Editing Foundations of mathematics (section)

{{Short description|Basic framework of mathematics}}
{{distinguish|Foundations of Mathematics (book)}}
{{Math topics TOC|expanded=Pure mathematics}}
{{mcn|date=May 2024}}

'''Foundations of mathematics''' are the [[mathematical logic|logical]] and [[mathematics|mathematical]] framework that allows the development of mathematics without generating [[consistency|self-contradictory theories]], and to have reliable concepts of [[theorem]]s, [[proof (mathematics)|proofs]], [[algorithm]]s, etc. in particular. This may also include the [[philosophy of mathematics|philosophical]] study of the relation of this framework with [[reality]].<ref>[https://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics Joachim Lambek (2007), "Foundations of mathematics", ''Encyclopædia Britannica'']</ref> 

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|Euclid's ''Elements'']]. A mathematical assertion is considered as [[truth (mathematics)|truth]] only if it is a <em>theorem</em> that is <em>proved</em> from true [[premise]]s by means of a sequence of [[syllogism]]s ([[inference rule]]s), the premises being either already proved theorems or self-evident assertions called [[axiom]]s or [[postulate]]s.

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 function]]s, [[derivative]]s, [[limit (mathematics)|limit]]s) that were not well founded, but had astonishing consequences, such as the deduction from [[Newton's law of gravitation]] that the [[orbit]]s of the planets are [[ellipse]]s. 

During the 19th century, progress was made towards elaborating precise definitions of the basic concepts of infinitesimal calculus, notably the [[natural number|natural]] and [[real number|real]] numbers. This led to a series of seemingly [[paradoxical]] mathematical results near the end of the 19th century 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 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 [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]].

It results from this that the basic mathematical concepts, such as [[number]]s, [[point (geometry)|point]]s, [[line (geometry)|line]]s, and [[geometrical space]]s are not defined as abstractions from reality but from basic properties ([[axiom]]s). 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.