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In the philosophy of mathematics, one of the varieties of finitism is an extreme form of constructivism, according to which a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps. Another form of finitism was pursued by Hilbert and Bernays.

In her book Philosophy of Set Theory, Mary Tiles characterized those who allow countably infinite objects as classical finitists, and those who deny even countably infinite objects as strict finitists. Historically, the written history of mathematics was thus classically finitist until Cantor invented the hierarchy of transfinite cardinals in the end of the 19th century. Leopold Kronecker remained a strident opponent to Cantor's set theory^[1]:

God created the natural numbers, all else is the work of man.^[2]

In 1923, Thoralf Skolem published a paper in which he presented a semi-formal system, which is now known as primitive recursive arithmetic, that is widely taken to be a suitable background for finitist mathematics. This was adopted by David Hilbert and Paul Bernays as the "contentual" finitist system for metamathematics, in which a proof of the consistency of other mathematical systems (e.g. full Peano Arithmetic) was to be given. (See Hilbert's program.)

Reuben Goodstein is another proponent of finitism. Some of his work involved building up to analysis from finitist foundations. Although he denied it, much of Ludwig Wittgenstein's writing on mathematics has a strong affinity with finitism. If finitists are contrasted with transfinitists (proponents of e.g. Georg Cantor's hierarchy of infinities), then also Aristotle may be characterized as a strict finitist. Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity. (Note that Aristotle's actual infinity means simply an actualization of something never-ending in nature, when in contrast the Cantorist actual infinity means the transfinite cardinal and ordinal numbers, that have nothing to do with the things in nature):

But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.

—Aristotle, Physics, Book 3, Chapter 6

Even stronger than finitism is ultrafinitism (also known as ultraintuitionism), associated primarily with Alexander Esenin-Volpin.

See also[link]

• Temporal finitism

References[link]

External links[link]

• Finitism in Geometry, entry in the Stanford Encyclopedia of Philosophy

David Hilbert
David Hilbert (1912)
Born	(1862-01-23)January 23, 1862 Königsberg or Wehlau, Province of Prussia (today Znamensk, Kaliningrad Oblast, Russia)
Died	February 14, 1943(1943-02-14) (aged 81) Göttingen, Germany
Residence	Germany
Nationality	German
Fields	Mathematician and Philosopher
Institutions	University of Königsberg Göttingen University
Alma mater	University of Königsberg
Doctoral advisor	Ferdinand von Lindemann
Doctoral students	Wilhelm Ackermann Otto Blumenthal Werner Boy Richard Courant Haskell Curry Max Dehn Paul Funk Erich Hecke Hellmuth Kneser Robert König Emanuel Lasker Erhard Schmidt Hugo Steinhaus Teiji Takagi Hermann Weyl Ernst Zermelo
Known for	Hilbert's basis theorem Hilbert's axioms Hilbert's problems Hilbert's program Einstein–Hilbert action Hilbert space
Notable awards	FRS[1]

David Hilbert, FRS^[1] (German pronunciation: [ˈdaːvɪt ˈhɪlbɐt]; January 23, 1862 – February 14, 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces,^[2] one of the foundations of functional analysis.

Hilbert adopted and warmly defended Georg Cantor's set theory and transfinite numbers. A famous example of his leadership in mathematics is his 1900 presentation of a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.^[3]

Life[link]

Hilbert, the first of two children of Otto and Maria Therese (Erdtmann) Hilbert, was born in the Province 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.^[4] In the fall of 1872, he entered the Friedrichskolleg Gymnasium (Collegium fridericianum, the same school that Immanuel Kant had attended 140 years before), but after an unhappy duration he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium.^[5] Upon graduation he enrolled (autumn 1880) at the University of Königsberg, the "Albertina". In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Königsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters),^[6] returned to Königsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski."^[7] In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius, i.e., an associate professor. An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert 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 professor from 1886 to 1895. In 1892, Hilbert married Käthe Jerosch (1864–1945), "the daughter of a Konigsberg merchant, an outspoken young lady with an independence of mind that matched his own".^[8] While at Königsberg they had their one child Franz Hilbert (1893–1969). In 1895, as a result of intervention on his behalf by Felix Klein he obtained the position of Chairman of Mathematics at the University of Göttingen, at that time the best research center for mathematics in the world and where he remained for the rest of his life.

His son Franz suffered his entire life from an (undiagnosed) mental illness, his inferior intellect was a terrible disappointment to his father and this tragedy was a matter of distress to the mathematicians and students at Göttingen.^[9] Sadly, Minkowski — Hilbert's "best and truest friend"^[10] — died prematurely of a ruptured appendix in 1909.

The Mathematical Institute in Göttingen. Its new building, constructed with funds from the Rockefeller Foundation, was opened by Hilbert and Courant in 1930.

The Göttingen school[link]

Among the students of Hilbert 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).^[11] Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time.

"Good, he did not have enough imagination to become a mathematician".

—Hilbert's response upon hearing that one of his students had dropped out to study poetry.^[12]

Later years[link]

Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Göttingen in 1933.^[13] 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.

About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, "How is mathematics in Göttingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Göttingen? There is really none any more."^[14]

Hilbert's tomb:
Wir müssen wissen
Wir werden wissen

By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, in as much 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.^[15] News of his death only became known to the wider world six months after he had died.

The epitaph on his tombstone in Göttingen is the famous lines he had spoken at the conclusion of his retirement address to the Society of German Scientists and Physicians in the fall of 1930. The words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know, we cannot know":^[16]

Wir müssen wissen.

Wir werden wissen.

In English:

We must know.

We will 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 roundtable discussion during the Conference on Epistemology held jointly with the Society meetings—tentatively announced the first expression of his incompleteness theorem.^[17]

Hilbert solves Gordan's Problem[link]

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. In order 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^[18] 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, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

Das ist nicht Mathematik. Das ist Theologie.

(This is not Mathematics. This is Theology.)^[19]

Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein and by the comments of Gordan, Hilbert in a second article extended his method, 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.^[20]

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.^[21]

For all his successes, the nature of his proof stirred up more trouble than Hilbert could have imagined at the time. 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".^[21] Not all were convinced. While Kronecker would die soon after, his constructivist philosophy would continue with the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years.^[22] 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".^[23] Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert would respond:

Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.^[24]

Axiomatization of geometry[link]

The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. Independently and contemporaneously, a 19-year-old American student named Robert Lee Moore published an equivalent set of axioms. Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa.

Hilbert's approach signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Peano's work from 1889. 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 says, 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 planes), betweenness, congruence of pairs of points, and congruence of angles. The axioms unify both the plane geometry and solid geometry of Euclid in a single system.

The 23 Problems[link]

Hilbert put forth a most influential list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally reckoned 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 enlist in problems, which he had identified as crucial aspects of the areas of mathematics he took to be key.

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. Here is the introduction of the speech that Hilbert gave:

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?^[25]

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. 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 even continue to this day to remain a challenge for mathematicians.

Formalism[link]

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[link]

In 1920 he proposed explicitly a research project (in metamathematics, as it was then termed) 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[link]

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, was however to end 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 Alonzo Church and Alan Turing also grew directly out of this 'debate'.

Functional analysis[link]

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[link]

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 in 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.^[26] 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 with a confounding problem of putting the theory into final form.^[27] 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.^[28] Einstein received an enthusiastic reception at Göttingen.^[29] 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 David 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.^[30] 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 Neuman showed that if atomic states were understood as vectors in Hilbert space, then they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.^[31]

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher math, 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 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[link]

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.^[32] 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 on them were mostly proved by 1930, after work by Teiji Takagi.^[33]

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.

Miscellaneous talks, essays, and contributions[link]

• Hilbert's paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.
• He was a Foreign member of the Royal Society.^[1]
• He received the second Bolyai Prize in 1910.
• His collected works (Gesammelte Abhandlungen) have been published several times. The original versions of his papers contained “many technical errors of varying degree”;^[34] 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.^[35] The errors were nonetheless so numerous and significant that it took Olga Taussky-Todd three years to make the corrections.^[35]

Quotes[link]

• 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.^[36]

See also[link]

	Biography portal
	Logic portal

Notes[link]

References[link]

Primary literature in English translation[link]

• Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press.
  • 1918. "Axiomatic thought," 1115–14.
  • 1922. "The new grounding of mathematics: First report," 1115–33.
  • 1923. "The logical foundations of mathematics," 1134–47.
  • 1930. "Logic and the knowledge of nature," 1157–65.
  • 1931. "The grounding of elementary number theory," 1148–56.
  • 1904. "On the foundations of logic and arithmetic," 129–38.
  • 1925. "On the infinite," 367–92.
  • 1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464–89.
• Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
• David Hilbert; Cohn-Vossen, S. (1999). Geometry and Imagination. American Mathematical Society. ISBN 0-8218-1998-4.  - an accessible set of lectures originally for the citizens of Göttingen.
• David Hilbert (2004). Michael Hallett and Ulrich Majer. ed. David Hilbert's Lectures on the foundations of Mathematics and Physics, 1891–1933. Springer-Verlag Berlin Heidelberg. ISBN 3-540-64373-7. 

Secondary literature[link]

• 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.
• Folsing, Albrecht, 1998. Albert Einstein. Penguin.
• 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
• Mancosu, Paolo (1998). From Brouwer to Hilbert, The Debate on the Foundations of Mathematics in the 1920's. Oxford Univ. Press. ISBN 0-19-509631-2. 
• Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of Gravitation. Reidel.
• Piergiorgio Odifreddi, 2003. Divertimento Geometrico - Da Euclide ad 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.
• Reid, Constance, 1996. Hilbert, Springer, ISBN 0-387-94674-8. The biography in English.
• Rowe, D. E. (1989). "Klein, Hilbert, and the Gottingen Mathematical Tradition". Osiris 5: 186–213. DOI:10.1086/368687.  edit
• Sauer, Tilman, 1999, "The relativity of discovery: Hilbert's first note on the foundations of physics," Arch. Hist. Exact Sci. 53: 529-75.
• 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.

External links[link]

Wikisource has original works written by or about: David Hilbert

Wikiquote has a collection of quotations related to: David Hilbert

Wikimedia Commons has media related to: David Hilbert

• O'Connor, John J.; Robertson, Edmund F., "David Hilbert", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Hilbert.html .
• David Hilbert at the Mathematics Genealogy Project
• Hilbert Bernays Project
• Hilbert's 23 Problems Address
• Hilbert's Program
• Works by David Hilbert at Project Gutenberg
• Hilbert's radio speech recorded in Königsberg 1930 (in German), with English translation
• 'From Hilbert's Problems to the Future', lecture by Professor Robin Wilson, Gresham College, 27 February 2008 (available in text, audio and video formats).

Enrico Fermi
Enrico Fermi (1901–1954)
Born	(1901-09-29)29 September 1901 Rome, Italy
Died	28 November 1954(1954-11-28) (aged 53) Chicago, Illinois, USA
Citizenship	Italy (1901–1954) United States (1944–1954)
Fields	Physics
Institutions	Scuola Normale Superiore University of Göttingen Leiden University University of Florence Sapienza University of Rome Columbia University University of Chicago
Alma mater	Scuola Normale Superiore
Doctoral advisor	Luigi Puccianti
Doctoral students	Edoardo Amaldi Owen Chamberlain Geoffrey Chew Mildred Dresselhaus Jerome I. Friedman Marvin Leonard Goldberger Tsung-Dao Lee Ettore Majorana James Rainwater Marshall Rosenbluth Arthur H. Rosenfeld Emilio Segrè Jack Steinberger Sam Treiman
Other notable students	Richard Garwin Bruno Pontecorvo Leona Woods
Known for	New radioactive elements produced by neutron irradiation Controlled nuclear chain reaction, Fermi–Dirac statistics Theory of beta decay
Influenced	James Grier Miller
Notable awards	Matteucci Medal (1926) Nobel Prize for Physics (1938) Hughes Medal (1942) Franklin Medal (1947) Rumford Prize (1953)
Spouse	Laura Fermi
Signature

Enrico Fermi (Italian pronunciation: [enˈriko ˈfermi]; 29 September 1901 – 28 November 1954)^[1] was an Italian-born, naturalized American physicist particularly known for his work on the development of the first nuclear reactor, Chicago Pile-1, and for his contributions to the development of quantum theory, nuclear and particle physics, and statistical mechanics. He was awarded the 1938 Nobel Prize in Physics for his work on induced radioactivity.

Fermi is widely regarded as one of the leading scientists of the 20th century, highly accomplished in both theory and experiment.^[2] Along with J. Robert Oppenheimer,^[3] he is frequently referred to as "the father of the atomic bomb".^[4][5] He also held several patents related to the use of nuclear power.

Several awards, concepts, and institutions are named after Fermi, such as the Enrico Fermi Award,^[6] the Enrico Fermi Institute, the Fermi National Accelerator Laboratory, the Fermi Gamma-ray Space Telescope, the Enrico Fermi Nuclear Generating Station, a class of particles called fermions, the synthetic element fermium, and many more.

Biography[link]

Early years[link]

Enrico Fermi was born in Rome to Alberto Fermi, a Chief Inspector of the Ministry of Communications, and Ida de Gattis, an elementary school teacher of unusual intelligence.^[7] As a young boy, he shared his interests with his older brother, Giulio. They dismantled small engines and other parts. When Giulio died unexpectedly of a throat abscess in 1915, Enrico was distraught, and immersed himself in scientific study to distract himself. According to his own account, each day he would walk in front of the hospital in which Giulio died until he was inured to the pain.

One of the first sources for the study of physics was a book found at the local market of Campo de' Fiori in Roma. The 900-page book, titled Elementorum physicae mathematicae, written in Latin by Jesuit Father Andrea Caraffa, a professor at the Collegio Romano, covered subjects like mathematics, classical mechanics, astronomy, optics, and acoustics. Notes found in the book indicate that Fermi studied it intensely. Later, Enrico befriended another scientifically inclined student named Enrico Persico, and the two worked together on scientific projects such as building gyroscopes and measuring the Earth's magnetic field. Fermi's interest in physics was further encouraged by Adolfo Amidei, a friend of his father, who gave him several books on physics and mathematics, which he read and assimilated quickly.

Scuola Normale Superiore in Pisa[link]

In 1918, Fermi enrolled at the Scuola Normale Superiore in Pisa. In order to enter the Institute, candidates had to take a difficult entrance exam which included an essay. The given theme was Specific characteristics of Sounds (Italian: Caratteri distintivi dei suoni).^[8] The 17-year-old Enrico Fermi chose to derive and solve the partial differential equation for a vibrating rod, applying Fourier analysis. The examiner, Prof. Giuseppe Pittarelli, interviewed Fermi and concluded that his entry would have been commendable even for a doctoral degree. Enrico Fermi achieved first place in the classification of the entrance exam.

During his years at the Scuola Normale Superiore, Fermi teamed up with a fellow student named Franco Rasetti with whom he would indulge in light-hearted pranks and who would later become Fermi's close friend and collaborator. Besides attending the classes, Enrico Fermi found the time to work on his extracurricular activities, particularly with the help of his friend Enrico Persico, who remained in Rome to attend the university.

In Pisa, Fermi was advised by the director of the physics laboratory, Luigi Puccianti, who acknowledged that there was little that he could teach Fermi, and frequently asked Fermi to teach him something. Fermi's knowledge of quantum physics reached such a high level that Prof. Puccianti asked him to organize seminars about that topic. During this time he learned tensor calculus, a mathematical instrument invented by Gregorio Ricci and Tullio Levi-Civita, and needed to demonstrate the principles of general relativity. Between 1919 and 1923, Fermi studied general relativity, quantum mechanics, and atomic physics.

In 1921, his third year at the university, he published his first scientific works in the Italian journal Nuovo Cimento: the first was titled: On the dynamics of a rigid system of electrical charges in translational motion; the second: On the electrostatics of a uniform gravitational field of electromagnetic charges and on the weight of electromagnetic charges. At first glance, the first paper seemed to point out a contradiction between the electrodynamic theory and the relativistic one concerning the calculation of the electromagnetic masses. After one year with a work entitled Correction of severe discrepancy between electrodynamic theory and the relativistic one of electromagnetic charges. Inertia and weight of electricity, Enrico Fermi showed the correctness of his paper. This last publication was so successful that it was translated into German and published in the famous German scientific journal Physikalische Zeitschrift.

In 1922, he published his first important scientific work in the Italian journal I Rendiconti dell'Accademia dei Lincei entitled "On the phenomena occurring near a world line", where he introduces for the first time the so-called "Fermi coordinates", and proves that when close to the time line, space behaves as a euclidean one. In July of that year, Fermi submitted his doctoral thesis Un teorema di calcolo delle probabilità ed alcune sue applicazioni (A theorem on probability and some of its applications) to the Scuola Normale Superiore and received his Laurea from there at the unusually young age of 21.

In 1923, while writing the appendix for the Italian edition of the book The Mathematical Theory of Relativity by A. Kopff, Enrico Fermi pointed out, for the first time, that hidden inside the famous Einstein equation (E = mc²), there was an enormous amount of nuclear potential energy to be exploited.

In 1924, Fermi spent a semester at the University of Göttingen with Max Born, and then stayed for a few months in Leiden with Paul Ehrenfest. From January 1925 to the autumn of 1926, he stayed at the University of Florence. In this period, he wrote his work on the Fermi–Dirac statistics.

Professor in Rome[link]

Aged 24, Fermi took a professorship at the University of Rome (one of the first three in theoretical physics in Italy) ^[9] which he won in a competition whose selection committee was chaired by Professor Orso Mario Corbino, director of the Institute of Physics. ^[9] Corbino helped Fermi in selecting his team, which soon was joined by notable minds like Edoardo Amaldi, Bruno Pontecorvo, Franco Rasetti and Emilio Segrè. For the theoretical studies only, Ettore Majorana also took part in what was soon nicknamed "the Via Panisperna boys" (after the name of the street where the Physics Institute was). The group went on with its now famous experiments, but in 1933 Rasetti left Italy for Canada and the United States, Pontecorvo went to France and Segrè left to teach in Palermo.

Enrico Fermi and his students (the Via Panisperna boys) in the courtyard of Rome University's Physics Institute in Via Panisperna, about 1934. From Left to right: Oscar D'Agostino, Emilio Segrè, Edoardo Amaldi, Franco Rasetti and Enrico Fermi

During their time in Rome, Fermi and his group made important contributions to many practical and theoretical aspects of physics. These include the theory of beta decay, later referred to as the theory of the "weak interaction" (one of the 4 basic forces in nature, then brand new) with the inclusion of the neutrino postulated in 1930 by Wolfgang Pauli, and the discovery of slow neutrons, which was to prove pivotal for the working of nuclear reactors. His group systematically bombarded elements with slow neutrons, and during their experiments with uranium, narrowly missed observing nuclear fission. At that time, fission was thought to be improbable if not impossible, mostly on theoretical grounds. While people expected elements with higher atomic number to form from neutron bombardment of lighter elements, nobody expected neutrons to have enough energy to actually split a heavier atom into two light element fragments. However, the chemist Ida Noddack had criticised Fermi's work and had suggested that some of his experiments could have produced lighter elements. At the time, Fermi dismissed this possibility on the basis of calculations.

Fermi was well known for his simplicity in solving problems.^[10] He began his inquiries with the simplest lines of mathematical reasoning, then later produced complete solutions to the problems he deemed worth pursuing. His abilities as a great scientist, combining theoretical and applied nuclear physics, were acknowledged by all. He influenced many physicists who worked with him, such as Hans Bethe, who spent two semesters working with Fermi in the early 1930s. From the time he was a boy, Fermi meticulously recorded his calculations in notebooks, and later used them to solve many new problems that he encountered based on these earlier known problems.

When Fermi submitted his paper on beta decay^[11] to the prestigious journal Nature, the journal's editor turned it down because "it contained speculations which were too remote from reality". Thus Fermi saw the theory published in Italian and in German before it was published in English. Nature eventually did publish Fermi's report on beta decay on January 16, 1939.

Fermi remained in Rome until 1938.

Manhattan Project[link]

The Chicago Pile Team, including Fermi (bottom left), and Leó Szilárd (second from right on bottom)

In 1938, Fermi received the Nobel Prize in Physics at the age of 37 for his "demonstrations of the existence of new radioactive elements produced by neutron irradiation, and for his related discovery of nuclear reactions brought about by slow neutrons". After Fermi received the Nobel Prize in Stockholm, he, his wife Laura, and their children emigrated to New York. This was mainly because of the Manifesto of Race promulgated by the fascist regime of Benito Mussolini in order to bring Italian Fascism ideologically closer to German Nazism. The new laws threatened Laura, who was Jewish. Also, the new laws put most of Fermi's research assistants out of work. Soon after his arrival in New York, Fermi began working at Columbia University, where he had already given summer lectures in 1936 (preface to Thermodinamics, Dover Publications, Inc. NY).

In December 1938, the German chemists Otto Hahn and Fritz Strassmann sent a manuscript to Naturwissenschaften reporting they had detected the element barium after bombarding uranium with neutrons;^[12] simultaneously, they communicated these results to Lise Meitner. Meitner, and her nephew Otto Robert Frisch, correctly interpreted these results as being nuclear fission.^[13] Following an advice of George Placzek, Frisch confirmed this experimentally on 13 January 1939.^[14][15]

Fermi's ID badge photo from Los Alamos

Meitner's and Frisch's interpretation of the work of Hahn and Strassmann crossed the Atlantic Ocean with Niels Bohr, who was to lecture at Princeton University. Isidor Isaac Rabi and Willis Lamb, two Columbia University physicists working at Princeton, heard the news and carried it back to Columbia. Rabi said he told Enrico Fermi; Fermi gave credit to Lamb. Bohr soon thereafter went from Princeton to Columbia to see Fermi. Not finding Fermi in his office, Bohr went down to the cyclotron area and found Herbert L. Anderson. Bohr grabbed him by the shoulder and said: “Young man, let me explain to you about something new and exciting in physics.”^[10] It was clear to a number of scientists at Columbia that they should try to detect the energy released in the nuclear fission of uranium from neutron bombardment. On 25 January 1939, a Columbia University team conducted the first nuclear fission experiment in the United States,^[16] which was done in the basement of Pupin Hall; the members of the team were Herbert L. Anderson, Eugene T. Booth, John R. Dunning, Enrico Fermi, G. Norris Glasoe, and Francis G. Slack. The next day, the Fifth Washington Conference on Theoretical Physics began in Washington, D.C. under the joint auspices of The George Washington University and the Carnegie Institution of Washington. There, the news on nuclear fission was spread even further, which fostered many more experimental demonstrations.^[10]