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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 may also include the philosophical study of the relation of this framework with reality.[1]
The term "foundations of mathematics" was not coined before the end of the 19th century. However, there were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements. In short, a mathematical assertion is considered as truth only if it is a theorem that is proved from true premises by means of a sequence of syllogisms (inference rules), the premises being either already proved theorems or self-evident assertions called axioms or postulates.
These foundations seemed to be a definitive achievement until the 17th century, and the introduction of infinitesimal calculus by Isaac Newton and Gottfried Wilhelm Leibniz. This new area of mathematics involved new methods of reasoning and new basic concepts (continuous functions, derivatives, limits) that were not well founded, but have astonishing consequences, such as the fact that one can deduce from Newton's law of gravitation that the orbits of the planets are ellipses.
During the 19th century, several mathematicians worked for elaborating precise definitions for the basic concepts of infinitesimal calculus, including the definitions of natural and real numbers. This led, near the end of the 19th century, to a series of paradoxical mathematical results that challenged the general confidence in reliability and truth of mathematical results. This has been called the foundational crisis of mathematics.
The resolution of this crisis involved the rise of 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.