place boundaries and placeholders onto areas of investigation that are not understood, it is not reasonable to populate those regions with agents whose purpose is to obscure the unknown.
While a determined investigation of our physical environment appears to present a world that is complex and very alien to what we are accustomed to from our perception, there have been suggestions that this complexity might be the product of a very simple underlying rule based system. We exist inside this system and as a result we only observe the complexity that has arisen from the rules that govern the system, but we are unable to see the rules themselves. We classify and record the relationships within this complexity, but this complexity reveals little about the system itself. Other areas of study, such as the study of natural neural systems, indicate that sometimes studying the complexity that arises from simple systems is counterproductive. Cellular automata are examples of exceedingly simple systems which lead to unbounded complexity. The study of cellular automata has demonstrated that even the simplest rules can lead to extraordinary complexity [Wofram]. Cellular automata are the product of computation, and it may be shown that many of the systems which show complexity are sufficiently complex to themselves be able to support general computation. A study of common features found within systems of cellular automata shows that many of the patterns and regularities found in our physical environment can also be found in the complex patterns of cellular automata. This suggests that systems of computation may be at the heart of physics and that these systems might be a helpful guide to assist us in our attempts to understand of space and time.
Conway's Game of Life is a common well known example of cellular automata. It is a two-dimensional space of ‘cells’ governed by a set of four simple rules. Each cell has only two states; alive or dead (zero or one). The rules determine a cell's next state purely on the state of a cells immediate neighbours, with the rules being iterated repeatedly. Despite this simplicity, it is sufficiently complex to support general computation (having been shown to support structure whose function is Turing-complete). One dimensional cellular automata have also been shown to exhibit sufficient complexity to support general computation [Wolfram]. Cellular automata can also be defined in three or more dimensions. Because cellular automata are computed turn-by-turn it has a time dimension which is distinct from its spacial dimensions. The space of cells exists only in the moment of time when its state is calculated, and any residents of the space would conclude that time is a universal constant. For Conway's Game of Life time must flow forwards, it cannot be reversed. For a two-dimensional cellular automata like Conway's Game of Life, the past and the future may be presented as the pages of a book, with the curent page being the current state. The pages prior to the current page represent the past and the pages after the current page represent the future. Presented in this way, time may be seen as a further spatial dimension. If the game were to be constructed so that the entire book is represented rather than just the current page, then instead of simply computing the contents of the cells on the current page, then the entire space would be computed at each turn, including time dimension. In the case of Conway's Game of Life, the game may continue to propogate forwards into empty space in the time dimension as before but it would leave behind a past which might have its own unique structure. The Game of Life already has a model of this in the form of the puffer train, which propages forward into empty space but leaves behind a complex and active ‘debris’ field. Depending on the rules of the game, coherent structures within the game may propagate forwards, backwards or remain static in the time dimension. From the point of view of those structures, time is now not a universal external dimension but is instead just another spatial dimension. It is special only to those structures which happen to remain coherent and propogate forwards in the time dimension. Residents of the space would conclude that they exist within a space-time structure. Cellular automata, therefore, provide a convenient model to understand the four-dimensional space-time in which we appear to exist. Cellular automata demonstrate that that great complexity, possibly complexity too great for us to understand, can arise from systems that are composed of very simple rules that can easily be understood by us. The complexity that arises from simple systems is a unique feature of computation, and is illustrated both by the systems of computation which we ourselves design and make use of and the complexity exhibited by natural neural systems, which are systems of computation that have evolved in living organisms.
While it is unlikely that the rules which govern space-time are as simple as the rules which govern simple cellular automata such as Conway's Game of Life, it is possible to define simple rules which produce a space-time manifold that is more consistent with the observed properties of our space-time. If cellular automata could be found that is entirely consistent with the known properties of the space-time in which we exist then this may be considered a general theory of our physical environment. This theory is unlikely to be as simple as Conway's Game of Life because the space-time in which we exist is inconsistent with this type of simple cellular automata. We can be reasonably certain that our physics appears to be fundamentally probabilisitic whereas the rules of Game of Life are deterministic. Our space-time also appears to be flexible as opposed to the rigid manifold of cellular automata. Space-time seems to be able to expand, contract, distort and bend – and this seems to be inherent to space-time itself. Simple cellular automata such as Conway's Game of Life also has visible structure, which our space-time appears to have only indirectly. Matter seems to be the product of wave-like distortions of the manifold of space-time, which suggests greater precision than the binary on-off states of Conway's Game of Life. When these distortions propagate linearly they appear to us as electromagnetic radiation and with circular propagation it appears as matter. None of this contradicts what is possible with cellular automata, but it indicates that a simple binary cellular matrix might be insufficient to represent our physical environment.
Irrespective of whether our physical environment is a product of cellular automata, it is certain that cellular automata can help us understand our physical environment. The benefit of cellular automata is that is that we can understand it fully and as a result it is subject to logic and reason. It is not so much a theory of everything but it is rather a tool to help us to investigate and understand our physical environment. In particular, it allows us to reason about the nature of our physical environment rather than simply observing the mathematical relationships implicit within it. It can illustrate, for example, that the concept of causality is simply a mistaken product of our perception rather than a property of the physical environment. Cause and effect are reasonable assumptions in the seemingly deterministic macroscopic world in which we have evolved, but it holds little meaning in the microscopic world of probabilistic wave-like fluctuations of space-time. This can be seen even from deterministic cellular automata such as Conway's Game of Life. An observer within the Game of Life might see macroscopic structures such as beacons, pulsars and gliders, and conclude that a causal relationship exists in their interaction. However, although the Game of Life is deterministic and observations of cause and effect are valid, they are merely artefacts of the more fundamental rules which drive the game. The cause and effect interactions of the macroscopic structures can not only be arbitrarily complex but the rules of interaction are not universal. A study of the interaction will lead to an unbounded set of structures and their rules of interaction. This might be locally useful, but the unbounded complexity will obscure rather than reveal the simplicity of the underlying system. The underlying rules of Game of Life simply cannot be derived from studying the complexity that arises from the rules. After all, the purpose of the game is to demonstrate that unbounded complexity can arise from even the simplest of rules. In cellular automata such as Game of Life causality only has validity with respect to the observed structures and has no meaning with respect to the game itself.
Assuming that causality is universal can easily lead to erroneous concepts such as causal loops. In the Game of Life time is the product of iteration and as a result it is no more possible to reverse time than it is possible to exceed the speed of light – in fact, the computation that produces the state of each cell is not reversible and so the Game of Life could in principle not be run backwards. That means that even if an inhabitant of the Game of Life were to hypothetically ‘hack’ the system in an attempt to reverse time by running the game in reverse this could not be done. The computation is valid only in one direction and past states cannot be recovered from a current state. If we consider a modified Game of Life with a three dimensional simultaneous space-time, the past is known and time could be reversed (by changing a structure that is coherent forwards in time to an identical structure which propogates backwards in time) but because the time dimension is just one of the dimensions of space-time the past is not what we intuitively understand the past to be. The time dimension is simply one of the dimensions of space in which some of the coherent structures that exist propogate forwards. There may be other structures which propogate backwards (which we might call anti-structures), whereas other structures might propogate in a time-wise way in one of the other spatial dimensions. In this type of space, we could imagine that some form of portal exists which transports any forward moving structures backwards in what they perceive as the the time dimension, allowing the inhabitants of the space to travel back in time. These inhabitants might imagine, based on their everyday experience of time, that they would be rewinding time and would be able to causally interact with their own past, thereby potentially changing their present and future. But we know as external observers that cellular automata such as Game of Life cannot be rewound in the way that would seem intuitive to those who inhabit the game. As external observers we would know that the game progresses into empty space from an initial starting condition. A specific starting condition leads to the structures within the game that are stable, and the rules of their interaction would be the ‘physics’ that an inhabitant of the game would observe. A different starting condition might lead to an entirely different physics. That physics might be seen in the past, where the detritus of the present plays itself out. This might be a very hostile environment for a visitor accustomed to progressing into virgin space-time, or on the other hand the visitor might be only warmed only slightly by the background glow of strange types of matter in an otherwise empty universe. Whatever the traveller might encounter, there would be no possibility of any causal loops. Causality is simply something that does not apply in the space-time of cellular automata.
The concept of causality is a product of classical mechanics, which itself is a formalization of our everyday perceptual experience. As a result it ultimately reflects the mechanics of our perception more than it does the reality of our physical environment. Classical mechanics deals with the objects of our everyday perceptual experience. We have found that if we can observe the direction and velocity of an object then we can with seeming perfect precision predict its future. The mathematical relationships which predict the future are also valid in reverse and therefore we believe we can use the same formal system to predict the past. However, while we can test this empirically with respect to the future we cannot actually test the past. We can of course remember the past, but this is the past of our perception. It is not the past of space-time but merely the past as it appears to us. If the complexity of our environment is sufficient and it is sufficiently regular, we can also reconstruct the past from empirical information gathered from our environment and we can test our predictions with the indicators of this empirical past. But a reconstructed past is not the past of our current space-time, merely a partial record of past states that space-time has progressed through. The error in reasoning in respect of this lies in believing that the past we can calculate actually exists in our current space-time, rather than simply being a record of our progression through space-time. This distinction is made clear by contrasting the space-time of cellular automata, which is a product of computation, with the independent space and time of classical mechanics, in which all objects progress/regress in time by calculation (hence the slogan "Shut up and calculate!"). The mathematical relationships of the latter are valid in both directions of time, whereas the computation of cellular automata cannot be reversed. The inhabitants of a computational universe could certainly represent the past from available empirical evidence and they could describe the structures of their environment mathematically in a way that can help them predict the future. But this would be in the same spirit that we predict the weather. The predictions have a certain local validity, but no universality. Even if the computation that produces space-time is completely deterministic, the resulting complexity might still make it appear probabilistic to a local observer. At larger scales this apparant randomness might coalesce into mathematically predicatability, which is what would be observed by a local inhabitant of the space. The layer of apparent randomness at smaller scales would act as a firewall to local inhabitants of the space seeking to understand their environment. This environment might well be perfectly predictable, but not by a theory of physics cobbled together by simply observing the local environment. A theory of the physical environment would invariably be constructed from empirical observations of how the local objects are seen to interact, which would necessarily be at the respective scale at which the observations are carried out. The environment might, however, not be scale invariant. More subtly, what might be referred to as an object by a local inhabitant may not have any tangible existence in itself, much like a rainbow or a cloud might appear concrete when seen from a distance but disappear when approached. They are reflections of more complex phenomena and not things in themselves. It is not that the world as it is perceived by a local observer is an illusion, but rather it is epiphenomena produced by an underlying system that is not directly apparent. When space-time is viewed one slice at a time with respect of the direction in which we progress through it, it appears like the space and time of our everyday experience. But when viewed as a whole, the structures of space-time might appear very different. There might be stable structures which move backwards in time which we only interact with weakly, or there might be structures that are static in what we experience as the time dimension which subtly affect the space which we observe. What would become immediately clear is that whichever way stable structures might progress and interact that time cannot be reversed. What would become clear from being able to see space-time as a whole is that the everyday objects of our perception are merely shadows of the rules that produce space-time, and the mathematical rules about how they may be observed to interact have little meaning with respect to the computational rules which govern space-time. With respect to this computational space-time, the calculated mechanics of causality simply have no meaning. The rules which produce space-time may have been chosen to produce interesting stable structures, but however complex and rule-based the interactions of those stable structures might be they are necessarily a product of the underlying computational rules. The causal relations that may be observed locally are of course necessarily a product of those rules, but this relationship is not bi-directional. Where this is most evident is computation allows time to flow only in one direction whereas causality can be reversed.
This is because there is no explicit time dimension. Any stable structures within space-time only experience a time-like flow because they progress with clock-like regularity in one of the dimensions while having freedom to move and interact in the other dimensions. That movement or interaction might appear reversible but it is in fact a product of computation that is irreversible. time travel is fundamentally misconceived idea. Even if we were somehow able to travel backward in the time dimension we would not be travelling into our past of our memories and imagination but rather we would simply be moving in the opposite direction in the time dimension of space-time. What we might find there is simply the detritus
Causality
It may be that the fundamentals of our ability to investigate and understand the external world is limited by the tools of perception and understanding specific to our evolutionary history. This means the way we understand things may simply be an artifact of that evolutionary history and that it has no universality. Nevertheless, our confidence in those tools is increased by the success of areas of investigation such as computation and chemistry. A practially complete theory of chemistry allows us to build a complete understanding of the cellular mechanics that underpin all organic life on our planet. That same chemistry allows us to build devices which implement the mathematical ideas of general computation, devices which currently are of sufficient complexity to rival natural neural networks. Although the higher level function of natural neural networks remains a mystery, it is reasonably clear that this function lies within the realm of computation. The success of logic and reason in these areas of investigation suggests we are capable of achieving universality. While our ability to understand may be limited, universality means we can use our ability to reason to explore areas which we are inherently incapable of understanding. Even if we cannot understand what something is, we can use logical reasoning to determine what it is not.
While the success of rational thinking might indicate that it is the predominant means of trying to understand ourselves and our natural environment, it is in fact alien to our natural disposition. In fact, our evolutionary history predisposes us to a wide variety of error, much of which is well documented. One of the most insidious types of errors are short cuts in reasoning. In the same way that a sudoku puzzle can be solved slowly and step-by-step by the application of the rules of logic inherent to the design of the puzzle or it can be solved quickly by a simple constrained search, we are prone to reaching conclusions by a variety of short-cuts.
A perceptual system that uses logical induction casts its framework onto the world it perceives. This framework is necessarily hidden from within the world of perception, but nevertheless many of the principles it uses to function may be decyphered. Many subjects of study which appear to be ‘scientific’ (that is, the a study of the external environment based on experience derived from sensor input) may in fact simply be reflection of the underlying mechanics of the system by which we perceive. This is particularly true of mathematics and logic, which have developed as tools to explore and explain the empirical world. They may in fact simply reflect the fundamentals of perception. In particular, the concepts of logic and reason may simply be crude reflections (or abstracted reflections) of the central mechanism of inductive logic that our perceptual system relies on to translate sparse sensor input into a cohesive and comprehensive perceptual world. What is worrying is that when we examine the external physical world systematically using tools that extend the reach of our senses we find a world so strange and alien that it defies our ability to understand it. The ability to reason and to use logic is so fundamental to our desire to understand our environment that if they are merely artefacts
What cellular automata demonstrate is that although our universe might appear stranger that we can understand, this may be merely a byproduct of computational complexity. The universe may in fact not be more complex than we can think but be simpler than we would like to think.
Heisenberg: The Quantum and Cosmic Codes of the Universe, Sebahattin Tüzemen, Cambridge Scholars Publishing, 2019.