In Königsberg on 8 September 1930, David Hilbert addressed the yearly meeting of the Society of German Natural Scientists and Physicians (Gesellschaft der Deutschen Naturforscher und Ärzte). Generally regarded as the world’s leading mathematician, Hilbert was born and educated in Königsberg and spent the early years of his career there. Retiring at age sixtyeight from his professorship at Göttingen, he was being honored by his home city. Hilbert forcefully delineated a basic tenet of his research philosophy—that every mathematical problem is solvable—and countered a widespread but controversial opposing opinion. Shortly afterward, he read on German radio a fourminute version of the finale of his speech. This excerpt stands on its own as a dramatic poem or anthem that has inspired many mathematicians and will continue to do so, and it has the potential to reinforce public appreciation of the historical and present context of mathematical research.
On subsequent pages, you will have the opportunity to listen to Hilbert's radio address (Hilbert [1930] 1971) while reading a German transcription, an English translation, or a sidebyside transcription and translation.
Hilbert’s address did not spring forward in isolation. It climaxed a development that is evidenced in his earlier work, and reacted to broad and deep trends in philosophy and mathematical culture. The present article concludes with a discussion that will steer the reader to some of this lore. It identifies persons and explains terms that Hilbert mentioned, and provides both background for the occasion and a hint of subsequent related developments.
In Königsberg on 8 September 1930, David Hilbert addressed the yearly meeting of the Society of German Natural Scientists and Physicians (Gesellschaft der Deutschen Naturforscher und Ärzte). Generally regarded as the world’s leading mathematician, Hilbert was born and educated in Königsberg and spent the early years of his career there. Retiring at age sixtyeight from his professorship at Göttingen, he was being honored by his home city. Hilbert forcefully delineated a basic tenet of his research philosophy—that every mathematical problem is solvable—and countered a widespread but controversial opposing opinion. Shortly afterward, he read on German radio a fourminute version of the finale of his speech. This excerpt stands on its own as a dramatic poem or anthem that has inspired many mathematicians and will continue to do so, and it has the potential to reinforce public appreciation of the historical and present context of mathematical research.
On subsequent pages, you will have the opportunity to listen to Hilbert's radio address (Hilbert [1930] 1971) while reading a German transcription, an English translation, or a sidebyside transcription and translation.
Hilbert’s address did not spring forward in isolation. It climaxed a development that is evidenced in his earlier work, and reacted to broad and deep trends in philosophy and mathematical culture. The present article concludes with a discussion that will steer the reader to some of this lore. It identifies persons and explains terms that Hilbert mentioned, and provides both background for the occasion and a hint of subsequent related developments.
Listen to David Hilbert's 1930 radio address (4 minutes) while reading along in German. An English translation is on the next page.
Das Instrument, welches die Vermittlung bewirkt zwischen Theorie und Praxis, zwischen Denken und Beobachten, ist die Mathematik; sie baut die verbindende Brücke und gestaltet sie immer tragfähiger. Daher kommt es, dass unsere ganze gegenwärtige Kultur, soweit sie auf der geistigen Durchdringung und Dienstbarmachung der Natur beruht, ihre Grundlage in der Mathematik findet. Schon GALILEI sagt: Die Natur kann nur der verstehen der ihre Sprache und die Zeichen kennengelernt hat, in der sie zu uns redet; diese Sprache aber ist die Mathematik, und ihre Zeichen sind die mathematischen Figuren. KANT tat den Ausspruch: “Ich behaupte, dass in jeder besonderen Naturwissenschaft nur so viel eigentliche Wissenschaft angetroffen werden kann, als darin Mathematik enthalten ist.” In der Tat: Wir beherrschen nicht eher eine naturwissenschaftliche Theorie, als bis wir ihren mathematischen Kern herausgeschält und völlig enthüllt haben. Ohne Mathematik ist die heutige Astronomie und Physik unmöglich; diese Wissenschaften lösen sich in ihren theoretischen Teilen geradezu in Mathematik auf. Diese wie die zahlreichen weiteren Anwendungen sind es, denen die Mathematik ihr Ansehen verdankt, soweit sie solches im weiteren Publikum geniesst.
Trotzdem haben es alle Mathematiker abgelehnt, die Anwendungen als Wertmesser für die Mathematik gelten zu lassen. GAUSS spricht von dem zauberischen Reiz, der die Zahlentheorie zur Lieblingswissenschaft der ersten Mathematiker gemacht habe, ihres unerschöpflichen Reichtums nicht zu gedenken, woran sie alle anderen Teile der Mathematik so weit übertrifft. KRONECKER vergleicht die Zahlentheoretiker mit den Lotophagen, die, wenn sie einmal von dieser Kost etwas zu sich genommen haben, nie mehr davon lassen können.
Der grosse Mathematiker POINCARÉ wendet sich einmal in auffallender Schärfe gegen TOLSTOI, der erklärt hatte, dass die Forderung “die Wissenschaft der Wissenschaft wegen” töricht sei. Die Errungenschaften der Industrie zum Beispiel hätten nie das Licht der Welt erblickt, wenn die Praktiker allein existiert hätten und wenn diese Errungenschaften nicht von uninteressierten Toren gefördert worden wären.
Die Ehre des menschlichen Geistes, so sagte der berühmte Königsberger Mathematiker JACOBI, ist der einzige Zweck aller Wissenschaft.
Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heisse im Gegenteil unsere Losung:
Wir müssen wissen,
Wir werden wissen.
Listen to David Hilbert's 1930 radio address in German (4 minutes) while reading along in English. A sidebyside German transcription and English translation are on the next page.
The instrument that mediates between theory and practice, between thought and observation, is mathematics; it builds the connecting bridge and makes it stronger and stronger. Thus it happens that our entire presentday culture, insofar as it rests on intellectual insight into and harnessing of nature, is founded on mathematics. Already, GALILEO said: Only he can understand nature who has learned the language and signs by which it speaks to us; but this language is mathematics and its signs are mathematical figures. KANT declared, “I maintain that in each particular natural science there is only as much true science as there is mathematics.” In fact, we do not master a theory in natural science until we have extracted its mathematical kernel and laid it completely bare. Without mathematics today’s astronomy and physics would be impossible; in their theoretical parts, these sciences unfold directly into mathematics. These, like numerous other applications, give mathematics whatever authority it enjoys with the general public.
Nevertheless, all mathematicians have refused to let applications serve as the standard of value for mathematics. GAUSS spoke of the magical attraction that made number theory the favorite science for the first mathematicians, not to mention its inexhaustible richness, in which it so far surpasses all other parts of mathematics. KRONECKER compared number theorists with the Lotus Eaters, who, once they had sampled that delicacy, could never do without it.
With astonishing sharpness, the great mathematician POINCARÉ once attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish. The achievements of industry, for example, would never have seen the light of day had the practicalminded existed alone and had not these advances been pursued by disinterested fools.
The glory of the human spirit, so said the famous Königsberg mathematician JACOBI, is the single purpose of all science.
We must not believe those, who today with philosophical bearing and a tone of superiority prophesy the downfall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion even none whatever in natural science. In place of the foolish ignorabimus let stand our slogan:
We must know,
We will know.
Listen to David Hilbert's 1930 radio address (4 minutes) while reading along in German and English.
Das Instrument, welches die Vermittlung bewirkt zwischen Theorie und Praxis, zwischen Denken und Beobachten, ist die Mathematik; sie baut die verbindende Brücke und gestaltet sie immer tragfähiger. Daher kommt es, dass unsere ganze gegenwärtige Kultur, soweit sie auf der geistigen Durchdringung und Dienstbarmachung der Natur beruht, ihre Grundlage in der Mathematik findet.
The instrument that mediates between theory and practice, between thought and observation, is mathematics; it builds the connecting bridge and makes it stronger and stronger. Thus it happens that our entire presentday culture, insofar as it rests on intellectual insight into and harnessing of nature, is founded on mathematics.
Schon GALILEI sagt: Die Natur kann nur der verstehen der ihre Sprache und die Zeichen kennengelernt hat, in der sie zu uns redet; diese Sprache aber ist die Mathematik, und ihre Zeichen sind die mathematischen Figuren. KANT tat den Ausspruch: “Ich behaupte, dass in jeder besonderen Naturwissenschaft nur so viel eigentliche Wissenschaft angetroffen werden kann, als darin Mathematik enthalten ist.”
Already, GALILEO said: Only he can understand nature who has learned the language and signs by which it speaks to us; but this language is mathematics and its signs are mathematical figures. KANT declared, “I maintain that in each particular natural science there is only as much true science as there is mathematics.”
In der Tat: Wir beherrschen nicht eher eine naturwissenschaftliche Theorie, als bis wir ihren mathematischen Kern herausgeschält und völlig enthüllt haben. Ohne Mathematik ist die heutige Astronomie und Physik unmöglich; diese Wissenschaften lösen sich in ihren theoretischen Teilen geradezu in Mathematik auf. Diese wie die zahlreichen weiteren Anwendungen sind es, denen die Mathematik ihr Ansehen verdankt, soweit sie solches im weiteren Publikum geniesst.
In fact, we do not master a theory in natural science until we have extracted its mathematical kernel and laid it completely bare. Without mathematics today’s astronomy and physics would be impossible; in their theoretical parts, these sciences unfold directly into mathematics. These, like numerous other applications, give mathematics whatever authority it enjoys with the general public.
Trotzdem haben es alle Mathematiker abgelehnt, die Anwendungen als Wertmesser für die Mathematik gelten zu lassen. GAUSS spricht von dem zauberischen Reiz, der die Zahlentheorie zur Lieblingswissenschaft der ersten Mathematiker gemacht habe, ihres unerschöpflichen Reichtums nicht zu gedenken, woran sie alle anderen Teile der Mathematik so weit übertrifft. KRONECKER vergleicht die Zahlentheoretiker mit den Lotophagen, die, wenn sie einmal von dieser Kost etwas zu sich genommen haben, nie mehr davon lassen können.
Nevertheless, all mathematicians have refused to let applications serve as the standard of value for mathematics. GAUSS spoke of the magical attraction that made number theory the favorite science for the first mathematicians, not to mention its inexhaustible richness, in which it so far surpasses all other parts of mathematics. KRONECKER compared number theorists with the Lotus Eaters, who, once they had sampled that delicacy, could never do without it.
Der grosse Mathematiker POINCARÉ wendet sich einmal in auffallender Schärfe gegen TOLSTOI, der erklärt hatte, dass die Forderung “die Wissenschaft der Wissenschaft wegen” töricht sei. Die Errungenschaften der Industrie zum Beispiel hätten nie das Licht der Welt erblickt, wenn die Praktiker allein existiert hätten und wenn diese Errungenschaften nicht von uninteressierten Toren gefördert worden wären.
With astonishing sharpness, the great mathematician POINCARÉ once attacked TOLSTOY, who had suggested that pursuing “science for science’s sake” is foolish. The achievements of industry, for example, would never have seen the light of day had the practicalminded existed alone and had not these advances been pursued by disinterested fools.
Die Ehre des menschlichen Geistes, so sagte der berühmte Königsberger Mathematiker JACOBI, ist der einzige Zweck aller Wissenschaft.
The glory of the human spirit, so said the famous Königsberg mathematician JACOBI, is the single purpose of all science.
Wir dürfen nicht denen glauben, die heute mit philosophischer Miene und überlegenem Tone den Kulturuntergang prophezeien und sich in dem Ignorabimus gefallen. Für uns gibt es kein Ignorabimus, und meiner Meinung nach auch für die Naturwissenschaft überhaupt nicht. Statt des törichten Ignorabimus heisse im Gegenteil unsere Losung:
Wir müssen wissen,
Wir werden wissen.
We must not believe those, who today with philosophical bearing and a tone of superiority prophesy the downfall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion even none whatever in natural science. In place of the foolish ignorabimus let stand our slogan:
We must know,
We will know.
As he read his 1930 radio address, the mathematician David Hilbert (18621943) assumed that his listeners were familiar with many philosophical ideas and arguments that formed the foundation of German intellectual life, including some which played major roles in the development of mathematics. Was his assumption justified?
The historian David E. Rowe recently gave a sweeping but succinct explanation (Rowe 2013, 38)
... that pertains to the rise of higher mathematics in the German universities during the course of the nineteenth century … . [T]hese mathematicians were accustomed to pondering philosophical questions. Indeed, as members of the Bildungsbürgertum—the ‘educated citizenry’ who formed a special elite class within German society—philosophizing in the academic sense of the term was a natural part of their collective cultural identity.
Hilbert belonged to that culture. His 1885 minor doctoral thesis, for example, was a defense of the view of Königsberg’s most illustrious son, the eighteenthcentury philosopher Immanuel Kant (17241804), that arithmetic is founded on a priori knowledge, not derived from experience (Reid 1970, 17). The borderline between inborn and experiential knowledge was disputed long before Kant’s time and has been disputed ever since. Hilbert began his radio address by noting that mathematics provides the means to connect the two realms.
An earlier foray into this murky interface had been undertaken by the noted Berlin physiologist Emil Du BoisReymond (18181896) in an 1872 speech presented to the Society of German Natural Scientists and Physicians (Du BoisReymond [1872] 1874 [note 1]). Du BoisReymond's investigations of electrical properties of the nervous system had led him to longstanding fundamental questions, especially the nature of matter and force and the relationship between mental phenomena and their physical aspects. He recognized scientists’ general belief that when we do not know a solution—ignoramus in Latin—nevertheless, under certain circumstances, we could know. However, he countered, concerning riddles of the material world such as these two, we must decide in favor of a harder truth: ignorabimus—we shall never know. Du BoisReymond reported later that his 1872 speech had excited considerable controversy and his ignorabimus slogan had become a sort of shibboleth in natural philosophy. To clarify misunderstandings, he specified in more detail the questions to which it should apply in a speech delivered in 1880 (Du BoisReymond 1880, 1046, 1053–1057). Hilbert would have become very familiar with such controversy during his student years at Königsberg (Reid 1970, 13).


Emil Du BoisReymond circa 1885 (Source: Project Gutenberg) 
David Hilbert in 1886 (Source: Wikimedia Commons) 
Hilbert remained at the University of Königsberg for the first part of his career, rising through the ranks to full professor. His research concentrated on algebraic geometry and algebraic number theory: abstract mathematics at the highest level. He moved to the University of Göttingen in 1895 and helped make it the world’s leading center for mathematical research. During the 1890s he applied algebraic methods to establish solid foundations for the geometry that is familiar to highschool teachers and students. By 1900 he was recognized as the leading German research mathematician.
The 1900 Universal Exposition in Paris celebrated the beginning of a new century of industrial and scientific progress. The International Congress of Mathematicians met there concurrently and Hilbert presented a now famous lecture on outstanding open problems in mathematics. In his introduction he took explicit issue with the philosophy emphasized by Du BoisReymond thirty years before (Hilbert [1900] 1902, 444–445):
Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. ... It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported by a proof) that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. ... This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.
Hilbert’s career during the next decades was characterized by major research on new problems—integral equations, mathematical physics, and establishment of a sound logical foundation for mathematics via finitistic reasoning. (For more information about the latter, see, for instance, "Hilbert's Program" in the Stanford Encyclopedia of Philososophy.) And he practiced entrepreneurship on behalf of mathematical research and the program at Göttingen, including continued outreach to general audiences. He took many opportunities to emphasize the philosophy stated in his famous address of 1900. (See Hilbert [1922–1923] 1988, 132, and Hilbert [1925] 1969, 384, for instance.)
Königsberg in the 1930s. Six of the seven bridges of the famous Königsberg bridges problem (see Steinhaus [1950] 1983, 256–259) are visible in the photograph above. The seventh stands just beyond the photograph, on the lower right. Königsberg was the capital of East Prussia, Germany, until 1945, when it came under Soviet control. The Soviets renamed it Kaliningrad, and it remains part of the Russian Federation today. It is in an enclave of Russia, separate from the rest of that country and surrounded by Poland, Lithuania, and the Baltic Sea. (Source: Ostpreussen.net)
As he approached retirement in 1930, Hilbert’s home city of Königsberg planned to honor him, and he prepared to respond with an address about the nature of mathematical inquiry. There would be a large audience in Königsberg at the beginning of September, as four German professional societies related to his work would be meeting together. The area around the University and Rathaus, shown in the photograph above, would be crowded with mathematicians, philosophers, and scientists. Hilbert chose to address the meeting of the Society of German Natural Scientists and Physicians, probably because the ignorabimus slogan that he would discredit had originated at its meeting nearly six decades before. Another factor in his decision may have been that in 1922, to celebrate Hilbert's sixtieth birthday, that Society's journal had published an entire issue devoted to his colleagues' accounts of his life and work (Naturwissenschaften 10, issue 4) [note 2].
Hilbert’s presentation opened the fourth day of the September 1930 meetings in Königsberg. His full text was published in the December 1930 issue of the broadly circulated journal Natural Sciences (Naturwissenschaften; Hilbert [1930] 1996). The structure of Hilbert’s speech was similar to that of Du BoisReymond’s 1872 speech, but its tone was more optimistic, reflecting accelerating scientific progress. The climactic final lines of the two speeches, even in choice of language, offer stunning contrast:
Du BoisReymond 1872:  Hilbert 1930:  Wir müssen wissen.  
Ignorabimus.  Wir werden wissen. 
Hilbert's radio address took place very soon after his speech at the Königsberg meeting. Hilbert's biographer Constance Reid interviewed Kurt Reidemeister (18931971), who had accompanied Hilbert to the studio (Reid 1970, 196; 1999). A recording of the radio address on a 45rpm record disk accompanied the original 1971 edition of Reidemeister's Hilbert Gedenkband (Hilbert Memorial Volume), but was not included with the first printing of the 2012 reissue of this volume.
Besides the meeting of the Society of German Natural Scientists and Physicians, the other three conferences at Königsberg in early September of 1930 were
The first of these was the most momentous of the four, a major step in bringing the adherents of the Vienna Circle of philosophers to both inner agreement and public notice. Their program challenged and eventually helped supplant much of the type of philosophy discussed and developed in the German universities in the late 19th and early 20th centuries. On September 6, two days before Hilbert’s speech, the young Austrian logician Kurt Gödel (19061978) presented his completeness theorem, which filled a major gap in Hilbert’s finitist foundation of mathematics. In a roundtable discussion on the next day, the day before Hilbert spoke, Gödel modestly announced his first incompleteness theorem. (See Wang 2005, 85–86, which also points out that Hilbert and Gödel evidently never met directly.) Gödel’s incompleteness theorem entailed the impossibility of Hilbert’s proposed solution of the foundations problem. In a 1999 article vividly describing Hilbert’s address, Victor Vinnikov concentrated on this aspect of its legacy [note 3]. For a yet deeper, broader, and extremely effective treatment of these questions, see the 2005 book by Torkel Franzén.
Hilbert died in 1944. His famous slogan
Wir müssen wissen
Wir werden wissen
forms the epitaph on his gravestone.
1. Emil Du BoisReymond is not to be confused with his younger brother, the mathematician Paul du BoisReymond (18311889).
2. In 1908 Hilbert’s close friend, Hermann Minkowski (18641909), had presented before the Society of German Natural Scientists and Physicians his seminal paper that publicized fourdimensional spacetime as a framework for relativity theory.
3. Vinnikov also pointed out two grammatical errors in the radio address, which Hilbert corrected in the published paper.
David Hilbert’s full address to the Society of German Natural Scientists and Physicians on 8 September 1930 described many examples of startling scientific achievements in the years since Emil Du BoisReymond’s speech in 1872. Those all rested on mathematics, from elementary to very advanced methods. The concluding portion that he incorporated in the radio address included a series of references to opinions of noted scholars about philosophical ideas concerning the nature and purpose of mathematical inquiry. Their original words tell an additional, underlying story of their contrasting evaluations of the study of pure and applied mathematics. The following paragraphs summarize that story, and link to notes containing, in almost all cases, the original texts and, in all cases, English translations of those scholars’ words.
First, Hilbert referred to famous sentiments of the Tuscan physicist Galileo Galilei (1564–1642), the Königsberg philosopher Immanuel Kant (1724–1804), and the Göttingen mathematician Carl Friedrich Gauss (1777–1855)—towering figures of the seventeenth, eighteenth, and nineteenth centuries. They emphasized that science describes nature in mathematical language, that mathematics is moreover the essential ingredient of proper science, and that (higher) arithmetic, or number theory, is the “queen” of mathematics. (See Their Own Words: Galileo, Kant, and Gauss.)
Hilbert mentioned Gauss’s description of the "magical attraction" that made number theory the "favorite science" for the first mathematicians. (See Their Own Words: Gauss.) Gauss may have been alluding to the legendary Lotus Eaters featured in Homer’s Odyssey. The noted Berlin mathematician Leopold Kronecker (1823–1891) must have thought so, for he compared Gauss’s sentiments to the LotusEater legend in his 1887 paper on the concept of number. Hilbert referred to that comparison next. (See Their Own Words: Homer and Kronecker/Jacobi.) Although Homer decried seduction by Lotus Eaters, the mathematicians may be interpreted merely as ironically recognizing it. Kronecker was also quoting an 1846 letter by the pioneering Königsberg mathematician Carl Gustav Jacob Jacobi (1804–1851) that parodied a well known poem of Friedrich Schiller, Archimedes and the Schoolboy, which was known to be a favorite of Gauss. With elegance and wit, Schiller and Jacobi supported Gauss’s view that mathematics should not be valued merely for its applications. (See Their Own Words: Gauss and Schiller/Jacobi.)
Concluding, Hilbert recounted that the leading French mathematician Henri Poincaré (1854–1912) had criticized philosophical remarks of the great Russian novelist Lev Nikolayevich Tolstoy (1828–1910). In the 1906 essay The Choice of Facts, Poincaré railed against Tolstoy’s 1887 essay On the Significance of Science and Art. The latter is emotional and illorganized, but maintained that the highest goal of science has always been “the true welfare of each man and of all men” (Tolstoy 1899, 228). Poincaré argued that had scientists always been guided only by immediate utility, they never would have achieved the deep results on which modern applications rest. Hilbert settled the dispute with the words of Jacobi, who had long before posited, “The glory of the human spirit is the single purpose of all science” (Jacobi [1830] 1881, 454). Poincaré’s attack could be an example of what Jacobi had called “the frenzy which those lotoseaters sink into when they suspect that their cult has been neglected or is valued only for its practical applications.” (See Their Own Words: Poincaré, Tolstoy, and Kronecker/Jacobi.)
Hilbert’s radio address included a series of references to opinions of noted scholars about philosophical ideas concerning the nature and value of mathematical inquiry. Their original words tell an additional, underlying story of their contrasting evaluations of the pursuit of pure and applied mathematics. The discussion above summarizes that story and provides links to these notes, which report these scientists’ original words with English translations. Because they tell that story, it is also helpful to read them in sequence.
Hilbert paraphrased a passage from The Assayer, an essay that Galileo Galilei wrote in 1623 in response to the work Libra Astronomica ac Philosophica, which the Jesuit scholar Orazio Grassi (15831654) had published in 1619 under the pseudonym Lotario Sarsi Sigensano. Galileo was attacking Grassi’s adherence to the theories and work of the Danish astronomer Tycho Brahe (1546–1601). Here is a translation of Galileo’s passage by the Canadian historian Stillman Drake:
Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering about in a dark labyrinth (Galilei [1623] 1960, 183–184).
And here is the original Italian text:
La Filosofia è scritta in questo grandissimo libro, che continuamente ci stà aperto innanzi à gli occhi (io dico l’universo) ma non si può intendere se prima non s’impara à intender la lingua, e conoscere i caratteri nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, & altre figure Geometriche, senza i quali mezi è impossibile à intenderne umanamente parola; senza questi è un aggirarsi vanamente per un’oscuro laberinto (Galilei [1623] 1864, 60).
Hilbert paraphrased a sentence from the preface to the 1786 treatise Metaphysical Foundations of Natural Science by Immanuel Kant. Here is the passage containing that text, followed by an English translation by the present author.
Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist. Denn nach dem Vorhergehenden erfordert eigentliche Wissenschaft, vornehmlich der Natur, einen reinen Theil, der dem empirischen zum Grunde liegt, und der auf Erkenntniss der Naturdinge a priori beruht (Kant [1786] 1903, 470).
I maintain however that in every special natural doctrine only so much proper science can be encountered as mathematics is to be found there. For in accordance with the foregoing, proper science, especially of nature, requires a pure part that underlies the empirical, and is based upon a priori knowledge of natural things.
Hilbert referred to Carl Friedrich Gauss’s famous characterization of number theory as the “queen of mathematics.” This is found in the 1856 biography of Gauss by the geologist Wolfgang Sartorius von Waltershausen (18091876). Here is a translation of that passage by the present author, followed by the original German text.
How many thoughts may have emerged from the unbelievable productivity in this powerful mind, and submerged again, which are lost, at least for science! Gauss said of himself that he is entirely a mathematician; he did not allow himself to aspire to anything else at the expense of mathematics; but natural science was not excluded. On the occasion when he wrote down the motto mentioned above, I heard him say that it would be a statement appropriate for a natural scientist. Gauss held mathematics, to use his own words, to be the queen of the sciences and arithmetic to be the queen of mathematics. She would often condescend to provide service to astronomy and other natural sciences, but she should bear in any case the first rank.
Wie viele Gedanken mögen bei dieser unglaublichen Productivität in diesem mächtigen Gehirne aufgetaucht und wieder untergegangen sein, die wenigstens für erst der Wissenschaft verloren sind! Gauss sagte von sich dass er ganz Mathematiker sei; etwas anderes auf Kosten der Mathematik sein zu wollen lehnte er von sich ab; doch war die Naturwissenschaft nicht ausgeschlossen. Bei der Gelegenheit als er das oben angeführte Motto, welches er besonders hoch schätzte und liebte, niedergeschrieben hatte, hörte ich ihn sagen es sei ein geeigneter Ausspruch für einen Naturforscher. Die Mathematik hielt Gauss um seine eigenen Worte zu gebrauchen, für die Königin der Wissenschaften und die Arithmetik für die Königin der Mathematik. Diese lasse sich dann öfter herab der Astronomie und andern Naturwissenschaften einen Dienst zu erweisen, doch gebühre ihr unter allen Verhältnissen der erste Rang (Sartorius von Waltershausen 1856, 79).
Sartorius von Waltershausen claimed that Gauss’s motto “mentioned above” in this passage stemmed from the first lines of King Lear, act I, scene II: “Thou, nature, art my goddess; to thy law my services are bound” (Shakespeare [1608] 1997). He noted that Gauss changed “law” to “laws”, which has a different meaning.
Gauss alluded to the “magical attraction” of number theory (higher arithmetic) in a note on quadratic residues submitted in 1808 to the Royal Society of Sciences in Göttingen, which immediately published a German paraphrase. Here is the original passage (Gauss [1808] 1863, 152), followed by an English translation by the present author.
Die schönsten Lehrsätze der höhern Arithmetik, und namentlich auch diejenigen, wovon hier die Rede ist, haben das Eigne, dass sie durch Induction leicht entdeckt werden, ihre Beweise hingegen äusserst versteckt liegen, und nur durch sehr tief eindringende Untersuchungen aufgespürt werden können. Gerade diess ist es, was der höhern Arithmetik jenen zauberischen Reiz gibt, der sie zur Lieblingswissenschaft der ersten Geometer gemacht hat, ihres unerschöpflichen Reichtums nicht zugedenken, woran sie alle andere Theile der reinen Mathematik so weit übertrifft.
The most beautiful theorems of higher arithmetic, and in particular even those that are described here, have the property that they are discovered easily by induction, but their proofs lie deeply hidden, and can be teased out only by deeply penetrating investigations. It is just this that gives higher arithmetic that magical attraction, which made it the favorite science of the first geometers, not to mention its inexhaustible richness, in which it so far surpasses all other parts of pure mathematics.
Gauss referred to Schiller’s poem that same year in his inaugural address as director of the astronomical observatory in Göttingen. Here is the original passage (Gauss [1808] 1929, 192–193), followed by an English translation by the present author.
Ich kann nicht umhin, Sie hier an ARCHIMEDES zu erinnern, den seine Zeitgenossen am meisten nur wegen seiner künstlichen Maschinen, wegen der zauberhaft scheinenden Wirkungen derselben bewunderten, der aber auf alles dieses in Vergleichung mit seinen herrlichen Entdeckungen im Felde der reinen Mathematik, die an und für sich nach dem gewöhnlichen Sprachgebrauch wenigstens damals meistens gar keinen sichtbaren Nutzen hatten, einen so geringen Werth legte, dass er uns über jene nichts aufzeichnete, während er diese in seinen unsterblichen Werken mit Liebe entwickelt hat. Sie kennen gewiss alle das schöne Gedicht von SCHILLER [Archimedes und der Schüler]. Lassen Sie uns auch die erhabene Astronomie am liebsten aus diesem schönern Gesichtspunkte betrachten. Welches edlere Gemüth hat nicht schon früh beim Anblick des gestirnten Himmels den lebhaften Wunsch empfunden, mit diesem herrlichen Schauspiel näher bekannt zu werden, seine wunderbaren Phänomene und wo möglich selbst seine verborgenen Geheimnisse zu ergründen, so weit es wenigstens sein individueller Beruf und seine Verhältniss verstatten ... ?
I cannot refrain from reminding you here about ARCHIMEDES, whom his contemporaries admired mostly for his ingenious machines because of their apparently magical efficacy, but who attached such little value to all that, in comparison with his beautiful discoveries in the field of pure mathematics, which in and of itself at that time,at least in common discourse generally had no visible uses at all, that he recorded nothing about that for us, although he developed these [latter] with love in his immortal works. All of you certainly know SCHILLER’s pretty poem “Archimedes and the Scholar.” What more noble soul has not experienced quite early with the sight of the starry heavens the strong desire to become better acquainted with this beautiful drama, to fathom its wonderful phenomena and when possible even its hidden secrets, at least as far as his individual calling and his circumstances permit ... ?
Hilbert mentioned Gauss’s description of the magical attraction that made number theory the favorite science for the first mathematicians. Gauss may have been alluding to the legendary Lotus Eaters featured in Homer’s Odyssey. Ten years after the Trojan War, soon after departing from Troy to return home to Ithaca, Odysseus lost his way:
Thence for nine days I drifted before the deadly winds along the swarming sea; but on the tenth we touched the land of Lotuseaters, men who make food of flowers. So here we went ashore and drew us water, and soon by the swift ships my men prepared their dinner. Then after we had tasted food and drink, I sent some sailors forth to go and learn what men who live by bread dwelt in the land, selecting two, and joining with them a herald as a third. These straightway went and mingled with the Lotuseaters, and yet the Lotuseaters had no thought of harm against our men; indeed, they gave them lotus to taste; but whosoever of them ate the lotus’ honeyed fruit wished to bring tidings back no more and never to leave the place, but with the Lotuseaters there desired to stay, to feed on lotus and forget his going home. These men I brought back weeping to the ships by very force, and dragging them under the benches of our hollow ships I bound them fast, and bade my other trusty men to hasten and embark on the swift ships, that none of them might eat the lotus and forget his going home. Quickly they came aboard, took places at the pins, and sitting in order smote the foaming water with their oars (Homer [1884] 1949, 131–132).
The most likely source for Hilbert’s reference to Leopold Kronecker is the following passage near the beginning of Kronecker’s 1887 paper, On the Concept of Number, in which he quoted Carl Gustav Jacob Jacobi. Kronecker suggested that preliminary work be done in basic fields so that the concepts of number, space, and time would be ready for use in special subjects. He continued,
So soll dies hier in Beziehung auf den Zahlbegriff geschehen, den einfachsten jener drei Begriffe, dessen dominirende Stellung Jacobi in einem seiner Briefe an Alexander v. Humboldt sehr schön hervorgehoben hat.
“Ein Alter”—so beginnt einer dieser Briefe—“vergleicht die Mathematiker mit den Lotophagen. Wer einmal, sagt er, die Süssigkeit der mathematischen Ideen gekostet, kann nicht mehr davon ablassen. Schreiben Sie also meinen vorigen Brief der Raserei zu, in welche jene Lotosfresser versinken, wenn sie den Cultus jener Ideen vernachlässigt oder sie nur ihrer zufälligen Anwendungen wegen geschätzt glauben. ...”
In dieser geistvollen Parodie des Schillerschen Gedichts “Archimedes und der Schüler” bezeichnet Jacobi die Stellung des Zahlbegriffs in der gesammten Mathematik echt poetisch aber auch genau zutreffend und ganz ähnlich wie Gauss in den Worten: “Die Mathematik sei die Königin der Wissenschaften und die Arithmetik die Königin der Mathematik. Diese lasse sich dann öfter herab, der Astronomie und andern Naturwissenschaften einen Dienst zu erweisen, doch gebühre ihr unter allen Verhältnissen der erste Rang” (Kronecker 1887, 337–338).
Here is a translation of that passage by William B. Ewald:
This shall be done here for the concept of number—the simplest of those three concepts, whose dominant position Jacobi beautifully stressed in one of his letters to Alexander v. Humboldt.
“An old man”—thus begins one of these letters—“compares mathematicians to the lotophagi. Whoever once, he says, tastes the sweetness of mathematical ideas can never desist. So ascribe my earlier letter to the frenzy which those lotoseaters sink into when they suspect that their cult has been neglected or is valued only for its practical applications. ...”
In this witty parody of Schiller’s poem “Archimedes and the Student,” Jacobi characterizes the position of the number concept in the whole of mathematics, and he does it with genuine poetry as well as with exact truth— precisely like Gauss in the words: “Mathematics is the queen of the sciences, and arithmetic the queen of mathematics. She frequently condescends to render a service to astronomy and other natural sciences, but in all circumstances the first rank is due to her” (Kronecker [1887] 1996, 948–949).
The ellipsis in the text by Leopold Kronecker that is displayed in the preceding note (see In Their Words: Kronecker/Jacobi, above) represents a quotation of the parody by Jacobi of Friedrich Schiller’s poem that is mentioned there. Kronecker did not mention how fond Gauss was of that little work by one of Germany’s greatest writers. Gauss referred to Schiller’s poem in his 1808 inaugural address as director of the astronomical observatory in Göttingen. The editors of the published version of that address indicated that Gauss had used the same material in his lectures on astronomy (Gauss [1808] 1929, 192–193, 198–199). Here is its text (Schiller 1795), alongside a translation by the present author:
Archimedes und der Schüler
Archimedes and the Schoolboy
Zu Archimedes kam ein wissbegieriger Jüngling,
To Archimedes came a young man eager to know.
“Weihe mich,” sprach er zu ihm, “ein in die
göttliche Kunst,
“Instruct me,” said he to him, “in the divine art,
Die so herrliche Frucht dem Vaterlande
getragen
which has borne such glorious fruit to the
fatherland
Und die Mauern der Stadt vor der Sambuca
beschützt!”
and which protects the walls of the city from the
sambuca!”
“Göttlich nennst du die Kunst? Sie ists,”
versetzte der Weise,
“You call art divine? She is that,” replied the sage,
“Aber das war sie, mein Sohn, eh sie dem Staat
noch gedient,
“But she was so, my son, before she ever served
the state.
Willst du nur Früchte von ihr, die kann auch die
Sterbliche zeugen;
If you want only fruits from her, mortals can also
bear those.
Wer um die Göttin freit, suche in ihr nicht das
Weib.”
Who courts the goddess, seek not in her the wife."
“Sambuca” is the name of a siege engine that the Roman soldier Marcellus used against Syracuse, the city of Archimedes. Here is the text of Jacobi’s parody (Ewald 1996, 948), alongside a translation by the present author.
Archimedes und der Jüngling
Archimedes and the Young Man
Zu Archimedes kam ein wissbegieriger Jüngling,
To Archimedes came a young man eager to know.
“Weihe mich,” sprach er zu ihm, “ein in die
göttliche Kunst,
“Instruct me,” said he to him, “in the divine art,
Die so herrliche Dienste der Sternenkunde
geleistet,
That renders such glorious service to astronomy,
Hinter dem Uranos noch einen Planet entdeckt!”
and reveals another planet beyond Uranus!”
“Göttlich nennst du die Kunst? Sie ist’s,”
versetzte der Weise,
“You call art divine? She is that,” replied the sage,
“Aber sie war es, bevor noch sie den Kosmos
erforscht,
“But she was so, even before she probed the
cosmos,
Ehe sie herrliche Dienste der Sternenkunde
geleistet,
Before she rendered glorious service to astronomy,
Hinter dem Uranos noch einen Planeten
entdeckt.
and revealed another planet beyond Uranus.
Was Du im Kosmos erblickst, ist nur der
Göttlichen Abglanz,
What you see in the cosmos is just the reflection
of the Divine,
In der Olympier Schar thronet die ewige Zahl.”
In the Olympian host is enthroned the eternal
number.”
Uranus is visible to the eye and had been recognized as a planet since before 1700. Mathematical analysis of its motion led in 1846 to discovery of the more distant planet Neptune. The correspondence mentioned between Jacobi and the Prussian scientist Alexander von Humboldt (1769–1859) dates from that year (Ewald 1996, 948).
Concluding the argument in his radio address, Hilbert explicitly paraphrased an earlier remark of Jacobi from an 1830 letter to the mathematician AdrienMarie Legendre (1752–1833). Here is the original French text from Jacobi [1830] 1881, 454, followed by an English translation by the present author:
Il est vrai que M. Fourier avait l’opinion que le but principal des mathématiques était l’utilité publique et l’explication des phénomènes naturels; mais un philosophe comme lui aurait dû savoir que le but unique de la science, c’est l’honneur de l’esprit humain, et que sous ce titre, une question de nombres vaut autant qu’une question du système du monde.
It is true that Fourier held the opinion that that the principal goal of mathematics should be utility to the public and explication of natural phenomena; but a philosopher such as he should have known that the unique goal of science is the honor of the human spirit, and that under this title, a question about numbers is worth as much as a question about the system of the world.
Jacobi was discussing the theory of heat conduction published in 1822 by the French mathematician Joseph Fourier (1768–1830).
Hilbert referred to the essay The Choice of Facts that became the first chapter of the book Science and Method by Henri Poincaré (1854–1912). Here is the American mathematician George Bruce Halstead’s translation of the relevant passage:
Tolstoy somewhere explains why “science for its own sake” is in his eyes an absurd conception. We cannot know all facts, since their number is practically infinite. It is necessary to choose; then we may let this choice depend on the pure caprice of our curiosity. Would it not be better to let ourselves be guided by utility, by our practical and above all by our moral needs? Have we nothing better to do than count the number of ladybugs on our planet?
It is clear the word “utility” has not for him the sense men of affairs give it, and following them most of our contemporaries. Little cares he for industrial applications, for the marvels of electricity or of automobilism, which he regards rather as obstacles to moral progress; utility for him is solely what can make man better. ...
[If] our choice can only be determined by caprice or by immediate utility, there can be no science for its own sake, and consequently no science. But ... scientists believe that there is a hierarchy of facts and that a judicious choice may be made among them. They are right, since otherwise there would be no science, and science exists. One need only open his eyes to see that the conquests of industry which have enriched so many practical men would never have seen the light, if these practical men alone had existed and if they had not been preceded by unselfish devotees who died poor, who never thought of utility, and yet had a guide far other than caprice (Poincaré [1906] 1909, 231–232).
And Poincaré’s original text:
Tolstoï explique quelque part pourquoi “la Science pour la Science” est à ses yeux une conception absurde. Nous ne pouvons connaître tous les faits, puisque leur nombre est pratiquement infini. Il faut choisir; dès lors, pouvonsnous régler ce choix sur le simple caprice de notre curiosité; ne vautil pas mieux nous laisser guider par l’utilité, par nos besoins pratiques et surtout moraux; n’avonsnous pas mieux à faire que de compter le nombre de coccinelles qui existent sur notre planète?
Il est clair que le mot utilité n’a pas pour lui le sens que lui attribuent les hommes d’affaires, et derrière eux la plupart de nos contemporains. Il se soucie peu des applications de l’industrie, des merveilles de l’électricité ou de l’automobilisme qu’il regarde plutôt comme des obstacles au progrès moral; l’utile, c’est uniquement ce qui peut rendre l’homme meilleur. ...
[Si] notre choix ne peut être déterminé que par le caprice ou par l’utilité immédiate, il ne peut y avoir de science pour la science, ni par conséquent de science. ...Mais les savants croient qu’il y a une hiérarchie des faits et qu’on peut faire entre eux un choix judicieux; ils ont raison, puisque sans cela il n’y aurait pas de science et que la science existe. Il suffit d’ouvrir les yeux pour voir que les conquêtes de l’industrie qui ont enrichi tant d’hommes pratiques n’auraient jamais vu le jour si ces hommes pratiques avaient seuls existé, et s’ils n’avaient été devancés par des fous désintéressés qui sont morts pauvres, qui ne pensaient jamais à l’utile, et qui pourtant avaient un autre guide que leur caprice (Poincaré 1909, 7–9).
Hilbert recounted that Poincaré had railed against Leo Tolstoy’s 1899 essay On the Significance of Science and Art (see In Their Words: Poincaré, above). There was much to criticize in Tolstoy's emotional, illorganized, and repetitive essay, but it is not clear that Poincaré’s criticism was on target. Were they really discussing the same ideas? Poincaré’s attack could be an example of what Jacobi called “the frenzy which those lotoseaters sink into when they suspect that their cult has been neglected or is valued only for its practical applications” (see In Their Words: Kronecker/Jacobi, above). Poincaré did not point to any specific passage of Tolstoy. Here is a selection of excerpts that seem relevant.
Modern science is also occupied exclusively with facts: it studies facts. But what facts? Why such facts and not others?
The men of modern science are very fond of speaking with a solemn assurance, “We study facts alone,” imagining that these words have some meaning.
To study facts alone is quite impossible, because the number of facts which may be objects of our study is countless, in the strict sense of the word.
Before beginning to study facts, one must have some theory, according to which facts are studied; that is, these or those being selected from the countless number of facts. And this theory indeed exists, and is even very definitely expressed, though many of the agents of modern science ignore it ... (Tolstoy [1887] 1904, 189190).
Such is the false tendency of science which deprives it of the possibility to fulfil its duty in serving the people. ... Science may point to its stupid excuse that science is acting for science, and that, when fully developed, it will become accessible to the people ... (ibid., 222).
And, therefore, the highest wisdom of men has always consisted in finding out the clue according to which must be arranged the information of men, and by which decided what kinds of information are more, and what are less, important. And this, which has directed all other knowledge, men have always called science in the strictest sense of the word. ...This science has always had for its object the inquiry as to what was the destiny, and therefore the true welfare, of each man and of all men (ibid., 227228).
The old secure justifications are all destroyed; and the new ephemeral justifications of the progress of science for science’s sake ... will not bear the light of plain common sense (ibid., 261).
Some cited works have appeared in several versions. Information about the first often has historical interest, even when reference to a later one is more appropriate. In such cases, both dates are given.
Du BoisReymond, Emil. [1872] 1874. On the limits of natural science. Popular Science Monthly 5: 17–32. (This American journal was founded in 1872; in altered form it is still published.) Translation by Joseph Fitzgerald of “Über die Grenzen des Naturerkennens,” a lecture presented to the meeting of the Society of German Natural Scientists and Physicians in Leipzig in 1872, and published there that year by Veit & Comp.
Du BoisReymond, Emil. 1880. Die sieben Welträthsel (The seven worldriddles). Monatsberichten der Königlich Preussischen Akademie der Wissenschaften zu Berlin: 1045–1072. Speech presented on 8 July to the Academy’s Leibniz Celebration. The title was added for later editions.
Ewald, William B., editor. 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Two volumes. Oxford: Clarendon Press. Contains Hilbert [1930] 1996 and Kronecker [1887] 1996.
Franzén, Torkel. 2005. Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley: A K Peters.
Galilei, Galileo. [1623] 1960. The Assayer. Translated by Stillman Drake. The Controversy on the Comets of 1618, by Galileo Galilei et al., 163–336. Philadelphia: University of Pennsylvania Press. Originally published as Il saggiatore in Rome, by Giacomo Mascardi.
Gauss, Carl Friedrich. [1808] 1863. Theorematis arithmetici demonstratio nova (New proof of an arithmetical theorem). In Gauss 1863–1929, volume 2, 151–154. Presented to the Society on 15 January 1808 in handwritten form, this German paraphrase was published in Göttingische gelehrte Anzeigen on 12 May 1808.
Gauss, Carl Friedrich. [1808] 1929. Astronomische Antrittsvorlesung (Inaugural lecture). In Gauss 1863–1929, volume 12, 177–199.
Gauss, Carl Friedrich. 1863–1929. Werke. Twelve volumes. Edited and published by the Gesellschaft der Wissenschaften zu Göttingen (Society of Sciences in Göttingen). Contains items Gauss [1808] 1863 and [1808] 1929.
Hilbert, David. [1900] 1902. Mathematical problems. Bulletin of the American Mathematical Society 8 (1902), no. 10: 437–479. Translation by Mary Winston Newson of “Mathematische Probleme,” an address presented to the International Congress of Mathematicians (ICM) in Paris in 1900 and published in Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Mathematischphysikalische Klasse 1900: 253–297. (Editor’s note: Like many items in this Bibliography, Mary Winston Newson's translation into English of Hilbert's address to the 1900 ICM in Paris is available online, including at the AMS website as a fulltext pdf of Mathematical Problems: Lecture Delivered Before the International Congress of Mathematicians at Paris in 1900 and at the website of David Joyce of Clark University as Hilbert's Address of 1900 and His 23 Mathematical Problems.)
Hilbert, David. [1922–1923] 1988. Wissen und mathematisches Denken: Vorlesung (Science and Mathematical Thought: Lecture). Prepared by Wilhelm Ackermann. Göttingen: Mathematisches Institut der Universität.
Hilbert, David. [1925] 1969. On the infinite. In Van Heijenoort 1969, 367–392. Translated by Stefan BauerMengelberg. Originally published as “Über das Unendliche,” Mathematische Annalen 95: 161–190.
Hilbert, David. [1930] 1971. Radio address. Fortyfiverpm disk recording of Hilbert reading material excerpted from the original version of Hilbert [1930] 1996. Distributed with Reidemeister 1971.
Hilbert, David. [1930] 1996. Logic and the knowledge of nature. Translated by William B. Ewald. In Ewald 1996, volume 2, 1157–1165. Originally published as “Naturerkennen und Logik,” Naturwissenschaften 18: 959–963.
Homer. [1884] 1949. Odyssey. Translated by George Herbert Palmer. Cambridge, Massachusetts: Houghton Mifflin Company.
Jacobi, Carl Gustav Jacob. [1830] 1881. Letter to AdrienMarie Legendre, 2 July. In Gesammelte Werke, volume 1, edited by Karl W. Borchardt, 453–455. Berlin: Georg Reimer.
Kant, Immanuel. [1786] 1903. Metaphysische Anfangsgründe der Naturwissenschaft (Metaphysical Foundations of Natural Science). In Kant’s gesammelte Schriften, edited by the Royal Prussian Academy of Sciences, volume 4, 465–566. Berlin: Georg Reimer.
Kronecker, Leopold. [1887] 1996. On the concept of number. Translated by William B. Ewald. In Ewald 1996, 947–955. Originally published as “Ueber den Zahlbegriff,” Journal für die reine und angewandte Mathematik 101: 337–355.
Minkowski, Hermann. [1908] 1923. Space and time. The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity, by H. A. Lorenz et al., with notes by A. Sommerfeld, 73–91. London: Methuen and Company. Translation by W. Perrett and G. B. Jeffrey of “Raum und Zeit,” an address presented to the Society of German Natural Scientists and Physicians in Cologne in 1908 and published in Jahresbericht der Deutschen MathematikerVereinigung 10: 75–88.
Naturwissenschaften. 1922. Celebratory issue honoring David Hilbert on his sixtieth birthday. Naturwissenschaften 10 (issue 4, January 27).
Poincaré, Henri. [1906] 1909. The choice of facts. Monist 19: 231–239. Translated by George Bruce Halsted. Address presented at the University of Paris in 1906. There are many more recent versions.
Poincaré, Henri. 1909. Science et méthode. Paris: Ernest Flammarion.
Reid, Constance. 1970. Hilbert. With an appreciation of Hilbert’s mathematical work by Hermann Weyl. New York: SpringerVerlag New York.
Reid, Constance. 1999. Down the rabbit hole, or Abenteuer im Wunderland (Adventure in Wonderland). Mitteilungen der Deutschen MathematikerVereinigung 7: 26–34.
Reidemeister, Kurt, editor. 1971. Hilbert Gedenkband (Hilbert Memorial Volume). Berlin: Springer Verlag. Contains Hilbert [1930] 1971. Reissued in 2012 but, at that time, without Hilbert [1930] 1971.
Rowe, David E. 2013. History quiz: Who linked Hegel’s philosophy with the history of mathematics? Mathematical Intelligencer 35: 38–41. While the title question may not be directly relevant to the present paper, Rowe’s background discussion certainly is.
Sartorius von Waltershausen, Wolfgang. 1856. Gauss zum Gedächtniss. Leipzig: S. Hirzel. An English translation, Carl Friedrich Gauss, a Memorial, by Helen Worthington Gauss, saw limited publication in 1967 by Colorado College in Colorado Springs.
Schiller, Friedrich. 1795. Archimedes und der Schüler. Die Horen 11: item 7. This journal was published by Schiller himself. It is accessible online. In English, the title means The Horæ (goddesses of the seasons). This poem is reprinted in many editions of Schiller’s works.
Steinhaus, Hugo. [1950] 1983. Mathematical Snapshots. With a preface by Morris Kline. Oxford: Oxford University Press.
Tolstoy, Lev Nikolayevich. 1899. What is to be done? Life. Translated by Isabel F. Hapgood. New York: Thomas Y. Crowell. A translation of a redacted version of the essay “What is to be done” was published in 1887. The material quoted in the present article, however, is from this revised, uncensored 1899 version. There are many more recent versions.
Van Heijenoort, Jean van. [1967] 1970. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Cambridge, Massachusetts: Harvard University Press. Contains Hilbert [1925] 1969.
Vinnikov, Victor. 1999. We shall know: Hilbert’s apology. Mathematical Intelligencer 21: 42–46.
Wang Hao. 1988. Reflections on Kurt Gödel. Cambridge, Massachusetts: MIT Press.
James T. Smith received the A.B. from Harvard College in 1961 and Ph.D. from the University of Saskatchewan, Regina, in 1970, in foundations of geometry. Since then he has worked primarily at San Francisco State University, mostly teaching and writing, and in software development. He is now retired but fully occupied with history, especially the legacies of Mario Pieri and Alfred Tarski.