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Foundations of mathematics is the study of the philosophical and logical and/or
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
ic basis of
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathematics. In this latter sense, the distinction between foundations of mathematics and
philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
turns out to be quite vague. Foundations of mathematics can be conceived as the study of the basic mathematical concepts (set, function, geometrical figure, number, etc.) and how they form hierarchies of more complex structures and concepts, especially the fundamentally important structures that form the language of mathematics (formulas, theories and their
models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
giving a meaning to formulas, definitions, proofs, algorithms, etc.) also called metamathematical concepts, with an eye to the philosophical aspects and the unity of mathematics. The search for foundations of mathematics is a central question of the philosophy of mathematics; the abstract nature of mathematical objects presents special philosophical challenges. The foundations of mathematics as a whole does not aim to contain the foundations of every mathematical topic. Generally, the ''foundations'' of a field of study refers to a more-or-less systematic analysis of its most basic or fundamental concepts, its conceptual unity and its natural ordering or hierarchy of concepts, which may help to connect it with the rest of human knowledge. The development,
emergence In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergenc ...
, and clarification of the foundations can come late in the history of a field, and might not be viewed by everyone as its most interesting part. Mathematics always played a special role in scientific thought, serving since ancient times as a model of truth and rigor for rational inquiry, and giving tools or even a foundation for other sciences (especially physics). Mathematics' many developments towards higher abstractions in the 19th century brought new challenges and paradoxes, urging for a deeper and more systematic examination of the nature and criteria of mathematical truth, as well as a unification of the diverse branches of mathematics into a coherent whole. The systematic search for the foundations of mathematics started at the end of the 19th century and formed a new mathematical discipline called
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, which later had strong links to
theoretical computer science computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the ...
. It went through a series of crises with paradoxical results, until the discoveries stabilized during the 20th century as a large and coherent body of mathematical knowledge with several aspects or components (
set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...
,
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 (math ...
,
proof theory Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, etc.), whose detailed properties and possible variants are still an active research field. Its high level of technical sophistication inspired many philosophers to conjecture that it can serve as a model or pattern for the foundations of other sciences.


Historical context


Ancient Greek mathematics

While the practice of mathematics had previously developed in other civilizations, special interest in its theoretical and foundational aspects was clearly evident in the work of the Ancient Greeks. Early Greek philosophers disputed as to which is more basic, arithmetic or geometry.
Zeno of Elea Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
(490 c. 430 BC) produced four paradoxes that seem to show the impossibility of change. The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist. The discovery of the irrationality of , the ratio of the diagonal of a square to its side (around 5th century BC), was a shock to them which they only reluctantly accepted. The discrepancy between rationals and reals was finally resolved by
Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...
(408–355 BC), a student of
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
, who reduced the comparison of two irrational ratios to comparisons of multiples of the magnitudes involved. His method anticipated that of the Dedekind cut in the modern definition of real numbers by
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
(1831–1916). In the ''
Posterior Analytics The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguis ...
'',
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
(384–322 BC) laid down the
axiomatic method In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
for organizing a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and theorems. Aristotle took a majority of his examples for this from arithmetic and from geometry. This method reached its high point with
Euclid Euclid (; grc-gre, Εὐκλείδης; 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 ...
'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 (though they do not always conform strictly to Aristotelian templates). Aristotle's syllogistic logic, together with the axiomatic method exemplified by Euclid's ''Elements'', are recognized as scientific achievements of ancient Greece.


Platonism as a philosophy of mathematics

Starting from the end of the 19th century, a Platonist view of mathematics became common among practicing mathematicians. The ''concepts'' or, as Platonists would have it, the ''objects'' of mathematics are abstract and remote from everyday perceptual experience: geometrical figures are conceived as idealities to be distinguished from effective drawings and shapes of objects, and numbers are not confused with the counting of concrete objects. Their existence and nature present special philosophical challenges: How do mathematical objects differ from their concrete representation? Are they located in their representation, or in our minds, or somewhere else? How can we know them? The ancient Greek philosophers took such questions very seriously. Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic.
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(424/423 BC 348/347 BC) insisted that mathematical objects, like other platonic ''Ideas'' (forms or essences), must be perfectly abstract and have a separate, non-material kind of existence, in a world of mathematical objects independent of humans. He believed that the truths about these objects also exist independently of the human mind, but is ''discovered'' by humans. In the ''
Meno ''Meno'' (; grc-gre, Μένων, ''Ménōn'') is a Socratic dialogue by Plato. Meno begins the dialogue by asking Socrates whether virtue is taught, acquired by practice, or comes by nature. In order to determine whether virtue is teachabl ...
'' Plato's teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval. Above the gateway to Plato's academy appeared a famous inscription: "Let no one who is ignorant of geometry enter here". In this way Plato indicated his high opinion of geometry. He regarded geometry as "the first essential in the training of philosophers", because of its abstract character. This philosophy of '' Platonist mathematical realism'' is shared by many mathematicians. Some authors argue that Platonism somehow comes as a necessary assumption underlying any mathematical work.Karlis Podnieks
Platonism, intuition and the nature of mathematics: 1. Platonism - the Philosophy of Working Mathematicians
/ref> In this view, the laws of nature and the laws of mathematics have a similar status, and the
effectiveness Effectiveness is the capability of producing a desired result or the ability to produce desired output. When something is deemed effective, it means it has an intended or expected outcome, or produces a deep, vivid impression. Etymology The ori ...
ceases to be unreasonable. Not our axioms, but the very real world of mathematical objects forms the foundation. Aristotle dissected and rejected this view in his
Metaphysics Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
. These questions provide much fuel for philosophical analysis and debate.


Aristotelian realism


Middle Ages and Renaissance

For over 2,000 years, Euclid's Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century. The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas):
Realism Realism, Realistic, or Realists may refer to: In the arts *Realism (arts), the general attempt to depict subjects truthfully in different forms of the arts Arts movements related to realism include: * Classical Realism *Literary realism, a mov ...
asserted their existence independently of perception; conceptualism asserted their existence within the mind only;
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 at least two main versions of nominalism. One version denies the existence of universalsthings ...
denied either, only seeing universals as names of collections of individual objects (following older speculations that they are words, "''logoi''").
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Ma ...
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'', he presents his method for obtaining clarity on any subject. ''La Géométr ...
'' (1637), aimed at reducing geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks used lengths to define the numbers that are presently 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s). Descartes' book became famous after 1649 and paved the way to infinitesimal calculus.
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
(1642–1727) in England and
Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
(1646–1716) in Germany independently developed the
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ari ...
on a basis that required new foundations. In particular, Leibniz described infinitesimals as numbers that are infinitely close to zero, a concept that does not fit in previous foundational framework of mathematics, and was not formalized before the 20th century. 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 whose primary achievement was the advancement of a theory he called "immate ...
(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, A Discourse Addressed to an Infidel Mathematician'' Leibniz also worked on logic but most of his writings on it remained unpublished until 1903. Then mathematics developed very rapidly and successfully in physical applications.


19th century

In the
19th century The 19th (nineteenth) century began on 1 January 1801 ( MDCCCI), and ended on 31 December 1900 ( MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social upheaval. Slavery was abolish ...
, mathematics became increasingly abstract. Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems.


Real analysis

Cauchy (1789–1857) started the project of formulating and proving the theorems of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", 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 ari ...
in a rigorous manner, rejecting the heuristic principle of the generality of algebra exploited by earlier authors. In his 1821 work '' Cours d'Analyse'' he defines infinitely small quantities in terms of decreasing sequences that converge to 0, which he then used to define continuity. But he did not formalize his notion of convergence. The modern (ε, δ)-definition of limit and continuous functions was first developed by
Bolzano Bolzano ( or ; german: Bozen, (formerly ); bar, Bozn; lld, Balsan or ) is the capital city of the province of South Tyrol in northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third ...
in 1817, but remained relatively unknown. It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, arguably resolving the Zeno paradoxes and Berkeley's arguments. Mathematicians such as Karl Weierstrass (1815–1897) discovered pathological functions such as continuous, nowhere-differentiable functions. Previous conceptions of a function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis, to axiomatize analysis using properties of the natural numbers. In 1858, Dedekind proposed a definition of the real numbers as cuts of rational numbers. This reduction of real numbers and continuous functions in terms of rational numbers, and thus of natural numbers, was later integrated by Cantor in his set theory, and axiomatized in terms of
second order arithmetic In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics. A precurs ...
by Hilbert and Bernays.


Group theory

For the first time, the limits of mathematics were explored. Niels Henrik Abel (1802–1829), a Norwegian, and
Évariste Galois Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radical ...
, (1811–1832) a Frenchman, investigated the solutions of various polynomial equations, and proved that there is no general algebraic solution to equations of degree greater than four ( Abel–Ruffini theorem). With these concepts, Pierre Wantzel (1837) proved that straightedge and compass alone cannot
trisect an arbitrary angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...
nor double a cube. In 1882,
Lindemann Lindemann is a German surname. Persons Notable people with the surname include: Arts and entertainment * Elisabeth Lindemann, German textile designer and weaver *Jens Lindemann, trumpet player * Julie Lindemann, American photographer * Maggie ...
building on the work of
Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. Her ...
showed that a straightedge and compass quadrature of the circle (construction of a square equal in area to a given circle) was also impossible by proving that is a
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
. Mathematicians had attempted to solve all of these problems in vain since the time of the ancient Greeks. Abel and Galois's works opened the way for the developments of
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
(which would later be used to study
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
in physics and other fields), and
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The ter ...
. Concepts of vector spaces emerged from the conception of barycentric coordinates by Möbius in 1827, to the modern definition of vector spaces and linear maps by Peano in 1888. Geometry was no more limited to three dimensions. These concepts did not generalize numbers but combined notions of functions and sets which were not yet formalized, breaking away from familiar mathematical objects.


Non-Euclidean geometries

After many failed attempts to derive the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...
from other axioms, the study of the still hypothetical
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
by
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Swiss or French, who made important contributions to the subject ...
(1728–1777) led him to introduce the hyperbolic functions and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician
Nikolai Lobachevsky Nikolai Ivanovich Lobachevsky ( rus, Никола́й Ива́нович Лобаче́вский, p=nʲikɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕɛfskʲɪj, a=Ru-Nikolai_Ivanovich_Lobachevsky.ogg; – ) was a Russian mathematician and geometer, ...
(1792–1856) established in 1826 (and published in 1829) the coherence of this geometry (thus the independence of the
parallel postulate In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ''If a line segmen ...
), in parallel with the Hungarian mathematician
János Bolyai János Bolyai (; 15 December 1802 – 27 January 1860) or Johann Bolyai, was a Hungarian mathematician, who developed absolute geometry—a geometry that includes both Euclidean geometry and hyperbolic geometry. The discovery of a consis ...
(1802–1860) in 1832, and with Gauss. Later in the 19th century, the German mathematician
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
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 point to mean a pair of antipodal points on a fixed sphere and line to mean a
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. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...
on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model for it.


Projective geometry

One of the traps in a deductive system is
circular reasoning Circular may refer to: * The shape of a circle * ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fallacy * Circula ...
, a problem that seemed to befall
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, ...
until it was resolved by Karl von Staudt. As explained by Russian historians: The purely geometric approach of von Staudt was based on the
complete quadrilateral In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and of the six ...
to express the relation of
projective harmonic conjugate In projective geometry, the harmonic conjugate point of an ordered triple of points on the real projective line is defined by the following construction: :Given three collinear points , let be a point not lying on their join and let any line ...
s. Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deducing the properties of a field can be found in either the book by
Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905; while this wa ...
and John Young, ''Projective Geometry'' (1938), or more recently in John Stillwell's ''Four Pillars of Geometry'' (2005). Stillwell writes on page 120 The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon
number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual number ...
s without worry about their basis. However, cross-ratio calculations use
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
features of geometry, features not admitted by purists. For instance, in 1961 Coxeter wrote ''Introduction to Geometry'' without mention of cross-ratio.


Boolean algebra and logic

Attempts of formal treatment of mathematics had started with Leibniz and Lambert (1728–1777), and continued with works by algebraists such as George Peacock (1791–1858). Systematic mathematical treatments of logic came with the British mathematician
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 ...
(1847) who devised an algebra that soon evolved into what is 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 variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas i ...
, in which the only numbers were 0 and 1 and logical combinations (conjunction, disjunction, implication and negation) are operations similar to the addition and multiplication of integers. Additionally, De Morgan published his
laws Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vari ...
in 1847. Logic thus became a branch of mathematics. Boolean algebra is the starting point of mathematical logic and has important applications in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
.
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for ...
built upon the work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. The German mathematician
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 p ...
(1848–1925) presented an independent development of logic with quantifiers in his
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept nota ...
(formula language) published in 1879, a work generally considered as marking a turning point in the history of logic. He exposed deficiencies in Aristotle's ''Logic'', and pointed out the three expected properties of a mathematical theory # Consistency: impossibility of proving contradictory statements. # Completeness: any statement is either provable or refutable (i.e. its negation is provable). # Decidability: there is a decision procedure to test any statement in the theory. He then showed in ''Grundgesetze der Arithmetik (Basic Laws of Arithmetic)'' how arithmetic could be formalised in his new logic. Frege's work was popularized by
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
near the turn of the century. But Frege's two-dimensional notation had no success. Popular notations were (x) for universal and (∃x) for existential quantifiers, coming from Giuseppe Peano and William Ernest Johnson until the ∀ symbol was introduced by Gerhard Gentzen in 1935 and became canonical in the 1960s. From 1890 to 1905, Ernst Schröder published ''Vorlesungen über die Algebra der Logik'' in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to
symbolic logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
as it was understood at the end of the 19th century.


Peano arithmetic

The formalization of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
(the theory of
natural numbers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
) as an axiomatic theory started with Peirce in 1881 and continued with
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
and Giuseppe Peano in 1888. This was still a second-order axiomatization (expressing induction in terms of arbitrary subsets, thus with an implicit use of
set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...
) as concerns for expressing theories in first-order logic were not yet understood. In Dedekind's work, this approach appears as completely characterizing natural numbers and providing recursive definitions of addition and multiplication from 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 ...
and
mathematical induction Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ...  all hold. Informal metaphors help ...
.


Foundational crisis

The foundational crisis of mathematics (in
German German(s) may refer to: * Germany (of or related to) **Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
''Grundlagenkrise der Mathematik'') was the early 20th century's term for the search for proper foundations for mathematics. Several schools of the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
ran into difficulties one after the other in the 20th century, as the assumption that mathematics had any foundation that could be consistently stated within mathematics itself was heavily challenged by the discovery of various paradoxes (such as Russell's paradox). The name ''"paradox"'' should not be confused with ''contradiction''. A contradiction in a formal theory is a formal proof of an absurdity inside the theory (such as ), showing that this theory is
inconsistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consiste ...
and must be rejected. But a paradox may be either a surprising but true result in a given formal theory, or an informal argument leading to a contradiction, so that a candidate theory, if it is to be formalized, must disallow at least one of its steps; in this case the problem is to find a satisfying theory without contradiction. Both meanings may apply if the formalized version of the argument forms the proof of a surprising truth. For instance, Russell's paradox may be expressed as "there is no set of all sets" (except in some marginal axiomatic set theories). Various schools of thought opposed each other. The leading school was that of the formalists, of which
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
was the foremost proponent, culminating in what is known as Hilbert's program, intending to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means. The main opponent of the formalist school was the intuitionist school, led by L. E. J. Brouwer, which resolutely discarded formalism as a meaningless game with symbols. The fight was acrimonious. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of ''
Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...
'', the leading mathematical journal of the time.


Philosophical views

At the beginning of the 20th century, three schools of philosophy of mathematics opposed each other: Formalism, Intuitionism and Logicism. The
Second Conference on the Epistemology of the Exact Sciences The Second Conference on the Epistemology of the Exact Sciences (german: 2. Tagung für Erkenntnislehre der exakten Wissenschaften in Königsberg) was held on 5–7 September 1930 in Königsberg, then located in East Prussia. It was at this conferen ...
held in
Königsberg Königsberg (, ) was the historic Prussian city that is now Kaliningrad, Russia. Königsberg was founded in 1255 on the site of the ancient Old Prussian settlement ''Twangste'' by the Teutonic Knights during the Northern Crusades, and was ...
in 1930 gave space to these three schools.


Formalism

It has been claimed that formalists, such as
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
(1862–1943), hold that mathematics is only a language and a series of games. 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, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, is a response to the paradoxes of
set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...
, and is based on formal logic. Virtually all mathematical
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
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 do "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 and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
would ask these very questions of 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 relating these classes to each other. A computational problem is a task solved ...
. As noted by Weyl, formal logical systems also run the risk of inconsistency; in
Peano arithmetic 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 nearl ...
, this arguably has already been settled with several proofs of consistency, but there is debate over whether or not they are sufficiently finitary to be meaningful. Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency. 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 f ...
'' or '' constructivism'', as exemplified in the extreme by
Brouwer Brouwer (also Brouwers and de Brouwer) is a Dutch and Flemish surname. The word ''brouwer'' means 'beer brewer'. Brouwer * Adriaen Brouwer (1605–1638), Flemish painter * Alexander Brouwer (b. 1989), Dutch beach volleyball player * Andries Brou ...
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 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 can be 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 ...
. Others try to create a
cognitive science of mathematics Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes ...
, 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 is a school of thought, and research programme, in the philosophy of mathematics, based on the thesis that mathematics is an extension of a 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 mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
and
Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applica ...
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 p ...
and influenced by
Richard Dedekind Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His ...
.


Set-theoretic Platonism

Many researchers in
axiomatic set theory Set theory is the branch of mathematical logic that studies 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 mathematics, is mostly concern ...
have subscribed to what is known as set-theoretic Platonism, 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 had an imm ...
. 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. 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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
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 The multiverse is a hypothetical group of multiple universes. Together, these universes comprise everything that exists: the entirety of space, time, matter, energy, information, and the physical laws and constants that describe them. The dif ...
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 statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialecti ...
by Willard Quine and
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, and computer scientist, and a major figure in analytic philosophy in the second half of the 20th century. He made significant contributions ...
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 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). This can be seen as giving a sort of justification to the Platonist view that the objects of our mathematical theories are real. More precisely, it shows that the mere assumption of the existence of the set of natural numbers as a totality (an actual infinity) suffices to imply the existence of a model (a world of objects) of any consistent theory. However several difficulties remain: * For any consistent theory this usually does not give just one world of objects, but an infinity of possible worlds that the theory might equally describe, with a possible diversity of truths between them. * In the case of set theory, none of the models obtained by this construction resemble the intended model, as they are countable while set theory intends to describe uncountable infinities. Similar remarks can be made in many other cases. For example, with theories that include arithmetic, such constructions generally give models that include non-standard numbers, unless the construction method was specifically designed to avoid them. * As it gives models to all consistent theories without distinction, it gives no reason to accept or reject any axiom as long as the theory remains consistent, but regards all consistent axiomatic theories as referring to equally existing worlds. It gives no indication on which axiomatic system should be preferred as a foundation of mathematics. * As claims of consistency are usually unprovable, they remain a matter of belief or non-rigorous kinds of justifications. Hence the existence of models as given by the completeness theorem needs in fact two philosophical assumptions: the actual infinity of natural numbers and the consistency of the theory. Another consequence of the completeness theorem is that it justifies the conception of infinitesimals as actual infinitely small nonzero quantities, based on the existence of non-standard models as equally legitimate to standard ones. This idea was formalized by Abraham Robinson into the theory of nonstandard analysis.


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 such ...
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, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
and ZF when the axiom of choice is excluded. *1920: Thoralf Skolem 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
urelement In set theory, a branch of mathematics, an urelement or ur-element (from the German prefix ''ur-'', 'primordial') is an object that is not a set, but that may be an element of a set. It is also referred to as an atom or individual. Theory There ...
s. *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 phil ...
, 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 part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
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 can not be completely determined and reduced to a purely
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A fo ...
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 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. Turing was highly influential in the development of theoretical ...
proved that a general algorithm to solve the
halting problem In 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 forever. Alan Turing proved in 1936 that a ...
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 and
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher, and theoretical biologist. Turing was highly influential in the development of theoretical ...
, respectively, published independent papers showing that a general solution to the Entscheidungsproblem 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). *1955:
Pyotr Novikov Pyotr Sergeyevich Novikov (russian: Пётр Серге́евич Но́виков; 15 August 1901, Moscow, Russian Empire – 9 January 1975, Moscow, Soviet Union) was a Soviet mathematician. Novikov is known for his work on combinatorial pr ...
showed that there exists a finitely presented group G such that the word problem for G is undecidable. *1963: Paul Cohen 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 person, 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 stat ...
results in set theory. *1964: Inspired by the fundamental randomness in physics, Gregory Chaitin starts publishing results on
algorithmic information theory Algorithmic information theory (AIT) is a branch of theoretical computer science that concerns itself with the relationship between computation and information of computably generated objects (as opposed to stochastically generated), such as str ...
(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 Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equ ...
is proven unsolvable: there is no recursive solution to decide whether a
Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a ...
(multivariable polynomial equation) has a solution in integers. *1971: Suslin's problem 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 clergy member who is entrusted with a position of authority and oversight in a religious institution. In Christianity, bishops are normally responsible for the governance of dioceses. The role or office of bishop is ...
's work in 1967 that constructive mathematics 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 don't 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 that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
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 formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
*
Brouwer–Hilbert controversy In a controversy over the foundations of mathematics, in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. The debate concerned fundamenta ...
*
Church–Turing thesis In 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) is a thesis about the nature of co ...
*
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 (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epi ...
* ''
Euclid's Elements The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postu ...
'' *
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 pro ...
*
Implementation of mathematics in set theory This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC (the dominant set theory) and in NFU, the version of Quine's New Found ...
* Liar paradox *
New Foundations In mathematical logic, New Foundations (NF) is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of '' Principia Mathematica''. Quine first proposed NF in a 1937 article titled "New Foundatio ...
*
Philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
* '' Principia Mathematica'' *
Quasi-empiricism in mathematics Quasi-empiricism in mathematics is the attempt in the philosophy of mathematics to direct philosophers' attention to mathematical practice, in particular, relations with physics, social sciences, and computational mathematics, rather than solely ...
* 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 f ...
and
Formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scien ...
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 MathematicsFoundations of Mathematics: past, present, and future
May 31, 2000, 8 pages.

by Gregory Chaitin. {{Foundations-footer Mathematical logic History of mathematics Philosophy of mathematics