Foundations of mathematics
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Foundations of mathematics are the
logical Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arg ...
and
mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s, proofs,
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
s, etc. in particular. This may also include the
philosophical Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
study of the relation of this framework with
reality Reality is the sum or aggregate of everything in existence; everything that is not imagination, imaginary. Different Culture, cultures and Academic discipline, academic disciplines conceptualize it in various ways. Philosophical questions abo ...
. 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 Ancient Greek philosophy arose in the 6th century BC. Philosophy was used to make sense of the world using reason. It dealt with a wide variety of subjects, including astronomy, epistemology, mathematics, political philosophy, ethics, metaphysics ...
under the name of
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
's logic and systematically applied in Euclid's ''Elements''. A mathematical assertion is considered as
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
only if it is a theorem that is proved from true
premise A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of a set of premises and a conclusion. An argument is meaningf ...
s by means of a sequence of
syllogism A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (defin ...
s (
inference rule Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the co ...
s), the premises being either already proved theorems or self-evident assertions called
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s or postulates. These foundations were tacitly assumed to be definitive until the introduction of
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
by
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
in the 17th century. This new area of mathematics involved new methods of reasoning and new basic concepts (
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s,
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s, limits) that were not well founded, but had astonishing consequences, such as the deduction from
Newton's law of gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the s ...
that the
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
s of the planets are
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...
s. 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 to a series of seemingly
paradoxical A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
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 Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...
. The resolution of this crisis involved the rise of a new mathematical discipline called
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
that includes
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
,
model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...
,
proof theory Proof theory is a major branchAccording to , proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. consists of four corresponding parts, with part D being about "Proof The ...
,
computability Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is c ...
and
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
, and more recently, parts of
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied 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 In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establis ...
and on set theory, specifically
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
. It results from this that the basic mathematical concepts, such as
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s, points, lines, and geometrical spaces are not defined as abstractions from reality but from basic properties (
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
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.


Ancient Greece

Most civilisations developed some mathematics, mainly for practical purposes, such as counting (merchants),
surveying Surveying or land surveying is the technique, profession, art, and science of determining the land, terrestrial Plane (mathematics), two-dimensional or Three-dimensional space#In Euclidean geometry, three-dimensional positions of Point (geom ...
(delimitation of fields), prosody,
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, and
astrology Astrology is a range of Divination, divinatory practices, recognized as pseudoscientific since the 18th century, that propose that information about human affairs and terrestrial events may be discerned by studying the apparent positions ...
. It seems that ancient Greek philosophers were the first to study the nature of mathematics and its relation with the real world.
Zeno of Elea Zeno of Elea (; ; ) was a pre-Socratic Greek philosopher from Elea, in Southern Italy (Magna Graecia). He was a student of Parmenides and one of the Eleatics. Zeno defended his instructor's belief in monism, the idea that only one single en ...
() produced several paradoxes he used to support his thesis that movement does not exist. These paradoxes involve mathematical infinity, a concept that was outside the mathematical foundations of that time and was not well understood before the end of the 19th century. The Pythagorean school of mathematics originally insisted that the only numbers are natural numbers and ratios of natural numbers. The discovery () that the ratio of the diagonal of a square to its side is not the ratio of two natural numbers was a shock to them which they only reluctantly accepted. A testimony of this is the modern terminology of
irrational number In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
for referring to a number that is not the quotient of two integers, since "irrational" means originally "not reasonable" or "not accessible with reason". The fact that length ratios are not represented by rational numbers was resolved by
Eudoxus of Cnidus Eudoxus of Cnidus (; , ''Eúdoxos ho Knídios''; ) was an Ancient Greece, ancient Greek Ancient Greek astronomy, astronomer, Greek mathematics, mathematician, doctor, and lawmaker. He was a student of Archytas and Plato. All of his original work ...
(408–355 BC), a student of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
, who reduced the comparison of two irrational ratios to comparisons of integer multiples of the magnitudes involved. His method anticipated that of
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
s in the modern definition of real numbers by Richard Dedekind (1831–1916); see . In the ''
Posterior Analytics The ''Posterior Analytics'' (; ) is a text from Aristotle's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguished as ''a syllogism productive of scientific knowledge'', while the de ...
'',
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
(384–322 BC) laid down the
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
for organizing a field of knowledge by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry, and his logic served as the foundation of mathematics for centuries. This method resembles the modern
axiomatic method In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establis ...
but with a big philosophical difference: axioms and postulates were supposed to be true, being either self-evident or resulting from
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
s, while no other truth than the correctness of the proof is involved in the axiomatic method. So, for Aristotle, a proved theorem is true, while in the axiomatic methods, the proof says only that the axioms imply the statement of the theorem. Aristotle's logic reached its high point with
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's ''Elements'' (300 BC), a treatise on mathematics structured with very high standards of rigor: Euclid justifies each proposition by a demonstration in the form of chains of
syllogisms A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. In its earliest form (define ...
(though they do not always conform strictly to Aristotelian templates). Aristotle's syllogistic logic, together with its exemplification by Euclid's ''Elements'', are recognized as scientific achievements of ancient Greece, and remained as the foundations of mathematics for centuries.


Before infinitesimal calculus

During
Middle Ages In the history of Europe, the Middle Ages or medieval period lasted approximately from the 5th to the late 15th centuries, similarly to the post-classical period of global history. It began with the fall of the Western Roman Empire and ...
, Euclid's ''Elements'' stood as a perfectly solid foundation for mathematics, and
philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
concentrated on the ontological status of mathematical concepts; the question was whether they exist independently of perception ( realism) or within the mind only ( conceptualism); or even whether they are simply names of collection of individual objects (
nominalism In metaphysics, nominalism is the view that universals and abstract objects do not actually exist other than being merely names or labels. There are two main versions of nominalism. One denies the existence of universals—that which can be inst ...
). In ''Elements'', the only numbers that are considered are
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s and ratios of lengths. This geometrical view of non-integer numbers remained dominant until the end of Middle Ages, although the rise of
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
led to consider them independently from geometry, which implies implicitly that there are foundational primitives of mathematics. For example, the transformations of equations introduced by
Al-Khwarizmi Muhammad ibn Musa al-Khwarizmi , or simply al-Khwarizmi, was a mathematician active during the Islamic Golden Age, who produced Arabic-language works in mathematics, astronomy, and geography. Around 820, he worked at the House of Wisdom in B ...
and the
cubic Cubic may refer to: Science and mathematics * Cube (algebra), "cubic" measurement * Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex ** Cubic crystal system, a crystal system w ...
and quartic formulas discovered in the 16th century result from algebraic manipulations that have no geometric counterpart. Nevertheless, this did not challenge the classical foundations of mathematics since all properties of numbers that were used can be deduced from their geometrical definition. In 1637,
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
published ''
La Géométrie ''La Géométrie'' () was published in 1637 as an appendix to ''Discours de la méthode'' ('' Discourse on the Method''), written by René Descartes. In the ''Discourse'', Descartes presents his method for obtaining clarity on any subject. ''La ...
'', in which he showed that geometry can be reduced to algebra by means of
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
, which are numbers determining the position of a point. This gives to the numbers that he called
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s a more foundational role (before him, numbers were defined as the ratio of two lengths). Descartes' book became famous after 1649 and paved the way to
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
.


Infinitesimal calculus

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
(1642–1727) in England and
Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to many ...
(1646–1716) in Germany independently developed the
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
for dealing with mobile points (such as planets in the sky) and variable quantities. This needed the introduction of new concepts such as
continuous function In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More preci ...
s,
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s and limits. For dealing with these concepts in a logical way, they were defined in terms of
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s that are hypothetical numbers that are infinitely close to zero. The strong implications of infinitesimal calculus on foundations of mathematics is illustrated by a pamphlet of the Protestant philosopher
George Berkeley George Berkeley ( ; 12 March 168514 January 1753), known as Bishop Berkeley (Bishop of Cloyne of the Anglican Church of Ireland), was an Anglo-Irish philosopher, writer, and clergyman who is regarded as the founder of "immaterialism", a philos ...
(1685–1753), who wrote " nfinitesimalsare neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?".''
The Analyst ''The Analyst'' (subtitled ''A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious ...
, A Discourse Addressed to an Infidel Mathematician''
Also, a lack of rigor has been frequently invoked, because infinitesimals and the associated concepts were not formally defined ( lines and planes were not formally defined either, but people were more accustomed to them). Real numbers, continuous functions, derivatives were not formally defined before the 19th century, as well as
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
. It is only in the 20th century that a formal definition of infinitesimals has been given, with the proof that the whole infinitesimal can be deduced from them. Despite its lack of firm logical foundations, infinitesimal calculus was quickly adopted by mathematicians, and validated by its numerous applications; in particular the fact that the planet trajectories can be deduced from the
Newton's law of gravitation Newton's law of universal gravitation describes gravity as a force by stating that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the s ...
.


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 of mathematics Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...
. The following subsections describe the main such foundational problems revealed during the 19th century.


Real analysis

Cauchy (1789–1857) started the project of giving rigorous bases to
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
. In particular, he rejected the heuristic principle that he called the generality of algebra, which consisted to apply properties of algebraic operations 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 is very small must be understood as "there is a (sufficiently large)
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
such that ". In the proofs he used this in a way that predated the modern
(ε, δ)-definition of limit Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundame ...
., collected i
Who Gave You the Epsilon?
pp. 5–13. Also available at: http://www.maa.org/pubs/Calc_articles/ma002.pdf
The modern
(ε, δ)-definition of limit Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundame ...
s and continuous functions was first developed by
Bolzano Bolzano ( ; ; or ) is the capital city of South Tyrol (officially the province of Bolzano), Northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third largest in historical Tyrol. The ...
in 1817, but remained relatively unknown, and Cauchy probably did know Bolzano's work.
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
(1815–1897) formalized and popularized the (ε, δ)-definition of limits, and discovered some pathological functions that seemed paradoxical at this time, such as 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 Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series ( ...
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 Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on ...
, Charles Méray, and
Eduard Heine Heinrich Eduard Heine (16 March 1821 – 21 October 1881) was a German mathematician. Heine became known for results on special functions and in real analysis. In particular, he authored an important treatise on spherical harmonics and Leg ...
, but this is only in 1872 that two independent complete definitions of real numbers were published: one by Dedekind, by means of
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
s; the other one by
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
as equivalence classes of Cauchy sequences. Several problems were left open by these definitions, which contributed to the
foundational crisis of mathematics Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...
. Firstly both definitions suppose that
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s and thus
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s are rigorously defined; this was done a few years later with
Peano axioms In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nea ...
. Secondly, both definitions involve
infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set ...
s (Dedekind cuts and sets of the elements of a Cauchy sequence), and Cantor's
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
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 First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...
; that is, quantifiers 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 In mathematics, a Cauchy sequence is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all excluding a finite number of elements of the sequence are le ...
, it has a limit that is a real number'', or as ''every subset of the real numbers that is 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 In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are m ...
s during the first half of the 20th century.


Non-Euclidean geometries

Before the 19th century, there were many failed attempts to derive the
parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...
from other axioms of geometry. In an attempt to prove that its negation leads to a contradiction,
Johann Heinrich Lambert Johann Heinrich Lambert (; ; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, at that time allied to the Switzerland, Swiss Confederacy, who made important contributions to the subjects of mathematics, phys ...
(1728–1777) started to build
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For a ...
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 Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
is also inconsistent and thus that the parallel postulate cannot be proved. This was proved by
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky (; , ; – ) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, kno ...
in 1826, János Bolyai (1802–1860) in 1832 and
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
(unpublished). Later in the 19th century, the German mathematician
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
developed
Elliptic geometry Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines ...
, 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 points on a sphere (or hypersphere), and lines as
great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
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 Truth or verity is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as ...
.


Synthetic vs. analytic geometry

Since the introduction of
analytic geometry In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and als ...
by
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
in the 17th century, there were two approaches to geometry, the old one called
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...
, and the new one, where everything is specified in terms of real numbers called
coordinates In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the Position (geometry), position of the Point (geometry), points or other geometric elements on a manifold such as ...
. 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 In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...
, the two sides accusing each other of mixing projective and metric concepts". Indeed, there is no concept of distance in a
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
, 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, 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 Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrians, Austrian mathematician of Armenians, Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number t ...
's book '' Geometric Algebra'' published in 1957. It was well known that, given a field , one may define
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
and projective spaces over in terms of -
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
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 such that the geometry is the same as the affine or projective geometry over .


Natural numbers

The work of making rigorous real analysis and the definition of real numbers, consisted of reducing everything to
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
s and thus to
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
. This was started with
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
in 1881 and Richard Dedekind in 1888, who defined a natural numbers as the
cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...
of a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...
. However, this involves
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, which was not formalized at this time.
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much Mathematical notati ...
provided in 1888 a complete axiomatisation based on the 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 is a set then" or "if \varphi is a predicate then". So, Peano's axioms induce a quantification on infinite sets, and this means that Peano arithmetic is what is presently called a
Second-order logic In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies on ...
. This was not well understood at that times, but the fact that
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
occurred in the definition of the natural numbers was a problem for many mathematicians of this time. For example,
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
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. This applies in particular to the use of the last Peano axiom for showing that the
successor function In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by ''S'', so ''S''(''n'') = ''n'' +1. For example, ''S''(1) = 2 and ''S''(2) = 3. The successor functio ...
generates all natural numbers. Also,
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
said "God made the integers, all else is the work of man". This may be interpreted as "the integers cannot be mathematically defined".


Infinite sets

Before the second half of the 19th century,
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
was a philosophical concept that did not belong to mathematics. However, with the rise of
infinitesimal calculus Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
, 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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
, an
infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
or a limit. The possibility of an actual infinity was the subject of many philosophical disputes.
Set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
s, and more specially
infinite set In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Properties The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only 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 Georg Ferdinand Ludwig Philipp Cantor ( ; ;  – 6 January 1918) was a mathematician who played a pivotal role in the creation of set theory, which has become a foundations of mathematics, fundamental theory in mathematics. Cantor establi ...
who was the first mathematician to systematically study infinite sets. In particular, he introduced
cardinal number In mathematics, a cardinal number, or cardinal for short, is what is commonly called the number of elements of a set. In the case of a finite set, its cardinal number, or cardinality is therefore a natural number. For dealing with the cas ...
s that measure the size of infinite sets, and
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
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 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 Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particul ...
. The crisis was amplified with the Russel's paradox that asserts that the phrase "the set of all sets" is self-contradictory. This contradiction introduced a doubt on the
consistency In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
of all mathematics. With the introduction of the
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
() 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, De Morgan published his
laws Law is a set of rules that are created and are law enforcement, enforceable by social or governmental institutions to regulate behavior, with its precise definition a matter of longstanding debate. It has been variously described as a Socia ...
and
George Boole George Boole ( ; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Ireland. H ...
devised an algebra, now called
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, that allows expressing Aristotle's logic in terms of formulas and algebraic operations. Boolean algebra is the starting point of mathematization
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
and the basis of
propositional calculus The propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. Sometimes, it is called ''first-order'' propositional logic to contra ...
Independently, in the 1870's,
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
and
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
extended propositional calculus by introducing quantifiers, for building
predicate logic First-order logic, also called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables ove ...
. Frege pointed out three desired properties of a logical theory:
consistency In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
(impossibility of proving contradictory statements), completeness (any statement is either provable or refutable; that is, its negation is provable), and decidability (there is a decision procedure to test every statement). By near the turn of the century,
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
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.


Foundational crisis

The foundational crisis of mathematics arose at the end of the 19th century and the beginning of the 20th century with the discovery of several
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
es or counter-intuitive results. The first one was the proof that the
parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...
cannot be proved. This results from a construction of a non-Euclidean geometry inside
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, whose inconsistency would imply the inconsistency of Euclidean geometry. A well known paradox is
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox published by the British philosopher and mathematician, Bertrand Russell, in 1901. Russell's paradox shows that every set theory that contains ...
, which shows that the phrase "the set of all sets that do not contain themselves" is self-contradictory. Other philosophical problems were the proof of the existence of
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s that cannot be computed or explicitly described, and the proof of the existence of theorems of
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
that cannot be proved with Peano arithmetic. Several schools of
philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
were challenged with these problems in the 20th century, and are described below. These problems were also studied by mathematicians, and this led to establish
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
as a new area of mathematics, consisting of providing mathematical definitions to logics (sets of inference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts. This led to unexpected results, such as
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
, which, roughly speaking, assert that, if a theory contains the standard arithmetic, it cannot be used to prove that it itself is not self-contradictory; and, if it is not self-contradictory, there are theorems that cannot be proved inside the theory, but are nevertheless true in some technical sense.
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
with the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
(ZFC) is a logical theory established by
Ernst Zermelo Ernst Friedrich Ferdinand Zermelo (; ; 27 July 187121 May 1953) was a German logician and mathematician, whose work has major implications for the foundations of mathematics. He is known for his role in developing Zermelo–Fraenkel set theory, Z ...
and Abraham Fraenkel. It became the standard foundation of modern mathematics, and, unless the contrary is explicitly specified, it is used in all modern mathematical texts, generally implicitly. Simultaneously, the
axiomatic method In mathematics and logic, an axiomatic system is a set of formal statements (i.e. axioms) used to logically derive other statements such as lemmas or theorems. A proof within an axiom system is a sequence of deductive steps that establis ...
became a de facto standard: the proof of a theorem must result from explicit
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s and previously proved theorems by the application of clearly defined inference rules. The axioms need not correspond to some reality. Nevertheless, it is an open philosophical problem to explain why the axiom systems that lead to rich and useful theories are those resulting from abstraction from the physical reality or other mathematical theory. In summary, the foundational crisis is essentially resolved, and this opens new philosophical problems. In particular, it cannot be proved that the new foundation (ZFC) is not self-contradictory. It is a general consensus that, if this would happen, the problem could be solved by a mild modification of ZFC.


Philosophical views

When the foundational crisis arose, there was much debate among mathematicians and logicians about what should be done for restoring confidence in mathematics. This involved philosophical questions about
mathematical truth Truth or verity is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as ...
, the relationship of mathematics with
reality Reality is the sum or aggregate of everything in existence; everything that is not imagination, imaginary. Different Culture, cultures and Academic discipline, academic disciplines conceptualize it in various ways. Philosophical questions abo ...
, the reality of
mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...
s, and the nature of mathematics. For the problem of foundations, there were two main options for trying to avoid paradoxes. The first one led to
intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
and constructivism, and consisted to restrict the logical rules for remaining closer to intuition, while the second, which has been called formalism, considers that a theorem is true if it can be deduced from
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s by applying inference rules (
formal proof In logic and mathematics, a formal proof or derivation is a finite sequence of sentences (known as well-formed formulas when relating to formal language), each of which is an axiom, an assumption, or follows from the preceding sentences in the s ...
), and that no "trueness" of the axioms is needed for the validity of a theorem.


Formalism

It has been claimed that formalists, such as
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
(1862–1943), hold that mathematics is only a language and a series of games. Hilbert insisted that formalism, called "formula game" by him, is a fundamental part of mathematics, but that mathematics must not be reduced to formalism. Indeed, he used the words "formula game" in his 1927 response to L. E. J. Brouwer's criticisms: Thus Hilbert is insisting that mathematics is not an ''arbitrary'' game with ''arbitrary'' rules; rather it must agree with how our thinking, and then our speaking and writing, proceeds. The foundational philosophy of formalism, as exemplified by
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
, is a response to the paradoxes of
set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, and is based on
formal logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
. Virtually all mathematical
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
s today can be formulated as theorems of set theory. The truth of a mathematical statement, in this view, is represented by the fact that the statement can be derived from the axioms of set theory using the rules of formal logic. Merely the use of formalism alone does not explain several issues: why we should use the axioms we do and not some others, why we should employ the logical rules we do and not some others, why "true" mathematical statements (e.g., the laws of arithmetic) appear to be true, and so on.
Hermann Weyl Hermann Klaus Hugo Weyl (; ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist, logician and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, ...
posed these very questions to Hilbert: In some cases these questions may be sufficiently answered through the study of formal theories, in disciplines such as reverse mathematics and
computational complexity theory In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem ...
. As noted by Weyl, formal logical systems also run the risk of inconsistency; in Peano arithmetic, this arguably has already been settled with several proofs of
consistency In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
, but there is debate over whether or not they are sufficiently
finitary In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values. In standard mathematics, an operat ...
to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own
consistency In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences ...
. What Hilbert wanted to do was prove a logical system ''S'' was consistent, based on principles ''P'' that only made up a small part of ''S''. But Gödel proved that the principles ''P'' could not even prove ''P'' to be consistent, let alone ''S''.


Intuitionism

Intuitionists, such as L. E. J. Brouwer (1882–1966), hold that mathematics is a creation of the human mind. Numbers, like fairy tale characters, are merely mental entities, which would not exist if there were never any human minds to think about them. The foundational philosophy of ''
intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
'' or '' constructivism'', as exemplified in the extreme by Brouwer and Stephen Kleene, requires proofs to be "constructive" in nature the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absur ...
is suspect. Some modern theories in the philosophy of mathematics deny the existence of foundations in the original sense. Some theories tend to focus on mathematical practice, and aim to describe and analyze the actual working of mathematicians as a
social group In the social sciences, a social group is defined as two or more people who interact with one another, share similar characteristics, and collectively have a sense of unity. Regardless, social groups come in a myriad of sizes and varieties. F ...
. Others try to create a cognitive science of mathematics, focusing on human cognition as the origin of the reliability of mathematics when applied to the real world. These theories would propose to find foundations only in human thought, not in any objective outside construct. The matter remains controversial.


Logicism

Logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that – for some coherent meaning of 'logic' – mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or al ...
is a school of thought, and research programme, in the philosophy of mathematics, based on the thesis that mathematics is an extension of logic or that some or all mathematics may be derived in a suitable formal system whose axioms and rules of inference are 'logical' in nature.
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British philosopher, logician, mathematician, and public intellectual. He had influence on mathematics, logic, set theory, and various areas of analytic ...
and
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as process philosophy, which has been applied in a wide variety of disciplines, inclu ...
championed this theory initiated by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philos ...
and influenced by Richard Dedekind.


Set-theoretic Platonism

Many researchers in
axiomatic set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
have subscribed to what is known as set-theoretic
Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...
, exemplified by
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
. Several set theorists followed this approach and actively searched for axioms that may be considered as true for heuristic reasons and that would decide the
continuum hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...
. Many
large cardinal In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
axioms were studied, but the hypothesis always remained independent from them and it is now considered unlikely that CH can be resolved by a new large cardinal axiom. Other types of axioms were considered, but none of them has reached consensus on the continuum hypothesis yet. Recent work by Hamkins proposes a more flexible alternative: a set-theoretic multiverse allowing free passage between set-theoretic universes that satisfy the continuum hypothesis and other universes that do not.


Indispensability argument for realism

This
argument An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
by
Willard Quine Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth centur ...
and
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
says (in Putnam's shorter words), However, Putnam was not a Platonist.


Rough-and-ready realism

Few mathematicians are typically concerned on a daily, working basis over logicism, formalism or any other philosophical position. Instead, their primary concern is that the mathematical enterprise as a whole always remains productive. Typically, they see this as ensured by remaining open-minded, practical and busy; as potentially threatened by becoming overly-ideological, fanatically reductionistic or lazy. Such a view has also been expressed by some well-known physicists. For example, the Physics Nobel Prize laureate
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
said And Steven Weinberg:Steven Weinberg, chapter
Against Philosophy
' wrote, in ''Dreams of a final theory''
Weinberg believed that any undecidability in mathematics, such as the continuum hypothesis, could be potentially resolved despite the incompleteness theorem, by finding suitable further axioms to add to set theory.


Philosophical consequences of Gödel's completeness theorem

Gödel's completeness theorem establishes an equivalence in first-order logic between the formal provability of a formula and its truth in all possible models. Precisely, for any consistent first-order theory it gives an "explicit construction" of a model described by the theory; this model will be countable if the language of the theory is countable. However this "explicit construction" is not algorithmic. It is based on an iterative process of completion of the theory, where each step of the iteration consists in adding a formula to the axioms if it keeps the theory consistent; but this consistency question is only semi-decidable (an algorithm is available to find any contradiction but if there is none this consistency fact can remain unprovable).


More paradoxes

The following lists some notable results in metamathematics.
Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...
is the most widely studied axiomatization of set theory. It is abbreviated ZFC when it includes the
axiom of choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
and ZF when the axiom of choice is excluded. *1920:
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skole ...
corrected Leopold Löwenheim's proof of what is now called the downward Löwenheim–Skolem theorem, leading to Skolem's paradox discussed in 1922, namely the existence of countable models of ZF, making infinite cardinalities a relative property. *1922: Proof by Abraham Fraenkel that the axiom of choice cannot be proved from the axioms of Zermelo set theory with urelements. *1931: Publication of
Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
, showing that essential aspects of Hilbert's program could not be attained. It showed how to construct, for any sufficiently powerful and consistent recursively axiomatizable system such as necessary to axiomatize the elementary theory of
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
on the (infinite) set of natural numbers a statement that formally expresses its own unprovability, which he then proved equivalent to the claim of consistency of the theory; so that (assuming the consistency as true), the system is not powerful enough for proving its own consistency, let alone that a simpler system could do the job. It thus became clear that the notion of mathematical truth cannot be completely determined and reduced to a purely
formal system A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...
as envisaged in Hilbert's program. This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means (it was never made clear exactly what axioms were the "finitistic" ones, but whatever axiomatic system was being referred to, it was a 'weaker' system than the system whose consistency it was supposed to prove). *1936:
Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
proved his truth undefinability theorem. *1936:
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...
proved that a general algorithm to solve the
halting problem In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ...
for all possible program-input pairs cannot exist. *1938: Gödel proved the consistency of the axiom of choice and of the generalized continuum hypothesis. *1936–1937:
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
and
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...
, respectively, published independent papers showing that a general solution to the
Entscheidungsproblem In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid ...
is impossible: the universal validity of statements in first-order logic is not decidable (it is only semi-decidable as given by the
completeness theorem Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies ...
). *1955: Pyotr Novikov showed that there exists a finitely presented group G such that the word problem for G is undecidable. *1963:
Paul Cohen Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independent from Zermelo–Fraenkel set theory, for which he was awarded a F ...
showed that the Continuum Hypothesis is unprovable from ZFC. Cohen's proof developed the method of forcing, which is now an important tool for establishing
independence Independence is a condition of a nation, country, or state, in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of ...
results in set theory. *1964: Inspired by the fundamental randomness in physics,
Gregory Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentina, Argentine-United States, American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, ...
starts publishing results on algorithmic information theory (measuring incompleteness and randomness in mathematics). *1966: Paul Cohen showed that the axiom of choice is unprovable in ZF even without urelements. *1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a
Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...
(multivariable polynomial equation) has a solution in integers. *1971:
Suslin's problem In mathematics, Suslin's problem is a question about totally ordered sets posed by and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; showed that the statement can neith ...
is proven to be independent from ZFC.


Toward resolution of the crisis

Starting in 1935, the Bourbaki group of French mathematicians started publishing a series of books to formalize many areas of mathematics on the new foundation of set theory. The intuitionistic school did not attract many adherents, and it was not until
Bishop A bishop is an ordained member of the clergy who is entrusted with a position of Episcopal polity, authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance and administration of di ...
's work in 1967 that
constructive mathematics In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove th ...
was placed on a sounder footing. One may consider that Hilbert's program has been partially completed, so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hilbert's original ambitions. His ambitions were expressed in a time when nothing was clear: it was not clear whether mathematics could have a rigorous foundation at all. There are many possible variants of set theory, which differ in consistency strength, where stronger versions (postulating higher types of infinities) contain formal proofs of the consistency of weaker versions, but none contains a formal proof of its own consistency. Thus the only thing we do not have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF. In practice, most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the incompleteness and paradoxes of the underlying formal theories never played a role anyway, and in those branches in which they do or whose formalization attempts would run the risk of forming inconsistent theories (such as logic and category theory), they may be treated carefully. The development of
category theory Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
in the middle of the 20th century showed the usefulness of set theories guaranteeing the existence of larger classes than does ZFC, such as Von Neumann–Bernays–Gödel set theory or Tarski–Grothendieck set theory, albeit that in very many cases the use of large cardinal axioms or Grothendieck universes is formally eliminable. One goal of the reverse mathematics program is to identify whether there are areas of "core mathematics" in which foundational issues may again provoke a crisis.


See also

* Aristotelian realist philosophy of mathematics *
Mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
* Brouwer–Hilbert controversy *
Church–Turing thesis In Computability theory (computation), computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) ...
*
Controversy over Cantor's theory In mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical set theory, it has been criticized in several areas by mathematicians and philosophers ...
*
Epistemology Epistemology is the branch of philosophy that examines the nature, origin, and limits of knowledge. Also called "the theory of knowledge", it explores different types of knowledge, such as propositional knowledge about facts, practical knowle ...
* ''
Euclid's Elements The ''Elements'' ( ) is a mathematics, mathematical treatise written 300 BC by the Ancient Greek mathematics, Ancient Greek mathematician Euclid. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. Drawing on the w ...
'' *
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...
* Implementation of mathematics in set theory * Liar paradox * New Foundations *
Philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
* ''
Principia Mathematica The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by the mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1 ...
'' * Quasi-empiricism in mathematics * Mathematical thought of Charles Peirce


Notes


References

* Avigad, Jeremy (2003)
Number theory and elementary arithmetic
', Philosophia Mathematica Vol. 11, pp. 257–284 * Eves, Howard (1990), ''Foundations and Fundamental Concepts of Mathematics Third Edition'', Dover Publications, INC, Mineola NY, (pbk.) cf §9.5 Philosophies of Mathematics pp. 266–271. Eves lists the three with short descriptions prefaced by a brief introduction. * Goodman, N.D. (1979),
Mathematics as an Objective Science
, in Tymoczko (ed., 1986). * Hart, W.D. (ed., 1996), ''The Philosophy of Mathematics'', Oxford University Press, Oxford, UK. * Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986). * Hilbert, D. (1922), "Neubegründung der Mathematik. Erste Mitteilung", ''Hamburger Mathematische Seminarabhandlungen'' 1, 157–177. Translated, "The New Grounding of Mathematics. First Report", in (Mancosu 1998). * Katz, Robert (1964), ''Axiomatic Analysis'', D. C. Heath and Company. * : In Chapter III ''A Critique of Mathematic Reasoning, §11. The paradoxes'', Kleene discusses
Intuitionism In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fu ...
and Formalism in depth. Throughout the rest of the book he treats, and compares, both Formalist (classical) and Intuitionist logics with an emphasis on the former. Extraordinary writing by an extraordinary mathematician. * Mancosu, P. (ed., 1998), ''From Hilbert to Brouwer. The Debate on the Foundations of Mathematics in the 1920s'', Oxford University Press, Oxford, UK. * Putnam, Hilary (1967), "Mathematics Without Foundations", ''Journal of Philosophy'' 64/1, 5–22. Reprinted, pp. 168–184 in W.D. Hart (ed., 1996). * —, "What is Mathematical Truth?", in Tymoczko (ed., 1986). * * Troelstra, A. S. (no date but later than 1990)
"A History of Constructivism in the 20th Century"
A detailed survey for specialists: §1 Introduction, §2 Finitism & §2.2 Actualism, §3 Predicativism and Semi-Intuitionism, §4 Brouwerian Intuitionism, §5 Intuitionistic Logic and Arithmetic, §6 Intuitionistic Analysis and Stronger Theories, §7 Constructive Recursive Mathematics, §8 Bishop's Constructivism, §9 Concluding Remarks. Approximately 80 references. * Tymoczko, T. (1986), "Challenging Foundations", in Tymoczko (ed., 1986). * —,(ed., 1986),
New Directions in the Philosophy of Mathematics
', 1986. Revised edition, 1998. * van Dalen D. (2008), "Brouwer, Luitzen Egbertus Jan (1881–1966)", in Biografisch Woordenboek van Nederland. URL:http://www.inghist.nl/Onderzoek/Projecten/BWN/lemmata/bwn2/brouwerle 008-03-13* Weyl, H. (1921), "Über die neue Grundlagenkrise der Mathematik", ''Mathematische Zeitschrift'' 10, 39–79. Translated, "On the New Foundational Crisis of Mathematics", in (Mancosu 1998). * Wilder, Raymond L. (1952), ''Introduction to the Foundations of Mathematics'', John Wiley and Sons, New York, NY.


External links

* *
Logic and Mathematics
*Harvey M. Friedman
Foundations of Mathematics: past, present, and future
May 31, 2000, 8 pages.
A Century of Controversy over the Foundations of Mathematics
by Gregory Chaitin. {{Foundations-footer Mathematical logic History of mathematics Philosophy of mathematics