Foundations of mathematics

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In a narrow sense, foundations of mathematics is the logical and mathematical framework that allows developing mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This includes also the philosophical study of the relation of this framework with the reality.^[1]

In a broader sense, foundations of mathematics include the philosophical theories concerning the nature of mathematics.^[2] In this latter sense, the distinction between foundations of mathematics and philosophy of mathematics turns out to be vague.

The foundations of mathematics were rarely studied before the end of the 19th century, except by the ancient Greek philosophers who established logic and its inference rules, as well as the metamathematical concepts of theorems and proofs. At the end of the 19th century, a series of paradoxical mathematical results challenged the general confidence in reliability and truth of mathematicsl results, leading to the foundational crisis of mathematics. This led to a new mathematical discipline called mathematical logic that includes set theory, model theory, proof theory, computability and computational complexity theory, and more recently, several parts of computer science. During the 20th century, the discoveries done in this area stabilized the foundations of mathematics into a coherent framework valid for all mathematics, that is based on ZFC, the Zermelo–Fraenkel set theory with the axiom of choice, and on a systematic use of axiomatic method.

It results from this that the basic mathematical concepts, such as numbers, points, lines, and geometrical spaces are no more defined as abstractions from reality; they are defined by their basic properties (axioms) only. Their adequation with their physical origin does not belong to mathematics anymore, although their relation with the physical reality, is still used by mathematicians for the choice of the axioms, to find which theorems are interesting to prove, and to get indications on possible proofs; in short the relation with reality is used for guiding mathematical intuition.