## Table of contents

**Mathematics** (from the Greek μάθημα (máthema), which can be translated as “science”, “knowledge”, or “learning”; μαθηματικός (mathematikós) means “inclined to learn”) is the discipline that studies quantities, numbers, space, structures, and calculations.

The term mathematics usually refers to the discipline (and its body of knowledge) that studies problems involving quantities, spatial extensions and figures, movements of bodies, and all the structures that allow us to treat these aspects in a general way. Mathematics makes extensive use of the tools of logic and develops its knowledge in the framework of hypothetical-deductive systems that, starting from rigorous definitions and axioms concerning properties of defined objects (results of a process of abstraction, such as triangles, functions, vectors, etc.), reach new certainties, by means of demonstrations, around less intuitive properties of the objects themselves (expressed by theorems).

The power and generality of the results of mathematics has given it the appellation of queen of the sciences: every scientific or technical discipline, from physics to engineering, from economics to computer science, makes extensive use of the tools of analysis, calculation and modeling offered by mathematics.

## Branches of mathematics

- Number theory (arithmetic)
- Geometry
- Algebra
- Calculus and analysis
- Discrete mathematics
- Mathematical logic and set theory
- Applied mathematics
- Statistics and other decision sciences
- Computational mathematics

## Historical notes

### From the origins to the nineteenth century

The problem of the foundations of mathematics has been present since antiquity and in this sense the problem should be framed in the broader theory of knowledge. But in the nineteenth century the rise of non-Euclidean geometry, on the one hand, and the tendency of mathematicians of that century to make arithmetic and analysis more and more rigorous, on the other hand, determined a new interest and a deeper approach to the problem of the foundations of mathematics, which no longer involved simply the problem of the insertion of this science in a wider philosophical context, but the mathematical work itself.

The need for greater rigor towards arithmetic and analysis were translated into the need for their axiomatization and the use of the concepts of the theory of natural numbers as a basis for defining the concepts of arithmetic and analysis. In this way, the problem of justifying the principles and the assertions of mathematics in a rigorous way came to the foreground; that is, it was necessary to realize what was meant when it was stated that certain principles were evident, to explain the reasons why principles that were not completely evident were accepted and to find and then abandon those principles that could not be justified.

Contributing to the new attitude towards mathematics was the end of the dominance of geometry, due to the loss of the absolute and descriptive character of its axioms and the reformulation on abstract bases of notions, such as those of continuity and real number, previously formulated on geometric bases. Thus, towards the end of the century, two contrasting positions emerged. On one side were the works of R. Dedekind, the set theory of G. Cantor, the attempt of G. Frege to found mathematics on purely logical bases. On the other was L. Kronecker and his constructivist conception of mathematics.

For Dedekind, Cantor, etc.. it was possible to consider as mathematically meaningful any notion specifiable in terms of abstract set theory (think of the theory of cardinals and ordinals of Cantor, that of ideals of Dedekind); for Kronecker, on the contrary, all these developments were meaningless, mere verbal games, as it was not possible for all these theories to provide a translation in terms of properties of natural numbers.

### The twentieth century

At the beginning of the twentieth century, the discovery of antinomies highlighted how the notions of class and set, which were at the base of the foundational attempts of Cantor and Frege, led to contradictions if used without restrictions. Thus opened the crisis of the foundations as it was questioned both the reconstruction on logical bases of mathematics, made by Frege, and the foundation of an abstract mathematics Cantor hinged precisely on the notion of set. The research then turned to address what were the problems raised by the antinomies and their possible elimination, but without achieving satisfactory results.

Then emerged three main lines of research whose point of contact can be identified in the attempt to avoid those assumptions that led to the emergence of antinomies. They were: logicism, which especially through the work of B. Russell resumed Frege’s program, overcoming the antinomies with the construction of the theory of types; intuitionism, supported by L. E. J. Brouwer, who, taking up the reservations of Kronecker, set himself the task of building a mathematics in many ways different from the traditional one and based on the concept of mental construction (defined step by step); formalism, whose greatest exponent in the twenties was D. Hilbert, who tried to justify classical mathematics on the basis of the concept of formal system.

But even these attempts to find a final solution to the problems of the foundations of mathematics were frustrated by the discovery of the incompleteness theorem by K. Gödel (1931). This result put an end to the general programs of founding mathematics in the sense of (formal) mathematical demonstration of non-contradictory formal theories. The research on the various topics contained in them continued and it can be seen that in the logical research after 1930 the problem of foundations passes in the background, but without dissolving. In particular, the antinomies of set theory are satisfactorily overcome by the axiomatic theories of Zermelo-Fraenkel and von Neumann-Bernays.