Tag Archives: Language

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.


[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.

Natural language: A paradigm of axiomatic processing

The Theory of Computation (TC) aims to establish computational models and determine the limits of what is computable and the complexity of a problem when it is computable. The formal models established by TC are based on abstract systems ranging from simple models, such as automatons, to the general computer model established by the Turing Machine (TM).

Formally, the concept of algorithm is based on TM, so that each of the possible implementations will perform a specific function that we call algorithm. The TC demonstrates that it is possible to build an idealized machine, called Universal Turing Machine (UTM), capable of executing all possible computable algorithms. In the case of commercial computers, these are equivalent to UTM, with the difference that their memory and runtime are limited. On the contrary, in the UTM these resources are unlimited.

But the question we can ask is: What does this have to do with language? The answer is simple. In TC, an L(TM) language is defined as the set of bit sequences that “accepts” a given TM. In which the term “accept” means that the TM analyzes the input sequence and reaches the Halt state. Consequently, a language is the set of mathematical objects accepted by a given TM.

Without going into details that can be consulted in the specialized literature, the TC classifies the languages into two basic types, as shown in the figure. Thus, a language is Turing-decidable (DEC) when the TM accepts the sequences belonging to the language and rejects the rest, reaching the Halt state in both cases. On the contrary, a language is Turing-recognizable or RE if it is the language of a TM. This means that, for the set of languages belonging to RE but not belonging to DEC, TM does not reach the Halt state when the input sequence does not correspond to the language.

It is necessary to emphasize that there are sequences that are not recognized by any TM. Therefore, if the formal definition of language is taken into account, they should not be considered as such, although in general they are defined as non-RE languages. It is important to note that the latter concept is equivalent to Gödel’s incompleteness theorem. As a consequence, they are the set of undecidable or unsolvable problems, that is, they have a cardinality superior to the one of the natural numbers.

Within DEC languages, two types, regular ​​and context-free (CFL) can be identified. Regular languages ​​are those composed of a set of sequences on which the TM can decide individually, so they do not have a grammatical structure. Examples of these are the languages ​​of the automatons we handle every day, such as elevators, device controls, etc. CFLs are those that have a formal structure (grammar) in which language elements can be nested recursively. In general, we can consider CFLs to programming languages, such as JAVA, C ++. This is not strictly true, but it will facilitate the exposure of certain concepts.

But the question is: What does this have to do with natural language? The answer is easy again. Natural language is, in principle, a Turing-decidable language. The proof of this is trivial. Maybe a few decades ago this was not so, but nowadays information technology shows it us clearly, without the need for theoretical knowledge. On the one hand, natural language is a sequence of bits, since both spoken and written language are coded as bit sequences in audio and text files, respectively. On the other hand, humans do not loop when we get a message, at least permanently ;-).

However, it can be argued that we did not reach the Halt state either. But in this context, this does not mean that we literally end our existence, although there are messages that kill! This means that information processing concludes and that, as a result, we can make a decision and tackle a new task.

Therefore, from an operational or practical point of view, natural language is Turing-decidable. But we can find arguments that can be in conflict with this and that materialize in the form of contradictions. Although it may seem surprising, this also happens with programming languages, since their grammar may be context sensitive (CSG). But for now, we are going to leave aside this aspect, in order to make the reasoning easier.

What can intuitively be seen is a clear parallel between the TM model and the human communication model, as shown in the figure. This can be extended to other communication models, such as body language, physicochemical language between molecules, etc.

In the case of TC, the input and output objects to the TM are language elements, which is very suitable because the practical objective is human-to-machine or machine-to-machine communication. But this terminology varies with the context. Thus, from an abstract point of view, objects have a purely mathematical nature. However, in other contexts such as physics, we talk about concepts such as space-time, energy, momentum, etc.

What seems to be clear, from the observable models, is that a model of reality is equivalent to bit sequences processed by a TM. In short, a model of reality is equivalent to an axiomatic processing of information, where the axioms are embedded in the TM. It should be clear that an axiom is not self-evident, and therefore does not need proof. On the contrary, an axiom is a proposition assumed within a theoretical body. Possibly, this misunderstanding is originated by the apparent simplicity of some axiomatic systems, produced by our perception of reality. This is obvious, for example, in Euclidean geometry based on five postulates or axioms, in which such postulates seem to us evident, due to our perception of space. On this, we will continue to insist since the axiomatic processing is surely one of the great mysteries that nature encloses.

Returning to natural language, it should be possible to establish a parallelism between it and the axiomatic processing determined by TM, as suggested in the figure. As with programming languages, the structure of natural language is defined by a grammar, which establishes a set of axiomatic rules that determine the categories (verb, predicate) of the elements of language (lexicon) and how they are combined to form expressions (sentences). Both the elements of language and the resulting expressions have a meaning, which is known as semantics of language. The pertinent question is: What is the axiomatic structure of a natural language?

To answer, let’s reorient the question: How is the semantics of natural language defined? To do this, we can begin by analyzing the definition of the lexicon of a language, collected in the dictionary. In it we can find the definition of the meaning of each word in different contexts. But we soon find a formal problem, since the definitions are based on one another in a circular fashion. What is the same, the defined is part of the definition, so it is not possible to establish the semantics of language from the linguistic information.

For example, according to the Oxford dictionary:

  • Word: A single distinct meaningful element of speech or writing, used with others (or sometimes alone) to form a sentence and typically shown with a space on either side when written or printed.
  • Write: Mark (letters, words, or other symbols) on a surface, typically paper, with a pen, pencil, or similar implement. 
  • Sentence: A set of words that is complete in itself, typically containing a subject and predicate, conveying a statement, question, exclamation, or command, and consisting of a main clause and sometimes one or more subordinate clauses. 
  • Statement: A definite or clear expression of something in speech or writing
  • Expression: A word or phrase, especially an idiomatic one, used to convey an idea. 
  • Phrase: A small group of words standing together as a conceptual unit, typically forming a component of a clause


  • Word: A single distinct … or marks (letters, words, or other symbols) on … to form a set of words that … conveying a definite or clear word or a small group of words standing together … or marking (letters, words, …. ) …

In this way, we could continue recursively replacing the meaning of each component within the definition, arriving at the conclusion that natural language as an isolated entity has no meaning. So it is necessary to establish an axiomatic basis external to the language itself. By the way: What will happen if we continue to replace each component of the sentence?

Consequently, we can rise what will be the result of an experiment in which an entity of artificial intelligence disconnected from all reality, except from the information on which the written language is based, analyzes the information. That is, the entity will have access to grammar, dictionary, written works, etc. What will be the result of the experiment? To what conclusions will the entity arrive?

If we mentally perform this experiment, we will see that the entity can come to understand the reality of language, and all the stories based on it, provided that it has an axiomatic basis. Otherwise, the entity will experience what in information theory is known as “information without meaning”. This explains the impossibility of deciphering archaic scripts without having cross-references to other languages ​​or other forms of expression. In the case of humans, the axiomatic basis is acquired from cognitive experiences external to the language itself.

To clarify the idea of what the axiomatic processing means, we can use simple examples related to programming languages. So, let’s analyze the semantics of the “if… then” statement. If we consult the programming manual we can determine its semantics, since in our brain we have implemented rules or axioms to decipher the written message. This is equivalent to what happens in the execution of program sentences, in which it is the TM that executes those expressions axiomatically. In the case of both the brain and TM, axioms are defined in the fields of biochemistry and physics, respectively, and therefore outside the realm of language.

This shows once again how reality is structured in functional layers, which can be seen as independent entities by means of the axiomatic processing, as has been analyzed in:

But this issue, as well as the analysis of the existence of linguistic contradictions, will be addressed in later posts.