Foundations of mathematics is the study of the

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

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.

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 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.

Logic and Mathematics

Foundations of Mathematics: past, present, and future

May 31, 2000, 8 pages.

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

philosophical
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

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
The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) w ...

(formulas, theories and their models 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.
Emergen ...

, 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
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as beliefs ...

, 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 formal ...

, 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 t ...

.
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 formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...

, 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 parts ...

, 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 f ...

(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
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. ...

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 institutio ...

, 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 distinguish ...

'', 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 ph ...

(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
A syllogism ( grc-gre, συλλογισμός, ''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 ...

(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 institutio ...

(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'' 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 PodnieksPlatonism, 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 consci ...

. 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 asserted their existence independently of perception;conceptualism
In metaphysics, conceptualism is a theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between nominalism and realism, the conceptualist view approaches the metaphysical ...

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 th ...

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. Mathem ...

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 Géométrie ...

'' (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, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...

(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 mathem ...

(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 arith ...

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 the19th 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
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He ...

(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 arith ...

in a rigorous manner, rejecting the heuristic principle of the generality of algebra In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange,. particularly ...

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
Although the function (sin ''x'')/''x'' is not defined at zero, as ''x'' becomes closer and closer to zero, (sin ''x'')/''x'' becomes arbitrarily close to 1. In other words, the limit of (sin ''x'')/''x'', as ''x'' approaches z ...

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
Karl Theodor Wilhelm Weierstrass (german: link=no, 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 mathematic ...

(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
The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.
History
Kronecker originally introduced the term ''arithmetization of analysis'', by which he meant its c ...

, 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 by Hilbert and Bernays.
Group theory

For the first time, the limits of mathematics were explored.Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvi ...

(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 Nth root, ...

, (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
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means t ...

). With these concepts, Pierre Wantzel
Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.
In a paper from 1837, Wantzel pro ...

(1837) proved that straightedge and compass alone cannot trisect an arbitrary angle nor double a cube. In 1882, Lindemann building on the work of Hermite 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 as ...

(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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

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 theparallel 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 segment ...

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 (1728–1777) led him to introduce the hyperbolic functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...

and compute the area of a hyperbolic triangle
In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three points called ''angles'' or ''vertices''.
Just as in the Euclidean case, three ...

(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, kn ...

(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 segment ...

), 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 consiste ...

(1802–1860) in 1832, and with Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...

.
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 rig ...

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
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

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 geometry ...

on the sphere. At that time, the main method for proving the consistency of a set of axioms was to provide a model
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 ...

for it.
Projective geometry

One of the traps in adeductive 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 for ...

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
* Circular ...

, 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, pro ...

until it was resolved by Karl von Staudt
Karl Georg Christian von Staudt (24 January 1798 – 1 June 1867) was a German mathematician who used synthetic geometry to provide a foundation for arithmetic.
Life and influence
Karl was born in the Free Imperial City of Rothenburg, which is ...

. As explained by Russian historians:
The purely geometric approach of von Staudt was based on the complete quadrilateral 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 t ...

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 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 numbers c ...

s without worry about their basis. However, cross-ratio calculations use metric features of geometry, features not admitted by purists. For instance, in 1961 Coxeter
Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.
Biography
Coxeter was born in Kensington to ...

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 mathematicianGeorge 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 I ...

(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 in ...

, 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 practical disciplines (includi ...

.
Charles Sanders Peirce 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 phil ...

(1848–1925) presented an independent development of logic with quantifiers in his Begriffsschrift (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
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 consistent ...

: 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, ar ...

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
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 notation. The sta ...

and William Ernest Johnson until the ∀ symbol was introduced by Gerhard Gentzen
Gerhard Karl Erich Gentzen (24 November 1909 – 4 August 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died ...

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 as it was understood at the end of the 19th century.
Peano arithmetic

The formalization ofarithmetic
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 19t ...

(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
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 notation. The sta ...

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
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

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 ''Grundlagenkrise der Mathematik'') was the early 20th century's term for the search for proper foundations for mathematics. Several schools of thephilosophy 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 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 premises, leads to a seemingly self-contradictory or a logically u ...

es (such as Russell's paradox
In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contain ...

).
The name ''"paradox"'' should not be confused with ''contradiction''. A contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...

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 a ...

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 held inKönigsberg
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 nam ...

in 1930 gave space to these three schools.
Formalism

It has been claimed that formalists, such asDavid 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 a ...

(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 a ...

, 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
Logic is the study of correct reasoning. It includes both Mathematical logic, formal and informal logic. Formal logic is the science of Validity (logic), deductively valid inferences or of logical truths. It is a formal science investigating h ...

. 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 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
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in con ...

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 b ...

. 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 nearly ...

, this arguably has already been settled with several proofs of consistency
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 consistent ...

, 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
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 consistent ...

. 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'' or '' constructivism'', as exemplified in the extreme byBrouwer 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 Bro ...

and Stephen Kleene
Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch ...

, 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 arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...

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, 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, ar ...

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 appli ...

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 phil ...

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 inaxiomatic 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 concer ...

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 of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...

, 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 im ...

.
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, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that
or equivalently, that
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to ...

. Many large cardinal 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 d ...

allowing free passage between set-theoretic universes that satisfy the continuum hypothesis and other universes that do not.
Indispensability argument for realism

Thisargument
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 dialectic ...

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 laureateRichard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superflu ...

said
And Steven Weinberg
Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interac ...

: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 byAbraham Robinson
Abraham Robinson (born Robinsohn; October 6, 1918 – April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were reincor ...

into the theory of nonstandard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...

.
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 o ...

and ZF when the axiom of choice is excluded.
*1920: Thoralf Skolem corrected Leopold Löwenheim
Leopold Löwenheim le:o:pɔl̩d ˈlø:vɛnhaɪm(26 June 1878 in Krefeld – 5 May 1957 in Berlin) was a German mathematician doing work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was conside ...

'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 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 19t ...

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 for ...

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. Turing was highly influential in the development of theoretical c ...

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 g ...

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 mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scienc ...

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 c ...

, 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 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 statu ...

results in set theory.
*1964: Inspired by the fundamental randomness in physics, Gregory Chaitin
Gregory John Chaitin ( ; born 25 June 1947) is an Argentine- American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer- ...

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 ...

(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 (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 untilBishop
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, cate ...

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
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a coll ...

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
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in con ...

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 formal ...

* 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 fundament ...

* Church–Turing 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 philosophe ...

* 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.
Episte ...

* ''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, postula ...

''
* 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
* Liar paradox
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the trut ...

* New Foundations
* 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
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...

''
* 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 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

Foundations of Mathematics: past, present, and future

May 31, 2000, 8 pages.

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