Category Archives: Mathematics

The unreasonable effectiveness of mathematics

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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 objects〈xyz〉(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?

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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.

Perception of complexity

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In previous posts, the nature of reality and its complexity has been approached from the point of view of Information Theory. However, it is interesting to make this analysis from the point of view of human perception and thus obtain a more intuitive view.

Obviously, making an exhaustive analysis of reality from this perspective is complex due to the diversity of the organs of perception and the physiological and neurological aspects that develop over them. In this sense, we could explain how the information perceived is processed, depending on each of the organs of perception. Especially the auditory and visual systems, as these are more culturally relevant. Thus, in the post dedicated to color perception it has been described how the physical parameters of light are encoded by the photoreceptor cells of the retina.

However, in this post the approach will consist of analyzing in an abstract way how knowledge influences the interpretation of information, in such a way that previous experience can lead the analysis in a certain direction. This behavior establishes a priori assumptions or conditions that limit the analysis of information in all its extension and that, as a consequence, prevent to obtain certain answers or solutions. Overcoming these obstacles, despite the conditioning posed by previous experience, is what is known as lateral thinking.

To begin with, let’s consider the case of series math puzzles in which a sequence of numbers, characters, or graphics is presented, asking how the sequence continues. For example, given the sequence “IIIIIIIVVV”, we are asked to determine which the next character is. If the Roman culture had not developed, it could be said that the next character is “V”, or also that the sequence has been made by little scribblers. But this is not the case, so the brain begins to engineer determining that the characters can be Roman and that the sequence is that of the numbers “1,2,3,…”.  Consequently, the next character must be “I”.

In this way, it can be seen how the knowledge acquired conditions the interpretation of the information perceived by the senses. But from this example another conclusion can be drawn, consisting of the ordering of information as a sign of intelligence. To expose this idea in a formal way let’s consider a numerical sequence, for example the Fibonacci series “0,1,2,3,5,8,…”. Similarly to the previous case, the following number should be 13, so that the general term can be expressed as fn=fn-1+fn-2. However, we can define another discrete mathematical function that takes the values “0,1,2,3,5,8” for n = 0,1,2,3,4,5, but differs for the rest of the values of n belonging to the natural numbers, as shown in the following figure. In fact, with this criterion it is possible to define an infinite number of functions that meet this criterion.

The question, therefore, is: What is so special about the Fibonacci series in relation to the set of functions that meet the condition defined above?

Here we can make the argument already used in the case of the Roman number series. So that mathematical training leads to identifying the series of numbers as belonging to the Fibonacci series. But this poses a contradiction, since any of the functions that meet the same criterion could have been identified. To clear up this contradiction, Algorithmic Information Theory (AIT) should be used again.

Firstly, it should be stressed that culturally the game of riddles implicitly involves following logical rules and that, therefore, the answer is free from arbitrariness. Thus, in the case of number series the game consists of determining a rule that justifies the result. If we now try to identify a simple mathematical series that determines the sequence “0,1,2,3,5,8,…” we see that the expression fn=fn-1+fn-2 fulfills these requirements. In fact, it is possible that this is the simplest within this type of expressions. The rest are either complex, arbitrary or simple expressions that follow different rules from the implicit rules of the puzzle.

From the AIT point of view, the solution that contains the minimum information and can therefore be expressed most readily will be the most likely response that the brain will give in identifying a pattern determined by a stimulus. In the example above, the description of the predictable solution will be the one composed of:

  • A Turing machine.
  • The information to code the calculus rules.
  • The information to code the analytical expression of the simplest solution. In the example shown it corresponds to the expression of the Fibonacci series.

Obviously, there are solutions of similar or even less complexity, such as the one performed by a Turing machine that periodically generates the sequence “0,1,2,3,5,8”. But in most cases the solutions will have a more complex description, so that, according to the AIT, in most cases their most compact description will be the sequence itself, which cannot be compressed or expressed analytically.

For example, it is easy to check that the function:

generates for integer values of n the sequence “0,1,1,2,3,5,8,0,-62,-279,…”, so it could be said that the quantities following the proposed series are “…,0,-62,-279,… Obviously, the complexity of this sequence is higher than that of the Fibonacci series, as a result of the complexity of the description of the function and the operations to be performed.

Similarly, we can try to define other algorithms that generate the proposed sequence, which will grow in complexity. This shows the possibility of interpreting the information from different points of view that go beyond the obvious solutions, which are conditioned by previous experiences.

If, in addition to all the above, it is considered that, according to Landauer’s principle, information complexity is associated with greater energy consumption, the resolution of complex problems not only requires a greater computational effort, but also a greater energy effort.

This may explain the feeling of satisfaction produced when a certain problem is solved, and the tendency to engage in relaxing activities that are characterized by simplicity or monotony. Conversely, the lack of response to a problem produces frustration and restlessness.

This is in contrast to the idea that is generally held about intelligence. Thus, the ability to solve problems such as the ones described above is considered a sign of intelligence. But on the contrary, the search for more complex interpretations does not seem to have this status. Something similar occurs with the concept of entropy, which is generally interpreted as disorder or chaos and yet from the point of view of information it is a measure of the amount of information.

Another aspect that should be highlighted is the fact that the cognitive process is supported by the processing of information and, therefore, subject to the rules of mathematical logic, whose nature is irrefutable. This nuance is important, since emphasis is generally placed on the physical and biological mechanisms that support the cognitive processes, which may eventually be assigned a spiritual or esoteric nature.

Therefore, it can be concluded that the cognitive process is subject to the nature and structure of information processing and that from the formal point of view of the Theory of Computability it corresponds to a Turing machine. In such a way that nature has created a processing structure based on the physics of emerging reality – classical reality -, materialized in a neural network, which interprets the information coded by the perception senses, according to the algorithmic established by previous experience. As a consequence, the system performs two fundamental functions, as shown in the figure:

  • Interact with the environment, producing a response to the input stimuli.
  • Enhance the ability to interpret, acquiring new skills -algorithmic- as a result of the learning capacity provided by the neural network. 

But the truth is that the input stimuli are conditioned by the sensory organs, which constitute a first filter of information and therefore they condition the perception of reality. The question that can be raised is: What impact does this filtering have on the perception of reality?

On the complexity of PI (π)

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Introduction

There is no doubt that since the origins of geometry humans have been seduced by the number π. Thus, one of its fundamental characteristics is that it determines the relationship between the length of a circumference and its radius. But this does not stop here, since this constant appears systematically in mathematical and scientific models that describe the behavior of nature. In fact, it is so popular that it is the only number that has its own commemorative day. The great fascination around π has raised speculations about the information encoded in its figures and above all has unleashed an endless race for its determination, having calculated several tens of billions of figures to date.

Formally, the classification of real numbers is done according to the rules of calculus. In this way, Cantor showed that real numbers can be classified as countable infinities and uncountable infinities, what are commonly called rational and irrational. Rational numbers are those that can be expressed as a quotient of two whole numbers. While irrational numbers cannot be expressed this way. These in turn are classified as algebraic numbers and transcendent numbers. The former correspond to the non-rational roots of the algebraic equations, that is, the roots of polynomials. On the contrary, transcendent numbers are solutions of transcendent equations, that is, non-polynomial, such as exponential and trigonometric functions.

Georg Cantor. Co-creator of Set Theory

Without going into greater detail, what should catch our attention is that this classification of numbers is based on positional rules, in which each figure has a hierarchical value. But what happens if the numbers are treated as ordered sequences of bits, in which the position is not a value attribute.  In this case, the Algorithmic Information Theory (AIT) allows to establish a measure of the information contained in a finite sequence of bits, and in general of any mathematical object, and that therefore is defined in the domain of natural numbers.

What does the AIT tell us?

This measure is based on the concept of Kolmogorov complexity (KC). So that, the Kolmogorov complexity K(x) of a finite object x is defined as the length of the shortest effective binary description of x. Where the term “effective description” connects the Kolmogorov complexity with the Theory of Computation, so that K(x) would correspond to the length of the shortest program that prints x and enters the halt state. To be precise, the formal definition of K(x) is:

K(x) = minp,i{K(i) + l(p):Ti (p) = x } + O(1)

Where Ti(p) is the Turing machine (TM) i that executes p and prints x, l(p) is the length of p, and K(i) is the complexity of Ti. Therefore, object p is a compressed representation of object x, relative to Ti, since x can be retrieved from p by the decoding process defined by Ti, so it is defined as meaningful information. The rest is considered as meaningless, redundant, accidental or noise (meaningless information). The term O(1) indicates that K(i) is a recursive function and in general it is non-computable, although by definition it is machine independent, and whose result has the same order of magnitude in each one of the implementations. In this sense, Gödel’s incompleteness theorems, Turing machine and Kolmogorov complexity lead to the same conclusion about undecidability, revealing the existence of non-computable functions.

KC shows that information can be compressed, but does not establish any general procedure for its implementation, which is only possible for certain sequences. In effect, from the definition of KC it is demonstrated that this is an intrinsic property of bitstreams, in such a way that there are sequences that cannot be compressed. Thus, the number of n-bit sequences that can be encoded by m bits is less than 2m, so the fraction of n-bit sequences with K(x) ≥ n-k is less than 2-k. If the n-bit possible sequences are considered, each one of them will have a probability of occurrence equal to 2-n, so the probability that the complexity of a sequence is K(x) ≥ n-k is equal to or greater than (1-2-k). In short, most bit sequences cannot be compressed beyond their own size, showing a high complexity as they do not present any type of pattern. Applied to the field of physics, this behavior justifies the ergodic hypothesis. As a consequence, this means that most of the problems cannot be solved analytically, since they can only be represented by themselves and as a consequence they cannot be described in a compact way by means of formal rules.

It could be thought that the complexity of a sequence can be reduced at will, by applying a coding criterion that modifies the sequence into a less complex sequence. In general, this only increases the complexity, since in the calculation of K(x) we would have to add the complexity of the coding algorithm that makes it grow as n2. Finally, add that the KC is applicable to any mathematical object, integers, sets, functions, and it is demonstrated that, as the complexity of the mathematical object grows, K(x) is equivalent to the entropy H defined in the context of Information Theory. The advantage of AIT is that it performs a semantic treatment of information, being an axiomatic process, so it does not require having a priori any type of alphabet to perform the measurement of information.

What can be said about the complexity of π?

According to its definition, KC cannot be applied to irrational numbers, since in this case the Turing machine does not reach the halt state, and as we know these numbers have an infinite number of digits. In other words, and to be formally correct, the Turing machine is only defined in the field of natural numbers (it must be noted that their cardinality is the same as that of the rationals), while irrational numbers have a cardinality greater than that of rational numbers. This means that KC and the equivalent entropy H of irrational numbers are undecidable and therefore non-computable.

To overcome this difficulty we can consider an irrational number X as the concatenation of a sequence of bits composed of a rational number x and a residue δx, so that in numerical terms X=x+δx, but in terms of information X={x,δx}. As a consequence, δx is an irrational number δx→0, and therefore δx is a sequence of bits with an undecidable KC and hence non-computable. In this way, it can be expressed:

K(X) = K(x)+K(δx)

The complexity of X can be assimilated to the complexity of x. A priori this approach may seem surprising and inadmissible, since the term K(δx) is neglected when in fact it has an undecidable complexity. But this is similar to the approximation made in the calculation of the entropy of a continuous variable or to the renormalization process used in physics, in order to circumvent the complexity of the underlying processes that remain hidden from observable reality.

Consequently, the sequence p, which runs the Turing machine i to get x, will be composed of the concatenation of:

  • The sequence of bits that encode the rules of calculus in the Turing machine i.
  • The bitstream that encodes the compressed expression of x, for example a given numerical series of x.
  • The length of the sequence x that is to be decoded and that determines when the Turing machine should reach the halt state, for example a googol (10100).

In short, it can be concluded that the complexity K(x) of known irrational numbers, e.g. √2, π, e,…, is limited. For this reason, the challenge must be to obtain the optimum expression of K(x) and not the figures that encode these numbers, since according to what has been said, their uncompressed expression, or the development of their figures, has a high degree of redundancy (meaningless information).

What in theory is a surprising and questionable fact is in practice an irrefutable fact, since the complexity of δx will always remain hidden, since it is undecidable and therefore non-computable.

Another important conclusion is that it provides a criterion for classifying irrational numbers into two groups: representable and non-representable. The former correspond to irrational numbers that can be represented by mathematical expressions, which would be the compressed expression of these numbers. While non-representable numbers would correspond to irrational numbers that could only be expressed by themselves and are therefore undecidable. In short, the cardinality of representable irrational numbers is that of natural numbers. It should be noted that the previous classification criterion is applicable to any mathematical object.

On the other hand, it is evident that mathematics, and calculus in particular, de facto accepts the criteria established to define the complexity K(x). This may go unnoticed because, traditionally in this context, numbers are analyzed from the perspective of positional coding, in such a way that the non-representable residue is filtered out through the concept of limit, in such a way that δx→0. However, when it comes to evaluating the informative complexity of a mathematical object, it may be required to apply a renormalization procedure.