text:
David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).

Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set a course for mathematical research of the 20th century.

Hilbert and his students contributed to establishing rigor and developed important tools used in modern mathematical physics. Hilbert was one of the founders of proof theory and mathematical logic.

Early life and education
Hilbert, the first of two children and only son of Otto, a county judge, and Maria Therese Hilbert (née Erdtmann), the daughter of a merchant, was born in the Province of Prussia, Kingdom of Prussia, either in Königsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Königsberg where his father worked at the time of his birth. His paternal grandfather was David Hilbert, a judge and Geheimrat. His mother Maria had an interest in philosophy, astronomy and prime numbers, while his father Otto taught him Prussian virtues. After his father became a city judge, the family moved to Königsberg. David's sister, Elise, was born when he was six. He began his schooling aged eight, two years later than the usual starting age.

In late 1872, Hilbert entered the Friedrichskolleg Gymnasium (Collegium fridericianum, the same school that Immanuel Kant had attended 140 years before); but, after an unhappy period, he transferred to (late 1879) and graduated from (early 1880) the more science-oriented Wilhelm Gymnasium. Upon graduation, in autumn 1880, Hilbert enrolled at the University of Königsberg, the "Albertina". In early 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but had gone to Berlin for three semesters), returned to Königsberg and entered the university. Hilbert developed a lifelong friendship with the shy, gifted Minkowski.

Career
In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (i.e., an associate professor). An intense and fruitful scientific exchange among the three began, and Minkowski and Hilbert especially would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen ("On the invariant properties of special binary forms, in particular the spherical harmonic functions").

Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. In 1895, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen. During the Klein and Hilbert years, Göttingen became the preeminent institution in the mathematical world. He remained there for the rest of his life.

Göttingen school
Among Hilbert's students were Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert was surrounded by a social circle of some of the most important mathematicians of the 20th century, such as Emmy Noether and Alonzo Church.

Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), and Wilhelm Ackermann (1925). Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. He was elected an International Member of the United States National Academy of Sciences in 1907.

Personal life

In 1892, Hilbert married Käthe Jerosch (1864–1945), who was the daughter of a Königsberg merchant, "an outspoken young lady with an independence of mind that matched [Hilbert's]." While at Königsberg they had their one child, Franz Hilbert (1893–1969). Franz suffered throughout his life from mental illness, and after he was admitted into a psychiatric clinic, Hilbert said, "From now on, I must consider myself as not having a son." His attitude toward Franz brought Käthe considerable sorrow.

Hilbert considered the mathematician Hermann Minkowski to be his "best and truest friend".

Hilbert was baptized and raised a Calvinist in the Prussian Evangelical Church. He later left the Church and became an agnostic. David Hilbert seemed to be agnostic and had nothing to do with theology proper or even religion. Constance Reid tells a story on the subject:"The Hilberts had by this time [around 1902] left the Reformed Protestant Church in which they had been baptized and married. It was told in Göttingen that when [David Hilbert's son] Franz had started to school he could not answer the question, 'What religion are you?' (1970, p. 91)" In the 1927 Hamburg address, Hilbert asserted: "mathematics is pre-suppositionless science (die Mathematik ist eine voraussetzungslose Wissenschaft)" and "to found it I do not need a good God ([z]u ihrer Begründung brauche ich weder den lieben Gott)" (1928, S. 85; van Heijenoort, 1967, p. 479). However, from Mathematische Probleme (1900) to Naturerkennen und Logik (1930) he placed his quasi-religious faith in the human spirit and in the power of pure thought with its beloved child– mathematics. He was deeply convinced that every mathematical problem could be solved by pure reason: in both mathematics and any part of natural science (through mathematics) there was "no ignorabimus" (Hilbert, 1900, S. 262; 1930, S. 963; Ewald, 1996, pp. 1102, 1165). That is why finding an inner absolute grounding for mathematics turned into Hilbert's life-work. He never gave up this position, and it is symbolic that his words "wir müssen wissen, wir werden wissen" ("we must know, we shall know") from his 1930 Königsberg address were engraved on his tombstone. Here, we meet a ghost of departed theology (to modify George Berkeley's words), for to absolutize human cognition means to identify it tacitly with a divine one. — He also argued that mathematical truth was independent of the existence of God or other a priori assumptions. When Galileo Galilei was criticized for failing to stand up for his convictions on the Heliocentric theory, Hilbert objected: "But [Galileo] was not an idiot. Only an idiot could believe that scientific truth needs martyrdom; that may be necessary in religion, but scientific results prove themselves in due time."

Later years
Like Albert Einstein, Hilbert had closest contacts with the Berlin Group whose leading founders had studied under Hilbert in Göttingen (Kurt Grelling, Hans Reichenbach and Walter Dubislav).

Around 1925, Hilbert developed pernicious anemia, a then-untreatable vitamin deficiency whose primary symptom is exhaustion; his assistant Eugene Wigner described him as subject to "enormous fatigue" and how he "seemed quite old," and that even after eventually being diagnosed and treated, he "was hardly a scientist after 1925, and certainly not a Hilbert."

Hilbert was elected to the American Philosophical Society in 1932.

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933. Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him the important book Grundlagen der Mathematik (which eventually appeared in two volumes, in 1934 and 1939). This was a sequel to the Hilbert–Ackermann book Principles of Mathematical Logic from 1928. Hermann Weyl's successor was Helmut Hasse.

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked whether "the Mathematical Institute really suffered so much because of the departure of the Jews." Hilbert replied, "Suffered? It doesn't exist any longer, does it?"

Death
By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Königsberg. News of his death only became known to the wider world several months after he died.

The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930. The words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know and we shall not know":

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel—in a round table discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem. "The Conference on Epistemology of the Exact Sciences ran for three days, from 5 to 7 September" (Dawson 1997:68). "It ... was held in conjunction with and just before the ninety-first annual meeting of the Society of German Scientists and Physicians ... and the sixth Assembly of German Physicists and Mathematicians.... Gödel's contributed talk took place on Saturday, 6 September [1930], from 3 until 3:20 in the afternoon, and on Sunday the meeting concluded with a round table discussion of the first day's addresses. During the latter event, without warning and almost offhandedly, Gödel quietly announced that "one can even give examples of propositions (and in fact of those of the type of Goldbach or Fermat) that, while contentually true, are unprovable in the formal system of classical mathematics [153]" (Dawson:69) "... As it happened, Hilbert himself was present at Königsberg, though apparently not at the Conference on Epistemology. The day after the roundtable discussion he delivered the opening address before the Society of German Scientists and Physicians – his famous lecture Naturerkennen und Logik (Logic and the knowledge of nature), at the end of which he declared: 'For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either. ... The true reason why [no-one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignorabimus, our credo avers: We must know, We shall know [159]'"(Dawson:71). Gödel's paper was received on November 17, 1930 (cf Reid p. 197, van Heijenoort 1976:592) and published on 25 March 1931 (Dawson 1997:74). But Gödel had given a talk about it beforehand... "An abstract had been presented in October 1930 to the Vienna Academy of Sciences by Hans Hahn" (van Heijenoort:592); this abstract and the full paper both appear in van Heijenoort:583ff. Gödel's incompleteness theorems show that even elementary axiomatic systems such as Peano arithmetic are either self-contradicting or contain logical propositions that are impossible to prove or disprove within that system.

Solving Gordan's Problem
Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as Gordan's Problem, Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem, showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof—it did not display "an object"—but rather, it was an existence proof and relied on use of the law of excluded middle in an infinite extension.

Hilbert sent his results to the Mathematische Annalen. Gordan, the house expert on the theory of invariants for the Mathematische Annalen, could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the Annalen. After having read the manuscript, Klein wrote to him, saying:

"Without doubt this is the most important work on general algebra that the Annalen has ever published."

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

"I have convinced myself that even theology has its merits."

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". Not all were convinced. While Kronecker would die soon afterwards, his constructivist philosophy would continue with the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:

"Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists."

The Hilbert curve

In 1890, Giuseppe Peano had published an article in the Mathematische Annalen describing the historically first space-filling curve. In response, Hilbert designed his own construction of such a curve, which is now called Hilbert curve. Approximations to this curve are constructed iteratively according to the replacement rules in the first picture of this section. The curve itself is then the pointwise limit.

Axiomatization of geometry
The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert without referring to the publication history of the Grundlagen since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. New editions followed the 7th, but the main text was essentially not revised.

Hilbert's approach signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Moritz Pasch's work from 1882. Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert is reported to have said to Schoenflies and Kötter, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed.

Hilbert first enumerates the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points (line segments), and congruence of angles. The axioms unify both the plane geometry and solid geometry of Euclid in a single system.

The 23 problems
Hilbert put forth a highly influential list consisting of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.

After re-working the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed, however, from the later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki, and from his contemporary Giuseppe Peano. The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians held in Paris. The introduction of the speech that Hilbert gave said:

"Who among us would not be happy to lift the veil behind which is hidden the future; to gaze at the coming developments of our science and at the secrets of its development in the centuries to come? What will be the ends toward which the spirit of future generations of mathematicians will tend? What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?"

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert. See also Hilbert's twenty-fourth problem. The full text is important, since the exegesis of the questions still can be a matter of inevitable debate, whenever it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society.

 * 1. Cantor's problem of the cardinal number of the continuum.
 * 2. The compatibility of the arithmetical axioms.
 * 3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
 * 4. Problem of the straight line as the shortest distance between two points.
 * 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
 * 6. Mathematical treatment of the axioms of physics.
 * 7. Irrationality and transcendence of certain numbers.
 * 8. Problems of prime numbers (The "Riemann Hypothesis").
 * 9. Proof of the most general law of reciprocity in any number field.
 * 10. Determination of the solvability of a Diophantine equation.
 * 11. Quadratic forms with any algebraic numerical coefficients
 * 12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
 * 13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
 * 14. Proof of the finiteness of certain complete systems of functions.
 * 15. Rigorous foundation of Schubert's enumerative calculus.
 * 16. Problem of the topology of algebraic curves and surfaces.
 * 17. Expression of definite forms by squares.
 * 18. Building up of space from congruent polyhedra.
 * 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
 * 20. The general problem of boundary values (Boundary value problems in PDE's).
 * 21. Proof of the existence of linear differential equations having a prescribed monodromy group.
 * 22. Uniformization of analytic relations by means of automorphic functions.
 * 23. Further development of the methods of the calculus of variations.

Formalism
In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought. There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense.

Hilbert's program
In 1920, Hilbert proposed a research project in metamathematics that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that:

 * 1) all of mathematics follows from a correctly chosen finite system of axioms; and
 * 2) that some such axiom system is provably consistent through some means such as the epsilon calculus.

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the ignorabimus, still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond.

This program is still recognizable in the most popular philosophy of mathematics, where it is usually called formalism. For example, the Bourbaki group adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the axiomatic method as a research tool. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

"We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."

Hilbert published his views on the foundations of mathematics in the 2-volume work, Grundlagen der Mathematik.

Gödel's work
Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

Gödel demonstrated that any non-contradictory formal system, which was comprehensive enough to include at least arithmetic, cannot demonstrate its completeness by way of its own axioms. In 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely finitary.

Nevertheless, the subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s. The basis for later theoretical computer science, in the work of Alonzo Church and Alan Turing, also grew directly out of this "debate".

Functional analysis
Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

Physics
Until 1912, Hilbert was almost exclusively a pure mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar on the subject in 1905.

In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself. He started studying kinetic gas theory and moved on to elementary radiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of Albert Einstein and others were followed closely.

By 1907, Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years to put the theory into its final form. By early summer 1915, Hilbert's interest in physics had focused on general relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject. Einstein received an enthusiastic reception at Göttingen. Over the summer, Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts. During November 1915, Einstein published several papers culminating in The Field Equations of Gravitation (see Einstein field equations). Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–Hilbert action). Hilbert fully credited Einstein as the originator of the theory and no public priority dispute concerning the field equations ever arose between the two men during their lives. See more at priority.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, and his namesake Hilbert space plays an important part in quantum theory. In 1926, von Neumann showed that, if quantum states were understood as vectors in Hilbert space, they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher mathematics, physicists tended to be "sloppy" with it. To a pure mathematician like Hilbert, this was both ugly, and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found – most importantly in the area of integral equations. When his colleague Richard Courant wrote the now classic Methoden der mathematischen Physik (Methods of Mathematical Physics) including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant-Hilbert book made it easier for them.

Number theory
Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal. Ernst Hellinger, a student of Hilbert, once told André Weil that Hilbert had announced in his seminar in the early 1900s that he expected the proof of the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.

Works
His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained "many technical errors of varying degree"; when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the continuum hypothesis. The errors were nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the corrections.

Concepts

 * List of things named after David Hilbert
 * Foundations of geometry
 * Hilbert C*-module
 * Hilbert cube
 * Hilbert curve
 * Hilbert matrix
 * Hilbert metric
 * Hilbert–Mumford criterion
 * Hilbert number
 * Hilbert ring
 * Hilbert–Poincaré series
 * Hilbert series and Hilbert polynomial
 * Hilbert space
 * Hilbert spectrum
 * Hilbert system
 * Hilbert transform
 * Hilbert's arithmetic of ends
 * Hilbert's paradox of the Grand Hotel
 * Hilbert–Schmidt operator
 * Hilbert–Smith conjecture

Theorems

 * Hilbert–Burch theorem
 * Hilbert's irreducibility theorem
 * Hilbert's Nullstellensatz
 * Hilbert's theorem (differential geometry)
 * Hilbert's Theorem 90
 * Hilbert's syzygy theorem
 * Hilbert–Speiser theorem

Other

 * Brouwer–Hilbert controversy
 * Direct method in the calculus of variations
 * Entscheidungsproblem
 * Geometry and the Imagination
 * General relativity priority dispute

Primary literature in English translation

 * 1918. "Axiomatic thought," 1114–1115.
 * 1922. "The new grounding of mathematics: First report," 1115–1133.
 * 1923. "The logical foundations of mathematics," 1134–1147.
 * 1930. "Logic and the knowledge of nature," 1157–1165.
 * 1931. "The grounding of elementary number theory," 1148–1156.
 * 1904. "On the foundations of logic and arithmetic," 129–138.
 * 1925. "On the infinite," 367–392.
 * 1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–489.
 * 1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–489.

Secondary literature

 * , available at Gallica. The "Address" of Gabriel Bertrand of 20 December 1943 at the French Academy: he gives biographical sketches of the lives of recently deceased members, including Pieter Zeeman, David Hilbert and Georges Giraud.
 * Bottazzini Umberto, 2003. Il flauto di Hilbert. Storia della matematica. UTET, ISBN 88-7750-852-3
 * Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," Science 278: nn-nn.
 * Dawson, John W. Jr 1997. Logical Dilemmas: The Life and Work of Kurt Gödel. Wellesley MA: A. K. Peters. ISBN 1-56881-256-6.
 * Grattan-Guinness, Ivor, 2000. The Search for Mathematical Roots 1870–1940. Princeton Univ. Press.
 * Gray, Jeremy, 2000. The Hilbert Challenge. ISBN 0-19-850651-1
 * Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of Gravitation. Reidel.
 * Piergiorgio Odifreddi, 2003. Divertimento Geometrico. Le origini geometriche della logica da Euclide a Hilbert. Bollati Boringhieri, ISBN 88-339-5714-4. A clear exposition of the "errors" of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry.
 * The definitive English-language biography of Hilbert.
 * Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 981-99. (in English)
 * Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.
 * Georg von Wallwitz: Meine Herren, dies ist keine Badeanstalt. Wie ein Mathematiker das 20. Jahrhundert veränderte. Berenberg Verlag, Berlin 2017, ISBN 978-3-946334-24-8. The definitive German-language biography of Hilbert.
 * Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 981-99. (in English)
 * Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.
 * Georg von Wallwitz: Meine Herren, dies ist keine Badeanstalt. Wie ein Mathematiker das 20. Jahrhundert veränderte. Berenberg Verlag, Berlin 2017, ISBN 978-3-946334-24-8. The definitive German-language biography of Hilbert.
 * Sieg, Wilfried, and Ravaglia, Mark, 2005, "Grundlagen der Mathematik" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: 981-99. (in English)
 * Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0-393-31276-3.
 * Georg von Wallwitz: Meine Herren, dies ist keine Badeanstalt. Wie ein Mathematiker das 20. Jahrhundert veränderte. Berenberg Verlag, Berlin 2017, ISBN 978-3-946334-24-8. The definitive German-language biography of Hilbert.

title: David Hilbert


text:
Down syndrome

Down syndrome (United States) or Down's syndrome (United Kingdom and other English-speaking nations), also known as trisomy 21, is a genetic disorder caused by the presence of all or part of a third copy of chromosome 21. It is usually associated with developmental delays, mild to moderate intellectual disability, and characteristic physical features. There are three types of Down syndrome, the most common being trisomy 21. Mosaic Down syndrome accounts for two per cent of Down syndrome cases, and Translocation Down syndrome accounts for three per cent of cases.

The parents of the affected individual are usually genetically normal. The probability increases from less than 0.1% in 20-year-old mothers to 3% in those of age 45. The extra chromosome is provided at conception as the egg and sperm combine. A very small percentage of 1–2% gets the additional chromosome in the embryo stage and it only impacts some of the cells in the body; this is known as Mosaic Down syndrome. Usually, babies get 23 chromosomes from each parent for a total of 46, whereas in Down syndrome, a third 21st chromosome is attached. It is believed to occur by chance, with no known behavioral activity or environmental factor that changes the probability. Down syndrome can be identified during pregnancy by prenatal screening, followed by diagnostic testing, or after birth by direct observation and genetic testing. Since the introduction of screening, Down syndrome pregnancies are often aborted (rates varying from 50 to 85% depending on maternal age, gestational age, and maternal race/ethnicity).

There is no cure for Down syndrome. Education and proper care have been shown to provide good quality of life. Some children with Down syndrome are educated in typical school classes, while others require more specialized education. Some individuals with Down syndrome graduate from high school, and a few attend post-secondary education. In adulthood, about 20% in the United States do paid work in some capacity, with many requiring a sheltered work environment. Support in financial and legal matters is often needed. Life expectancy is around 50 to 60 years in the developed world, with proper health care. Regular screening for health issues common in Down syndrome is recommended throughout the person's life.

Down syndrome is the most common chromosomal abnormality. It occurs in about 1 in 1,000 babies born each year. In the US this figure is given as one in 700 births. In 2015, Down syndrome was present in 5.4 million individuals globally and resulted in 27,000 deaths, down from 43,000 deaths in 1990. It is named after British physician John Langdon Down, who fully described the syndrome in 1866. Some aspects of the condition were described earlier by French psychiatrist Jean-Étienne Dominique Esquirol in 1838 and French physician Édouard Séguin in 1844. The genetic cause of Down syndrome was discovered in 1959.

Signs and symptoms
Those with Down syndrome nearly always have physical and intellectual disabilities. As adults, their mental abilities are typically similar to those of an 8- or 9-year-old. At the same time, their emotional and social awareness is very high. They can have poor immune function and generally reach developmental milestones at a later age. They have an increased risk of a number of health concerns, such as congenital heart defect, epilepsy, leukemia, and thyroid diseases.

Physical
People with Down syndrome may have these physical characteristics: a small chin, epicanthic folds, low muscle tone, a flat nasal bridge, a single crease of the palm, and a protruding tongue. A protruding tongue is caused by low tone and weak facial muscles, and often corrected with myofunctional exercises. Some characteristic airway features can lead to obstructive sleep apnea in around half of those with Down syndrome. Other common features include: excessive joint flexibility, extra space between big toe and second toe, single palm lines and short fingers.

Instability of the atlantoaxial joint occurs in about 1–2%. Atlantoaxial instability may cause myelopathy due to cervical spinal cord compression later in life, this often manifests as new onset weakness, problems with coordination, bowel or bladder incontinence, and gait dysfunction. Serial imaging cannot reliably predict future cervical cord compression, but changes can be seen on neurological exam. The condition is surgically corrected with spine surgery.

Growth in height is slower, resulting in adults who tend to have short stature—the average height for men is 154 cm, and for women is 142 cm. Individuals with Down syndrome are at increased risk for obesity as they age due to hypothyroidism, other medical issues and lifestyle. Growth charts have been developed specifically for children with Down syndrome.

Neurological
This syndrome causes about a third of cases of intellectual disability. Many developmental milestones are delayed with the ability to crawl typically occurring around 8–22 months rather than 6–12 months, and the ability to walk independently typically occurring around 1–4 years rather than 9–18 months. Walking is acquired in 50% of children after 24 months.

Most individuals with Down syndrome have mild (IQ: 50–69) or moderate (IQ: 35–50) intellectual disability with some cases having severe (IQ: 20–35) difficulties. Those with mosaic Down syndrome typically have IQ scores 10–30 points higher than that. As they age, the gap tends to widen between people with Down syndrome and their same-age peers.

Commonly, individuals with Down syndrome have better language understanding than ability to speak. Babbling typically emerges around 15 months on average. 10–45% of those with Down syndrome have either a stutter or rapid and irregular speech, making it difficult to understand them. After reaching 30 years of age, some may lose their ability to speak.

They typically do fairly well with social skills. Behavior problems are not generally as great an issue as in other syndromes associated with intellectual disability. In children with Down syndrome, mental illness occurs in nearly 30% with autism occurring in 5–10%. People with Down syndrome experience a wide range of emotions. While people with Down syndrome are generally happy, symptoms of depression and anxiety may develop in early adulthood.

Children and adults with Down syndrome are at increased risk of epileptic seizures, which occur in 5–10% of children and up to 50% of adults. This includes an increased risk of a specific type of seizure called infantile spasms. Many (15%) who live 40 years or longer develop Alzheimer's disease. In those who reach 60 years of age, 50–70% have the disease.

Down syndrome regression disorder is a sudden regression with neuropsychiatric symptoms such as catatonia, possibly caused by an autoimmune disease. It primarily appears in teenagers and younger adults.

Senses
Hearing and vision disorders occur in more than half of people with Down syndrome.

Ocular findings
Brushfield spots (small white or grayish/brown spots on the periphery of the iris), upward slanting palpebral fissures (the opening between the upper and lower lids) and epicanthal folds (folds of skin between the upper eyelid and the nose) are clinical signs at birth suggesting the diagnosis of Down syndrome especially in the Western World. None of these requires treatment.

Visually significant congenital cataracts (clouding of the lens of the eye) occur more frequently with Down syndrome. Neonates with Down syndrome should be screened for cataract because early recognition and referral reduce the risk of vision loss from amblyopia. Dot-like opacities in the cortex of the lens (cerulean cataract) are present in up to 50% of people with Down syndrome, but may be followed without treatment if they are not visually significant.

Strabismus, nystagmus and nasolacrimal duct obstruction occur more frequently in children with Down syndrome. Screening for these diagnoses should begin within six months of birth. Strabismus is more often acquired than congenital. Early diagnosis and treatment of strabismus reduces the risk of vision loss from amblyopia. In Down syndrome, the presence of epicanthal folds may give the false impression of strabismus, referred to as pseudostabismus. Nasolacrimal duct obstruction, which causes tearing (epiphora), is more frequently bilateral and multifactorial than in children without Down syndrome.

Refractive error is more common with Down syndrome, though the rate may not differ until after twelve months of age compared to children without Down syndrome. Early screening is recommended to identify and treat significant refractive error with glasses or contact lenses. Poor accommodation (ability to focus on close objects) is associated with Down syndrome, which may mean bifocals are indicated.

In keratoconus, the cornea progressively thins and bulges into a cone shape, causing visual blurring or distortion. Keratoconus first presents in the teen years and progresses into the thirties. Down syndrome is a strong risk factor for developing keratoconus, and onset may be occur at a younger age than in those without Down syndrome. Eye rubbing is also a risk factor for developing keratoconus. It is speculated that chronic eye irritation from blepharitis may increase eye rubbing in Down syndrome, contributing to the increased prevalence of keratoconus.

An association between glaucoma and Down syndrome is often cited. Glaucoma in children with Down syndrome is uncommon, with a prevalence of less than 1%. It is currently unclear if the prevalence of glaucoma in those with Down syndrome differs from that in the absence of Down syndrome.

Estimates of prevalence of ocular findings in Down Syndrome vary widely depending on the study. Some prevalence estimates follow. Vision problems have been observed in 38–80% of cases. Brushfield spots are present in 38–85% of individuals. Between 20 and 50% have strabismus. Cataracts occur in 15%, and may be present at birth. Keratoconus may occur in as many as 21–30%.

Hearing Loss
Hearing problems are found in 50–90% of children with Down syndrome. This is often the result of otitis media with effusion which occurs in 50–70% and chronic ear infections which occur in 40–60%. Ear infections often begin in the first year of life and are partly due to poor eustachian tube function. Excessive ear wax can also cause hearing loss due to obstruction of the outer ear canal. Even a mild degree of hearing loss can have negative consequences for speech, language understanding, and academics. It is important to rule out hearing loss as a factor in social and cognitive deterioration. Age-related hearing loss of the sensorineural type occurs at a much earlier age and affects 10–70% of people with Down syndrome.

Heart
The rate of congenital heart disease in newborns with Down syndrome is around 40%. Of those with heart disease, about 80% have an atrial septal defect or ventricular septal defect with the former being more common. Congenital heart disease can also put individuals at a higher risk of pulmonary hypertension, where arteries in the lungs narrow and cause inadequate blood oxygenation. Some of the genetic contributions to pulmonary hypertension in individuals with Down Syndrome are abnormal lung development, endothelial dysfunction, and proinflammatory genes. Mitral valve problems become common as people age, even in those without heart problems at birth. Other problems that may occur include tetralogy of Fallot and patent ductus arteriosus. People with Down syndrome have a lower risk of hardening of the arteries.

Cancer
Although the overall risk of cancer in Down syndrome is not changed, the risk of testicular cancer and certain blood cancers, including acute lymphoblastic leukemia (ALL) and acute megakaryoblastic leukemia (AMKL) is increased while the risk of other non-blood cancers is decreased. People with Down syndrome are believed to have an increased risk of developing cancers derived from germ cells whether these cancers are blood- or non-blood-related. In 2008, the World Health Organization (WHO) introduced a distinct classification for myeloid proliferation in individuals with Down syndrome.

Blood cancers
Leukemia is 10 to 15 times more common in children with Down syndrome. In particular, acute lymphoblastic leukemia is 20 times more common and the megakaryoblastic form of acute myeloid leukemia (acute megakaryoblastic leukemia), is 500 times more common. Acute megakaryoblastic leukemia (AMKL) is a leukemia of megakaryoblasts, the precursors cells to megakaryocytes which form blood platelets. Acute lymphoblastic leukemia in Down syndrome accounts for 1–3% of all childhood cases of ALL. It occurs most often in those older than nine years or having a white blood cell count greater than 50,000 per microliter and is rare in those younger than one year old. ALL in Down syndrome tends to have poorer outcomes than other cases of ALL in people without Down syndrome. In short, the likelihood of developing acute myeloid leukemia (AML) and acute lymphoblastic leukemia (ALL) is higher in children with Down syndrome compared to those without Down syndrome.

Myeloid leukemia typically precedes Down syndrome and is accompanied by a condition known as transient abnormal myelopoiesis (TAM), which generally disrupts the differentiation of megakaryocytes and erythrocytes. In Down syndrome, AMKL is typically preceded by transient myeloproliferative disease (TMD), a disorder of blood cell production in which non-cancerous megakaryoblasts with a mutation in the GATA1 gene rapidly divide during the later period of pregnancy. GATA1 mutations combined with trisomy 21 contribute to a predisposition to TAM. In trisomy 21, the process of leukemogenesis starts in early fetal life, with genetic factors, including GATA1 mutations, contributing to the development of TAM on the preleukemic pathway. The condition affects 3–10% of babies with Down. While it often spontaneously resolves within three months of birth, it can cause serious blood, liver, or other complications. In about 10% of cases, TMD progresses to AMKL during the three months to five years following its resolution.

Non-blood cancers
People with Down syndrome have a lower risk of all major solid cancers, including those of lung, breast, and cervix, with the lowest relative rates occurring in those aged 50 years or older. This low risk is thought to be due to an increase in the expression of tumor suppressor genes present on chromosome 21. One exception is testicular germ cell cancer which occurs at a higher rate in Down syndrome.

Endocrine
Problems of the thyroid gland occur in 20–50% of individuals with Down syndrome. Low thyroid is the most common form, occurring in almost half of all individuals. Thyroid problems can be due to a poorly or nonfunctioning thyroid at birth (known as congenital hypothyroidism) which occurs in 1% or can develop later due to an attack on the thyroid by the immune system resulting in Graves' disease or autoimmune hypothyroidism. Type 1 diabetes mellitus is also more common.

Gastrointestinal
Constipation occurs in nearly half of people with Down syndrome and may result in changes in behavior. One potential cause is Hirschsprung's disease, occurring in 2–15%, which is due to a lack of nerve cells controlling the colon. Other congenital problems can include duodenal atresia, imperforate anus and gastroesophageal reflux disease. Celiac disease affects about 7–20%

Teeth
People with Down syndrome tend to be more susceptible to gingivitis as well as early, severe periodontal disease, necrotising ulcerative gingivitis, and early tooth loss, especially in the lower front teeth. While plaque and poor oral hygiene are contributing factors, the severity of these periodontal diseases cannot be explained solely by external factors. Research suggests that the severity is likely a result of a weakened immune system. The weakened immune system also contributes to increased incidence of yeast infections in the mouth (from Candida albicans).

People with Down syndrome also tend to have a more alkaline saliva resulting in a greater resistance to tooth decay, despite decreased quantities of saliva, less effective oral hygiene habits, and higher plaque indexes.

Higher rates of tooth wear and bruxism are also common. Other common oral manifestations of Down syndrome include enlarged hypotonic tongue, crusted and hypotonic lips, mouth breathing, narrow palate with crowded teeth, class III malocclusion with an underdeveloped maxilla and posterior crossbite, delayed exfoliation of baby teeth and delayed eruption of adult teeth, shorter roots on teeth, and often missing and malformed (usually smaller) teeth. Less common manifestations include cleft lip and palate and enamel hypocalcification (20% prevalence).

Taurodontism, an elongation of the pulp chamber, has a high prevalence in people with DS.

Fertility
Males with Down syndrome usually do not father children, while females have lower rates of fertility relative to those who are unaffected. Fertility is estimated to be present in 30–50% of females. Menopause usually occurs at an earlier age. The poor fertility in males is thought to be due to problems with sperm development; however, it may also be related to not being sexually active. As of 2006, three instances of males with Down syndrome fathering children and 26 cases of females having children have been reported. Without assisted reproductive technologies, around half of the children of someone with Down syndrome will also have the syndrome.

Cause
The cause of the extra full or partial chromosome is still unknown. Most of the time, Down syndrome is caused by a random mistake in cell division during early development of the fetus, but not inherited, and there is no scientific research which shows that environmental factors or the parents' activities contribute to Down syndrome. The only factor that has been linked to the increased chance of having a baby with Down syndrome is advanced parental age. This is mostly associated with advanced maternal age but about 10 per cent of cases are associated with advanced paternal age. Down syndrome is caused by having three copies of the genes on chromosome 21, rather than the usual two. The parents of the affected individual are typically genetically normal. Those who have one child with Down syndrome have about a 1% possibility of having a second child with the syndrome, if both parents are found to have normal karyotypes.

The extra chromosome content can arise through several different ways. The most common cause (about 92–95% of cases) is a complete extra copy of chromosome 21, resulting in trisomy 21. In 1–2.5% of cases, some of the cells in the body are normal and others have trisomy 21, known as mosaic Down syndrome. The other common mechanisms that can give rise to Down syndrome include: a Robertsonian translocation, isochromosome, or ring chromosome. These contain additional material from chromosome 21 and occur in about 2.5% of cases. An isochromosome results when the two long arms of a chromosome separate together rather than the long and short arm separating together during egg or sperm development.

Trisomy 21
Down syndrome (also known by the karyotype 47,XX,+21 for females and 47,XY,+21 for males) is mostly caused by a failure of the 21st chromosome to separate during egg or sperm development, known as nondisjunction. As a result, a sperm or egg cell is produced with an extra copy of chromosome 21; this cell thus has 24 chromosomes. When combined with a normal cell from the other parent, the baby has 47 chromosomes, with three copies of chromosome 21. About 88% of cases of trisomy 21 result from nonseparation of the chromosomes in the mother, 8% from nonseparation in the father, and 3% after the egg and sperm have merged.

Mosaic Down syndrome
Mosaic Down syndrome is diagnosed when there is a mixture of two types of cells: some cells have three copies of chromosome 21 but some cells have the typical two copies of chromosome 21. This type is the least common form of Down syndrome and accounts for only about 1% of all cases. Children with mosaic Down syndrome may have the same features as other children with Down syndrome. However, they may have fewer characteristics of the condition due to the presence of some (or many) cells with a typical number of chromosomes.

Translocation Down syndrome
The extra chromosome 21 material may also occur due to a Robertsonian translocation in 2–4% of cases. In this translocation Down syndrome, the long arm of chromosome 21 is attached to another chromosome, often chromosome 14. In a male affected with Down syndrome, it results in a karyotype of 46XY,t(14q21q). This may be a new mutation or previously present in one of the parents. The parent with such a translocation is usually normal physically and mentally; however, during production of egg or sperm cells, a higher chance of creating reproductive cells with extra chromosome 21 material exists. This results in a 15% chance of having a child with Down syndrome when the mother is affected and a less than 5% probability if the father is affected. The probability of this type of Down syndrome is not related to the mother's age. Some children without Down syndrome may inherit the translocation and have a higher probability of having children of their own with Down syndrome. In this case it is sometimes known as familial Down syndrome.

Mechanism
The extra genetic material present in Down syndrome results in overexpression of a portion of the 310 genes located on chromosome 21. This overexpression has been estimated at 50%, due to the third copy of the chromosome present. Some research has suggested the Down syndrome critical region is located at bands 21q22.1–q22.3, with this area including genes for the amyloid precursor protein, superoxide dismutase, and likely the ETS2 proto oncogene. Other research, however, has not confirmed these findings. MicroRNAs are also proposed to be involved.

The dementia that occurs in Down syndrome is due to an excess of amyloid beta peptide produced in the brain and is similar to Alzheimer's disease, which also involves amyloid beta build-up. Amyloid beta is processed from amyloid precursor protein, the gene for which is located on chromosome 21. Senile plaques and neurofibrillary tangles are present in nearly all by 35 years of age, though dementia may not be present. It is hypothesized that those with Down syndrome lack a normal number of lymphocytes and produce less antibodies which is said to present an increased risk of infection.

Epigenetics
Down syndrome is associated with an increased risk of some chronic diseases that are typically associated with older age such as Alzheimer's disease. It is believed that accelerated aging occurs and increases the biological age of tissues, but molecular evidence for this hypothesis is sparse. According to a biomarker of tissue age known as epigenetic clock, it is hypothesized that trisomy 21 increases the age of blood and brain tissue (on average by 6.6 years).

Screening before birth
Guidelines recommend screening for Down syndrome to be offered to all pregnant women, regardless of age. A number of tests are used, with varying levels of accuracy. They are typically used in combination to increase the detection rate. None can be definitive; thus, if screening predicts a high possibility of Down syndrome, either amniocentesis or chorionic villus sampling is required to confirm the diagnosis.

Ultrasound
Prenatal ultrasound can be used to screen for Down syndrome. Findings that indicate increased chances when seen at 14 to 24 weeks of gestation include a small or no nasal bone, large ventricles, nuchal fold thickness, and an abnormal right subclavian artery, among others. The presence or absence of many markers is more accurate. Increased fetal nuchal translucency (NT) indicates an increased possibility of Down syndrome picking up 75–80% of cases and being falsely positive in 6%.

Blood tests
Several blood markers can be measured to predict the chances of Down syndrome during the first or second trimester. Testing in both trimesters is sometimes recommended and test results are often combined with ultrasound results. In the second trimester, often two or three tests are used in combination with two or three of: α-fetoprotein, unconjugated estriol, total hCG, and free βhCG detecting about 60–70% of cases.

Testing of the mother's blood for fetal DNA is being studied and appears promising in the first trimester. The International Society for Prenatal Diagnosis considers it a reasonable screening option for those women whose pregnancies are at a high likelihood of trisomy 21. Accuracy has been reported at 98.6% in the first trimester of pregnancy. Confirmatory testing by invasive techniques (amniocentesis, CVS) is still required to confirm the screening result.

Efficacy
For combinations of ultrasonography and non-genetic blood tests, screening in both the first and second trimesters is better than just screening in the first trimester. The different screening techniques in use are able to pick up 90–95% of cases, with a false-positive rate of 2–5%. If Down syndrome occurs in one in 500 pregnancies with a 90% detection rate and the test used has a 5% false-positive rate, this means, of 20 women who test positive on screening, only one will not have a fetus with Down syndrome confirmed. If the screening test has a 2% false-positive rate, this means, of 50 women who test positive on screening, one will not have a fetus with Down syndrome.

Invasive genetic testing
Amniocentesis and chorionic villus sampling are more reliable tests, but they increase the risk of miscarriage by between 0.5–1%. The risk of limb problems may be increased in the offspring if chorionic villus sampling is performed before 10 weeks. The risk from the procedure is greater the earlier it is performed, thus amniocentesis is not recommended before 15 weeks gestational age and chorionic villus sampling before 10 weeks gestational age.

Abortion rates
About 92% of pregnancies in Europe with a diagnosis of Down syndrome are terminated. As a result, there is almost no one with Down syndrome in Iceland and Denmark, where screening is commonplace. In the United States, the termination rate after diagnosis is around 75%, but varies from 61 to 93%, depending on the population surveyed. Rates are lower among women who are younger and have decreased over time. When asked if they would have a termination if their fetus tested positive, 23–33% said yes, when high-risk pregnant women were asked, 46–86% said yes, and when women who screened positive are asked, 89–97% say yes.

After birth
The diagnosis can often be suspected based on the child's physical appearance at birth. Informing the parents of a diagnosis needs to be made as soon as possible, with care and sensitivity. Even an uncertain diagnosis needs to be informed of in the same way. This allows for a longer time for processing the information.

An analysis of the child's chromosomes is needed to confirm the diagnosis, and to determine if a translocation is present, as this may help determine the chances of the child's parents having further children with Down syndrome.

Management
Efforts such as early childhood intervention, therapies, screening for common medical issues, a good family environment, and work-related training can improve the development of children with Down syndrome and provide good quality of life. Common therapies utilized include physical therapy, occupational therapy and speech therapy. Education and proper care can provide a positive quality of life. Typical childhood vaccinations are recommended.

Health screening
A number of health organizations have issued recommendations for screening those with Down syndrome for particular diseases. This is recommended to be done systematically.

At birth, all children should get an electrocardiogram and ultrasound of the heart. Surgical repair of heart problems may be required as early as three months of age. Heart valve problems may occur in young adults, and further ultrasound evaluation may be needed in adolescents and in early adulthood. Due to the elevated risk of testicular cancer, some recommend checking the person's testicles yearly.

Cognitive development
Some people with Down syndrome experience hearing loss. In this instance, hearing aids or other amplification devices can be useful for language learning. Speech therapy may be useful and is recommended to be started around nine months of age. As those with Down syndrome typically have good hand-eye coordination, learning sign language is a helpful communication tool. Augmentative and alternative communication methods, such as pointing, body language, objects, or pictures, are often used to help with communication. Behavioral issues and mental illness are typically managed with counseling or medications.

Education programs before reaching school age may be useful. School-age children with Down syndrome may benefit from inclusive education (whereby students of differing abilities are placed in classes with their peers of the same age), provided some adjustments are made to the curriculum. In the United States, the Individuals with Disabilities Education Act of 1975 requires public schools generally to allow attendance by students with Down syndrome.

Individuals with Down syndrome may learn better visually. Drawing may help with language, speech, and reading skills. Children with Down syndrome still often have difficulty with sentence structure and grammar, as well as developing the ability to speak clearly. Several types of early intervention can help with cognitive development. Efforts to develop motor skills include physical therapy, speech and language therapy, and occupational therapy. Physical therapy focuses specifically on motor development and teaching children to interact with their environment. Speech and language therapy can help prepare for later language. Lastly, occupational therapy can help with skills needed for later independence.

Other
Tympanostomy tubes are often needed and often more than one set during the person's childhood. Tonsillectomy is also often done to help with sleep apnea and throat infections. Surgery does not correct every instance of sleep apnea and a continuous positive airway pressure (CPAP) machine may be useful in those cases.

Efforts to prevent respiratory syncytial virus (RSV) infection with human monoclonal antibodies should be considered, especially in those with heart problems. In those who develop dementia there is no evidence for memantine, donepezil, rivastigmine, or galantamine.

Prognosis
Between 5–15% of children with Down syndrome in Sweden attend regular school. Some graduate from high school; however, most do not. Of those with intellectual disability in the United States who attended high school about 40% graduated. Many learn to read and write and some are able to do paid work. In adulthood about 20% in the United States do paid work in some capacity. In Sweden, however, less than 1% have regular jobs. Many are able to live semi-independently, but they often require help with financial, medical, and legal matters. Those with mosaic Down syndrome usually have better outcomes.

Individuals with Down syndrome have a higher risk of early death than the general population. This is most often from heart problems or infections. Following improved medical care, particularly for heart and gastrointestinal problems, the life expectancy has increased. This increase has been from 12 years in 1912, to 25 years in the 1980s, to 50 to 60 years in the developed world in the 2000s. Data collected between the 1985–2003 showed between 4–12% infants with Down syndrome die in the first year of life. The probability of long-term survival is partly determined by the presence of heart problems. From research at the turn of the century, it tracked those with congenital heart problems, showing 60% survived to at least 10 years and 50% survived to at least 30 years of age. The research failed to track further aging beyond 30 years. In those without heart problems, 85% studied survived to at least 10 years and 80% survived to at least 30 years of age. It is estimated that 10% lived to 70 years of age in the early 2000s. Much of this data is outdated and life expectancy has drastically improved with more equitable healthcare and continuous advancement of surgical practice. The National Down Syndrome Society provides information regarding raising a child with Down syndrome.

Epidemiology
Down syndrome is the most common chromosomal abnormality in humans. Globally,, Down syndrome occurs in about 1 per 1,000 births and results in about 17,000 deaths. More children are born with Down syndrome in countries where abortion is not allowed and in countries where pregnancy more commonly occurs at a later age. About 1.4 per 1,000 live births in the United States and 1.1 per 1,000 live births in Norway are affected. In the 1950s, in the United States, it occurred in 2 per 1,000 live births with the decrease since then due to prenatal screening and abortions. The number of pregnancies with Down syndrome is more than two times greater with many spontaneously aborting. It is the cause of 8% of all congenital disorders.

Maternal age affects the chances of having a pregnancy with Down syndrome. At age 20, the chance is 1 in 1,441; at age 30, it is 1 in 959; at age 40, it is 1 in 84; and at age 50 it is 1 in 44. Although the probability increases with maternal age, 70% of children with Down syndrome are born to women 35 years of age and younger, because younger people have more children. The father's older age is also a risk factor in women older than 35, but not in women younger than 35, and may partly explain the increase in risk as women age.

History

English physician John Langdon Down first described Down syndrome in 1862, recognizing it as a distinct type of mental disability, and again in a more widely published report in 1866. Édouard Séguin described it as separate from cretinism in 1844. By the 20th century, Down syndrome had become the most recognizable form of mental disability.

Due to his perception that children with Down syndrome shared facial similarities with those of Blumenbach's Mongoloid race, John Langdon Down used the term "mongoloid". He felt that the existence of Down syndrome confirmed that all peoples were genetically related. In the 1950s with discovery of the underlying cause as being related to chromosomes, concerns about the race-based nature of the name increased.

In 1961, a group of nineteen scientists suggested that "mongolism" had "misleading connotations" and had become "an embarrassing term". The World Health Organization (WHO) dropped the term in 1965 after a request by the delegation from the Mongolian People's Republic. While this terminology continued to be used until the late twentieth century, it is now considered unacceptable and is no longer in common use.

In antiquity, many infants with disabilities were either killed or abandoned. In June 2020, the earliest incidence of Down syndrome was found in genomic evidence from an infant that was buried before 3200 BC at Poulnabrone dolmen in Ireland. Researchers believe that a number of historical pieces of art portray Down syndrome, including pottery from the pre-Columbian Tumaco-La Tolita culture in present-day Colombia and Ecuador, and the 16th-century painting The Adoration of the Christ Child.

In the 20th century, many individuals with Down syndrome were institutionalized, few of the associated medical problems were treated, and most people died in infancy or early adulthood. With the rise of the eugenics movement, 33 of the then 48 U.S. states and several countries began programs of forced sterilization of individuals with Down syndrome and comparable degrees of disability. Action T4 in Nazi Germany saw the systematic murder of people with Down syndrome made public policy.

With the discovery of karyotype techniques in the 1950s it became possible to identify abnormalities of chromosomal number or shape. In 1959 Jérôme Lejeune reported the discovery that Down syndrome resulted from an extra chromosome. However, Lejeune's claim to the discovery has been disputed, and in 2014 the Scientific Council of the French Federation of Human Genetics unanimously awarded its Grand Prize to his colleague Marthe Gautier for her role in this discovery. The discovery took place in the laboratory of Raymond Turpin at the Hôpital Trousseau in Paris, France. Jérôme Lejeune and Marthe Gautier were both his students.

As a result of this discovery, the condition became known as trisomy 21. Even before the discovery of its cause, the presence of the syndrome in all races, its association with older maternal age, and its rarity of recurrence had been noticed. Medical texts had assumed it was caused by a combination of inheritable factors that had not been identified. Other theories had focused on injuries sustained during birth.

Name
Down syndrome is named after John Langdon Down. He was the first person to provide an accurate description of the syndrome. His research that was published in 1866 earned him the recognition as the Father of the syndrome. While others had previously recognized components of the condition, John Langdon Down described the syndrome as a distinct, unique medical condition.

In 1975, the United States National Institutes of Health (NIH) convened a conference to standardize the naming and recommended replacing the possessive form, "Down's syndrome", with "Down syndrome". However, both the possessive and nonpossessive forms remain in use by the general population. The term "trisomy 21" is also commonly used.

Ethics
Obstetricians routinely offer antenatal screenings for various conditions, including Down syndrome. As a medically reasonable procedure, requiring informed consent, people should be given information about it. It will then be the woman's choice, based on her personal beliefs, how much or how little screening she wishes. When results from testing become available, it is considered an ethical requirement to share the results with the patient.

Some bioethicists deem it reasonable for parents to select a child who would have the highest well-being. One criticism of this reasoning is that it often values those with disabilities less. Some parents argue that Down syndrome should not be prevented or cured and that eliminating Down syndrome amounts to genocide. The disability rights movement does not have a position on screening, although some members consider testing and abortion discriminatory. Some in the United States who are anti-abortion support abortion if the fetus is disabled, while others do not. Of a group of 40 mothers in the United States who have had one child with Down syndrome, half agreed to screening in the next pregnancy.

Within the US, some Protestant denominations see abortion as acceptable when a fetus has Down syndrome while Orthodox Christianity and Roman Catholicism do not. Some of those against screening refer to it as a form of eugenics. Disagreement exists within Islam regarding the acceptability of abortion in those carrying a fetus with Down syndrome. Some Islamic countries allow abortion, while others do not. Parents may be stigmatized whichever decision they make.

Advocacy groups
Advocacy groups for individuals with Down syndrome began to be formed after the Second World War. These were organizations advocating for the inclusion of people with Down syndrome into the general school system and for a greater understanding of the condition among the general population, as well as groups providing support for families with children living with Down syndrome. Before this individuals with Down syndrome were often placed in mental hospitals or asylums. Organizations included the Royal Society for Handicapped Children and Adults founded in the UK in 1946 by Judy Fryd, Kobato Kai founded in Japan in 1964, the National Down Syndrome Congress founded in the United States in 1973 by Kathryn McGee and others, and the National Down Syndrome Society founded in 1979 in the United States. The first Roman Catholic order of nuns for women with Down Syndrome, Little Sisters Disciples of the Lamb, was founded in 1985 in France.

The first World Down Syndrome Day was held on 21 March 2006. The day and month were chosen to correspond with 21 and trisomy, respectively. It was recognized by the United Nations General Assembly in 2011.

Special21.org, founded in 2015, advocates the need for a specific classification category to enable Down syndrome swimmers the opportunity to qualify and compete at the Paralympic Games. The project began when International Down syndrome swimmer Filipe Santos broke the world record in the 50m butterfly event, but was unable to compete at the Paralympic Games.

Paralympic Swimming
International Paralympic Committee Para-swimming classification codes are based upon single impairment only, whereas Down syndrome individuals have both physical and intellectual impairments.

Although Down syndrome swimmers are able to compete in the Paralympic Swimming S14 intellectual impairment category (provided they score low in IQ tests), they are often outmatched by the superior physicality of their opponents.

At present there is no designated Paralympic category for swimmers with Down syndrome, meaning they have to compete as intellectually disadvantaged athletes. This disregards their physical disabilities.

A number of advocacy groups globally have been lobbying for the inclusion of a distinct classification category for Down syndrome swimmers within the IPC Classification Codes framework.

Despite ongoing advocacy, the issue remains unresolved, and swimmers with Down syndrome continue to face challenges in accessing appropriate classification pathways.

Research
Efforts are underway to determine how the extra chromosome 21 material causes Down syndrome, as currently this is unknown, and to develop treatments to improve intelligence in those with the syndrome. Two efforts being studied are the use stem cells and gene therapy. Other methods being studied include the use of antioxidants, gamma secretase inhibition, adrenergic agonists, and memantine. Research is often carried out on an animal model, the Ts65Dn mouse.

Other hominids
Down syndrome may also occur in hominids other than humans. In great apes chromosome 22 corresponds to the human chromosome 21 and thus trisomy 22 causes Down syndrome in apes. The condition was observed in a common chimpanzee in 1969 and a Bornean orangutan in 1979, but neither lived very long. The common chimpanzee Kanako (born around 1993, in Japan) has become the longest-lived known example of this condition. Kanako has some of the same symptoms that are common in human Down syndrome. It is unknown how common this condition is in chimps, but it is plausible it could be roughly as common as Down syndrome is in humans.

Fossilized remains of a Neanderthal aged approximately 6 at death were described in 2024. The child, nicknamed Tina, suffered from a malformation of the inner ear that only occurs in people with Down syndrome, and would have caused hearing loss and disabling vertigo. The fact that a Neanderthal with such a condition survived to such an age was taken as evidence of compassion and extra-maternal care among Neanderthals.

Individuals

 * Jamie Brewer is an American actress and model. She is best known for her roles in the FX horror anthology television series American Horror Story. In its first season, Murder House, she portrayed Adelaide "Addie" Langdon; in the third season, Coven, she portrayed Nan, an enigmatic and clairvoyant witch; in the fourth season Freak Show, she portrayed Chester Creb's vision of his doll, Marjorie; in the seventh season Cult, she portrayed Hedda, a member of the 'SCUM' crew, led by feminist Valerie Solanas; and she also returned to her role as Nan in the eighth season, Apocalypse. In February 2015, Brewer became the first woman with Down syndrome to walk the red carpet at New York Fashion Week, for designer Carrie Hammer.
 * Sofía Jirau is a Puerto Rican model with Down syndrome, working with top designers and renowned media outlets such Vogue Mexico, People, Hola!, among others.  In February 2020, Jirau made her debut at New York Fashion Week. Then in February 2022, she became the first-ever model with Down Syndrome to be hired by the American retail company Victoria's Secret. She walked the LA Fashion Week runway in 2022. Jirau launched a campaign in 2021 called Sin Límites or No Limits "which seeks to make visible the challenges facing the Down syndrome community, demonstrate our ability to achieve our goals, and raise awareness about the condition throughout the world."
 * Chris Nikic is the first person with Down syndrome to finish an Ironman Triathlon. He was awarded the Jimmy V Award for Perseverance at the 2021 ESPY Awards. Nikic continues to run races around the world, using his platform to promote his 1% Better message and bring awareness to the endless possibilities for people with Down syndrome.
 * Grace Strobel is an American model and the first person with Down Syndrome to represent an American skin-care brand. She first joined Obagi in 2020, and continues to be an Ambassador for the brand as of 2022. She walked the runway representing Tommy Hilfiger for Runway of Dreams New York Fashion Week 2020 and Atlantic City Fashion Week. Strobel has been featured in Forbes, on The Today Show, Good Morning America, by Rihanna's Fenty Beauty, Lady Gaga's Kindness Channel, and many more. She is also a public speaker and gives a presentation called #TheGraceEffect about what it is like to live with Down syndrome.

Television and film

 * Life Goes On is an American drama television series that aired on ABC from September 12, 1989, to May 23, 1993. The show centers on the Thatcher family living in suburban Chicago: Drew, his wife Libby, and their children Paige, Rebecca and Charles. Charles, called Corky on the show and portrayed by Chris Burke, was the first major character on a television series with Down syndrome. Burke's revolutionary role conveyed a realistic portrayal of people with Down syndrome and changed the way audiences viewed people with disabilities.
 * Struck by Lightning, an Australian film by Jerzy Domaradzki and starring Garry McDonald, is a comedy-drama depicting the efforts by a newly appointed physical education teacher to introduce soccer to a specialized school for youths with Down syndrome.
 * Champions (2023) is a film starring four main actors with Down syndrome: Madison Tevlin, Kevin Iannucci, Matthew Von Der Ahe and James Day Keith. It is an American sports comedy film directed by Bobby Farrelly in his solo directorial debut, from a screenplay written by Mark Rizzo. The film stars Woody Harrelson as a temperamental minor-league basketball coach who after an arrest must coach a team of players with intellectual disabilities as community service; Kaitlin Olson, Ernie Hudson, and Cheech Marin also star.
 * Born This Way is an American reality television series produced by Bunim/Murray Productions featuring seven adults with Down syndrome with work hard to achieve goals and overcome obstacles. The show received a Television Academy Honor in 2016.
 * The Peanut Butter Falcon is a 2019 American comedy-drama film written and directed by Tyler Nilson and Michael Schwartz, in their directorial film debut, and starring Zack Gottsagen, Shia LaBeouf, Dakota Johnson and John Hawkes. The plot follows a young man with Down syndrome who escapes from an assisted living facility, in order to follow his dream of being a wrestler, and befriends a wayward fisherman on the run. As the two men form a rapid bond, a social worker attempts to track them.

Music

 * The Devo song "Mongoloid" is about someone with Down syndrome.
 * The Amateur Transplants song "Your Baby" is about a fetus with Down syndrome.

Toys

 * Mattell released a Barbie doll with characteristics of a person having Down syndrome as a way to promote diversity.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Down syndrome