Tag Archives: Physics

The unreasonable effectiveness of mathematics

In the post “What is the nature of mathematics“, the dilemma of whether mathematics is discovered or invented by humans has been exposed, but so far no convincing evidence has been provided in either direction.

A more profound way of approaching the issue is as posed by Eugene P. Wigner [1], asking about the unreasonable effectiveness of mathematics in the natural sciences. 

According to Roger Penrose this poses three mysteries [2] [3], identifying three distinct “worlds”: the world of our conscious perception, the physical world and the Platonic world of mathematical forms. Thus:

  • The world of physical reality seems to obey laws that actually reside in the world of mathematical forms.  
  • The perceiving minds themselves – the realm of our conscious perception – have managed to emerge from the physical world.
  • Those same minds have been able to access the mathematical world by discovering, or creating, and articulating a capital of mathematical forms and concepts.

The effectiveness of mathematics has two different aspects. An active one in which physicists develop mathematical models that allow them to accurately describe the behavior of physical phenomena, but also to make predictions about them, which is a striking fact.

Even more extraordinary, however, is the passive aspect of mathematics, such that the concepts that mathematicians explore in an abstract way end up being the solutions to problems firmly rooted in physical reality.

But this view of mathematics has detractors especially outside the field of physics, in areas where mathematics does not seem to have this behavior. Thus, the neurobiologist Jean-Pierre Changeux notes [4], “Asserting the physical reality of mathematical objects on the same level as the natural phenomena studied in biology raises, in my opinion, a considerable epistemological problem. How can an internal physical state of our brain represent another physical state external to it?”

Obviously, it seems that analyzing the problem using case studies from different areas of knowledge does not allow us to establish formal arguments to reach a conclusion about the nature of mathematics. For this reason, an abstract method must be sought to overcome these difficulties. In this sense, Information Theory (IT) [5], Algorithmic Information Theory (AIT) [6] and Theory of Computation (TC) [7] can be tools of analysis that help to solve the problem.

What do we understand by mathematics?

The question may seem obvious, but mathematics is structured in multiple areas: algebra, logic, calculus, etc., and the truth is that when we refer to the success of mathematics in the field of physics, it underlies the idea of physical theories supported by mathematical models: quantum physics, electromagnetism, general relativity, etc.

However, when these mathematical models are applied in other areas they do not seem to have the same effectiveness, for example in biology, sociology or finance, which seems to contradict the experience in the field of physics.

For this reason, a fundamental question is to analyze how these models work and what are the causes that hinder their application outside the field of physics. To do this, let us imagine any of the successful models of physics, such as the theory of gravitation, electromagnetism, quantum physics or general relativity. These models are based on a set of equations defined in mathematical language, which determine the laws that control the described phenomenon, which admit analytical solutions that describe the dynamics of the system. Thus, for example, a body subjected to a central attractive force describes a trajectory defined by a conic.

This functionality is a powerful analysis tool, since it allows to analyze systems under hypothetical conditions and to reach conclusions that can be later verified experimentally. But beware! This success scenario masks a reality that often goes unnoticed, since generally the scenarios in which the model admits an analytical solution are very limited. Thus, the gravitational model does not admit an analytical solution when the number of bodies is n>=3 [8], except in very specific cases such as the so-called Lagrange points. Moreover, the system has a very sensitive behavior to the initial conditions, so that small variations in these conditions can produce large deviations in the long term.

This is a fundamental characteristic of nonlinear systems and, although the system is governed by deterministic laws, its behavior is chaotic. Without going into details that are beyond the scope of this analysis, this is the general behavior of the cosmos and everything that happens in it.

One case that can be considered extraordinary is the quantum model which, according to the Schrödinger equation or the Heisenberg matrix model, is a linear and reversible model. However, the information that emerges from quantum reality is stochastic in nature.  

In short, the models that describe physical reality only have an analytical solution in very particular cases. For complex scenarios, particular solutions to the problem can be obtained by numerical series, but the general solution of any mathematical proposition is obtained by the Turing Machine (TM) [9].

This model can be represented in an abstract form by the concatenation of three mathematical objectsxyz〉(bit sequences) which, when executed in a Turing machine TM(〈xyz〉), determine the solution. Thus, for example, in the case of electromagnetism, the object z will correspond to the description of the boundary conditions of the system, y to the definition of Maxwell’s equations and x to the formal definition of the mathematical calculus. TM is the Turing machine defined by a finite set of states. Therefore, the problem is reduced to the treatment of a set of bits〈xyz〉 according to axiomatic rules defined in TM, and that in the optimal case can be reduced to a machine with three states (plus the HALT state) and two symbols (bit).

Nature as a Turing machine

And here we return to the starting point. How is it possible that reality can be represented by a set of bits and a small number of axiomatic rules?

Prior to the development of IT, the concept of information had no formal meaning, as evidenced by its classic dictionary definition. In fact, until communication technologies began to develop, words such as “send” referred exclusively to material objects.

However, everything that happens in the universe is interaction and transfer, and in the case of humans the most elaborate medium for this interaction is natural language, which we consider to be the most important milestone on which cultural development is based. It is perhaps for this reason that in the debate about whether mathematics is invented or discovered, natural language is used as an argument.

But TC shows that natural language is not formal, not being defined on axiomatic grounds, so that arguments based on it may be of questionable validity. And it is here that IT and TC provide a broad view on the problem posed.

In a physical system each of the component particles has physical properties and a state, in such a way that when it interacts with the environment it modifies its state according to its properties, its state and the external physical interaction. This interaction process is reciprocal and as a consequence of the whole set of interactions the system develops a temporal dynamics.

Thus, for example, the dynamics of a particle is determined by the curvature of space-time which indicates to the particle how it should move and this in turn interacts with space-time, modifying its curvature.

In short, a system has a description that is distributed in each of the parts that make up the system. Thus, the system could be described in several different ways:

  • As a set of TMs interacting with each other. 
  • As a TM describing the total system.
  • As a TM partially describing the global behavior, showing emergent properties of the system.

The fundamental conclusion is that the system is a Turing machine. Therefore, the question is not whether the mathematics is discovered or invented or to ask ourselves how it is possible for mathematics to be so effective in describing the system. The question is how it is possible for an intelligent entity – natural or artificial – to reach this conclusion and even to be able to deduce the axiomatic laws that control the system.

The justification must be based on the fact that it is nature that imposes the functionality and not the intelligent entities that are part of nature. Nature is capable of developing any computable functionality, so that among other functionalities, learning and recognition of behavioral patterns is a basic functionality of nature. In this way, nature develops a complex dynamic from which physical behavior, biology, living beings, and intelligent entities emerge.

As a consequence, nature has created structures that are able to identify its own patterns of behavior, such as physical laws, and ultimately identify nature as a Universal Turing Machine (UTM). This is what makes physical interaction consistent at all levels. Thus, in the above case of the ability of living beings to establish a spatio-temporal map, this allows them to interact with the environment; otherwise their existence would not be possible. Obviously this map corresponds to a Euclidean space, but if the living being in question were able to move at speeds close to light, the map learned would correspond to the one described by relativity.

A view beyond physics

While TC, IT and AIT are the theoretical support that allows sustaining this view of nature, advances in computer technology and AI are a source of inspiration, showing how reality can be described as a structured sequence of bits. This in turn enables functions such as pattern extraction and recognition, complexity determination and machine learning.

Despite this, fundamental questions remain to be answered, in particular what happens in those cases where mathematics does not seem to have the same success as in the case of physics, such as biology, economics or sociology. 

Many of the arguments used against the previous view are based on the fact that the description of reality in mathematical terms, or rather, in terms of computational concepts does not seem to fit, or at least not precisely, in areas of knowledge beyond physics. However, it is necessary to recognize that very significant advances have been made in areas such as biology and economics.

Thus, knowledge of biology shows that the chemistry of life is structured in several overlapping languages:

  • The language of nucleic acids, consisting of an alphabet of 4 symbols that encodes the structure of DNA and RNA.
  • The amino acid language, consisting of an alphabet of 64 symbols that encodes proteins. The transcription process for protein synthesis is carried out by means of a concordance between both languages.
  • The language of the intergenic regions of the genome. Their functionality is still to be clarified, but everything seems to indicate that they are responsible for the control of protein production in different parts of the body, through the activation of molecular switches. 

On the other hand, protein structure prediction by deep learning techniques is a solid evidence that associates biology to TC [10]. To emphasize also that biology as an information process must verify the laws of logic, in particular the recursion theorem [11], so DNA replication must be performed at least in two phases by independent processes.

In the case of economics there have been relevant advances since the 80’s of the twentieth century, with the development of computational finance [12]. But as a paradigmatic example we will focus on the financial markets, which should serve to test in an environment far from physics the hypothesis that nature has the behavior of a Turing machine. 

Basically, financial markets are a space, which can be physical or virtual, through which financial assets are exchanged between economic agents and in which the prices of such assets are defined.

A financial market is governed by the law of supply and demand. In other words, when an economic agent wants something at a certain price, he can only buy it at that price if there is another agent willing to sell him that something at that price.

Traditionally, economic agents were individuals but, with the development of complex computer applications, these applications now also act as economic agents, both supervised and unsupervised, giving rise to different types of investment strategies.

This system can be modeled by a Turing machine that emulates all the economic agents involved, or as a set of Turing machines interacting with each other, each of which emulates an economic agent.

The definition of this model requires implementing the axiomatic rules of the market, as well as the functionality of each of the economic agents, which allow them to determine the purchase or sale prices at which they are willing to negotiate. This is where the problem lies, since this depends on very diverse and complex factors, such as the availability of information on the securities traded, the agent’s psychology and many other factors such as contingencies or speculative strategies.

In brief, this makes emulation of the system impossible in practice. It should be noted, however, that brokers and automated applications can gain a competitive advantage by identifying global patterns, or even by insider trading, although this practice is punishable by law in suitably regulated markets.

The question that can be raised is whether this impossibility of precise emulation invalidates the hypothesis put forward. If we return to the case study of Newtonian gravitation, determined by the central attractive force, it can be observed that, although functionally different, it shares a fundamental characteristic that makes emulation of the system impossible in practice and that is present in all scenarios. 

If we intend to emulate the case of the solar system we must determine the position, velocity and angular momentum of all celestial bodies involved, sun, planets, dwarf planets, planetoids, satellites, as well as the rest of the bodies located in the system, such as the asteroid belt, the Kuiper belt and the Oort cloud, as well as the dispersed mass and energy. In addition, the shape and structure of solid, liquid and gaseous bodies must be determined. It will also be necessary to consider the effects of collisions that modify the structure of the resulting bodies. Finally, it will be necessary to consider physicochemical activity, such as geological, biological and radiation phenomena, since they modify the structure and dynamics of the bodies and are subject to quantum phenomena, which is another source of uncertainty.  And yet the model is not adequate, since it is necessary to apply a relativistic model.

This makes accurate emulation impossible in practice, as demonstrated by the continuous corrections in the ephemerides of GPS satellites, or the adjustments of space travel trajectories, where the journey to Pluto by NASA’s New Horizons spacecraft is a paradigmatic case.

Conclusions

From the previous analysis it can be hypothesized that the universe is an axiomatic system governed by laws that determine a dynamic that is a consequence of the interaction and transference of the entities that compose it.

As a consequence of the interaction and transfer phenomena, the system itself can partially and approximately emulate its own behavior, which gives rise to learning processes and finally gives rise to life and intelligence. This makes it possible for living beings to interact in a complex way with the environment and for intelligent entities to observe reality and establish models of this reality.

This gave rise to abstract representations such as natural language and mathematics. With the development of IT [5] it is concluded that all objects can be represented by a set of bits, which can be processed by axiomatic rules [7] and which optimally encoded determine the complexity of the object, defined as Kolmogorov complexity [6].

The development of TC establishes that these models can be defined as a TM, so that in the limit it can be hypothesized that the universe is equivalent to a Turing machine and that the limits of reality can go beyond the universe itself, in what is defined as multiverse and that it would be equivalent to a UTM. Esta concordancia entre un universo y una TM  permite plantear la hipótesis de que el universo no es más que información procesada por reglas axiomáticas.

Therefore, from the observation of natural phenomena we can extract the laws of behavior that constitute the abstract models (axioms), as well as the information necessary to describe the cases of reality (information). Since this representation is made on a physical reality, its representation will always be approximate, so that only the universe can emulate itself. Since the universe is consistent, models only corroborate this fact. But reciprocally, the equivalence between the universe and a TM implies that the deductions made from consistent models must be satisfied by reality.

However, everything seems to indicate that this way of perceiving reality is distorted by the senses, since at the level of classical reality what we observe are the consequences of the processes that occur at this functional level, appearing concepts such as mass, energy, inertia.

But when we explore the layers that support classical reality, this perception disappears, since our senses do not have the direct capability for its observation, in such a way that what emerges is nothing more than a model of axiomatic rules that process information, and the physical sensory conception disappears. This would justify the difficulty to understand the foundations of reality.

It is sometimes speculated that reality may be nothing more than a complex simulation, but this poses a problem, since in such a case a support for its execution would be necessary, implying the existence of an underlying reality necessary to support such a simulation [13].

There are two aspects that have not been dealt with and that are of transcendental importance for the understanding of the universe. The first concerns irreversibility in the layer of classical reality. According to the AIT, the amount of information in a TM remains constant, so the irreversibility of thermodynamic systems is an indication that these systems are open, since they do not verify this property, an aspect to which physics must provide an answer.

The second is related to the non-cloning theorem. Quantum systems are reversible and, according to the non-cloning theorem, it is not possible to make exact copies of the unknown quantum state of a particle. But according to the recursion theorem, at least two independent processes are necessary to make a copy. This would mean that in the quantum layer it is not possible to have at least two independent processes to copy such a quantum state. An alternative explanation would be that these quantum states have a non-computable complexity.

Finally, it should be noted that the question of whether mathematics was invented or discovered by humans is flawed by an anthropic view of the universe, which considers humans as a central part of it. But it must be concluded that humans are a part of the universe, as are all the entities that make up the universe, particularly mathematics.

References

[1]E. P. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences.,” Communications on Pure and Applied Mathematics, vol. 13, no. 1, pp. 1-14, 1960.
[2]R. Penrose, The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics, Oxford: Oxford University Press, 1989.
[3]R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, London: Jonathan Cape, 2004.
[4]J.-P. Changeux and A. Connes, Conversations on Mind, Matter, and Mathematics, Princeton N. J.: Princeton University Press, 1995.
[5]C. E. Shannon, “A Mathematical Theory of Communication,” The Bell System Technical Journal, vol. 27, pp. 379-423, 1948.
[6]P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[7]M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[8]H. Poincaré, New Methods of Celestial Mechanics, Springer, 1992.
[9]A. M. Turing, “On computable numbers, with an application to the Entscheidungsproblem.,” Proceedings, London Mathematical Society, pp. 230-265, 1936.
[10]A. W. Senior, R. Evans and e. al., “Improved protein structure prediction using potentials from deep learning,” Nature, vol. 577, pp. 706-710, Jan 2020.
[11]S. Kleene, “On Notation for ordinal numbers,” J. Symbolic Logic, no. 3, p. 150–155, 1938.
[12]A. Savine, Modern Computational Finance: AAD and Parallel Simulations, Wiley, 2018.
[13]N. Bostrom, “Are We Living in a Computer Simulation?,” The Philosophical Quarterly, vol. 53, no. 211, p. 243–255, April 2003.

What is the nature of mathematics?

The ability of mathematics to describe the behavior of nature, particularly in the field of physics, is a surprising fact, especially when one considers that mathematics is an abstract entity created by the human mind and disconnected from physical reality.  But if mathematics is an entity created by humans, how is this precise correspondence possible?

Throughout centuries this has been a topic of debate, focusing on two opposing ideas: Is mathematics invented or discovered by humans?

This question has divided the scientific community: philosophers, physicists, logicians, cognitive scientists and linguists, and it can be said that not only is there no consensus, but generally positions are totally opposed. Mario Livio in the essay “Is God a Mathematician? [1] describes in a broad and precise way the historical events on the subject, from Greek philosophers to our days.

The aim of this post is to analyze this dilemma, introducing new analysis tools  such as Information Theory (IT) [2], Algorithmic Information Theory (AIT) [3] and Computer Theory (CT) [4], without forgetting the perspective that shows the new knowledge about Artificial Intelligence (AI).

In this post we will make a brief review of the current state of the issue, without entering into its historical development, trying to identify the difficulties that hinder its resolution, for in subsequent posts to analyze the problem from a different perspective to the conventional, using the logical tools that offer us the above theories.

Currents of thought: invented or discovered?

In a very simplified way, it can be said that at present the position that mathematics is discovered by humans is headed by Max Tegmark, who states in “Our Mathematical Universe” [5] that the universe is a purely mathematical entity, which would justify that mathematics describes reality with precision, but that reality itself is a mathematical entity.

On the other extreme, there is a large group of scientists, including cognitive scientists and biologists who, based on the fact of the brain’s capabilities, maintain that mathematics is an entity invented by humans.

Max Tegmark: Our Mathematical Universe

In both cases, there are no arguments that would tip the balance towards one of the hypotheses. Thus, in Max Tegmark’s case he maintains that the definitive theory (Theory of Everything) cannot include concepts such as “subatomic particles”, “vibrating strings”, “space-time deformation” or other man-made constructs. Therefore, the only possible description of the cosmos implies only abstract concepts and relations between them, which for him constitute the operative definition of mathematics.

This reasoning assumes that the cosmos has a nature completely independent of human perception, and its behavior is governed exclusively by such abstract concepts. This view of the cosmos seems to be correct insofar as it eliminates any anthropic view of the universe, in which humans are only a part of it. However, it does not justify that physical laws and abstract mathematical concepts are the same entity.  

In the case of those who maintain that mathematics is an entity invented by humans, the arguments do not usually have a formal structure and it could be said that in many cases they correspond more to a personal position and sentiment. An exception is the position maintained by biologists and cognitive scientists, in which the arguments are based on the creative capacity of the human brain and which would justify that mathematics is an entity created by humans.

For these, mathematics does not really differ from natural language, so mathematics would be no more than another language. Thus, the conception of mathematics would be nothing more than the idealization and abstraction of elements of the physical world. However, this approach presents several difficulties to be able to conclude that mathematics is an entity invented by humans.

On the one hand, it does not provide formal criteria for its demonstration. But it also presupposes that the ability to learn is an attribute exclusive to humans. This is a crucial point, which will be addressed in later posts. In addition, natural language is used as a central concept, without taking into account that any interaction, no matter what its nature, is carried out through language, as shown by the TC [4], which is a theory of language.

Consequently, it can be concluded that neither current of thought presents conclusive arguments about what the nature of mathematics is. For this reason, it seems necessary to analyze from new points of view what is the cause for this, since physical reality and mathematics seem intimately linked.

Mathematics as a discovered entity

In the case that considers mathematics the very essence of the cosmos, and therefore that mathematics is an entity discovered by humans, the argument is the equivalence of mathematical models with physical behavior. But for this argument to be conclusive, the Theory of Everything should be developed, in which the physical entities would be strictly of a mathematical nature. This means that reality would be supported by a set of axioms and the information describing the model, the state and the dynamics of the system.

This means a dematerialization of physics, something that somehow seems to be happening as the development of the deeper structures of physics proceeds. Thus, the particles of the standard model are nothing more than abstract entities with observable properties. This could be the key, and there is a hint in Landauer’s principle [6], which establishes an equivalence between information and energy.

But solving the problem by physical means or, to be more precise, by contrasting mathematical models with reality presents a fundamental difficulty. In general, mathematical models describe the functionality of a certain context or layer of reality, and all of them have a common characteristic, in such a way that these models are irreducible and disconnected from the underlying layers. Therefore, the deepest functional layer should be unraveled, which from the point of view of AIT and TC is a non-computable problem.

Mathematics as an invented entity

The current of opinion in favor of mathematics being an entity invented by humans is based on natural language and on the brain’s ability to learn, imagine and create. 

But this argument has two fundamental weaknesses. On the one hand, it does not provide formal arguments to conclusively demonstrate the hypothesis that mathematics is an invented entity. On the other hand, it attributes properties to the human brain that are a general characteristic of the cosmos.

The Hippocampus: A paradigmatic example of the dilemma discovered or invented

To clarify this last point, let us take as an example the invention of whole numbers by humans, which is usually used to support this view. Let us now imagine an animal interacting with the environment. Therefore, it has to interpret spacetime accurately as a basic means of survival. Obviously, the animal must have learned or invented the space-time map, something much more complex than natural numbers.

Moreover, nature has provided or invented the hippocampus [7], a neuronal structure specialized in acquiring long-term information that forms a complex convolution, forming a recurrent neuronal network, very suitable for the treatment of the space-time map and for the resolution of trajectories. And of course this structure is physical and encoded in the genome of higher animals. The question is: Is this structure discovered or invented by nature?

Regarding the use of language as an argument, it should be noted that language is the means of interaction in nature at all functional levels. Thus, biology is a language, the interaction between particles is formally a language, although this point requires a deeper analysis for its justification. In particular, natural language is in fact a non-formal language, so it is not an axiomatic language, which makes it inconsistent.

Finally, in relation to the learning capability attributed to the brain, this is a fundamental characteristic of nature, as demonstrated by mathematical models of learning and evidenced in an incipient manner by AI.

Another way of approaching the question about the nature of mathematics is through Wigner’s enigma [8], in which he asks about the inexplicable effectiveness of mathematics. But this topic and the topics opened before will be dealt with and expanded in later posts.

References

[1] M. Livio, Is God a Mathematician?, New York: Simon & Schuster Paperbacks, 2009.
[2] C. E. Shannon, «A Mathematical Theory of Communication,» The Bell System Technical Journal, vol. 27, pp. 379-423, 1948. 
[3] P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[4] M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[5] M. Tegmark, Our Mathematical Universe: My Quest For The Ultimate Nature Of Reality, Knopf Doubleday Publishing Group, 2014.
[6] R. Landauer, «Irreversibility and Heat Generation in Computing Process,» IBM J. Res. Dev., vol. 5, pp. 183-191, 1961.
[7] S. Jacobson y E. M. Marcus, Neuroanatomy for the Neuroscientist, Springer, 2008.
[8] E. P. Wigner, «The unreasonable effectiveness of mathematics in the natural sciences.,» Communications on Pure and Applied Mathematics, vol. 13, nº 1, pp. 1-14, 1960.

Reality as an information process

The purpose of physics is the description and interpretation of physical reality based on observation. To this end, mathematics has been a fundamental tool to formalize this reality through models, which in turn have allowed predictions to be made that have subsequently been experimentally verified. This creates an astonishing connection between reality and abstract logic that makes suspect the existence of a deep relationship beyond its conceptual definition. In fact, the ability of mathematics to accurately describe physical processes can lead us to think that reality is nothing more than a manifestation of a mathematical world.

But perhaps it is necessary to define in greater detail what we mean by this. Usually, when we refer to mathematics we think of concepts such as theorems or equations. However, we can have another view of mathematics as an information processing system, in which the above concepts can be interpreted as a compact expression of the behavior of the system, as shown by the algorithmic information theory [1].

In this way, physical laws determine how the information that describes the system is processed, establishing a space-time dynamic. As a consequence, a parallelism is established between the physical system and the computational system that, from an abstract point of view, are equivalent. This equivalence is somewhat astonishing, since in principle we assume that both systems belong to totally different fields of knowledge.

But apart from this fact, we can ask what consequences can be drawn from this equivalence. In particular, computability theory [2] and information theory [3] [1] provide criteria for determining the computational reversibility and complexity of a system [4]. In particular:

  • In a reversible computing system (RCS) the amount of information remains constant throughout the dynamics of the system.
  • In a non-reversible computational system (NRCS) the amount of information never increases along the dynamics of the system.
  • The complexity of the system corresponds to the most compact expression of the system, called Kolmogorov complexity and is an absolute measure.

It is important to note that in an NRCS system information is not lost, but is explicitly discarded. This means that there is no fundamental reason why such information should not be maintained, as the complexity of an RCS system remains constant. In practice, the implementation of computer systems is non-reversible in order to optimize resources, as a consequence of the technological limitations for its implementation. In fact, the energy currently needed for its implementation is much higher than that established by the Landauer principle [5].

If we focus on the analysis of reversible physical systems, such as quantum mechanics, relativity, Newtonian mechanics or electromagnetism, we can observe invariant physical magnitudes that are a consequence of computational reversibility. These are determined by unitary mathematical processes, which mean that every process has an inverse process [6]. But the difficulties in understanding reality from the point of view of mathematical logic seem to arise immediately, with thermodynamics and quantum measurement being paradigmatic examples.

In the case of quantum measurement, the state of the system before the measurement is made is in a superposition of states, so that when the measurement is made the state collapses in one of the possible states in which the system was [7]. This means that the quantum measurement scenario corresponds to that of a non-reversible computational system, in which the information in the system decreases when the superposition of states disappears, making the system non-reversible as a consequence of the loss of information.

This implies that physical reality systematically loses information, which poses two fundamental contradictions. The first is the fact that quantum mechanics is a reversible theory and that observable reality is based on it. The second is that this loss of information contradicts the systematic increase of classical entropy, which in turn poses a deeper contradiction, since in classical reality there is a spontaneous increase of information, as a consequence of the increase of entropy.

The solution to the first contradiction is relatively simple if we eliminate the anthropic vision of reality. In general, the process of quantum measurement introduces the concept of observer, which creates a certain degree of subjectivity that is very important to clarify, as it can lead to misinterpretations. In this process there are two clearly separated layers of reality, the quantum layer and the classical layer, which have already been addressed in previous posts. The realization of quantum measurement involves two quantum systems, one that we define as the system to be measured and another that corresponds to the measurement system, which can be considered as a quantum observer, and both have a quantum nature. As a result of this interaction, classical information emerges, where the classical observer is located, who can be identified e.g. with a physicist in a laboratory. 

Now consider that the measurement is structured in two blocks, one the quantum system under observation and the other the measurement system that includes the quantum observer and the classical observer. In this case it is being interpreted that the quantum system under measurement is an open quantum system that loses quantum information in the measurement process and that as a result a lesser amount of classical information emerges. In short, this scenario offers a negative balance of information.

But, on the contrary, in the quantum reality layer the interaction of two quantum systems takes place which, it can be said, mutually observe each other according to unitary operators, so that the system is closed producing an exchange of information with a null balance of information. As a result of this interaction, the classical layer emerges. But then there seems to be a positive balance of information, as classical information emerges from this process. But what really happens is that the emerging information, which constitutes the classical layer, is simply a simplified view of the quantum layer. For this reason we can say that the classical layer is an emerging reality.

So, it can be said that the quantum layer is formed by subsystems that interact with each other in a unitary way, constituting a closed system in which the information and, therefore, the complexity of the system is invariant. As a consequence of these interactions, the classical layer emerges as an irreducible reality of the quantum layer.

As for the contradiction produced by the increase in entropy, the reasons justifying this behavior seem more subtle. However, a first clue may lie in the fact that this increase occurs only in the classical layer. It must also be considered that, according to the algorithmic information theory, the complexity of a system, and therefore the amount of information that describes the system, is the set formed by the processed information and the information necessary to describe the processor itself. 

A physical scenario that can illustrate this situation is the case of the big bang [8], in which it is considered that the entropy of the system in its beginning was small or even null. This is so because the microwave background radiation shows a fairly homogeneous pattern, so the amount of information for its description and, therefore, its entropy is small. But if we create a computational model of this scenario, it is evident that the complexity of the system has increased in a formidable way, which is incompatible from the logical point of view. This indicates that in the model not only the information is incomplete, but also the description of the processes that govern it. But what physical evidence do we have to show that this is so?

Perhaps the clearest sample of this is cosmic inflation [9], so that the space-time metric changes with time, so that the spatial dimensions grow with time. To explain this behavior the existence of dark energy has been postulated as the engine of this process [10], which in a physical form recognizes the gaps revealed by mathematical logic. Perhaps one aspect that is not usually paid attention is the interaction between vacuum and photons, which produces a loss of energy in photons as space-time expands. This loss supposes a decrease of information that necessarily must be transferred to space-time.

This situation causes the vacuum, which in the context of classical physics is nothing more than an abstract metric, to become a fundamental physical piece of enormous complexity. Aspects that contribute to this conception of vacuum are the entanglement of quantum particles [11], decoherence and zero point energy [12].  

From all of the above, a hypothesis can be made as to what the structure of reality is from a computational point of view, as shown in the following figure. If we assume that the quantum layer is a unitary and closed structure, its complexity will remain constant. But the functionality and complexity of this remains hidden from observation and it is only possible to model it through an inductive process based on experimentation, which has led to the definition of physical models, in such a way that these models allow us to describe classical reality. As a consequence, the quantum layer shows a reality that constitutes the classical layer and that is a partial vision and, according to the theoretical and experimental results, extremely reduced of the underlying reality and that makes the classical reality an irreducible reality.  

The fundamental question that can be raised in this model is whether the complexity of the classical layer is constant or whether it can vary over time, since it is only bound by the laws of the underlying layer and is a partial and irreducible view of that functional layer. But for the classical layer to be invariant, it must be closed and therefore its computational description must be closed, which is not verified since it is subject to the quantum layer. Consequently, the complexity of the classical layer may change over time.

Consequently, the question arises as to whether there is any mechanism in the quantum layer that justifies the fluctuation of the complexity of the classical layer. Obviously one of the causes is quantum decoherence, which makes information observable in the classical layer. Similarly, cosmic inflation produces an increase in complexity, as space-time grows. On the contrary, attractive forces tend to reduce complexity, so gravity would be the most prominent factor.

From the observation of classical reality we can answer that currently its entropy tends to grow, as a consequence of the fact that decoherence and inflation are predominant causes. However, one can imagine recession scenarios, such as a big crunch scenario in which entropy decreased. Therefore, the entropy trend may be a consequence of the dynamic state of the system.

In summary, it can be said that the amount of information in the quantum layer remains constant, as a consequence of its unitary nature. On the contrary, the amount of information in the classical layer is determined by the amount of information that emerges from the quantum layer. Therefore, the challenge is to determine precisely the mechanisms that determine the dynamics of this process. Additionally, it is possible to analyze specific scenarios that generally correspond to the field of thermodynamics. Other interesting scenarios may be quantum in nature, such as the one proposed by Hugh Everett on the Many-Worlds Interpretation (MWI).  

Bibliography

[1] P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[2] M. Sipser, Introduction to the Theory of Computation, Course Technology, 2012.
[3] C. E. Shannon, “A Mathematical Theory of Communication,” vol. 27, pp. 379-423, 623-656, 1948.
[4] M. A. Nielsen and I. L. Chuang, Quantum computation and Quantum Information, Cambridge University Press, 2011.
[5] R. Landauer, «Irreversibility and Heat Generation in Computing Process,» IBM J. Res. Dev., vol. 5, pp. 183-191, 1961.
[6] J. Sakurai y J. Napolitano, Modern Quantum Mechanics, Cambridge University Press, 2017.
[7] G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific, 2001.
[8] A. H. Guth, The Inflationary Universe, Perseus, 1997.
[9] A. Liddle, An Introduction to Modern Cosmology, Wiley, 2003.
[10] P. J. E. Peebles and Bharat Ratra, “The cosmological constant and dark energy,” arXiv:astro-ph/0207347, 2003.
[11] A. Aspect, P. Grangier and G. Roger, “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Phys. Rev. Lett., vol. 47, pp. 460-463, 1981.
[12] H. B. G. Casimir and D. Polder, “The Influence of Retardation on the London-van der Waals Forces,” Phys. Rev., vol. 73, no. 4, pp. 360-372, 1948.

A macroscopic view of the Schrödinger cat

From the analysis carried out in the previous post, it can be concluded that, in general, it is not possible to identify the macroscopic states of a complex system with its quantum states. Thus, the macroscopic states corresponding to the dead cat (DC) or to the living cat (AC) cannot be considered quantum states, since according to quantum theory the system could be expressed as a superposition of these states. Consequently, as it has been justified, for macroscopic systems it is not possible to define quantum states such as |DC⟩ and |DC⟩. On the other hand, the states (DC) and (AC) are an observable reality, indicating that the system presents two realities, a quantum reality and an emerging reality that can be defined as classical reality.

Quantum reality will be defined by its wave function, formed by the superposition of the quantum subsystems that make up the system and which will evolve according to the existing interaction between all the quantum elements that make up the system and the environment. For simplicity, if the CAT system is considered isolated from the environment, the succession of its quantum state can be expressed as:

            |CAT[n]⟩ = |SC1[n]⟩ ⊗|SC2[n]⟩ ⊗…⊗|SCi[n]⟩ ⊗…⊗|SCk[n][n]⟩.

Expression in which it has been taken into account that the number of non-entangled quantum subsystems k also varies with time, so it is a function of the sequence n, considering time as a discrete variable. 

The observable classical reality can be described by the state of the system that, if for the object “cat” is defined as (CAT[n]), from the previous reasoning it is concluded that (CAT[n]) ≢ |CAT[n]⟩. In other words, the quantum and classical states of a complex object are not equivalent. 

The question that remains to be justified is the irreducibility of the observable classical state (CAT) from the underlying quantum reality, represented by the quantum state |CAT⟩. This can be done if it is considered that the functional relationship between states |CAT⟩ and (CAT) is extraordinarily complex, being subject to the mathematical concepts on which complex systems are based, such as they are:

  • The complexity of the space of quantum states (Hilbert space).
  • The random behavior of observable information emerging from quantum reality.
  • The enormous number of quantum entities involved in a macroscopic system.
  • The non-linearity of the laws of classical physics.

Based on Kolmogorov complexity [1], it is possible to prove that the behavior of systems with these characteristics does not support, in most cases, an analytical solution that determines the evolution of the system from its initial state. This also implies that, in practice, the process of evolution of a complex object can only be represented by itself, both on a quantum and a classical level.

According to the algorithmic information theory [1], this process is equivalent to a mathematical object composed of an ordered set of bits processed according to axiomatic rules. In such a way that the information of the object is defined by the Kolmogorov complexity, in a manner that it remains constant throughout time, as long as the process is an isolated system. It should be pointed out that the Kolmogorov complexity makes it possible to determine the information contained in an object, without previously having an alphabet for the determination of its entropy, as is the case in the information theory [2], although both concepts coincide at the limit.

From this point of view, two fundamental questions arise. The first is the evolution of the entropy of the system and the second is the apparent loss of information in the observation process, through which classical reality emerges from quantum reality. This opens a possible line of analysis that will be addressed later.

But going back to the analysis of what is the relationship between classic and quantum states, it is possible to have an intuitive view of how the state (CAT) ends up being disconnected from the state |CAT⟩, analyzing the system qualitatively.

First, it should be noted that virtually 100% of the quantum information contained in the state |CAT⟩ remains hidden within the elementary particles that make up the system. This is a consequence of the fact that the physical-chemical structure [3] of the molecules is determined exclusively by the electrons that support its covalent bonds. Next, it must be considered that the molecular interaction, on which molecular biology is based, is performed by van der Waals forces and hydrogen bonds, creating a new level of functional disconnection with the underlying layer.

Supported by this functional level appears a new functional structure formed by cellular biology  [4], from which appear living organisms, from unicellular beings to complex beings formed by multicellular organs. It is in this layer that the concept of living being emerges, establishing a new border between the strictly physical and the concept of perception. At this level the nervous tissue [5] emerges, allowing the complex interaction between individuals and on which new structures and concepts are sustained, such as consciousness, culture, social organization, which are not only reserved to human beings, although it is in the latter where the functionality is more complex.

But to the complexity of the functional layers must be added the non-linearity of the laws to which they are subject and which are necessary and sufficient conditions for a behavior of deterministic chaos [6] and which, as previously justified, is based on the algorithmic information theory [1]. This means that any variation in the initial conditions will produce a different dynamic, so that any emulation will end up diverging from the original, this behavior being the justification of free will. In this sense, Heisenberg’s uncertainty principle [7] prevents from knowing exactly the initial conditions of the classical system, in any of the functional layers described above. Consequently, all of them will have an irreducible nature and an unpredictable dynamic, determined exclusively by the system itself.

At this point and in view of this complex functional structure, we must ask what the state (CAT) refers to, since in this context the existence of a classical state has been implicitly assumed. The complex functional structure of the object “cat” allows a description at different levels. Thus, the cat object can be described in different ways:

  • As atoms and molecules subject to the laws of physical chemistry.
  • As molecules that interact according to molecular biology.
  • As complex sets of molecules that give rise to cell biology.
  • As sets of cells to form organs and living organisms.
  • As structures of information processing, that give rise to the mechanisms of perception and interaction with the environment that allow the development of individual and social behavior.

As a result, each of these functional layers can be expressed by means of a certain state. So to speak of, the definition of a unique macroscopic state (CAT) is not correct. Each of these states will describe the object according to different functional rules, so it is worth asking what relationship exists between these descriptions and what their complexity is. Analogous to the arguments used to demonstrate that the states |CAT⟩ and (CAT) are not equivalent and are uncorrelated with each other, the states that describe the “cat” object at different functional levels will not be equivalent and may to some extent be disconnected from each other.

This behavior is a proof of how reality is structured in irreducible functional layers, in such a way that each one of the layers can be modeled independently and irreducibly, by means of an ordered set of bits processed according to axiomatic rules.

Refereces

[1] P. Günwald and P. Vitányi, “Shannon Information and Kolmogorov Complexity,” arXiv:cs/0410002v1 [cs:IT], 2008.
[2] C. E. Shannon, «A Mathematical Theory of Communication,» The Bell System Technical Journal, vol. 27, pp. 379-423, 1948.
[3] P. Atkins and J. de Paula, Physical Chemestry, Oxford University Press, 2006.
[4] A. Bray, J. Hopkin, R. Lewis and W. Roberts, Essential Cell Biology, Garlan Science, 2014.
[5] D. Purves and G. J. Augustine, Neuroscience, Oxford Univesisty press, 2018.
[6] J. Gleick, Chaos: Making a New Science, Penguin Books, 1988.
[7] W. Heisenberg, «The Actual Content of Quantum Theoretical Kinematics and Mechanics,» Zeit-schrift fur Physik. Translation: NASA TM-77379., vol. 43, nº 3-4, pp. 172-198, 1927.

Reality as an irreducible layered structure

Note: This post is the first in a series in which macroscopic objects will be analyzed from a quantum and classical point of view, as well as the nature of the observation. Finally, all of them will be integrated into a single article.

Introduction

Quantum theory establishes the fundamentals of the behavior of particles and their interaction with each other. In general, these fundamentals apply to microscopic systems formed by a very limited number of particles. However, nothing indicates that the application of quantum theory cannot be applied to macroscopic objects, since the emerging properties of such objects must be based on the underlying quantum reality. Obviously, there is a practical limitation established by the increase in complexity, which grows exponentially as the number of elementary particles increases. 

The initial reference to this approach was made by Schrödinger [1], indicating that the quantum superposition of states did not represent any contradiction at the macroscopic level. To do this, he used what is known as Schrödinger’s cat paradox in which the cat could be in a superposition of states, one in which the cat was alive and another in which the cat was dead. Schrödinger’s original motivation was to raise a discussion about the EPR paradox [2], which revealed the incompleteness of quantum theory. This has finally been solved by Bell’s theorem [3] and its experimental verification by Aspect [4], making it clear that the entanglement of quantum particles is a reality on which quantum computation is based [5]. A summary of the aspects related to the realization of a quantum system that emulates Schrödinger cat has been made by Auletta [6], although these are restricted to non-macroscopic quantum systems.

But the question that remains is whether quantum theory can be used to describe macroscopic objects and whether the concept of quantum entanglement applies to these objects as well. Contrary to Schrödinger’s position, Wigner argued, through the friend paradox, that quantum mechanics could not have unlimited validity [7]. Recently, Frauchiger and Renner [8] have proposed a virtual experiment (Gedankenexperiment) that shows that quantum mechanics is not consistent when applied to complex objects. 

The Schrödinger cat paradigm will be used to analyze these results from two points of view, with no loss of generality, one as a quantum object and the other as a macroscopic object (in a next post). This will allow their consistency and functional relationship to be determined, leading to the establishment of an irreducible functional structure. As a consequence of this, it will also be necessary to analyze the nature of the observer within this functional structure (also in a later posts). 

Schrödinger’s cat as a quantum reality

In the Schrödinger cat experiment there are several entities [1], the radioactive particle, the radiation monitor, the poison flask and the cat. For simplicity, the experiment can be reduced to two quantum variables: the cat, which we will identify as CAT, and the system formed by the radioactive particle, the radiation monitor and the poison flask, which we will define as the poison system PS. 


Schrödinger Cat. (Source: Doug Hatfield https://commons.wikimedia.org/wiki/File:Schrodingers_cat.svg)

These quantum variables can be expressed as [9]: 

            |CAT⟩ = α1|DC⟩ + β1|LC⟩. Quantum state of the cat: dead cat |DC⟩, live cat |LC⟩.

            |PS⟩ = α2|PD⟩ + β2|PA⟩. Quantum state of the poison system: poison deactivated |PD⟩, poison activated |PA⟩.

The quantum state of the Schrödinger cat experiment SCE as a whole can be expressed as: 
               |SCE⟩ = |CAT⟩⊗|PS⟩= α1α2|DC⟩|PD⟩+α1β2|DC⟩|PA⟩+β1α2|LC⟩|PD⟩+β1β2|LC⟩|PA⟩.

Since for a classical observer the final result of the experiment requires that the states |DC⟩|PD⟩ and |LC⟩|PA⟩ are not compatible with observations,  the experiment must be prepared in such a way that the quantum states |CAT⟩ and |PS⟩ are entangled [10] [11], so that the wave function of the experiment must be: 

               |SCE⟩ = α|DC⟩|PA⟩ + β|LC⟩|PD⟩. 

As a consequence, the observation of the experiment [12] will result in a state:

            |SCE⟩ = |DC⟩|PA⟩, with probability α2, (poison activated, dead cat). 

or:

            |SCE⟩ =|LC⟩|PD⟩, with probability β2, (poison deactivated, live cat). 

Although from the formal point of view of quantum theory the approach of the experiment is correct, for a classical observer the experiment presents several objections. One of these is related to the fact that the experiment requires establishing “a priori” the requirement that the PS and CAT systems are entangled. Something contradictory, since from the point of view of the preparation of the quantum experiment there is no restriction, being able to exist results with quantum states |DC⟩|PD⟩, or |LC⟩|PA⟩, something totally impossible for a classical observer, assuming in any case that the poison is effective, that it is taken for granted in the experiment. Therefore, the SCE experiment is inconsistent, so it is necessary to analyze the root of the incongruence between the SCE quantum system and the result of the observation. 

Another objection, which may seem trivial, is that for the SCE experiment to collapse in one of its states the OBS observer must be entangled with the experiment, since the experiment must interact with it. Otherwise, the operation performed by the observer would have no consequence on the experiment. For this reason, this aspect will require more detailed analysis. 

Returning to the first objection, from the perspective of quantum theory it may seem possible to prepare the PS and CAT systems in an entangled superposition of states. However, it should be noted that both systems are composed of a huge number of non-entangled quantum subsystems Ssubject to continuous decoherence [13] [14]. It should be noted that the Si subsystems will internally have an entangled structure. Thus, the CAT and PS systems can be expressed as: 

            |CAT⟩ = |SC1⟩ ⊗ |SC2⟩ ⊗…⊗ |SCi⟩ ⊗…⊗ |SCk⟩,

            |PS⟩= |SP1⟩⊗|SP2⟩⊗…⊗|SPi⟩⊗…⊗|SPl⟩, 

in such a way that the observation of a certain subsystem causes its state to collapse, producing no influence on the rest of the subsystems, which will develop an independent quantum dynamics. This makes it unfeasible that the states |LC⟩ and |DC⟩ can be simultaneous and as a consequence the CAT system cannot be in a superposition of these states. An analogous reasoning can be made of the PS system, although it imay seem obvious that functionally it is much simpler. 

In short, from a theoretical point of view it is possible to have a quantum system equivalent to the SCE, for which all the subsystems must be fully entangled with each other, and in addition the system will require an “a priori” preparation of its state. However, the emerging reality differs radically from this scenario, so that the experiment seems to be unfeasible in practice. But the most striking fact is that, if the SCE experiment is generalized, the observable reality would be radically different from the observed reality. 

To better understand the consequences of the quantum state of the ECS system having to be prepared “a priori”, imagine that the supplier of the poison has changed its contents to a harmless liquid. As a result of this, the experiment will be able to kill the cat without cause. 

From these conclusions the question can be raised as to whether quantum theory can explain in a general and consistent way the observable reality at the macroscopic level. But perhaps the question is also whether the assumptions on which the SCE experiment has been conducted are correct. Thus, for example: Is it correct to use the concepts of live cat or dead cat in the domain of quantum physics? Which in turn raises other kinds of questions, such as: Is it generally correct to establish a strong link between observable reality and the underlying quantum reality? 

The conclusion that can be drawn from the contradictions of the SCE experiment is that the scenario of a complex quantum system cannot be treated in the same terms as a simple system. In terms of quantum computation these correspond, respectively, to systems made up of an enormous number and a limited number of qubits [5]. As a consequence of this, classical reality will be an irreducible fact, which based on quantum reality ends up being disconnected from it. This leads to defining reality in two independent and irreducible functional layers, a quantum reality layer and a classical reality layer. This would justify the criterion established by the Copenhagen interpretation [15] and its statistical nature as a means of functionally disconnecting both realities. Thus, quantum theory would be nothing more than a description of the information that can emerge from an underlying reality, but not a description of that reality. At this point, it is important to emphasize that statistical behavior is the means by which the functional correlation between processes can be reduced or eliminated [16] and that it would be the cause of irreducibility

References

[1] E. Schrödinger, «Die gegenwärtige Situation in der Quantenmechanik,» Naturwissenschaften, vol. 23, pp. 844-849, 1935.
[2] A. Einstein, B. Podolsky and N. Rose, “Can Quantum-Mechanical description of Physical Reality be Considered Complete?,” Physical Review, vol. 47, pp. 777-780, 1935.
[3] J. S. Bell, «On the Einstein Podolsky Rosen Paradox,» Physics,vol. 1, nº 3, pp. 195-290, 1964.
[4] A. Aspect, P. Grangier and G. Roger, “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Phys. Rev. Lett., vol. 47, pp. 460-463, 1981.
[5] M. A. Nielsen and I. L. Chuang, Quantum computation and Quantum Information, Cambridge University Press, 2011.
[6] G. Auletta, Foundations and Interpretation of Quantum Mechanics, World Scientific, 2001.
[7] E. P. Wigner, «Remarks on the mind–body question,» in Symmetries and Reflections, Indiana University Press, 1967, pp. 171-184.
[8] D. Frauchiger and R. Renner, “Quantum Theory Cannot Consistently Describe the Use of Itself,” Nature Commun., vol. 9, no. 3711, 2018.
[9] P. Dirac, The Principles of Quantum Mechanics, Oxford University Press, 1958.
[10] E. Schrödinger, «Discussion of Probability Relations between Separated Systems,» Mathematical Proceedings of the Cambridge Philosophical Society, vol. 31, nº 4, pp. 555-563, 1935.
[11] E. Schrödinger, «Probability Relations between Separated Systems,» Mathematical Proceedings of the Cambridge Philosophical Society, vol. 32, nº 3, pp. 446­-452, 1936.
[12] M. Born, «On the quantum mechanics of collision processes.,» Zeit. Phys.(D. H. Delphenich translation), vol. 37, pp. 863-867, 1926.
[13] H. D. Zeh, «On the Interpretation of Measurement in Quantum Theory,» Found. Phys., vol. 1, nº 1, pp. 69-76, 1970.
[14] W. H. Zurek, «Decoherence, einselection, and the quantum origins of the classical,» Rev. Mod. Phys., vol. 75, nº 3, pp. 715-775, 2003.
[15] W. Heisenberg, Physics and Philosophy. The revolution in Modern Science, Harper, 1958.
[16] E. W. Weisstein, «MathWorld,» [En línea]. Available http://mathworld.wolfram.com/Covariance.html.

Why the rainbow has 7 colors?

Published on OPENMIND August 8, 2018

Color as a physical concept

Visible light, heat, radio waves and other types of radiation all have the same physical nature and are constituted by a flow of particles called photons. The photon or “light quantum” was proposed by Einstein, for which he was awarded the Nobel Prize in 1921 and is one of the elementary particles of the standard model, belonging to the boson family. The fundamental characteristic of a photon is its capacity to transfer energy in quantized form, which is determined by its frequency, according to the expression E=h∙ν, where h is the Planck constant and ν the frequency of the photon.

Electromagnetic spectrum

Thus, we can find photons of very low frequencies located in the band of radio waves, to photons of very high energy called gamma rays, as shown in the following figure, forming a continuous frequency range that constitutes the electromagnetic spectrum. Since the photon can be modeled as a sinusoid traveling at the speed of light c, the length of a complete cycle is called the photon wavelength λ, so the photon can be characterized either by its frequency or its wavelength, since λ=c/ν. But it is common to use the term color as a synonym for frequency, since the color of light perceived by humans is a function of frequency. However, as we are going to see, this is not strictly physical but a consequence of the process of measuring and interpreting information, which makes color an emerging reality of another underlying reality, sustained by the physical reality of electromagnetic radiation.

Structure of an electromagnetic wave

But before addressing this issue, it should be considered that to detect photons efficiently it is necessary to have a detector called an antenna, whose size must be similar to the wavelength of the photons.

Color perception by humans

The human eye is sensitive to wavelengths ranging from deep red (700nm, nanometers=10-9 meters) to violet (400nm).  This requires receiving antennas of the order of hundreds of nanometres in size! But for nature this is not a big problem, as complex molecules can easily be this size. In fact, the human eye, for color vision, is endowed with three types of photoreceptor proteins, which produce a response as shown in the following figure.

Response of photoreceptor cells of the human retina

Each of these types configures a type of photoreceptor cell in the retina, which due to its morphology are called cones. The photoreceptor proteins are located in the cell membrane, so that when they absorb a photon they change shape, opening up channels in the cell membrane that generate a flow of ions. After a complex biochemical process, a flow of nerve impulses is produced that is preprocessed by several layers of neurons in the retina that finally reach the visual cortex through the optic nerve, where the information is finally processed.

But in this context, the point is that the retinal cells do not measure the wavelength of the photons of the stimulus. On the contrary, what they do is convert a stimulus of a certain wavelength into three parameters called L, M, S, which are the response of each of the types of photoreceptor cells to the stimulus. This has very interesting implications that need to be analyzed. In this way, we can explain aspects such as:

  • The reason why the rainbow has 7 colors.
  • The possibility of synthesizing the color by means of additive and subtractive mixing.
  • The existence of non-physical colors, such as white and magenta.
  • The existence of different ways of interpreting color according to the species.

To understand this, let us imagine that they provide us with the response of a measurement system that relates L, M, S to the wavelength and ask us to establish a correlation between them. The first thing we can see is that there are 7 different zones in the wavelength, 3 ridges and 4 valleys. 7 patterns! This explains why we perceive the rainbow composed of 7 colors, an emerging reality as a result of information processing that transcends physical reality.

But what answer will a bird give us if we ask it about the number of colors of the rainbow? Possibly, though unlikely, it will tell us nine! This is because the birds have a fourth type of photoreceptor positioned in the ultraviolet, so the perception system will establish 9 regions in the light perception band. And this leads us to ask: What will be the chromatic range perceived by our hypothetical bird, or by species that only have a single type of photoreceptor? The result is a simple case of combinatorial!

On the other hand, the existence of three types of photoreceptors in the human retina makes it possible to synthesize the chromatic range in a relatively precise way, by means of the additive combination of three colors, red, green and blue, as it is done in the video screens. In this way, it is possible to produce an L,M,S response at each point of the retina similar to that produced by a real stimulus, by means of the weighted application of a mixture of photons of red, green and blue wavelengths.

Similarly, it is possible to synthesize color by subtractive or pigmentary mixing of three colors, magenta, cyan and yellow, as in oil paint or printers. And this is where the virtuality of color is clearly shown, since there are no magenta photons, since this stimulus is a mixture of blue and red photons. The same happens with the white color, as there are no individual photons that produce this stimulus, since white is the perception of a mixture of photons distributed in the visible band, and in particular by the mixture of red, green and blue photons.

In short, the perception of color is a clear example of how reality emerges as a result of information processing. Thus, we can see how a given interpretation of the physical information of the visible electromagnetic spectrum produces an emerging reality, based a much more complex underlying reality.

In this sense, we could ask ourselves what an android with a precise wavelength measurement system would think of the images we synthesize in painting or on video screens. It would surely answer that they do not correspond to the original images, something that for us is practically imperceptible. And this connects with a subject, which may seem unrelated, as is the concept of beauty and aesthetics. The truth is that when we are not able to establish patterns or categories in the information we perceive it as noise or disorder.  Something unpleasant or unsightly!

What is the nature of the information?

Published on OPENMIND May 7, 2018

A historical perspective

Classically, information is considered to be human-to-human transactions. However, throughout history this concept has been expanded, not so much by the development of mathematical logic but by technological development. A substantial change occurred with the arrival of the telegraph at the beginning of the 19th century. Thus, “send” went from being strictly material to a broader concept, as many anecdotes make clear. Among the most frequent highlights the intention of many people to send material things by means of telegrams, or the anger of certain customers arguing that the telegraph operator had not sent the message when he returned them the message note.

Currently, “information” is an abstract concept based on the theory of information, created by Claude Shannon in the mid-twentieth century. However, computer technology is what has contributed most to the concept of “bit” being something totally familiar. Moreover, concepts such as virtual reality, based on the processing of information, have become everyday terms.

The point is that information is ubiquitous in all natural processes, physics, biology, economics, etc., in such a way that these processes can be described by mathematical models and ultimately by information processing. This makes us wonder: What is the relationship between information and reality? 

Information as a physical entity

It is evident that information emerges from physical reality, as computer technology demonstrates. The question is whether information is fundamental to physical reality or simply a product of it. In this sense, there is evidence of the strict relationship between information and energy.

Claude Elwood Shannon was a mathematician, electrical engineer and American cryptographer remembered as «the father of information theory» / Image: DobriZheglov

Thus, the Shannon-Hartley theorem of information theory establishes the minimum amount of energy required to transmit a bit, known as the Bekenstein bound. In a different way and in order to determine the energy consumption in the computation process, Rolf Landauer established the minimum amount of energy needed to erase a bit, a result known as Landauer principle, and its value exactly coincides with the Bekenstein bound, which is a function of the absolute temperature of the medium.

These results allow determining the maximum capacity of a communication channel and the minimum energy required by a computer to perform a given task. In both cases, the inefficiency of current systems is evidenced, whose performance is extremely far from theoretical limits. But in this context, the really important thing is that Shannon-Hartley’s theorem is a strictly mathematical development, in which the information is finally coded on physical variables, leading us to think that information is something fundamental in what we define as reality.

Both cases show the relationship between energy and information, but are not conclusive in determining the nature of information. What is clear is that for a bit to emerge and be observed on the scale of classical physics requires a minimum amount of energy determined by the Bekenstein bound. So, the observation of information is something related to the absolute temperature of the environment.

This behavior is fundamental in the process of observation, as it becomes evident in the experimentation of physical phenomena. A representative example is the measurement of the microwave background radiation produced by the big bang, which requires that the detector located in the satellite be cooled by liquid helium. The same is true for night vision sensors, which must be cooled by a Peltier cell. On the contrary, this is not necessary in a conventional camera since the radiation emitted by the scene is much higher than the thermal noise level of the image sensor.

Cosmic Microwave Background (CMB). NASA’s WMAP satellite

This proofs that information emerges from physical reality. But we can go further, as information is the basis for describing natural processes. Therefore, something that cannot be observed cannot be described. In short, every observable is based on information, something that is clearly evident in the mechanisms of perception.

From the emerging information it is possible to establish mathematical models that hide the underlying reality, suggesting a functional structure in irreducible layers. A paradigmatic example is the theory of electromagnetism, which accurately describes electromagnetism without relying itself on the photon’s existence, and the existence of photos cannot be inferred from it. Something that is generally extendable to all physical models.

Another indication that information is a fundamental entity of what we call reality is the impossibility of transferring information faster than light. This would make reality a non-causal and inconsistent system. Therefore, from this point of view information is subject to the same physical laws as energy. And considering a behavior such as particle entanglement, we can ask: How does information flow at the quantum level?

Is information the essence of reality?

Based on these clues, we could hypothesize that information is the essence of reality in each of the functional layers in which it is manifested. Thus, for example, if we think of space-time, its observation is always indirect through the properties of matter-energy, so we could consider it to be nothing more than the emergent information of a more complex underlying reality. This gives an idea of ​​why the vacuum remains one of the great enigmas of physics. This kind of argument leads us to ask: What is it and what do we mean by reality?

Space-Time perception

From this perspective, we can ask what conclusions we could reach if we analyze what we define as reality from the point of view of information theory and, in particular, from  the algorithmic information theory and the theory of computability. All this without losing sight of the knowledge provided by the different areas that study reality, especially physics.

 

A classic example of axiomatic processing

In the article “Reality and information: Is information a physical entity?” what we mean by information is analyzed. This is a very general review of the development of the theoretical and practical aspects that occurred throughout the twentieth century to the present day and which have led to the current vision of what information is.

The article “Reality and information: What is the nature of information?” goes deeper into this analysis. This is made from a more theoretical perspective based on the computation theory, information theory (IT) and algorithmic information theory (AIT).

But in this post, we will leave aside the mathematical formalism and expose some examples that will give us a more intuitive view of what information is and its relation to reality. And above all try to expose what the axiomatic process of information means. This should help to understand the concept of information beyond what is generally understood as a set of bits. And this is what I consider one of the obstacles to establishing a strong link between information and reality.

Nowadays, information and computer technology offers countless examples of how what we observe as reality can be represented by a set of bits. Thus, videos, images, audio and written information can be encoded, compressed, stored and reproduced as a set of bits. This is possible since they are all mathematical objects, which can be represented by numbers subject to axiomatic rules and can, therefore, be represented by a set of bits. However, the number of bits needed to encode the object depends on the coding procedure (axiomatic rules), so that the AIT determines its minimum value defined as the entropy of the object. However, the AIT does not provide any criteria for the implementation of the compression process, so in general they are based on practical criteria, for example statistical criteria, psychophysical, etc.

The AIT establishes a formal definition of the complexity of mathematical objects, called the Kolmogorov complexity K(x). For a finite object x, K(x) is defined as the length of the shortest effective binary description of x, and is an intrinsic property of the object and not a property of the evaluation process. Without entering into theoretical details, the AIT determines that only a small part of n-bit mathematical objects can be compressed and encoded in m bits n>m, which means that most of them have a great complexity and can only be represented by themselves.

The compression and decompression of video, images, audio, etc., are a clear example of axiomatic processing. Imagine a video content x which, by means of a compression process C, has generated a content C(x) , so that by means of a decompression process D we can retrieve the original content x=D(y) . In this context, both C and D are axiomatic processes, understanding as axiom a proposition assumed within a theoretical body. This may seem shocking to the idea that an axiom is an obvious and accepted proposition without requiring demonstration. To clarify this point I will develop this idea in another post, for which I will use as an example the structure of natural languages.

In this context, the term axiomatic is totally justified theoretically, since the AIT does not establish any criteria for the implementation of the compression process. And, as already indicated, most mathematical objects are not compressible.

This example reveals an astonishing result of IT, defined as “information without meaning”. In such a way that a bit string has no meaning unless a process is applied that interprets the information and transforms it into knowledge. Thus, when we say that x is a video content we are assuming that it responds to a video coding system, according to the visual perception capabilities of humans.

And here we come to a transcendental conclusion regarding the nexus between information and reality. Historically, the development of IT has created the tendency to establish this nexus by considering the information as a sequence of bits exclusively. But AIT shows us that we must understand information as a broader concept, made up of axiomatic processes and bit strings. But for this, we must define it in a formal way.

Thus, both C and D are mathematical objects that in practice are embodied in a set consisting of a processor and programs that encode the functions of compression and decompression. If we define a processor as T() and c and d the bit strings that encode the compression and decompression algorithms, we can express:

         y=T(<c,x>)

         x=T(<d,y>)

where <,> is the concatenation of bit sequences.

Therefore, the axiomatic processing would be determined by the processor T(). And if we use any of the implementations of the universal Turing machine we will see that the number of axiomatic rules is very small. This may seem surprising if one considers that the above is extendable to the  definition of any mathematical model of reality.

Thus, any mathematical model that describes an element of reality can be formalized by means of a Turing machine. The result of the model can be enumerable or Turing computable, in which case the Halt state will be reached, concluding the process. On the contrary, the problem can be undecidable or non-computable, so that the Halt state is never reached, continuing the execution of the process forever.

For example, let us weigh in the Newtonian mechanics determined by the laws of the dynamics and the attraction exerted by the masses. In this case, the system dynamics will be determined by the recursive process w=T(<x,y,z>). Where x is the bit string encoding the laws of calculus, y the bit sequence encoding the laws of Newtonian mechanics and z the initial conditions of the masses constituting the system.

It is frequent, as a consequence of the numerical calculus, to think that the processes are nothing more than numerical simulations of the models. However, in the above example, both x and y can be the analytic expressions of the model and w=T(<x,y,z>) the analytical expression of the solution. Thus, if z specifies that the model is composed of only two massive bodies, w=T(<x,y,z>) will produce an analytical expression of the two ellipses corresponding to the ephemeris of both bodies. However, if z specifies more than two massive bodies, in general, the process will not be able to produce any result, not reaching the Halt state. This is because the Newtonian model has no analytical solution for three or more orbiting bodies, except for very particular cases, and is known as the three-body problem.

But we can make x and y encode the functions of numerical calculus, corresponding respectively to the mathematical calculus and to the computational functions of the Newtonian model. In this case, w=T(<x,y,z>) will produce recursively the numerical description of the ephemeris of the massive bodies. However, the process will not reach the Halt state, except in very particular cases in which the process may decide that the ephemeris is a closed trajectory.

This behaviour shows that the Newtonian model is not computable or undecidable. This is extendable to all models of nature established by physics since they are all non-linear models. If we consider the complexity of the y sequence corresponding to the Newtonian model, both in the analytical or in the numerical version, it is evident that the complexity K(x) is small. However, the complexity of w=T(<x,y,z>) is, in general, non-computable which justifies that it cannot be expressed analytically. If this were possible it would mean that w is an enumerable expression, which is in contradiction with the fact that it is a non-computable expression.

What is surprising is that from an enumerable expression <x, y, z> we can get a non-computable result. But this will be addressed another post.

What do we mean by reality?

In the article “Reality and information: Is information a physical entity?” we analyze what we mean by reality, for which the models established by physics are taken as a reference since they have reached a level of formal definition not attained so far in other areas of knowledge.

One of the conclusions of this analysis is that physical models are axiomatic mathematical structures that describe an emerging reality layer without the need of connection with the underlying reality. This means that models describe reality at a given functional level. This makes reality closely linked to observation, which justifies our view of reality determined by our perception capabilities.

Consequently, reality can be structured into irreducible functional layers, and only when one looks at the edges or boundaries of the models describing the functionality of each emergent layer are there signs of another more complex underlying reality.

In this sense, physics aims to reveal the ultimate foundation of reality and has materialized in the development of quantum physics and in particular in the standard model of particles, although the questions raised by these suggest a more complex reality. However, the structure of layers could have no end and according to Gödel’s incompleteness theorem be an undecidable problem, that is, an unsolvable problem.

All this is very abstract, but with an example, we can understand it better. Thus, let us suppose the system of human color perception, based on three types of photoreceptors tuned in the bands of red, green or blue. Due to Heisenberg’s uncertainty principle, the response of these photoreceptors also responds to stimuli of near frequencies (in the future we could discuss it in detail), as shown in the figure. As a consequence, the photoreceptors do not directly measure the frequency of color stimuli, but instead translate their frequency into three parameters (L, M, S) corresponding to the excitation level of each type of photoreceptors.

This makes possible the synthesis of color by three components, red, green and blue in the case of additive synthesis, and yellow, cyan and magenta for subtractive synthesis. In this way, if the synthesized image is analyzed by means of spectroscopy the perception of the image in relation to color would have very little to do with the original. In the case of birds, the rainbow must have, hypothetically, 9 colors, since they are equipped with a fourth type of photoreceptor sensitive to ultraviolet.

One of the consequences of this measurement system, designed by natural evolution, is that the rainbow is composed of seven colors, determined by the three summits and the four valleys produced by the superposition of the photoreceptor response. In addition, the system creates the perception of additional virtual colors, such as magenta and white. In the case of magenta, this is the result of the simultaneous stimulation of the bands above the blue and below the red. In the case of white is the result of simultaneous stimulation of the red, green and blue bands.

From the physical point of view, this color structure does not exist, since the physical parameter that characterizes a photon is its frequency (or its wavelength λ= 1 / f). Therefore, it can be concluded that color perception is an emergent structure of a more complex structure, determined by an axiomatic observational system. But for the moment, the analysis of the term “axiomatic” will be left for later!

This is an example of how reality emerges from more complex underlying structures, so we can say that reality and observation are inseparable terms. And make no mistake! Although the example refers to the perception of color by humans, this is materialized in a mathematical model of information processing.

Now the question is: How far can we look into this layered structure? In the above case, physics shows by means of electromagnetism that the spectrum is continuous and includes radio waves, microwaves, heat, infrared, visible light, ultraviolet, etc. But electromagnetism is nothing more than an emergent model of a more complex underlying reality, as quantum physics shows us. So that, electromagnetic waves are a manifestation of a flow of quantum particles: photons.

And here appears a much more complex reality in which a photon seems to follow simultaneously multiple paths or to have multiple frequencies simultaneously, even infinite, until it is observed, being determined its position, energy, trajectory, etc., with a precision established by the Heisenberg’s uncertainty principle. And all this described by an abstract mathematical model contrasted by observation….

The search for the ultimate reasons behind things has led physics to deepen, with remarkable success, in the natural processes hidden to our systems of perception. For this purpose, there have been designed experiments and developed detectors that expand our capacity for perception and that have resulted in models such as the standard particle model.

The point is that, despite having increased our capacity for perception and as a result of our knowledge, it seems that we are again in the same situation. The result is that we have new, much more complex, underlying abstract reality models described in mathematical language. This is a clear sign that we can not find an elementary entity that can explain the foundation of reality since these models presuppose the existence of complex entities. Thus, everything seems to indicate that we enter an endless loop, in which from a greater perception of reality we define a new abstract model that in turn opens a new horizon of reality and therefore the need to go deeper into it.

As we can see, we are referring to abstract models to describe reality. For this reason, the second part of the article is dedicated to this. But we will discuss this later!