Editing Foundations of mathematics (section)

== 19th century ==
In the 19th century, mathematics developed quickly in many directions. Several of the problems that were considered led to questions on the foundations of mathematics. Frequently, the proposed solutions led to further questions that were often simultaneously of philosophical and mathematical nature. All these questions led, at the end of the 19th century and the beginning of the 20th century, to debates which have been called the [[#Foundational crisis|foundational crisis of mathematics]]. The following subsections describe the main such foundational problems revealed during the 19th century.

=== Real analysis === 
{{See also|Mathematical analysis#History}}

[[Cauchy]] (1789–1857) started the project of giving rigorous bases to [[infinitesimal calculus]]. In particular, he rejected the heuristic principle that he called the [[generality of algebra]], which consisted to apply properties of [[algebraic operation]]s to [[infinite sequences]] without proper proofs. In his ''[[Cours d'Analyse]]'' (1821), he considered ''very small quantities'', which could presently be called "sufficiently small quantities"; that is, a sentence such that "if {{mvar|x}} is very small {{nowrap|then ..."}} must be understood as "there is a (sufficiently large) [[natural number]] {{mvar|n}} such that {{math|{{abs|''x''}} < 1/''n''}}". In the proofs he used this in a way that predated the modern [[(ε, δ)-definition of limit]].<ref name="Grabiner1983">{{Citation
| last = Grabiner | first = Judith V.
| title = Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus
| journal = [[American Mathematical Monthly]]
| volume = 90
| issue = 3
| pages = 185–194
| doi = 10.2307/2975545
| year = 1983
| jstor = 2975545
}}, collected in [http://www.maa.org/ebooks/spectrum/WGE.html Who Gave You the Epsilon?], {{isbn|978-0-88385-569-0}} pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf</ref>

The modern [[(ε, δ)-definition of limit]]s and [[continuous functions]] was first developed by [[Bernard Bolzano|Bolzano]] in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work.

[[Karl Weierstrass]] (1815–1897) formalized and popularized the (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as [[Weierstrass function|continuous, nowhere-differentiable functions]]. Indeed, such functions contradict previous conceptions of a function as a rule for computation or a smooth graph. 

At this point, the program of [[arithmetization of analysis]] (reduction of [[mathematical analysis]] to arithmetic and algebraic operations) advocated by Weierstrass was essentially completed, except for two points.

Firstly, a formal definition of real numbers was still lacking. Indeed, beginning with [[Richard Dedekind]] in 1858, several mathematicians worked on the definition of the real numbers, including [[Hermann Hankel]], [[Charles Méray]], and [[Eduard Heine]], but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means of [[Dedekind cut]]s; the other one by [[Georg Cantor]] as equivalence classes of [[Cauchy sequences]].<ref>{{MacTutor| class=HistTopics| id=Real_numbers_2| title=The real numbers: Stevin to Hilbert|date=October 2005}}</ref> 

Several problems were left open by these definitions, which contributed to the [[foundational crisis of mathematics]]. Firstly both definitions suppose that [[rational number]]s and thus [[natural number]]s are rigorously defined; this was done a few years later with [[Peano axioms]]. Secondly, both definitions involve [[infinite set]]s (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's [[set theory]] was published several years later.

The third problem is more subtle: and is related to the foundations of logic: classical logic is a [[first-order logic]]; that is, [[quantifier (logic)|quantifier]]s apply to variables representing individual elements, not to variables representing (infinite) sets of elements. The basic property of the [[completeness of the real numbers]] that is required for defining and using real numbers involves a quantification on infinite sets. Indeed, this property may be expressed either as ''for every infinite sequence of real numbers, if it is a [[Cauchy sequence]], it has a limit that is a real number'', or as ''every subset of the real numbers that is [[bounded set|bounded]] has a [[least upper bound]] that is a real number''. This need of quantification over infinite sets is one of the motivation of the development of [[higher-order logic]]s during the first half of the 20th century.

=== Non-Euclidean geometries ===
{{See also|Non-Euclidean geometry#History}}

Before the 19th century, there were many failed attempts to derive the [[parallel postulate]] from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction, [[Johann Heinrich Lambert]] (1728–1777) started to build [[hyperbolic geometry]] and introduced the [[hyperbolic functions]] and computed the area of a [[hyperbolic triangle]] (where the sum of angles is less than 180°). 

Continuing the construction of this new geometry, several mathematicians proved independently that if it is [[inconsistent]], then [[Euclidean geometry]] is also inconsistent and thus that the parallel postulate cannot be proved. This was proved by [[Nikolai Lobachevsky]] in 1826, [[János Bolyai]] (1802–1860) in 1832 and [[Carl Friedrich Gauss]] (unpublished).

Later in the 19th century, the German mathematician [[Bernhard Riemann]] developed [[Elliptic geometry]], another [[non-Euclidean geometry]] where no parallel can be found and the sum of angles in a triangle is more than 180°. It was proved consistent by defining points as pairs of [[antipodal point]]s on a sphere (or [[hypersphere]]), and lines as [[great circle]]s on the sphere. 

These proofs of unprovability of the parallel postulate lead to several philosophical problems, the main one being that before this discovery, the parallel postulate and all its consequences were considered as ''true''. So, the non-Euclidean geometries challenged the concept of [[mathematical truth]].

=== Synthetic vs. analytic geometry ===

Since the introduction of [[analytic geometry]] by [[René Descartes]] in the 17th century, there were two approaches to geometry, the old one called [[synthetic geometry]], and the new one, where everything is specified in terms of real numbers called [[coordinates]].

Mathematicians did not worry much about the contradiction between these two approaches before the mid-nineteenth century, where there was "an acrimonious controversy between the proponents of synthetic and analytic methods in [[projective geometry]], the two sides accusing each other of mixing projective and metric concepts".<ref>Laptev, B.L. & B.A. Rozenfel'd (1996) ''Mathematics of the 19th Century: Geometry'', page 40, [[Springer Science+Business Media|Birkhäuser]] {{ISBN|3-7643-5048-2}}</ref> Indeed, there is no concept of distance in a [[projective space]], and the [[cross-ratio]], which is a number, is a basic concept of synthetic projective geometry. 

[[Karl von Staudt]] developed a purely geometric approach to this problem by introducing "throws" that form what is presently called a [[field (mathematics)|field]], in which the cross ratio can be expressed.

Apparently, the problem of the equivalence between analytic and synthetic approach was completely solved only with [[Emil Artin]]'s book ''[[Geometric Algebra (book)|Geometric Algebra]]'' published in 1957. It was well known that, given a [[field (mathematics)|field]] {{mvar|k}}, one may define [[affine space|affine]] and projective spaces over {{mvar|k}} in terms of {{mvar|k}}-[[vector space]]s. In these spaces, the [[Pappus hexagon theorem]] holds. Conversely, if the Pappus hexagon theorem is included in the axioms of a plane geometry, then one can define a field {{mvar|k}} such that the geometry is the same as the affine or projective geometry over {{mvar|k}}.

=== Natural numbers ===
{{Main|Peano arithmetic}}

The work of [[#Real analysis|making rigorous real analysis and the definition of real numbers]], consisted of reducing everything to [[rational number]]s and thus to [[natural number]]s, since positive rational numbers are fractions of natural numbers. There was therefore a need of a formal definition of natural numbers, which imply as [[axiomatic theory]] of [[arithmetic]]. This was started with [[Charles Sanders Peirce]] in 1881 and [[Richard Dedekind]] in 1888, who defined a natural numbers as the [[cardinality]] of a [[finite set]].<ref>{{cite book |last1=Dedekind |first1=Richard |title=What Are and What Should the Numbers Be? Continuity and Irrational Numbers |publisher=Springer |isbn=978-3-662-70059-4}}</ref> However, this involves [[set theory]], which was not formalized at this time.

[[Giuseppe Peano]] provided in 1888 a complete axiomatisation based on the [[ordinal numeral|ordinal]] property of the natural numbers. The last Peano's axiom is the only one that induces logical difficulties, as it begin with either "if {{mvar|S}} is a set then" or "if <math>\varphi</math> is a [[predicate (mathematical logic)|predicate]] then". So, Peano's axioms induce a [[quantification (logic)|quantification]] on infinite sets, and this means that Peano arithmetic is what is presently called a [[Second-order logic]].

This was not well understood at that times, but the fact that [[infinity]] occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example, [[Henri Poincaré]] stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.<ref>{{cite book |last1=Poincaré |first1=Henri|translator1-first=William John|translator1-last= Greenstreet |title=La Science et l'hypothèse|trans-title=Science and Hypothesis|orig-date=1902|date=1905|chapter=On the nature of mathematical reasoning|chapter-url=https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_1|at=VI}}</ref> This applies in particular to the use of the last Peano axiom for showing that the [[successor function]] generates all natural numbers. Also, [[Leopold Kronecker]] said "God made the integers, all else is the work of man".{{efn|The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."<ref>{{cite book
 |last=Gray |first=Jeremy |author-link=Jeremy Gray
 |year=2008
 |title=Plato's Ghost: The modernist transformation of mathematics
 |page=153
 |publisher=Princeton University Press
 |isbn=978-1-4008-2904-0
 |via=Google Books
 |url=https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22
 |url-status=live
 |archive-url=https://web.archive.org/web/20170329150904/https://books.google.com/books?id=ldzseiuZbsIC&q=%22God+made+the+integers%2C+all+else+is+the+work+of+man.%22#v=snippet&q=%22God%20made%20the%20integers%2C%20all%20else%20is%20the%20work%20of%20man.%22&f=false
 |archive-date=29 March 2017
}}</ref><ref>{{cite book
 |last=Weber |first=Heinrich L.
 |year=1891–1892
 |chapter=Kronecker
 |chapter-url=http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6 
 |archive-url=https://web.archive.org/web/20180809110042/http://www.digizeitschriften.de/dms/img/?PPN=PPN37721857X_0002&DMDID=dmdlog6
 |archive-date=9 August 2018
 |title=''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |trans-title=Annual report of the German Mathematicians Association
 |pages=2:5–23. (The quote is on p.&nbsp;19)
 |postscript=;
}} {{cite web
 |title=access to ''Jahresbericht der Deutschen Mathematiker-Vereinigung''
 |url=http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-url=https://web.archive.org/web/20170820201100/http://www.digizeitschriften.de/dms/toc/?PPN=PPN37721857X_0002
 |archive-date=20 August 2017 
}}</ref>}} This may be interpreted as "the integers cannot be mathematically defined".

===Infinite sets===
Before the second half of the 19th century, [[infinity]] was a philosophical concept that did not belong to mathematics. However, with the rise of [[infinitesimal calculus]], mathematicians became accustomed  to infinity, mainly through [[potential infinity]], that is, as the result of an endless process, such as the definition of an [[infinite sequence]], an [[infinite series]] or a [[limit (mathematics)|limit]]. The possibility of an [[actual infinity]] was the subject of many philosophical disputes.

[[Set (mathematics)|Set]]s, and more specially [[infinite set]]s were not considered as a mathematical concept; in particular, there was no fixed term for them. A dramatic change arose with the work of [[Georg Cantor]] who was the first mathematician to systematically study infinite sets. In particular, he introduced [[cardinal number]]s that measure the size of infinite sets, and [[ordinal number]]s that, roughly speaking, allow one to continue to count after having reach infinity. One of his major results is the discovery that there are strictly more real numbers than natural numbers (the cardinal of the [[continuum (set theory)|continuum]] of the real numbers is greater than that of the natural numbers).

These results were rejected by many mathematicians and philosophers, and led to debates that are a part of the [[#Foundational crisis of mathematics|foundational crisis of mathematics]].

The crisis was amplified with the [[Russel's paradox]] that asserts that the phrase "the set of all sets" is self-contradictory. This condradiction introduced a doubt on the [[consistency]] of all mathematics.

With the introduction of the [[Zermelo–Fraenkel set theory]] ({{circa|1925}}) and its adoption by the mathematical community, the doubt about the consistency was essentially removed, although consistency of set theory cannot be proved because of [[Gödel's incompleteness theorem]].

=== Mathematical logic ===

In 1847, [[Augustus De Morgan|De Morgan]] published his [[De Morgan's laws|laws]] and [[George Boole]] devised an algebra, now called [[Boolean algebra]],   that allows expressing [[Aristotle's]] logic in terms of formulas and [[algebraic operation]]s. Boolean algebra is the starting point of mathematization  [[logic]] and the basis of [[propositional calculus]]

Independently, in the 1870's, [[Charles Sanders Peirce]] and [[Gottlob Frege]] extended propositional calculus by introducing  [[Quantifier (logic)|quantifiers]], for building [[predicate logic]].

Frege pointed out three desired properties of a logical theory:{{Citation needed|date=April 2020|reason=Please provide a reference as these ideas are normally attributed to David Hilbert}}[[consistency]] (impossibility of proving contradictory statements), [[Completeness (logic)|completeness]] (any statement is either provable or refutable; that is, its negation is provable), and [[Decidability (logic)|decidability]] (there is a decision procedure to test every statement).

By near the turn of the century, [[Bertrand Russell]] popularized Frege's work and discovered [[Russel's paradox]] which implies that the phrase ''"the set of all sets"'' is self-contradictory. This paradox seemed to make the whole mathematics inconsistent  and is one of the major causes of the foundational crisis of mathematics.