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The fundamental nature of the concept automatic computations attracted a greatattention of mathematicians (and later of computer scientists) since 1930’s and more recent monographs. At that time, this strong impetus for understanding what is computable was actively supported

by David Hilbert who believed that all of Mathematics could be precisely axiomatized. Several mathematicians from around the world proposed their independent definitions of what it means to be computable and what it means an automatic computing machine. In order to perform a rigorous study of sequential computations, they worked with different mathematical models of computing

machines. Surprisingly, it has been discovered that all of these models were equivalent, e.g., anything

computable in the l-calculus is computable by a Turing machine.

In spite of the fact that the famous results of Church, Godel, and Turing have shown that Hilbert’s programme cannot be realized, the idea of finding an adequate set of axioms for one or another field of Mathematics continues to be among the most attractive goals for contemporary mathematicians as well. Usually, when it is necessary to define a concept or an object, logicians try to introduce a number of

axioms describing the object. However, this way is fraught with danger because of the following reasons.

First, when we describe a mathematical object or concept we are limited by the expressive capacity of the language we use to make this description. A richer language allows us to say more about the object and a weaker language – less. Thus, development of the mathematical (and not only mathematical) languages

leads to a continuous necessity of a transcription and specification of axiomatic systems. Second, there is no guarantee that the chosen axiomatic system defines ‘sufficiently well’ the required concept and a continuous comparison with practice is required in order to check the goodness of the accepted set of axioms. However, there cannot be again any guarantee that the new version will be the last and definitive

one. Finally, the third limitation already mentioned above has been discovered by Godel in his two famous incompleteness theorems.

In linguistics, the relativity of the language with respect to the world around us has been formulated in the form of the Sapir-Whorf thesis also known as the ‘linguistic relativity thesis’ (that has also interesting relations to the ideas of K.E. Iverson exposed in his Turing lecture). As becomes clear from its name, the thesis does not accept the idea of the universality of language and postulates that the nature of a particular language influences the thought of its speakers. The thesis challenges the possibility of perfectly representing the world with language, because it implies that the mechanisms of any language condition

the thoughts of its speakers.

In this paper, we study the relativity of mathematical languages in situations where they are used to observe and to describe automatic computations (we consider the traditional computational paradigm mainly following results of Turing whereas emerging computational paradigms. Let us illustrate the concept of the relativity of mathematical languages by the following example. In his study published in Science, Peter Gordon describes a primitive tribe living in Amazonia – Piraha – that uses a very simple numeral system1 for counting: one, two, ‘many’.

by David Hilbert who believed that all of Mathematics could be precisely axiomatized. Several mathematicians from around the world proposed their independent definitions of what it means to be computable and what it means an automatic computing machine. In order to perform a rigorous study of sequential computations, they worked with different mathematical models of computing

machines. Surprisingly, it has been discovered that all of these models were equivalent, e.g., anything

computable in the l-calculus is computable by a Turing machine.

In spite of the fact that the famous results of Church, Godel, and Turing have shown that Hilbert’s programme cannot be realized, the idea of finding an adequate set of axioms for one or another field of Mathematics continues to be among the most attractive goals for contemporary mathematicians as well. Usually, when it is necessary to define a concept or an object, logicians try to introduce a number of

axioms describing the object. However, this way is fraught with danger because of the following reasons.

First, when we describe a mathematical object or concept we are limited by the expressive capacity of the language we use to make this description. A richer language allows us to say more about the object and a weaker language – less. Thus, development of the mathematical (and not only mathematical) languages

leads to a continuous necessity of a transcription and specification of axiomatic systems. Second, there is no guarantee that the chosen axiomatic system defines ‘sufficiently well’ the required concept and a continuous comparison with practice is required in order to check the goodness of the accepted set of axioms. However, there cannot be again any guarantee that the new version will be the last and definitive

one. Finally, the third limitation already mentioned above has been discovered by Godel in his two famous incompleteness theorems.

In linguistics, the relativity of the language with respect to the world around us has been formulated in the form of the Sapir-Whorf thesis also known as the ‘linguistic relativity thesis’ (that has also interesting relations to the ideas of K.E. Iverson exposed in his Turing lecture). As becomes clear from its name, the thesis does not accept the idea of the universality of language and postulates that the nature of a particular language influences the thought of its speakers. The thesis challenges the possibility of perfectly representing the world with language, because it implies that the mechanisms of any language condition

the thoughts of its speakers.

In this paper, we study the relativity of mathematical languages in situations where they are used to observe and to describe automatic computations (we consider the traditional computational paradigm mainly following results of Turing whereas emerging computational paradigms. Let us illustrate the concept of the relativity of mathematical languages by the following example. In his study published in Science, Peter Gordon describes a primitive tribe living in Amazonia – Piraha – that uses a very simple numeral system1 for counting: one, two, ‘many’.

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