David Hilbert

David Hilbert (; German: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a prominent German mathematician and philosopher of mathematics, widely recognized as one of the most influential figures in the field during his era.

David Hilbert (; German: [ˈdaːvɪtˈhɪlbɐt]; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.

Hilbert's contributions encompassed the discovery and development of numerous foundational concepts, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators with its applications to integral equations, mathematical physics, and the foundations of mathematics, particularly proof theory. He was a staunch advocate for Georg Cantor's set theory and transfinite numbers. His presentation of a seminal collection of problems in 1900 significantly shaped the trajectory of mathematical research throughout the 20th century.

Together with his students, Hilbert played a crucial role in establishing mathematical rigor and devising essential tools utilized in contemporary mathematical physics. He is also recognized as a co-founder of both proof theory and mathematical logic.

Life

Early life and education

David Hilbert, the elder of two children and the sole son of Otto, a county judge, and Maria Therese Hilbert (née Erdtmann), a merchant's daughter, was born in the Province of Prussia, within the Kingdom of Prussia. His birthplace is recorded as either Königsberg (present-day Kaliningrad), based on Hilbert's personal account, or Wehlau (known as Znamensk since 1946), located near Königsberg, where his father was employed at the time of his birth. His paternal grandfather, also named David Hilbert, held positions as a judge and Geheimrat. Maria, his mother, cultivated interests in philosophy, astronomy, and prime numbers, while his father, Otto, instilled in him Prussian virtues. Following his father's appointment as a city judge, the family relocated to Königsberg. His sister, Elise, was born when he was six years old. Hilbert commenced his formal education at the age of eight, two years beyond the typical starting age.

In late 1872, Hilbert enrolled at the Friedrichskolleg Gymnasium (Collegium fridericianum), a school previously attended by Immanuel Kant 140 years prior. However, after an unsatisfactory period, he transferred in late 1879 and subsequently graduated in early 1880 from the Wilhelm Gymnasium, which offered a more science-focused curriculum. Following his graduation in autumn 1880, Hilbert matriculated at the University of Königsberg, known as the "Albertina." In early 1882, Hermann Minkowski, who was two years Hilbert's junior and also a Königsberg native (though he had spent three semesters in Berlin), returned to the city and joined the university. Hilbert subsequently formed a lifelong friendship with the reserved yet talented Minkowski.

Career

In 1884, Adolf Hurwitz joined the faculty from Göttingen as an Extraordinarius, equivalent to an associate professor. This marked the beginning of an intense and productive scientific collaboration among the three scholars, with Minkowski and Hilbert, in particular, exerting mutual influence throughout their respective scientific careers. Hilbert successfully defended his doctoral dissertation in 1885, supervised by Ferdinand von Lindemann. The thesis was titled Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen, which translates to "On the invariant properties of special binary forms, in particular the spherical harmonic functions."

Hilbert served as a Privatdozent (senior lecturer) at the University of Königsberg from 1886 to 1895. In 1895, through the advocacy of Felix Klein, he secured the position of Professor of Mathematics at the University of Göttingen. The period during which Klein and Hilbert were active transformed Göttingen into the foremost institution in the global mathematical community. He continued his tenure there for the remainder of his life.

Göttingen school

Notable individuals among Hilbert's students included Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann served as his assistant. At the University of Göttingen, Hilbert was part of a distinguished intellectual community that comprised several of the 20th century's most significant mathematicians, including Emmy Noether and Alonzo Church.

Of his 69 doctoral students at Göttingen, many subsequently achieved renown as mathematicians, including (with thesis completion year): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), and Wilhelm Ackermann (1925). From 1902 to 1939, Hilbert held the position of editor for the Mathematische Annalen, which was then the foremost mathematical journal. In 1907, he was elected an International Member of the United States National Academy of Sciences.

Personal life

In 1892, Hilbert married Käthe Jerosch (1864–1945), the daughter of a Königsberg merchant, who was characterized as "an outspoken young lady with an independence of mind that matched [Hilbert's]." During their time in Königsberg, they had one son, Franz Hilbert (1893–1969). Franz experienced lifelong mental illness, and following his admission to a psychiatric clinic, Hilbert reportedly stated, "From now on, I must consider myself as not having a son." This stance deeply distressed Käthe.

Hilbert regarded the mathematician Hermann Minkowski as his closest and most trusted friend.

Hilbert was baptized and raised as a Calvinist within the Prussian Evangelical Church. He subsequently departed from the Church and adopted an agnostic worldview. He further contended that mathematical truth existed independently of divine existence or other a priori presuppositions. Responding to criticisms of Galileo Galilei for not upholding his heliocentric convictions, Hilbert asserted: "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 Life

Similar to Albert Einstein, Hilbert maintained close associations with the Berlin Group, whose principal founders, including Kurt Grelling, Hans Reichenbach, and Walter Dubislav, had been his students in Göttingen.

Approximately in 1925, Hilbert contracted pernicious anemia, a vitamin deficiency that was then untreatable and primarily manifested as exhaustion. His assistant, Eugene Wigner, characterized Hilbert as experiencing "enormous fatigue" and appearing "quite old." Wigner further noted that even after a diagnosis and subsequent treatment, Hilbert "was hardly a scientist after 1925, and certainly not a Hilbert."

In 1932, Hilbert was elected as a member of the American Philosophical Society.

Hilbert witnessed the Nazi regime's purge of numerous distinguished faculty members from the University of Göttingen in 1933. Among those dismissed were Hermann Weyl, who had assumed Hilbert's chair upon his retirement in 1930; Emmy Noether; and Edmund Landau. Paul Bernays, another individual compelled to leave Germany, had collaborated with Hilbert on mathematical logic and co-authored the significant work Grundlagen der Mathematik, which was eventually published in two volumes in 1934 and 1939. This publication served as a continuation of the Hilbert–Ackermann volume, Principles of Mathematical Logic (1928). Helmut Hasse succeeded Hermann Weyl.

Approximately one year subsequent to the purge, Hilbert attended a banquet where he was seated beside Bernhard Rust, the newly appointed Minister of Education. Rust inquired whether "the Mathematical Institute really suffered so much because of the departure of the Jews." Hilbert's poignant response was: "Suffered? It doesn't exist any longer, does it?"

Death

By Hilbert's death in 1943, the Nazi regime had almost entirely replaced the university's faculty, largely due to the dismissal of individuals who were Jewish or married to Jews. His funeral was sparsely attended, with fewer than a dozen individuals present, including only two fellow academics, one of whom was Arnold Sommerfeld, a theoretical physicist and Königsberg native. Public awareness of his passing emerged only several months after his death.

The epitaph inscribed on Hilbert's tombstone in Göttingen features the renowned statements he delivered at the culmination of his retirement address to the Society of German Scientists and Physicians on September 8, 1930. These words were offered as a rejoinder to the Latin maxim: "Ignoramus et ignorabimus," which translates to "We do not know and we shall not know":

On the day preceding Hilbert's pronouncement of these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel, during a roundtable discussion at the Conference on Epistemology held concurrently with the Society meetings, provisionally presented the initial formulation of his incompleteness theorem. Gödel's incompleteness theorems demonstrate that even fundamental axiomatic systems, such as Peano arithmetic, are either inherently self-contradictory or encompass logical propositions that cannot be proven or disproven within the confines of that system.

Contributions to Mathematics and Physics

Resolution of Gordan's Problem

Hilbert's initial research on invariant functions culminated in 1888 with the presentation of his renowned finiteness theorem. Two decades prior, Paul Gordan had established the theorem concerning the finiteness of generators for binary forms, employing an intricate computational methodology. Attempts to extend Gordan's approach to functions involving more than two variables proved unsuccessful due to the immense computational complexity. To address what became known in certain academic circles as Gordan's Problem, Hilbert recognized the necessity of adopting an entirely different strategy. Consequently, he formulated Hilbert's basis theorem, which demonstrated the existence of a finite set of generators for the invariants of quantics across any number of variables. However, this proof was abstract, establishing existence without providing a constructive method to identify such a set; it relied on the law of excluded middle within an infinite extension.

Hilbert submitted his findings to the journal Mathematische Annalen. Gordan, who served as the journal's resident authority on invariant theory for Mathematische Annalen, failed to grasp the groundbreaking nature of Hilbert's theorem and subsequently rejected the manuscript, citing an insufficiently comprehensive exposition. His commentary stated:

In contrast, Klein acknowledged the significance of the work and guaranteed its publication without any revisions. Encouraged by Klein, Hilbert expanded his methodology in a subsequent article, offering estimations for the maximum degree of the minimal set of generators, and resubmitted it to the Annalen. Upon reviewing the manuscript, Klein conveyed to Hilbert:

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

Subsequently, after the utility of Hilbert's method gained universal acceptance, Gordan himself remarked:

I have convinced myself that even theology has its merits.

Despite his successes, the inherent nature of Hilbert's proof generated unforeseen challenges. Although Kronecker eventually conceded, Hilbert later addressed similar criticisms by asserting that "many different constructions are subsumed under one fundamental idea"—or, as Reid articulated, "Through a proof of existence, Hilbert had been able to obtain a construction"; thus, "the proof" (i.e., the written symbols) was "the object." This perspective did not universally persuade. While Kronecker's death followed soon after, his constructivist philosophy persisted through the emerging intuitionist "school" led by the young Brouwer, causing considerable distress to Hilbert in his later years. Indeed, Hilbert witnessed his "gifted pupil" Weyl embrace intuitionism, a development that "disturbed Hilbert by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker." Brouwer, as an intuitionist, specifically opposed the application of the Law of Excluded Middle to infinite sets, a principle Hilbert had employed. Hilbert's retort was:

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

Nullstellensatz

In algebra, a field is defined as algebraically closed if every polynomial defined over it possesses a root within that field. Building upon this concept, Hilbert established a criterion to determine when a set of ${\displaystyle (p_{\lambda })_{\lambda \in \Lambda }}$ polynomials in ${\displaystyle n}$ variables shares a common root. This condition holds precisely when there are no polynomials ${\displaystyle q_{1},\ldots ,q_{k}}$ and indices ${\displaystyle \lambda _{1},\ldots ,\lambda _{k}}$ satisfying the following equation:

§67§ = ∑ j = §1920§ k p λ j ( x → ) q j ( x → ) {\displaystyle 1=\sum _{j=1}^{k}p_{\lambda _{j}}({\vec {x}})q_{j}({\vec {x}})} .

This significant finding is formally recognized as the Hilbert root theorem, also known by its German designation, "Hilberts Nullstellensatz." Furthermore, Hilbert demonstrated a bijective correspondence between vanishing ideals and their associated vanishing sets, specifically linking affine varieties with radical ideals within ${\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]}$.

Curve

In 1890, Giuseppe Peano introduced the first historically documented space-filling curve in an article published in the Mathematische Annalen. Subsequently, Hilbert developed his own variant of this curve, which is presently known as the Hilbert curve. Iterative approximations of this curve are generated based on the replacement rules illustrated in the initial figure of this section. The curve itself is defined as the pointwise limit of these approximations.

Axiomatization of Geometry

In 1899, Hilbert published Grundlagen der Geometrie, translated as Foundations of Geometry, which proposed a formal set of axioms, known as Hilbert's axioms, to replace Euclid's traditional postulates. These new axioms addressed weaknesses identified in Euclid's work, which was still widely used as a textbook at the time. Precisely defining Hilbert's axioms necessitates referencing the publication history of Grundlagen, as Hilbert revised and modified them multiple times. The initial monograph was swiftly followed by a French translation, to which Hilbert appended V.2, the Completeness Axiom. An English translation, authorized by Hilbert and copyrighted in 1902 by E.J. Townsend, incorporated the changes from the French edition and is thus considered a translation of the second edition. Hilbert continued to introduce alterations to the text, resulting in several German editions, with the seventh being the last published during his lifetime. Subsequent editions appeared after the seventh, though the core text remained largely unrevised.

Hilbert's methodology marked a pivotal shift toward the modern axiomatic approach, a development anticipated by Moritz Pasch's work in 1882. Under this paradigm, axioms are not regarded as self-evident truths. While geometry may concern itself with things that evoke strong intuitions, it is not essential to assign explicit meaning to undefined concepts. Elements such as points, lines, and planes, among others, could, as Hilbert reportedly suggested to Schoenflies and Kötter, be substituted by objects like tables, chairs, or glasses of beer. The focus, instead, lies on their defined relationships.

Hilbert initially enumerated the undefined concepts: point, line, plane, the relation of "lying on" (which applies between points and lines, points and planes, and lines and planes), betweenness, congruence of point pairs (line segments), and congruence of angles. These axioms integrate both Euclidean plane geometry and solid geometry into a unified system.

Twenty-Three Problems

At the International Congress of Mathematicians in Paris in 1900, Hilbert presented a highly influential list of 23 unsolved problems. This compilation is widely regarded as the most successful and profoundly considered collection of open problems ever formulated by an individual mathematician.

Following his foundational work in classical geometry, Hilbert could have extended his approach to the entirety of mathematics. His methodology diverged from the later "foundationalist" perspectives of Russell–Whitehead and the "encyclopedist" approach of Nicolas Bourbaki, as well as from his contemporary Giuseppe Peano. Hilbert's problems were designed to engage the broader mathematical community in crucial aspects of significant mathematical domains.

The problem set was introduced during a lecture titled "The Problems of Mathematics," delivered at the Second International Congress of Mathematicians in Paris. Hilbert's introductory remarks for this speech stated:

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries ? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive ? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

Hilbert presented fewer than half of these problems at the Congress, with their initial publication appearing in the Congress's proceedings. In a subsequent publication, he expanded this overview, leading to the definitive formulation of the now-canonical 23 Problems of Hilbert. The complete text remains significant, as the interpretation of these questions can still be a subject of debate regarding the number of problems that have been definitively solved.

Some of these problems were resolved relatively quickly. Others have been subjects of extensive discussion throughout the 20th century, with a few now considered too open-ended to achieve definitive closure. A subset of these problems continues to pose significant challenges.

The following are the headings for Hilbert's 23 problems as they appeared in the 1902 translation published 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.

The fourth problem addresses the concept of a straight line as the shortest distance between two points.

The fifth problem concerns Lie's theory of continuous transformation groups, specifically without presuming the differentiability of the functions that define these groups.

The sixth problem involves the mathematical formalization of physical axioms.

The seventh problem investigates the irrationality and transcendence properties of specific numbers.

The eighth problem focuses on prime number distribution, notably encompassing the Riemann Hypothesis.

The ninth problem seeks to establish a proof for the most generalized law of reciprocity within any number field.

The tenth problem aims to determine the solvability of Diophantine equations.

The eleventh problem addresses quadratic forms that incorporate arbitrary algebraic numerical coefficients.

The twelfth problem involves extending Kronecker's theorem, which pertains to Abelian fields, to encompass any algebraic domain of rationality.

The thirteenth problem explores the impossibility of solving the general seventh-degree equation using functions restricted to only two arguments.

The fourteenth problem requires demonstrating the finiteness of specific complete systems of functions.

The fifteenth problem calls for a rigorous foundational framework for Schubert's enumerative calculus.

The sixteenth problem concerns the topology of algebraic curves and surfaces.

The seventeenth problem involves expressing definite forms as sums of squares.

The eighteenth problem investigates the construction of space using congruent polyhedra.

The nineteenth problem questions whether solutions to regular problems in the calculus of variations are invariably analytic.

The twentieth problem addresses the general theory of boundary values, specifically boundary value problems in partial differential equations.

The twenty-first problem seeks to prove the existence of linear differential equations possessing a predefined monodromy group.

The twenty-second problem involves the uniformization of analytic relations through the application of automorphic functions.

The twenty-third problem proposes the further advancement of methodologies within the calculus of variations.

Formalism

By mid-century, Hilbert's influential problem set was widely recognized as a foundational manifesto, paving the way for the emergence of the formalist school, a prominent mathematical philosophy of the 20th century. Formalists posit that mathematics constitutes the manipulation of symbols governed by established formal rules, thereby representing an autonomous intellectual endeavor.

Program

In 1920, Hilbert introduced a metamathematical research initiative, subsequently termed Hilbert's program, which aimed to establish mathematics upon a robust and comprehensive logical framework. He theorized that this objective could be achieved by demonstrating two key principles:

1. First, that the entirety of mathematics could be derived from a precisely selected finite axiomatic system; and
2. Second, that such an axiomatic system could be demonstrably consistent through methods like the epsilon calculus.

Hilbert's formulation of this proposal appears to have been motivated by both technical and philosophical considerations. It notably reflected his opposition to the concept known as "ignorabimus," a significant intellectual debate in contemporary German thought, which originated with Emil du Bois-Reymond.

This program remains identifiable within the predominant philosophy of mathematics, commonly referred to as formalism. For instance, the Bourbaki group implemented a modified and selective iteration of this program, deeming it suitable for their dual objectives: (a) compiling comprehensive foundational texts, and (b) advocating for the axiomatic method as a research instrument. While this approach proved successful and impactful concerning Hilbert's contributions to algebra and functional analysis, it did not similarly resonate with his engagements in physics and logic.

In 1919, Hilbert articulated:

We are not discussing arbitrariness in any context. Mathematics does not resemble a game where tasks are defined by arbitrarily established rules. Instead, it constitutes a conceptual system endowed with an inherent necessity, which dictates its nature and precludes any alternative.

Hilbert's perspectives on the fundamental principles of mathematics were disseminated in his two-volume publication, *Grundlagen der Mathematik*.

Gödel's Contributions

Hilbert and his collaborators were deeply committed to this ambitious undertaking. However, his endeavor to underpin axiomatized mathematics with conclusive principles, intended to eliminate theoretical ambiguities, ultimately proved unsuccessful.

Gödel conclusively demonstrated that any consistent formal system capable of expressing fundamental arithmetic cannot establish its own completeness solely through its intrinsic axioms and rules of inference. His 1931 incompleteness theorem revealed that Hilbert's comprehensive program, as originally conceived, was unattainable. Specifically, the second tenet of Hilbert's program cannot be coherently integrated with the first, provided the axiomatic system is genuinely finitary.

Nevertheless, the subsequent advancements in proof theory significantly clarified the concept of consistency, particularly concerning theories central to mathematical inquiry. Hilbert's foundational work initiated this trajectory of clarification in logic. Subsequently, the imperative to comprehend Gödel's contributions spurred the evolution of recursion theory, which then established mathematical logic as an autonomous academic discipline in the 1930s. Furthermore, the foundational principles for later theoretical computer science, notably through the contributions of Alonzo Church and Alan Turing, emerged directly from this intellectual discourse.

Functional Analysis

Approximately in 1909, Hilbert dedicated his efforts to investigating differential and integral equations, yielding direct implications for significant areas within modern functional analysis. To facilitate these investigations, Hilbert conceptualized an infinite-dimensional Euclidean space, subsequently designated as Hilbert space. His endeavors in this analytical domain provided a crucial foundation for substantial contributions to the mathematics of physics over the ensuing two decades, albeit from an unforeseen perspective. Later, Stefan Banach expanded upon this concept by defining Banach spaces. Hilbert spaces constitute a pivotal class of entities within functional analysis, particularly relevant to the spectral theory of self-adjoint linear operators, a field that developed around them throughout the 20th century.

Physics

Prior to 1912, Hilbert primarily functioned as a pure mathematician. When Hermann Minkowski, a fellow mathematician and friend, planned a Indeed, Minkowski appears to have been instrumental in most of Hilbert's physics explorations before 1912, including their collaborative seminar on the subject in 1905.

In 1912, three years following Minkowski's demise, Hilbert shifted his academic focus almost exclusively to physics. He arranged for a personal "physics tutor" and commenced studies in kinetic gas theory, progressing to elementary radiation theory and the molecular theory of matter. Even after the outbreak of war in 1914, he maintained seminars and classes that meticulously examined the works of Albert Einstein and other contemporary physicists.

By 1907, Einstein had articulated the fundamental tenets of the theory of gravity but subsequently labored for nearly eight years to finalize its complete formulation. His meeting with Emmy Noether at Göttingen proved pivotal for this breakthrough. By early summer 1915, Hilbert's interest in physics had converged on general relativity, prompting him to invite Einstein to Göttingen for a week-long series of lectures on the topic. Einstein was met with an enthusiastic reception. During the summer, Einstein learned of Hilbert's parallel work on the field equations, which intensified his own research efforts. In November 1915, Einstein published several papers culminating in The Field Equations of Gravitation. Almost concurrently, Hilbert published "The Foundations of Physics," which presented an axiomatic derivation of the field equations. Hilbert consistently acknowledged Einstein as the original conceptualizer of the theory, and no public dispute regarding priority for the field equations ever arose between the two scholars during their lifetimes.

Furthermore, Hilbert's research anticipated and facilitated several advancements in the mathematical formalization of quantum mechanics. His contributions were central to Hermann Weyl's and John von Neumann's work on demonstrating the mathematical equivalence between Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation. Moreover, the eponymous Hilbert space holds a significant role in quantum theory. In 1926, von Neumann conclusively demonstrated that if quantum states were conceptualized as vectors within Hilbert space, they would align with both Schrödinger's wave function theory and Heisenberg's matrices.

Hilbert dedicated himself to instilling mathematical rigor within the field of physics. Despite physics' heavy reliance on advanced mathematics, practitioners often exhibited a lack of precision in its application. For a pure mathematician like Hilbert, this imprecision was both aesthetically displeasing and intellectually opaque. As he deepened his understanding of physics and the mathematical methods employed by physicists, he formulated a cohesive mathematical theory for his observations, particularly in the domain of integral equations. When his colleague Richard Courant authored the seminal work Methoden der mathematischen Physik (Methods of Mathematical Physics), incorporating some of Hilbert's concepts, he included Hilbert's name as a co-author, notwithstanding Hilbert's lack of direct contribution to the manuscript. Hilbert famously remarked, "Physics is too hard for physicists," implying that the requisite mathematical sophistication often exceeded their grasp; the Courant–Hilbert publication subsequently facilitated their engagement with these complex mathematical tools.

Number Theory

Hilbert significantly advanced the unification of algebraic number theory through his 1897 treatise, Zahlbericht (literally, "report on numbers"). He also successfully resolved a substantial number-theory problem initially posed by Waring in 1770. Similar to his finiteness theorem, Hilbert employed an existence proof, demonstrating the certainty of solutions without providing a constructive method for their derivation. Following this, his subsequent publications on the subject were limited; however, the emergence of Hilbert modular forms in a student's dissertation further associated his name with a prominent area of research.

He proposed a series of conjectures pertaining to class field theory. These concepts proved profoundly influential, and Hilbert's enduring contributions are recognized through the nomenclature of the Hilbert class field and the Hilbert symbol within local class field theory. The majority of these results were substantiated by 1930, largely due to the work of Teiji Takagi.

Although Hilbert did not concentrate on the core areas of analytic number theory, his name is associated with the Hilbert–Pólya conjecture, a connection rooted in anecdotal origins. Ernst Hellinger, a former student of Hilbert, once recounted to André Weil that Hilbert had declared in a seminar during the early 1900s his expectation that the proof of the Riemann Hypothesis would emerge as a consequence of Fredholm's research on integral equations featuring a symmetric kernel.

Works

His collected scholarly works, titled Gesammelte Abhandlungen, have undergone multiple publications. The initial versions of his papers contained numerous technical inaccuracies of varying severity. Upon the collection's first publication, these errors were rectified, and it was determined that such corrections could be implemented without necessitating major alterations to the theorems' statements, with the singular exception of a purported proof for the continuum hypothesis. Nevertheless, the errors were sufficiently pervasive and significant that Olga Taussky-Todd required three years to complete the necessary revisions.

Concepts

Citations

Primary Literature in English Translation

Primary literature in English translation

• Ewald, William B., ed. (1996). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Oxford, UK: Oxford University Press.
• 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 commentary by Weyl and an Appendix by Bernays, 464–489.