foundations of mathematics

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Foundations of mathematics is the study of the
philosophical Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, Epistemology, knowledge, Ethics, values, Philosophy of mind, mind, and Philosophy of language, language. Such quest ...
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 Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
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 Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such quest ...
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 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. Emergence ...
, 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 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 quantiti ...
theoretical computer science Theoretical 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 circumsc ...
. 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 Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
,
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. Jon Barwise, Barwise (1978) consists of four correspo ...
, 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, wikt:Ζήνων, 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 i ...
(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. T ...
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 Greece, ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though som ...
(408–355 BC), a student of
Plato Plato ( ; grc-gre, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greeks, Greek philosopher born in Athens during the Classical Greece, Classical period in Ancient Greece. He founded the Platonist school of thou ...
, 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 (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 (teaching), demonstration, definition, and scientific knowledge. The demon ...
'',
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
(384–322 BC) laid down the axiomatic method 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, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
'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 Argument, logical argument that applies deductive reasoning to arrive at a Logical consequence, conclusion based on two propositions that are as ...
(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, wikt:Πλάτων, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greeks, Greek philosopher born in Athens during the Classical Greece, Classical period in Ancient Greece. He founded the Platonist school of thou ...
(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, wikt:Μένων, Μένων, ''Ménōn'') is a Socratic dialogue by Plato. Meno (general), Meno begins the dialogue by asking Socrates whether virtue is taught, acquired by practice, or comes by nature. In order to determ ...
'' 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.

## 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 asserted their existence within the mind only;
nominalism In metaphysics, nominalism is the view that Universal (metaphysics), 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 existen ...
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 ; ; Latinisation of names, Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French people, French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of m ...
published '' 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 mathema ...
(1646–1716) in Germany independently developed the infinitesimal calculus 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 "immateri ...
(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 (Roman numerals, MDCCCI), and ended on 31 December 1900 (Roman numerals, MCM). The 19th century was the ninth century of the 2nd millennium. The 19th century was characterized by vast social ...
, 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 w ...
(1789–1857) started the project of formulating and proving the theorems of infinitesimal calculus 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 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 mathematical analysis, analysis". Despite leaving university without a degree, ...
(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 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 solvin ...
(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 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 m ...
). 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 geometry, geometric problems were impossible to solve using only compass and straightedge. In a paper from 1837, Wa ...
(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 number, algebraic—that is, not the Zero of a function, root of a non-zero polynomial of finite degree with rational number, rational coefficients. The best known transcen ...
. 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 group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
(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 group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathe ...
. Concepts of
vector spaces In 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 ...
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, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...
from other axioms, the study of the still hypothetical
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For an ...
by
Johann Heinrich Lambert Johann Heinrich Lambert (, ''Jean-Henri Lambert'' in French language, French; 26 or 28 August 1728 – 25 September 1777) was a polymath from the Republic of Mulhouse, generally referred to as either Switzerland, Swiss or France, French, who made i ...
(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 unit circle, circle with a unit radius, the points form the right ha ...
and compute the area of a hyperbolic triangle (where the sum of angles is less than 180°). Then the Russian mathematician Nikolai Lobachevsky (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, Euclid's ''Elements'', is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: ' ...
), in parallel with the Hungarian mathematician János Bolyai (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 refer ...
. 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 mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly ...
developed
Elliptic geometry Elliptic geometry is an example of a geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathem ...
, 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 circle, circular Intersection (geometry), intersection of a sphere and a Plane (geometry), plane incidence (geometry), passing through the sphere's centre (geometry), center point. Any Circula ...
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 Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...
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), ''Circular'' (album), a 2006 album by Spanish singer Vega * Circular letter (disambiguation) ** Flyer (pamphlet), a form of advertisement * Circular reasoning, a type of logical fa ...
, 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. 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 conjugates. 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 was long ...
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 can ...
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 mathema ...
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 mathematician
George Boole George Boole (; 2 November 1815 – 8 December 1864) was a largely Autodidacticism, self-taught English people, English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics ...
(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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denote ...
, 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 law enforcement, enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. ...
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 t ...
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, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
(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 A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such proc ...
: 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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concer ...
and
William Ernest Johnson William Ernest Johnson, Fellow of the British Academy, FBA (23 June 1858 – 14 January 1931), usually cited as W. E. Johnson, was a British philosopher, logician and economic theorist.Zabell, S.L. (2008"Johnson, William Ernest (1858–1931)"I ...
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 logic, 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 for ...
as it was understood at the end of the 19th century.

### Peano arithmetic

The formalization of
arithmetic Arithmetic () is an elementary part 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 chang ...
(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 n ...
) as an axiomatic theory started with Peirce in 1881 and continued with Richard Dedekind and
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concer ...
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 Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
) 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 Quantifica ...
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 mathematical proof, 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. Infor ...
.

# 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 the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such quest ...
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 ...
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 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'', 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 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 named ...
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 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 Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, ...
, 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 (logic), statement that has been Mathematical proof, 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 th ...
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 assoc ...
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 by ...
. 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 people, Italian mathematician Giuseppe Peano. These axioms have be ...
, this arguably has already been settled with several proofs of consistency proof, consistency, but there is debate over whether or not they are sufficiently finitism, finitary to be meaningful. Gödel's incompleteness theorem, Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency proof, 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'' or ''constructivism (mathematics), constructivism'', as exemplified in the extreme by Luitzen Egbertus Jan Brouwer, Brouwer and Stephen Kleene, requires proofs to be "constructive" in nature the existence of an object must be demonstrated rather than inferred from a demonstration of the impossibility of its non-existence. For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect. Some modern theory, 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. 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 championed this theory initiated by
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, Mathematical logic, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the fath ...
and influenced by Richard Dedekind.

### Set-theoretic Platonism

Many researchers in axiomatic set theory have subscribed to what is known as set-theoretic Platonism#Modern Platonism, Platonism, exemplified by Kurt Gödel. 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 axioms were studied, but the hypothesis always remained Independence (mathematical logic), independent from them and it is now considered unlikely that CH can be resolved by a new large cardinal axiom. Other types of axioms were considered, but none of them has reached consensus on the continuum hypothesis yet. Recent work by Joel David Hamkins, Hamkins proposes a more flexible alternative: a set-theoretic multiverse allowing free passage between set-theoretic universes that satisfy the continuum hypothesis and other universes that do not.

### Indispensability argument for realism

This Quine–Putnam indispensability thesis, argument by Willard Quine and Hilary Putnam says (in Putnam's shorter words), However, Putnam was not a Platonist.

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.

The following lists some notable results in metamathematics. Zermelo–Fraenkel set theory is the most widely studied axiomatization of set theory. It is abbreviated ZFC when it includes the axiom of choice 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 urelements. *1931: Publication of Gödel's incompleteness theorems, 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 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 chang ...
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 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 Tarski's undefinability theorem, truth undefinability theorem. *1936: Alan Turing proved that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. *1938: Gödel proved the Constructible universe, consistency of the axiom of choice and of the generalized continuum hypothesis. *1936–1937: Alonzo Church and Alan Turing, 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 (mathematician), Paul Cohen showed that the Continuum Hypothesis is unprovable from Zermelo–Fraenkel set theory, ZFC. Cohen's proof developed the method of Forcing (mathematics), forcing, which is now an important tool for establishing Independence (mathematical logic), independence results in set theory. *1964: Inspired by the fundamental randomness in physics, Gregory Chaitin starts publishing results on algorithmic information theory (measuring incompleteness and randomness in mathematics). *1966: Paul Cohen showed that the axiom of choice is unprovable in ZF even without urelements. *1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a Diophantine equation (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 Nicolas Bourbaki, 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 Errett Bishop, Bishop's work in 1967 that constructivism (mathematics), constructive mathematics was placed on a sounder footing. One may consider that Hilbert's program#Hilbert's program after Gödel, 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 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, reverse mathematics program is to identify whether there are areas of "core mathematics" in which foundational issues may again provoke a crisis.

* Aristotelian realist philosophy of mathematics * Mathematical logic * Brouwer–Hilbert controversy * Church–Turing thesis * Controversy over Cantor's theory * Epistemology * ''Euclid's Elements'' * Hilbert's problems * Implementation of mathematics in set theory * Liar paradox * New Foundations * Philosophy of mathematics * ''Principia Mathematica'' * Quasi-empiricism in mathematics * Charles Sanders Peirce#Mathematics, Mathematical thought of Charles Peirce

# References

Number theory and elementary arithmetic
', Philosophia Mathematica Vol. 11, pp. 257–284 * Howard Eves, 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. * Nicholas D. Goodman, Goodman, N.D. (1979),
Mathematics as an Objective Science
, in Tymoczko (ed., 1986). * Wilbur Dyre Hart, Hart, W.D. (ed., 1996), ''The Philosophy of Mathematics'', Oxford University Press, Oxford, UK. * Reuben Hersh, Hersh, R. (1979), "Some Proposals for Reviving the Philosophy of Mathematics", in (Tymoczko 1986). * David Hilbert, 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 (mathematics), 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. * Hilary Putnam, 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). * * Anne Sjerp Troelstra, 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. * Thomas Tymoczko, 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 [2008-03-13] * Hermann Weyl, 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). * Raymond Louis Wilder, Wilder, Raymond L. (1952), ''Introduction to the Foundations of Mathematics'', John Wiley and Sons, New York, NY.