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Foundations of mathematics are the logical and mathematical framework that allows the development of 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, although foundations were first established by the ancient Greek philosophers under the name of Aristotle's logic and systematically applied in Euclid's Elements. 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 were tacitly assumed to be definitive until the introduction of infinitesimal calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. This new area of mathematics involved new methods of reasoning and new basic concepts (continuous functions, derivatives, limits) that were not well founded, but had astonishing consequences, such as the deduction from Newton's law of gravitation that the orbits of the planets are ellipses.
During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the natural and real numbers. This led, near the end of the 19th century, to a series of seemingly paradoxical mathematical results that challenged the general confidence in the 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, parts of computer science. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of axiomatic method and on set theory, specifically ZFC, the Zermelo–Fraenkel set theory with the axiom of choice.
It results from this that the basic mathematical concepts, such as numbers, points, lines, and geometrical spaces are not defined as abstractions from reality but from basic properties (axioms). Their adequation with their physical origins does not belong to mathematics anymore, although their relation with reality is still used for guiding mathematical intuition: physical reality is still used by mathematicians to choose axioms, find which theorems are interesting to prove, and obtain indications of possible proofs.