David Hilbert's Radio Address - Hilbert's Sources - Their Own Words

Hilbert's Sources

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 Lotus-Eater 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 ill-organized, 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 lotos-eaters 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.)

Their Own Words

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.

Galileo

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 (1583-1654) 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).

Kant

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.

Gauss

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 (1809-1876). 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 ... ?

Homer

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 Lotus-eaters, 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 Lotus-eaters, and yet the Lotus-eaters 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 Lotus-eaters 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).

Kronecker/Jacobi

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 lotos-eaters 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).

Schiller/Jacobi

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:

“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.

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 Adrien-Marie 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).

Poincaré

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 lady-bugs 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, pouvons-nous régler ce choix sur le simple caprice de notre curiosité; ne vaut-il pas mieux nous laisser guider par l’utilité, par nos besoins pratiques et surtout moraux; n’avons-nous 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).

Tolstoy

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, ill-organized, 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 lotos-eaters 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, 189-190).

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., 227-228).

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).