Hilbert and Physics (1900-1915)
by Leo Corry1
Some mathematical-physical theories look to me like a toy,
that a child has completely messed up and that every three minutes
needs to be fixed again, in order to keep it working.
David Hilbert, Tagebuch2
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
1
Physics and Geometry in Hilbert’s Early Career
5
Die Grundlagen der Geometrie
10
Physics in Hilbert’s 1900 List of Problems
14
Hilbert, Minkowski, and Physics in Göttingen: 1900-1909
16
Mechanics, Kinetic Theory, Radiation Theory: 1910-1914
32
Electrodynamics and General Relativity: 1913-1915
36
Concluding Remarks
Bibliography
42
1. Introduction
The name of David Hilbert (1862-1943) arises in connection with the history of
physics usually in three different, and rather circumscribed contexts. The first of these
contexts is the famous list of twenty three problems that Hilbert presented in 1900 in
Paris, with occasion of the Second International Congress of Mathematicians. The
sixth problem of this list deals specifically with physics. Only one year previous to the
Paris Congress Hilbert had published his seminal study on the foundations of
geometry, Die Grundlagen der Geometrie. The axiomatic approach adopted by Hilbert
in this book was to have an enormous influence on the development of twentieth-
1. Sections 1 to 5 of the present preprint include condensed versions of the contents of Corry
1997, 1997a.
2. “Manche math-physikalische Theorie erscheint mir wie ein Kinderspielzeug, dass in
Unordnung geraten ist und alle 3 Minuten wieder aufgerichtet werden muss, damit es weiter
geht.” The date of this quotation is unknown, but may be somewhat before 1900. The
manuscript of Hilbert’s Tagebuch is found in the Hilbert Nachlass, Staats- und
Universitätsbibliothek Göttingen, Cod MS Hilbert 600/3, see p. 134.
2
Hilbert and Physics:
century mathematics and on the way mathematicians looked at their science. Hilbert’s
sixth problem was the suggestion that an analysis similar to that performed for the case
of geometry in the Grundlagen should also be applied to individual physical
disciplines.
The problems in this list became an object of great mathematical and historical
interest over the following decades, with mathematicians of all specialties and of all
countries attempting to solve them, and with periodical accounts of the current state of
research on some, or on all of them. However, this particular problem, the sixth one,
received much less attention than any of the other twenty two in the list. Working
mathematicians directed in general relatively little effort to solve it.3 Neither have
historians greatly contributed to understand both its roots and whatever attempts have
been made to address it after 1900.4 This problem seems to have been largely
considered as an accessory surplus, that only artificially can be seen as part of Hilbert’s
otherwise comprehensive and harmonious image of mathematics.
The second context is the kinetic theory of gases. In 1912, at a time when most
of Hilbert’s energies were directed to his work on the theory of linear integral
equations, he solved the so-called Boltzmann equation. Although this represented a
major contribution to the development of this particular physical discipline, Hilbert is
often considered to have had no real interest in the kinetic theory as such. Rather, his
solution of the equation has been considered as an isolated, if important, furtive
incursion into this field. In his authoritative account of the development of the kinetic
theory, Stephen G. Brush dedicated one short section to describing Hilbert’s
contribution. Brush’s assessment of Hilbert’s motivations is expressed in the following
passage:
When Hilbert decided to include a chapter on kinetic theory in his treatise on integral
equations, it does not appear that he had any particular interest in the physical problems
associated with gases. He did not try to make any detailed calculations of gas properties, and
did not discuss the basic issues such as the nature of irreversibility and the validity of
mechanical interpretations which had exercised the mathematician Ernst Zermelo in his
debate with Boltzmann in 1896-97. A few years later, when Hilbert presented his views on the
contemporary problems of physics, he did not even mention kinetic theory. We must therefore
conclude that he was simply looking for another possible application of his mathematical
theories, and when he had succeeded in finding and characterizing a special class of solutions
(later called “normal”) ... his interest in the Boltzmann equation and in kinetic theory was
exhausted. (Brush 1976, 448)
Brush added that Hilbert did encourage some of his students to work on mathematical
problems connected with the theory, and that he seems to have also taught courses on
this issue. Yet these qualifications did not change Brush’s overall evaluation of
Hilbert’s motivations. It seems to me that Brush’s assessment manifests the widely
held beliefs about the nature of Hilbert’s contribution to physics.
The third context in which Hilbert’s name has been associated to a significant
development in physics is general relativity. On November 20, 1915, Hilbert presented
to the Royal Scientific Society in Göttingen his version of the field equations of
gravitation, in the framework of what he saw as an axiomatically formulated
3.
See, e.g., Wightman 1976, Gnedenko 1979.
4. For a general historical account of the problems in the list, their roots, and their place in the
development of twentieth centruy mathematics see Rowe 1996.
3
Hilbert and Physics:
foundation for the whole of physics. During that same month of November, Einstein
had been struggling with the final stages of his own effort to formulate the generally
covariant equations that lie at the heart of the general theory of relativity. His struggle
had spanned at least three years of intense work and included the publication of several
previous versions, each of which Einstein found inadequate for different reasons.5
Only in November of 1915 he presented three different versions at the weekly
meetings of the Prussian Academy of Sciences in Berlin, before attaining his final
version, on November 25, that is, five days after Hilbert had presented his theory.
Einstein had visited Göttingen in the summer of 1915, to lecture on the
progress and the difficulties encountered in his work. Hilbert was then in the audience
and Einstein was greatly impressed by him. He felt that in Göttingen his work had been
fully understood to the details.6 Hilbert’s involvement in the problems associated to
general relativity has in general been traced back not earlier than to this visit of
Einstein or, at best, to the years immediately preceding it. Although the axiomatic
character of Hilbert’s work on relativity has often been stressed—and by implication it
has been associated to his work on the foundations of geometry and of mathematics at
large—its actual place as part of his overall conception of mathematics and science has
not been fully clarified. Like in the case of the kinetic theory, this contribution of
Hilbert has mainly been seen as a furtive incursion in physics, aimed at illustrating the
power and the scope of validity of the “axiomatic method” (whatever is meant by that)
and as a test of Hilbert’s mathematical abilities.7
A source quoted very often when describing the various aspects of Hilbert’s
career—and which also refers to his work on physics—is a passage taken from
Hermann Weyl’s obituary of Hilbert. According to this passage, Hilbert’s
mathematical career covered five, clearly discernible periods during which his
attention focused on a single issue: (1) Theory of invariants (1885-1893); (2) Theory
of algebraic number fields (1893-1898); (3) Foundations, (a) of geometry (1898-1902),
(b) of mathematics in general (1922-1930); (4) Integral equations (1902-1912); (5)
Physics (1910-1922) (Weyl 1944, 619). According to Weyl, the passage from one
period to the next always signified a sharp departure from Hilbert’s past topic of
interest in order to move into a completely new one. For instance, concerning his
research on the foundations of geometry Weyl wrote (1944, 635): “[T]here could not
have been a more complete break than the one dividing Hilbert’s last paper on the
theory of number fields from his classical book Grundlagen der Geometrie.” Thus,
even if Weyl conceded that Hilbert’s scientific interests focused during a relatively
long period of time on physics alone (twelve years, two of which overlap with his work
on linear integral equations), his account still suggests a clear separation between this
period and the rest of his long career.
5. See Norton 1984.
6. As he wrote to Sommerfeld upon returning from Göttingen. See Hermann (ed.) 1968, 30.
7. See, e.g., Earman and Glymour 1978, 293; Mehra 1974, 17 & 83. For a recent
reinterpretation of the actual contents of Hilbert’s lecture in November 1915, see Corry, Renn
& Stachel 1997.
4
Hilbert and Physics:
The periodization manifest in Weyl’s account reflects indeed rather faithfully
the distribution of Hilbert’s published work along the years, and what constituted his
main domains of interest at different times. However, the actual scope of Hilbert
current interests is much broader than such an account may suggest. A clear and
balanced perception of Hilbert’s mathematical world, and in particular of the actual
place he accorded to physics as part of it, necessitates a deeper examination of his
docent and institutional activities in Göttingen: his lectures, his seminars, the doctoral
dissertations he advised, the activities of the Göttingen Mathematical Society (GMG).8
As even the most cursory examination of the lists included in the appendix at the end
of this article immediately indicates, these activities included a long-standing interest
in physical issues that covered all his years at Göttingen. Physics was always accorded
a central place in the scientific agenda implemented at Göttingen by Hilbert, by his
colleagues and by his students.
Given the astonishing breadth of Hilbert’s scope of scientific interests and
knowledge, an examination of his scientific activities other than pure research and
standard publishing becomes fundamental in any attempt to understand his
mathematical views. Hilbert had the ability to attract extremely gifted students and to
communicate them the kind of deep open questions that in his opinion should be
investigated in various fields. He also provided very often the inspiration to solve those
problems. Hilbert directed no less than sixty eight doctoral dissertations, sixty of them
in the relatively short period between 1898 and 1914.9 Four or five of these
dissertations deal with issues that can be directly related to physics as well as to
Hilbert’s current scientific interests. Also his lectures and seminars in Göttingen were
never mere systematic presentations of well-established knowledge. On the contrary,
Hilbert considered the classroom and the seminar meetings as ideal settings for putting
forward new, untried ideas and to benefit from the feedback of his students. Later in
life Hilbert described the central place he conceded to his teaching in the following
terms:
The closest conceivable connection between research and teaching became a decisive feature
of my mathematical activity. The interchange of scientific ideas, the communication of what
one found by himself and the elaboration of what one had heard, was since my early years at
Königsberg a pivotal aspect of my scientific work ... In my lectures, and above all in the
seminars, my guiding principle was not to present material in a standard and as smooth as
possible way, so as to help the student keeping clean and ordered notebooks. Above all, I
always tried to illuminate the problems and difficulties and to offer a bridge leading to the
present open questions. It often happened, that in the course of a semester the program of an
advanced lecture was completely changed, because I wanted to discuss issues in which I was
currently involved as a researcher and which in no way had yet attained their definite
formulation. (Hilbert 1971, 79)
Thus, the notes that Hilbert prepared for his courses provide an essential source for
understanding the development of his ideas.
8. In fact, Weyl himself acknowledged in some occassions the centrality of Hilbert’s docent
activities and the impact of his influence as a teacher as main traits of his sceintific legacy. See
Sigurdsson 1994, 356-358.
9. Hilbert’s doctoral students are listed in Hilbert GA Vol. 3, 431-433. This list, however, is
incomplete.
5
Hilbert and Physics:
A detailed and comprehensive account of the contents of Hilbert’s unpublished
lectures on physics, their historical context and their influence is yet to be done.10 Such
an account will probably bring to light many unfinished ideas that Hilbert presented in
the classroom to his students, and which the latter further developed in their own
works thus leading to central contributions in the evolution of particular physical
disciplines. It will perhaps likewise bring to light many ideas that led nowhere. The
present paper surveys in general lines—using both published an unpublished
sources—the evolution of Hilbert’s interest in various physical domains until 1915,
and attempts to explain the place that Hilbert attributed to physical issues within his
overall conception of mathematics. I will claim that understanding these aspects of his
work is essential for understanding his view on mathematics. I will also suggest that,
given the centrality of Hilbert in the scientific world of Göttingen, and given the
centrality of Göttingen in the scientific world at large, understanding Hilbert’s
influence on other scientists working at that center may appear as fundamental for
better understanding the historical context of many central developments in early
twentieth-century physics.
Hilbert was undoubtedly among the most influential mathematicians of the
beginning of this century, if not the most influential one. His name is associated with
many results and concepts that play fundamental roles in several, distant fields of
mathematics such as algebra, number theory, functional analysis, mathematical
physics, metamathematics, and foundations of geometry. Even more pervasive is the
view that associates his name with the formalist approach that came to dominate a
considerable part of mathematical practice throughout the twentieth century. It was
only relatively recently, however, that historians of mathematics undertook a careful
analysis of his work and of his mathematical world, and a more balanced image of the
latter has begun to arise from these studies. The present article makes use of that recent
scholarship and is also intended as a further contribution to it.
2. Physics and Geometry in Hilbert’s Early Career
Crucial for understanding the place of physics in Hilbert’s overall conception of
science is his view of geometry as a natural science, similar in most essential aspects to
other physical disciplines. This conception—opposed perhaps to the largely accepted
view of Hilbert as the champion of a formalistic interpretation of the essence of
mathematics—is consistently manifest throughout his career, in spite of many other
changes recognizable in the latter. An interest in physics, in geometry and in the
relationship among them was present in Hilbert’s mind from very early on and until
well the end of his career.
Hilbert’s studied in his native city of Königsberg and there he also spent his
first years as a young professor. The University of Königsberg had a long tradition of
experimental physics and of mathematical analysis that went back to the days of Franz
Ernst Neumann (1798-1895) and Carl Gustav Jacobi (1804-1851).11 We have no direct
10. But for an initial contribution in that direction, see Corry 1997.
6
Hilbert and Physics:
evidence, however, to decide whether Hilbert took courses of physics as a student at
all. We do know that Hilbert attended courses given by the versatile mathematician
Heinrich Weber (1842-1913),12 whose fields of interest covered mathematical physics
as well, but it is unlikely that Hilbert came under his influence in this respect. Hilbert’s
doctoral advisor was Ferdinad Lindemann (1852-1939), who introduced him mainly
into the study of algebraic invariants; Hilbert’s interest in physics apparently
developed somewhat later. The main sources of overall influence on Hilbert during his
Königsberg years came from his friends Adolf Hurwitz (1859-1919) and Hermann
Minkowski (1864-1909); the scientific scope of interest of these three young
colleagues focused above all on pure mathematical domains, but gradually it also
extended as to concede physics an important place in it.
In 1893 Hilbert wrote his last paper on the theory of invariants and his major
research effort shifted now towards a new domain, the theory of algebraic number
fields which, however, was not completely apart from the former one. Yet, as Michael
Toepell’s study of the origins of Hilbert’s Grundlagen der Geometrie has shown,
Hilbert was simultaneously pursuing at this time in a systematic way an additional
field of interest, namely, the study of the foundations of geometry. This interest was
reflected in the courses he taught at that time.13 Hilbert lectured on geometry in
Königsberg for the first time in 1891 and planned to do so again in 1893. For lack of
students registered, however, this latter course was postponed until 1894. A
remarkable change occurred in Hilbert’s approach to geometry between 1891 and
1893, whereby he gradually moved towards the axiomatic treatment as an appropriate
way to address the foundational problems of this discipline. But even in his first
lectures of 1891 Hilbert clearly expressed his conception of geometry as a natural
science. In its essence, this view was not unlike that of other German geometers—even
those who, like Moritz Pasch (1843-1930), had adopted an axiomatic approach in their
presentations from very early on (Pasch 1882).14 The similarity between geometry and
physics, and the different character of the former as compared to other mathematical
domains recurrently appears in Hilbert’s lectures. In the introduction to his 1891
course, for instance, it is expressed as follows:
Geometry is the science dealing with the properties of space. It differs essentially from pure
mathematical domains such as the theory of numbers, algebra, or the theory of functions. The
results of the latter are obtained through pure thinking ... The situation is completely different
in the case of geometry. I can never penetrate the properties of space by pure reflection, much
the same as I can never recognize the basic laws of mechanics, the law of gravitation or any
other physical law in this way. Space is not a product of my reflections. Rather, it is given to
me through the senses.15
11. On the Königsberg school see Klein 1926-7 Vol. 1, 112-115 & 216-221; Lorey 1916, 5964; Volk 1967. The workings of the Königsberg physics seminar—initiated in 1834 by Franz
Neumann—and its enormous influence on nineteenth-century physics education in Germany
are described in great detail in Olesko 1991.
12. For more details on Weber (especially concerning his contributions to algebra) see Corry
1996, §§ 1.2 & 2.2.4.
13. See Toepell 1986. Toepell also describes Hilbert’s only gradual adoption of an axiomatic
approach to deal with these questions.
14. See Contro 1976, 284-289.
15. The German original is quoted in Toepell 1986, 21. Similar testimonies can be found in
many other manuscripts of Hilbert’s lectures. Cf., e.g., Toepell 1986, 58.
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Hilbert and Physics:
When rewriting his lecture notes of 1893 for teaching in 1894, Hilbert added
some remarks from which we learn that in the meantime he became acquainted with
the ideas put forward in Heinrich Hertz’s textbook on The Principles of Mechanics,
published that same year. Hilbert referred once again to the natural character of
geometry and explained the possible role of axioms in elucidating its foundations, by
making reference to Hertz’s characterization of a “correct” scientific image (Bild) or
theory. Thus Hilbert wrote:
Nevertheless the origin [of geometrical knowledge] is in experience. The axioms are, as Hertz
would say, images or symbols in our mind, such that consequents of the images are again
images of the consequences, i.e., what we can logically deduce from the images is itself valid
in nature.16
In the same context, Hilbert also pointed out the need of establishing the independence
of the axioms of geometry, while alluding to the kind of demand stipulated by Hertz.
Stressing once more the objective and factual character of geometry, Hilbert wrote:
The problem can be formulated as follows: What are the necessary, sufficient, and mutually
independent conditions that must be postulated for a system of things, in order that any of their
properties correspond to a geometrical fact and, conversely, in order that a complete
description and arrangement of all the geometrical facts be possible by means of this system of
things.17
It is difficult to determine whether Hilbert had indeed mastered the contents of Hertz’s
book at this stage, and even whether he actually got to read thoroughly its introduction.
However, it is likely that whatever acquaintance he made with the basic ideas
presented in this book so soon after its publication, this was suggested to Hilbert by his
friend Minkowski. Minkowski spent three semesters in Bonn still as a student, and he
returned to that city after receiving his doctorate in Königsberg in 1885. He stayed in
Bonn until 1894. While at Bonn, Minkowski became involved in both mathematical
and experimental physics.18 In 1888 he published an article on hydrodynamics that
was submitted to the Berlin Academy by Hermann von Helmholtz (Minkowski 1888).
During this period of his life, Minkowski’s greatest scientific source of inspiration
came from Hertz19 and he certainly must have communicated this enthusiasm to
Hilbert as well.
Hilbert arrived in Göttingen in 1895, at a time when he was completing his
major work on algebraic number theory, the Zahlbericht (Hilbert 1897). Algebraic
number fields continued to be his major field of publication during his first years at
this university, but at the same time he organized seminars and lectured on other topics.
Between 1895 and 1898 he held joint seminars with Klein on number theory,
mechanics and function theory.20 In 1899 he lectured in Göttingen on the foundations
of geometry for the first time. The notes of this course provided the basis for the
Grundlagen der Geometrie, published in June 1899 as part of a Festschrift issued in
16. Hilbert 1893/94, 10: “Dennoch der Ursprung aus der Erfahrung. Die Axiome sind, wie
Herz [sic] sagen würde, Bilde[r] oder Symbole in unserem Geiste, so dass Folgen der Bilder
wieder Bilder der Folgen sind d.h. was wir aus den Bildern logisch ableiten, stimmt wieder in
der Natur.”
17. Translated from the original passage quoted in Toepell 1986, 58-59.
18. See Rüdenberg and Zassenhaus (eds.) 1973, 39-42, and Hilbert 1909, 355.
19. At least according to Hilbert’s report (Hilbert 1909, 355)..
20. See Lorey 1916, 128.
8
Hilbert and Physics:
Göttingen with occasion of the unveiling of the Gauss-Weber monument. But before
that, back in the winter semester of 1898/99 Hilbert taught his first course on a
physical issue: mechanics. In the introduction to this course, Hilbert stressed once
again the essential affinity between geometry and the natural sciences, and also
explained the role that axiomatization should play in the mathematization of the latter.
He compared the two domains in the following terms:
Also geometry [like mechanics] emerges from the observation of nature, from experience. To
this extent, it is an experimental science.... But its experimental foundations are so irrefutably
and so generally acknowledged, they have been confirmed to such a degree, that no further
proof of them is deemed necessary. Moreover, all what is needed is to derive these foundations
from a minimal set of independent axioms and thus to construct the whole building of
geometry by purely logical means. In this way [i.e., by means of the axiomatic treatment]
geometry is turned into a pure mathematical science. In mechanics it is also the case that all
physicists recognize its most basic facts. But the arrangement of the basic concepts is still
subject to a change in perception ... and therefore mechanics cannot yet be described today as
a pure mathematical discipline, at least to the same extent that geometry is. We must strive
that it becomes one. We must ever stretch the limits of pure mathematics further afar, on
behalf not only of our mathematical interest, but rather of the interest of science in general.21
For Hilbert, then, the difference between geometry and other physical sciences—
mechanics in this case—was not one of essence, but rather one of historical stage of
development. He saw no reason of principle why an axiomatic analysis of the kind he
was then developing for geometry could not eventually be applied to mechanics with
similar, useful consequences. Eventually, that is to say, when mechanics would attain a
degree of development similar to the present one of geometry, in terms of the quantity
and certainty of known results, and in terms of an appreciation of what are really the
“basic facts” on which the theory is based.
Hilbert’s first course on mechanics was an elementary one, based on standard
presentations, in which he attempted to give a basic overview of the discipline. The
long bibliographical list that Hilbert recommended to his students for further reading
(Hilbert 1898-9, 4-5) may be helpful for reconstructing and assessing the degree of his
acquaintance with current knowledge on the field. However, one must exercise some
care before relying too heavily on these lists as a faithful indicator, given Hilbert’s
general tendency not to study thoroughly and comprehensively all the existing
literature on a topic he was pursuing. On the contrary, Hilbert—as David Rowe has
argued— “measured the quality of a mathematician’s work by the number of earlier
21. Hilbert 1898/9, 1-3 (Emphasis in the original): “Auch die Geometrie ist aus der
Betrachtung der Natur, aus der Erfahrung hervorgegangen und insofern eine
Experimentalwissenschaft. ... Aber diese experimentellen Grundlagen sind so unumstösslich
und so allgemein anerkannt, haben sich so überall bewährt, dass es einer weitere
experimentellen Prüfung nicht mehr bedarf und vielmehr alles darauf ankommt diese
Grundlagen auf ein geringstes Mass unabhängiger Axiome zurückzuführen und hierauf rein
logisch den ganzen Bau der Geometrie aufzuführen. Also Geometrie ist dadurch eine rein
mathematische Wiss. geworden. Auch in der Mechanik werden die Grundthatsachen von allen
Physikern zwar anerkannt. Aber die Anordnung der Grundbegriffe ist dennoch dem Wechsel
der Auffassungen unterworfen ... so dass die Mechanik auch heute noch nicht, jedenfalls nicht
in dem Maasse wie die Geometrie als eine rein mathematische Disciplin zu bezeichnen ist. Wir
müssen streben, dass sie es wird. Wir müssen die Grenzen echter Math. immer weiter ziehen
nicht nur in unserem math. Interesse sondern im Interesse der Wissenschaft überhaupt.”
9
Hilbert and Physics:
investigations it rendered obsolete”.22 Still, given the fact that Hilbert had no made no
specific contributions of his own to mechanics, a detailed account of this
bibliographical list, which include several rather uncommon titles, seems to be
instructive about his views and his knowledge.
First, Hilbert included in the bibliography four “classical works:” Lagrange’s
Méchanique analytique (1788); Jacobi’s, Dynamik (1843); Kirchhoff’s, Mechanik
(1877) and Thomson and Tait’s, Theoretische Physik (in German translations of 1871
and 1886). The textbooks he recommended were the following: Mechanik (1880), by
someone called Schell,23 a book he described as “somewhat antiquated but rich in
contents”; Kinematik, Statik, Dynamik (1884), by some Petersen, “short and easily
comprehensible”; Cours de Mechanique 2 Vols. (1884-86) by Despeyrous and
Darboux, “like Schell’s”; and Analytische Mechanik (1888), by Otto Rausenberg.24 In
a different section of the bibliographical list, Hilbert mentioned various “courses”:
Elementare Mechanik (1889), by the Göttingen physicist Woldemar Voigt,
“illuminating from the physical point of view”; Mach’s Principien der Mechanik
(1889); Mechanik (1890), by some Büdele, “like Schell and Desperyous-Darboux”;
Hertz’s Prinzipien der Mechanik (1894); “a memorial (Denkmal), in which this young
and brilliant physicists presented classical mechanics with Euclidean rigor”;
Helmtoltz’s Dynamik diskreter Massenpunkte (1894), reportedly a manuscript of the
latter’s university lectures; Boltzmann’s Prinzipien der Mechanik (1897), “develops
the atomistic point of view; opposed to Hertz”; Dynamik der Systeme starrer Körper
(1897-98), by Routh; and Traité de méchanique rationelle 3 Vols. (1893-98), by
Appell, “a comprehensive and systematic handbook”. Hilbert also included a section
with “historical” texts: Düring’s Principien der Mechanik (1873), and again Mach’s
book. Finally, his list included three collections of exercises: Aufgaben aus der
analytischen Mechanik (1879), by Fuhrmann; Aufgabe aus der theor. Mechanik
(1891), by Zieh; and Problèmes de méchanique (1867), by Tullien.25
The list is certainly impressive and interesting in itself but, as already said, it is
very difficult to know to what extent Hilbert was actually familiar with all the texts and
relied on them for his course. In any case, following his lectures of 1899 on the
foundations of geometry, Hilbert was to concentrate over the following two years
exclusively in this latter topic. Only in the winter semester of 1901-02, he taught again
a course on potential theory. But before retaking that thread, it is relevant to discuss
very briefly the contents of the Grundlagen and the views put forward in it.
22. Rowe 1994, 192.
23. In his well-informed history of the teaching of mathematics in the nineteenth century,
Wilhelm Lorey (1916, p. 135), mentions the name of Wilhelm Schell (1826-1904), who taught
in Marburg and Karlsruhe, and published a book on the theory of curves. Lorey doesn’t
mention any textbook on mechanics published by him.
24. Lorey 1916, 179, asserts that Rausenberg was born in 1852.
25. Consulting several books on German mathematics an physics in the nineteenth century
(Jungnickel & McCormmach 1986, Lorey 1916, Olesko 1991), I haven’t been able to gather
any infromation about the other authors mentioned by Hilbert: Petersen, Büdele, Routh,
Appell, Fuhrmann, Zieh or Tullien,
10
Hilbert and Physics:
3. Die Grundlagen der Geometrie
As already said, Hilbert had been involved since 1891 with questions pertaining the
foundations of geometry. In 1898 Friedrich Schur (1856-1932) proved the so-called
theorem of Pappus without using any continuity assumptions. In doing this he
reinforced a line of attack initiated in 1891 by Hermann Ludwig Wiener (1857-1939)
in this context, which Hilbert had been closely attentive to. Wiener had proved in 1891
that this theorem, together with the so-called theorem of Desargues, suffice to prove
the fundamental theorem of projective geometry. The role of continuity assumption in
his proofs, however, remained to be fully elucidated. Wiener’s ideas had been a main
factor in attracting Hilbert’s attention to the study of the foundations of geometry in the
first place. Schur’s proof, on the other hand, seems to have provided the definitive
trigger that made him redirect all his efforts to this field of research.26 Thereafter
Hilbert undertook a detailed elucidation of the logical interdependence of the various
fundamental theorems of projective and Euclidean geometry and, more generally, of
the structure of the various kinds of geometries that can be produced under various sets
of assumptions. A main question that stood at the center of Hilbert’s investigation was
one that had arisen from the attempts of Felix Klein (1849-1925), beginning in 1871,
to coordinatize projective geometry using ideas originally formulated by Arthur
Cayley (1821-1895).27 This coordinatization was essential for providing a connecting
link between synthetic and analytic geometry, and its realization depended on the same
assumptions necessary for proving the already mentioned fundamental theorems.
Following the work of Wiener and Schur, Hilbert focused on understanding the
specific role of continuity in those proofs. These kinds of questions provided the main
motivations behind the investigation undertook by Hilbert in the Grundlagen. The
experience gained while thinking on these issues when preparing his lectures on
geometry, on the one hand, and his acquaintance with the ideas put forward in Hertz’s
treatment of physics,28 on the other hand, indicated to Hilbert that the axiomatic
method would provide a powerful and effective tool with which to address these
crucial issues in the most effective way.
It would be well beyond the scope of the present article to discuss all the details
of the Grundlagen, and how the main foundational questions of geometry were
addressed therein.29 Neither can we discuss here the criticisms aroused by its
publication and the successive versions of the book, beginning in 1902, which
corrected certain gaps of the original edition and added new results. However, in order
to understand the basic ideas behind Hilbert’s axiomatic approach and its bearing on
physical theories, we must nevertheless discuss some of the main features of the
analysis put forward in the Grundlagen. Of particular interest are the kinds of
questions that Hilbert systematically pursued here for the first time, thus establishing a
standard for his future work.
26. For Hilbert’s acquaintance with the works of Wiener and Schur, and the events
surrounding the publication of Schur’s proof and its effect on Hilbert, see Toepell 1986, 114122.
27. For an account of Cayley’s contributions see Klein 1926-7 Vol. 1, 147-151.
28. In fact, Hertz’s was perhaps only one among several sources from within the physical
literature that influenced Hilbert’s initial inclination to, and then his fully adoption of, the
axiomatic method. See Corry 1997, 92-103.
29. Such a discussion can be found in Toepell 1986. See especially, pp. 143-236.
11
Hilbert and Physics:
The declared aim of the Grundlagen was to present a “simple” and “complete”
system of “mutually independent” axioms,30 from which all known theorems of
geometry might be deduced. The axioms were defined for three systems of undefined
objects (“points”, “lines” and “planes”), and they postulated relations that should be
satisfied by these objects. The axioms are divided into five groups (axioms of
incidence, of order, of congruence, of parallels and of continuity), but the groups have
no pure logical significance in themselves. Rather they reflect Hilbert’s actual
conception of the axioms as an expression of our spatial intuition: each of the groups
expresses a particular way in which these intuitions manifest themselves.
The requirements imposed by Hilbert on his system of axioms are remarkably
similar to the criteria put forward by Hertz as a basis for constructing and assessing
physical theories: permissibility, correctness, and appropriateness.31 Consider, in the
first place, Hilbert’s requirement for independence of the axioms. This requirement is
the most direct manifestation of the kind of foundational concerns that motivated
Hilbert’s research. When analyzing independence, Hilbert’s interest focused mainly on
the axioms of congruence, of continuity and of parallels, since this independence
would specifically explain how the various basic theorems of projective and of
Euclidean geometry are logically interrelated. In Hilbert’s early lectures on geometry,
this requirement had already appeared, albeit more vaguely formulated, as a direct
echo of Hertz’s demand for appropriateness. Now, in the Grundlagen, independence of
axioms not only appeared as a more clearly formulated requirement, but Hilbert also
provided the tools to prove systematically the mutual independence among the
individual axioms within the groups and among the various groups of axioms in the
system. He did so by introducing the method that has since become standard: he
constructed models of geometries which fail to satisfy a given axiom of the system but
satisfy all the others.
Also the requirement of simplicity had been explicitly put forward by Hertz
and it complements that of independence. It means, roughly, that an axiom should
contain ‘no more than a single idea.’ Beyond mentioning it in the introduction, this
requirement was neither explicitly formulated nor otherwise realized in any clearly
identifiable way in the Grundlagen. It appeared, however, in an implicit way and
remained here—as well as in other, later works—an aesthetic desiderata for axiomatic
systems, which was not transformed into a mathematically controllable feature.32
30. See Hilbert 1899, 1 (Italics in the original): “... ein einfaches und vollständiges System von
einander unabhängiger Axiome aufzustellen ...”
31. See Jesper Lützen’s contribution to this volume.
32. In his 1905 lectures on the axiomatization of physics, Hilbert explicitly demanded the
simplicity of the axioms for physical theories. It should also be remarked that in a series of
investigations conducted in the USA in the first decade of the present century under the
influence of the Grundlagen, a workable criterion for simplicity of axioms was still sought
after. For instance, Edward Huntington (1904, p. 290) included simplicity among his
requirements for axiomatic systems, yet he warned that “the idea of a simple statement is a very
elusive one which has not been satisfactorily defined, much less attained.”
12
Hilbert and Physics:
The “completeness” that Hilbert demanded for his system of axioms runs
parallel to Hertz’s demand for correctness.33 Very much like Hertz’s stipulation for
correct images, Hilbert required from any adequate axiomatization to allow for a
derivation of all the known theorems of the discipline in question. The axioms
formulated in the Grundlagen purportedly allowed to show how all the known results
of Euclidean, as well as of certain non-Euclidean, geometries could be elaborated from
scratch, depending on which groups of axioms were admitted.34 Thus, Hilbert
discussed in great detail the role of each of the groups of axioms in the proofs of two
crucial results: the theorems of Desargues and the theorem of Pappus (also called by
Hilbert the theorem of Pascal). Hilbert’s analysis allowed a clear understanding of the
actual premises necessary for coordinatizing projective geometry, which, as already
stressed, was a key step in building the bridge between the latter and other kinds of
geometry and a main concern of Hilbert.35
Although not explicitly mentioned in the introduction to the Grundlagen, the
question of the consistency of the various kinds of geometries was an additional
concern of Hilbert’s analysis, which he addressed in the Festschrift right after having
introduced all the groups of axioms and discussed their immediate consequences.
Although seen from the point of view of Hilbert’s later metamathematical research and
the developments that followed it the question of consistency appears as the most
important one undertaken in the Grundlagen, in the historical context of the evolution
of his ideas it certainly was not. In fact, the consistency of the axioms is discussed in
barely two pages, and it is not immediately obvious why Hilbert addressed this
question at all. It doesn’t seem likely that in 1899 Hilbert envisaged the possibility that
the body of theorems traditionally associated with Euclidean geometry might contain
contradictions. Euclidean geometry, after all, was for Hilbert a natural science whose
subject matter is the properties of physical space. Hilbert seems rather to have been
echoing here Hertz’s demands for scientific theories, in particular his demand for the
permissibility of images. In fact, a main point that Hilbert will stress in future
opportunities, following Hertz, is that the axiomatic analysis of physical theories was
meant to clear away any possible contradictions brought about over time by the
gradual addition of new hypotheses into a specific theory.36 Although this was not
likely to be the case for the well-established discipline of geometry, it might still
happen that the particular way in which the axioms had been formulated in order to
account for the theorems of this science led to statements that contradict each other.
33. This completeness should not be confused with the later, model-theoretical notion of
completeness, which is totally foreign to Hilbert’s early axiomatic approach.
34. Several important changes concerning the derivability of certain theorems appeared in the
successive editions of the Grundlagen. These are, however, not directly relevant to the main
concerns of this article.
35. However, there were many subsequent corrections and additions, by Hilbert as well as by
others, that sharpened still further the picture put forward by Hilbert in the first edition of the
Grundlagen. Toepell 1986, 252, presents a table summarizing the interconnections between
theorems and groups of axioms as known by 1907. See also Freudenthal 1957 for later
developments.
36. See for instance Hertz 1956, 7-8: “It is not by finding out more and fresh relations and
connections that [the question of the nature of force] can be answered; but by removing the
contradictions existing between those already known, and thus perhaps by reducing their
number.”
13
Hilbert and Physics:
The recent development of non-Euclidean geometries made this possibility only more
patent. Thus, Hilbert believed that in the framework of his system of axioms for
geometry he could also easily show that no such contradictory statements would
appear.
The publication of the Grundlagen was followed by many further
investigations into Hilbert’s technical arguments, as well as by more general,
methodological and philosophical discussions. One important such discussion
appeared in the oft-cited correspondence between Hilbert and Gottlob Frege (18461925).37 This interchange has drawn considerable attention of historians and
philosophers, especially for the debate it contains between Hilbert and Frege
concerning the nature of mathematical truth. But this frequently-emphasized issue is
only one side of a more complex picture advanced by Hilbert in his letters. In
particular, it is interesting to notice Hilbert’s explanation to Frege, concerning the main
motivations for undertaking his axiomatic analysis: the latter had arisen, in the first
place, from difficulties Hilbert had encountered when dealing with physical, rather
than mathematical theories. Echoing once more ideas found in the introduction to
Hertz’s textbook, Hilbert stressed the need to analyze carefully the process whereby
physicists continually add new assumptions to existing physical theories, without
properly checking whether or not the former contradict the latter, or consequences of
the latter. In a letter written on December 29, 1899, Hilbert wrote to Frege:
After a concept has been fixed completely and unequivocally, it is on my view completely
illicit and illogical to add an axiom—a mistake made very frequently, especially by physicists.
By setting up one new axiom after another in the course of their investigations, without
confronting them with the assumptions they made earlier, and without showing that they do
not contradict a fact that follows from the axioms they set up earlier, physicists often allow
sheer nonsense to appear in their investigations. One of the main sources of mistakes and
misunderstandings in modern physical investigations is precisely the procedure of setting up
an axiom, appealing to its truth (?), and inferring from this that it is compatible with the
defined concepts. One of the main purposes of my Festschrift was to avoid this mistake.38
In a different passage of the same letter, Hilbert commented on the possibility of
substituting the basic objects of an axiomatically formulated theory by a different
system of objects, provided the latter can be put in a one-to-one, invertible relation
with the former. In this case, the known theorems of the theory are equally valid for the
second system of objects. Concerning physical theories, Hilbert wrote:
All the statements of the theory of electricity are of course valid for any other system of things
which is substituted for the concepts magnetism, electricity, etc., provided only that the
requisite axioms are satisfied. But the circumstance I mentioned can never be a defect in a
theory [footnote: it is rather a tremendous advantage], and it is in any case unavoidable.
However, to my mind, the application of a theory to the world of appearances always requires
a certain measure of good will and tactfulness: e.g., that we substitute the smallest possible
bodies for points and the longest possible ones, e.g., light-rays, for lines. At the same time, the
further a theory has been developed and the more finely articulated its structure, the more
obvious the kind of application it has to the world of appearances, and it takes a very large
amount of ill will to want to apply the more subtle propositions of [the theory of surfaces] or
of Maxwell’s theory of electricity to other appearances than the ones for which they were
meant ...39
37. The relevant letters between Hilbert and Frege appear in Gabriel et al. (eds.) 1980, esp. pp.
34-51. For comments on this interchange see Boos 1985; Peckhaus 1990, 40-46; Resnik 1974.
38. Quoted in Gabriel et al. (eds.) 1980, 40. The question mark “(?)” appears in the German
original (after the word “Wahrheit”).
14
Hilbert and Physics:
Hilbert’s letters to Frege help understanding the important role played in the
development of his axiomatic point of view by his conception of the relation between
physical and mathematical theories. Hilbert’s axiomatic approach clearly did not
involve either an empty game with arbitrary systems of postulates nor a conceptual
break with the classical entities and problems of mathematics and empirical science.
Rather it sought after an improvement in the mathematician’s understanding of the
latter. This motto was to guide much of Hilbert’s incursions into several domains of
physics over the years to come.
4. Physics in Hilbert’s 1900 List of Problems
In 1900, speaking before the Second International Congress of Mathematicians in
Paris, Hilbert presented his famous list of twenty three problems. This list implied an
overarching research program that Hilbert was suggesting for the entire mathematical
community for years to come. Hilbert declared that a wealth of significant open
problems is a necessary condition for the healthy development of any mathematical
branch and, more generally, of the living organism that mathematics constitute.40
Empirical motivations appear in his conception of mathematics as a main source of life
for that organism. Stressing once more at this opportunity the close interrelation
between mathematics and the physical sciences, Hilbert stated that the quest for rigor
in analysis and arithmetic should be extended to cover geometry and the physics, not
only because it would perfect our understanding of the latter, but also because it would
eventually provide mathematics with ever new and fruitful ideas. Commenting on the
opinion that geometry, mechanics and other physical sciences are beyond the
possibility of a rigorous treatment, he wrote:
But what an important nerve, vital to mathematical science, would be cut by the extirpation of
geometry and mathematical physics! On the contrary I think that whenever from the side of
the theory of knowledge or in geometry, or from the theories of natural or physical science,
mathematical ideas come up, the problem arises for mathematical science to investigate the
principles underlying these ideas and so to establish them upon a simple and complete system
of axioms, that the exactness of the new ideas and their applicability to deduction shall be in
no respect inferior to those of the old arithmetic concepts. (Hilbert 1902, 442)
Hilbert described the development of mathematical ideas as an ongoing dialectical
interplay between the two poles of thought and experience; an interplay that brings to
light a “pre-established harmony” between nature and mathematics.41 Hilbert also
expressed here his celebrated opinion that every mathematical problem can indeed be
solved: “In mathematics there is no ignorabimus”. (p. 445)
The sixth problem of the list was a calling to undertake the axiomatization of
physical science. Hilbert formulated it as follows:
39. Quoted in Gabriel et al. (eds.) 1980, 41. I have substituted here “theory of surfaces” for
“Plane geometry”, which was the English translator’s original choice, since in the German
original the term used is “Flächentheorie.”
40. See especially the opening remarks in Hilbert 1902, 438. See also his remarks on p. 480.
41. The issue of the “pre-established harmony” between mathematics and nature was a very
central one among Göttingen scientists. This point has been discussed in Pyenson 1975.
15
Hilbert and Physics:
The investigations on the foundations of geometry suggest the problem: To treat in the same
manner, by means of axioms, those physical sciences in which mathematics plays an
important part; in the first rank are the theory of probabilities and mechanics. (Hilbert 1902,
454)
Hilbert mentioned several existing works as examples of what he had in mind here: the
fourth edition of Mach’s Die Mechanik in ihrer Entwicklung, Hertz’s Principles,
Boltzmann’s 1897 Vorlesungen über die Principe der Mechanik, and also Einführung
in das Studium der theoretischen Physik (1900) by the Königsberg physicist Paul
Volkmann (1856-1938), with whom Hilbert, while still at his native city, may have had
the opportunity to discuss the question of the role of axioms in physics.42 Together
with these well-known works on mechanics, Hilbert also mentioned a recent work by
the Göttingen actuarial mathematician Georg Bohlmann (1869-1928) on the
foundations of the calculus of probabilities.43 The latter was important for physics, he
said, for its application to the method of mean values and to the kinetic theory of gases.
Modeling this research on what had already been done for geometry meant that
not only such theories which are considered as closer to “describing reality” should be
investigated, but also other, logically possible ones. The mathematician undertaking
the axiomatization of physical theories should obtain a complete survey of all the
results derivable from the accepted premises. Moreover, echoing the concern already
found in Hertz and in Hilbert’s letters to Frege, a main task of the axiomatization
would be to avoid that recurrent situation in physical research, in which new axioms
are added to existing theories without properly checking to what extent the former are
compatible with the latter. This proof of compatibility, concluded Hilbert, is important
not only in itself, but also because it compels us to search for ever more precise
formulations of the axioms (p. 445).
Although the sixth problem is the only one in the list to refer directly to physics
as such, three additional ones concern mathematical issues intimately connected to the
classical problems of mathematical physics. The nineteenth problem concerns the
question whether all the solutions of the Lagrangian equations that arise in the context
of certain typical variational problems are necessarily analytic. The twentieth, closely
related to the former and at the same time to Hilbert’s long-standing interest in the
domain of validity of the Dirichlet principle, deals with the existence of solutions to
partial differential equations with given boundary conditions. Finally, the twenty-third
problem of the list is an appeal to extend and refine the existing methods of variational
calculus. All these three problems are also strongly connected to physics, though at
variance with the sixth, they are part of mainstream, traditional research concerns.44
The role of variational principles in Hilbert’s program for axiomatizing physics will be
further discussed below.
42. See Corry 1997, 101-103.
43. Bohlmann 1900. This article reproduced a series of lectures delivered by Bohlmann in a
Ferienkurs in Göttingen. In his article Bohlmann referred the readers, for more details, to the
chapter he had written for the Encyclopädie der mathematischen Wissenschaften on insurance
mathematics.
44. A detailed account of the place of variational principles in Hilbert’s work, see Blum 1994
(unpublished).
16
Hilbert and Physics:
But what has been said above suffices to recognize the natural place of the sixth
problem of Hilbert’s 1900 list in his overall conception of science. The task of
axiomatizing physical theories had arisen organically in conjunction with the very
consolidation of Hilbert’s view of the centrality of the axiomatic method for studying
the foundations of geometry. By 1900 his interest in physical theories had found a
natural place among his overall views of mathematics, its main methods and its
problems.
5. Hilbert, Minkowski, and Physics in Göttingen: 1900-1909
Following the publication of the Grundlagen, Hilbert’s main focus of attention
remained in the study of the foundations of geometry, until 1903. That year Bertrand
Russell published his discovery of a paradox arising in Frege’s logical system.
Although contradictory arguments of the kind discovered by Russell had been made
known in Göttingen a couple of years earlier by Ernst Zermelo (1871-1953),45 it seems
that Russell’s publication led Hilbert to attribute to the axiomatic analysis of logic and
of the foundations of set theory a much central role in establishing the consistency of
arithmetic than it had been the case until then. Beginning in 1903, an intense activity
was developed in Göttingen in this direction,46 whereby the systematic study of logic
and set theory as a central issue in the foundations of mathematics was initiated in
Hilbert’s mathematical circle. Zermelo was then working on the proof of the
consistency of arithmetic and on the axiomatization of set theory. After 1905 Hilbert
dedicated no further efforts to such foundational studies, and Zermelo was left alone at
this. In 1908 Zermelo published his well-known paper on the foundations of settheory.47 But as early as 1902 Hilbert had begun publishing in the new domain of
research that would concentrate his best efforts until 1912: the theory of linear integral
equations.
Still, during all these years, Hilbert interest in physical issues became only
more sustained. In 1901 and 1902 he lectured on potential theory and in 1902 and 1903
on continuum mechanics. That Hilbert considered these courses to have some original
and interesting content, rather than being a simple repetition of existing presentations,
is evident from the fact that the only two talks he gave in 1903 at the meetings of the
GMG were dedicated to report on what he did in those courses.48 In the winter
semester of 1904-05 he taught an exercise course on mechanics and later gave a
seminar on mechanics. Then, in the summer semester of 1905, in the framework of a
45. Peckhaus 1990, 48-49.
46. Peckhaus 1990, 56-57.
47. Zermelo 1908. A comprehensive account of the background, development and influence of
Zermelo’s axioms see Moore 1982. For an account of the years preceding the publication, see
esp. pp. 155 ff.
48. See the announcements in the Jahresbericht der Deutschen Mathematiker-Vereinigung
(JDMV). Vol. 12 (1993), 226 & 445. Earlier volumes of the JDMV do not contain
announcements of the activities of the GMG, and therefore it is not known whether he also
discussed his earlier courses there.
17
Hilbert and Physics:
course on the “Logical Principles of Mathematical Thinking”, he gave a long and
detailed overview of how the axiomatic approach should be applied in various
individual physical disciplines. In the next winter semester (1905-06) he lectured again
on mechanics, and then two more semesters on continuum mechanics.
In 1902 Minkowski arrived in Göttingen, following the creation of a third chair
of mathematics in that university, under Hilbert’s pressure on Klein to convince the
Prussian ministry. The renewed encounter between the two old friends was an
enormous source of intellectual stimulation for both. As usual, their mathematical
walks were an opportunity to discuss a wide variety of mathematical topics. This time,
however, physics became a more prominent common interest that it had been in the
past. Teaching in Zürich since 1894, Minkowski had kept alive his interest in
mathematical physics, and in particular in thermodynamics.49 While at Göttingen, he
further developed this interest. In 1906 Minkowski published an article on capillarity,
commissioned for the physics volume of the Encyclopädie der mathematischen
Wissenschaften, edited by Arnold Sommerfeld (Minkowski 1906). At several meetings
of the GMG he lectured on this, as well as on other physical issues such as Euler’s
equations of hydrodynamics and Nernst’s work on thermodynamics.50 He also taught
advanced seminars on physical topics and more basic courses on continuum
mechanics, exercises on mechanics and heat radiation.51 In 1905 Hilbert and
Minkowski organized, together with other Göttingen professors, an advanced seminar
that studied recent progress in the theories of the electron. This seminar—whose
details Lewis Pyenson has helped reconstructing52—exemplifies the vitality of
physical research at that university, and the role Hilbert and Minkowski played in
fostering it. Again in 1907, the two conducted a joint seminar on the equations of
electrodynamics.53 Finally, as it is well known, during the last years of his life,
Minkowski’s efforts were intensively dedicated to electrodynamics and the principle of
relativity. In order to gain a deeper understanding of Hilbert’s views on physical issues
during these years it is useful to discuss his 1905 course on the axiomatic method and
the work of Minkowski in electrodynamics.
In order to understand correctly the context of Hilbert’s course of 1905 it is
important to stress, that it started with a detailed treatment of the axioms of arithmetic
and of geometry, and that in its last section it attempted to develop a formalized
calculus for prepositional logic.54 At this time the reconstruction of the logical
foundations of mathematics was increasingly drawing Hilbert’s attention, and the
overall aim of the course was to discuss issues related to it. In exactly what fashion
49. See Rüdenberg and Zassenhaus (eds.) 1973, 110-114.
50. As registered in the JDMV, Vol. 12 (1903), 445 & 447; Vol. 15 (1906), 407.
51. See the announcement of his courses in JDMV Vol. 13 (1904), 492; Vol. 16 (1907), 171;
Vol. 17 (1908), 116.
52. Pyenson 1979.
53. Notes of this seminar were taken by Hermann Mierendorff, and they are kept in Hilbert’s
Nachlass (Cod Ms 570/5).
54. For a discussion of this part of the course see Peckhaus 1990, 61-75.
18
Hilbert and Physics:
Hilbert conceived the possible role of the axiomatic approach as part of this
reconstruction and as part of his general views on physics and on mathematics at this
time is illuminatingly condensed in the following passage taken from one of the
lectures:
The building of science is not raised like a dwelling, in which the foundations are first firmly
laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to
secure as soon as possible comfortable spaces to wander around and only subsequently, when
signs appear here and there that the loose foundations are not able to sustain the expansion of
the rooms, it sets to support and fortify them. This is not a weakness, but rather the right and
healthy path of development.55
Hilbert’s discussion of the axioms of physical science covered a surprisingly
varied range of domains: mechanics, thermodynamics, probability calculus, kinetic
theory of gases, insurance mathematics, electrodynamics, psychophysics. Without
entering into a detailed account of what Hilbert did for each and every domain he
considered, and of the background of his treatment, it is nevertheless convenient to
describe the general aims he pursued in his presentation and what he did for certain
domains.56 The first domain of physics that Hilbert discussed was mechanics. He
started with the axioms defining the addition of vectors, the main ideas of which he
took from recent works of Gaston Darboux (1842-1917), of Georg Hamel (18771954), and of the Göttingen student Rudolf Schimmack (1881-1912).57 The first three
of the axioms adopted by Hilbert stipulate the existence of a sum of any two vectors,
its commutativity and its associativity. The fourth axiom connects the sum-vector with
the directions of the factors as follows:
4. Let aA denote the vector (aAx,aAy, aAz), having the same direction as A. Then every real
number a defines the sum:
A + aA = (1 + a)A.
i.e., the addition of two vectors having the same direction is defined as the algebraic addition
of the longitudes along the straight line on which both vectors lie.58
The fifth one establishes the interchangeability of addition and rotation of vectors. The
sixth axiom concerns continuity:
6. Addition is a continuous operation, i.e., given a sufficiently small domain G around the endpoint of A + B one can always find domains G1 and G2, around A and B respectively, such that
the end-point of the sum of any two vectors belonging to each of these domains will always
fall inside G. (p. 124)
55. Hilbert 1905, 102: “Das Gebäude der Wissenschaft wird nicht aufgerichtet wie ein
Wohnhaus, wo zuerst die Grundmauern fest fundiert werden und man dann erst zum Auf- und
Ausbau der Wohnräume schreitet; die Wissenschaft zieht es vor, sich möglichst schnell
wohnliche Räume zu verschaffen, in denen sie schalten kann, und erst nachträglich, wenn es
sich zeigt, dass hier und da die locker gefügten Fundamente den Ausbau der Wohnräume nicht
zu tragen vermögen, geht sie daran, dieselben zu stützen und zu befestigen. Das ist kein
Mangel, sondern die richtige und gesunde Entwicklung.”
Other places where Hilbert uses the “building metaphor” are Hilbert 1897, 67; Hilbert 1918, 148.
56. For a fully detailed account see Corry 1997. The following passages condensate the
arguments put forward in that article.
57. Darboux 1875, Hamel 1905, Schimmack 1903. An additional related work, also mentioned
by Hilbert in the manuscript, is Schur 1903.
58. I have transcribed the relevant, original passages of this manuscript in Corry 1997, 132.
19
Hilbert and Physics:
This last axiom of continuity plays a very central role in Hilbert’s overall
conception of the axiomatization of natural science—geometry, of course, included. It
is part of the essence of things—said Hilbert in his lecture—that the axiom of
continuity should appear in every geometrical or physical system. Therefore it can be
formulated not just with reference to a specific domain, as was the case here for vector
addition, but in a much more general way. A very similar opinion had been advanced
by Hertz, who in the introduction to his textbook described the continuity of nature as
“an experience of the most general kind” ... “an experience which has crystallized into
firm conviction in the old proposition—Natura non facit saltus” (Hertz 1956, 36-37).
Hilbert formulated a general principle of continuity in the following terms:
If a sufficiently small degree of accuracy is prescribed in advance as our condition for the
fulfillment of a certain statement, then an adequate domain may be determined, within which
one can freely choose the arguments [of the function defining the statement], without however
deviating from the statement, more than allowed by the prescribed degree. (p. 125)
Experiment—continued Hilbert—compels us to place this axiom on the top of every
natural science, since it allows to assert the validity of our assumptions and claims. In
every special case, this general axiom must be given the appropriate version, as was
done in an earlier part of the lectures for geometry and here for vector addition.
H
H ( v , H ) An interesting example of Hilbert’s detailed treatment of a physical
theory appears in the section on thermodynamics, a domain whose
central concern Hilbert described as the elucidation of the two main theorems of the
theory of heat. Hilbert declared that the system of axioms he was about to discuss
provided a new foundation of thermodynamics, which would resemble closely the kind
of axiomatic treatment used earlier in his discussion of mechanics. His stress on the
mathematical elegance of the presentation led to an unusual order in the introduction
of the concepts, in which the immediate physical motivations are not directly manifest.
For simplicity he considered only homogenous bodies (a gas, a metal), denoting by v
the reciprocal of the density. If H denotes the entropy of the body, then these two
magnitudes are meant to fully characterize the elastic and the thermodynamical state of
the body. Hilbert introduced the energy function , meant to describe the state of matter.
The various possible states of a certain amount of matter are represented by the
combinations of values of v and H, and they determine the corresponding values of the
function e. This function then allows to provide a foundation of thermodynamics by
means of five axioms, as follows:
I. Two states 1,2 of a certain amount of matter are in elastic equilibrium with one another if
i.e., when they have the same pressure. By pressure we understand here the negative partial
derivative of the energy with respect to v
.
II. Two states 1,2 of matter are in thermal equilibrium when
20
Hilbert and Physics:
ª wH ( v , H ) º
«¬ wH »¼ v v1
H H1
ª wH ( v , H ) º
«¬ wH »¼ v v2
H H2
.
i.e., when they have the same temperature q. By temperature we understand here the
derivative of the energy with respect to the entropy:
T
wH ( v , H )
wH
T (v , H )
The definitions of pressure and temperature are among the examples of
Hilbert’s subordinating physical meaning of concepts to considerations of
mathematical convenience. Assume now that v and H are functions of time t, and call
the set of points in the v,H plane between any two states a path. One introduces two
new functions of the parameter t: Q(t) (heat) and A(t) (work). Given two states and a
2
path between them, the total heat acquired between the two states is
³ dQ
1
2
³
1
dQ
dt
dt
,
and similarly for the work. Hilbert added the following axiom involving these functions:
III. The sum of acquired work and heat on a given path between 1 and 2 equals the difference
of the energy-functions at the endpoints:
t1
t1
³ dQ ³ dA >H @
2
1
t2
H ( v 2 , H 2 ) H ( v1 , H1 ).
t2
This the law of conservation of energy, or of the mechanical equivalent.
The remaining axioms are:
IV. On a path with H = const., the total heat acquired is zero. A path of this kind (parallel to the
v-axis is called adiabatic).
V. On a path with v=const. the total work introduced is zero.
To these five the Hilbert added—as he had done before for geometry, for vector
addition, and for mechanics—the continuity axiom. For thermodynamics this axiom is
formulated as follows:
VI. Given two paths connecting the points 1,2, the quantities of heat added when moving
along those two paths may be made to diverge from one another less than any arbitrarily given
quantity, if the two paths are sufficiently close to one another in a uniform way (i.e., the two
lie in a sufficiently narrow strip around the other).
As an important feature of this system of six axioms Hilbert stressed its
symmetric treatment of work and heat. This clearly appears as a very convenient
feature from the perspective of Hilbert’s mathematical account of the theory, but one
wonders how physicists may have reacted to it, being opposed as it to a basic
dichotomy of thermodynamics, namely, that between reversible and irreversible
processes. Hilbert also discussed briefly the logical interdependence of the axioms.
From axioms VI. and III., for instance, one can deduce a continuity condition similar
21
Hilbert and Physics:
to VI., but valid for work rather than for heat. Finally, Hilbert showed how some of the
basic results of thermodynamics, such as the entropy formula, can actually be derived
from this system. Consider the curves of constant temperature (isothermals)
Q(v,H) = const. In order to move along one of this curves from the point H = 0, to the
point H, one uses a certain amount of heat, which depends only on the temperature Q
and on H:
ªH
º
« ³ dQ »
¬ H 0 ¼T ( v ,H )
f (T , H ).
T
The quantity of heat invested in moving through an isothermal line is given by the
function f (T , H )
. But what is the exact form of this function? Its determination, Hilbert said in this
lecture, is typical of the axiomatic method. It is the same problem as, in the case of geometry, the
determination of the function that represents the straight line; or, in the addition of vectors, the proof that the
components of the vector that represents the addition equal the additions of the components of the factors. In
all these cases, the idea is to decompose the properties of a certain function into small, directly evident
axioms, and from them to obtain its precise, analytical representation. In this way—he concluded—we
59
obtain directly from the axioms, the basic laws of the discipline investigated.
At least one well-known, published work on the foundations of
thermodynamics was directly influenced by these lectures of Hilbert: an article of 1909
by Constantin Carathéodory (1873-1950). An examination of Carathéodory’s axioms
and of his more general remarks in the article indicate very clearly the influence of
Hilbert’s ideas. In 1925 Carathéodory presented a second axiomatic treatment of
thermodynamics in which this influence is even more visible. Elaborating on a
suggestion of Planck, Carathéodory discussed the place of irreversible processes in
thermodynamics. He referred here once again to this earlier paper and explained what
he had tried to do in it. He thus wrote:
If one believes that geometry should be seen as the first chapter of mathematical physics, it
seems judicious to treat other portions of this discipline in the same manner as geometry. In
order to do so, we are in possession since ancient times of a method that leaves nothing to
desire in terms of clarity, and that is so perfect that it has been impossible ever since to
improve essentially on it. Newton felt this already when trying to present also his mechanics
in an external appearance that would fit the classical model of geometry. It is quite remarkable
that with even less effort than in mechanics, the classical thermodynamics can be treated by
the same methods as geometry.
This method consists in the following:
1. Create thought experiments, as in the case of geometry, constructing figures or moving
around spaces figures already constructed.
2. Apply to these thought experiments the axioms that the objects considered are supposed in
general to satisfy.
3. Extract the logical conclusion that follow from the given premises. (Carathéodory 1925,
176-177)
59. Hilbert 1905, 163: “Allemal handelt es sich darum, die Eigenschaften einer gewissen
Funktion im kleine unmittelbarer evidente Axiome zu zerlegen, und aus ihnen dann die
anlitysch Darstellung der Funktion herzuleiten; diese läßt dann die wesentlichen Eigenschaften
Sätze der vorliegenden Disziplin unmitelbar zu erkennen.”
22
Hilbert and Physics:
Carathéodory explained that in his 1909 article he had proceeded exactly in this way,
but, in his opinion, only in this later paper of 1925 the parallel application of the
axiomatic method to thermodynamics and geometry was more clearly manifest.
Carathéodory work had itself little impact among contemporary physicists.
This becomes evident from a paper published in 1921 by Max Born in the
Physikalische Zeitschrift, aimed precisely at making Carathéodory’s point of view
more widely known than it was. Born’s article, in turn, interestingly makes manifest
the influence of Hilbert on his own conception of the link between physics and
mathematics. In its traditional presentation, Born said, thermodynamics had not
attained the logical separation—so desirable, and in fact necessary, in the eyes of this
disciple of the Göttingen school—between the physical content and the mathematical
representation of the theory. Born’s characterization of the litmus test for identifying
the point when this separation is achieved brings us back directly to Hilbert’s 1905
lecture: a clear specification of the way to determine the form the entropy function
(Born 1921, 218).
In his article, Born reelaborated Carathédory’s presentation of
thermodynamics, in a way he thought more amenable to appreciation by physicists.
His article seems to have had no more noticeable influence than the one that inspired it.
But for the purposes of the present account it is very helpful for understanding the way
Hilbert conceived of the role of axiomatization on physical theories: starting from the
basic facts of experience, to strive after the formulation of an elaborate mathematical
theory in which the physical theorems are derived from simple axioms. This theory
may itself be different from the classical, more physically intuitive one, but the
mathematical presentation helps providing a more unified view of physics as a whole.
Hilbert next discussed the calculus of probability, which, as we saw above, he
considered to be among the natural sciences that deserved an axiomatic treatment.
Hilbert presented a system of axioms that he said to have taken from an article on
insurance mathematics that Georg Bohlmann published in the Encyclopädie der
mathematischen Wissenschaften (1901). Like Bohlmann in his article, beyond stating
the axioms as such Hilbert did not go any further. He did not comment on the
independence, consistency or “completeness” of these axioms. This system was a
rather crude one from the point of view of Hilbert’s conception; more elaborate ones
were attempted after Bohlmann.60
Much more interesting than the calculus of probabilities as such, however,
were for Hilbert its applications, among which he mentioned three: the theory of
compensations of errors (Ausgleichungsrechnung), the kinetic theory of gases, and
insurance mathematics. Of special interest for the present account is Hilbert’s
discussion of the kinetic theory. This theory will occupy much of Hilbert’s efforts
during the years 1911 to 1913, but his lectures of 1905 is perhaps the first place where
we find a clear evidence to the deep attraction that the theory exerted on him. Hilbert
60. For a review of later attempts to axiomatize the calculus of probabilities until 1933, see
Schneider (ed.) 1988, 353-358. A more detailed account appears in Von Plato 1994; see
especially pp. 179-278, for the foundational works of von Mises, Kolmogorov, and De Finetti.
23
Hilbert and Physics:
admired the way this theory combined the postulation of far-reaching assumptions
concerning the structure of matter with the use of the theory of probabilities, thus
leading to new and interesting physical results. Moreover, the particular historical
development of this theory offered a remarkable example of the kind of problematic
situation that Hilbert’s axiomatic analysis was meant to help overcoming. In fact,
along the years additional assumptions had been gradually added to the existing body
of knowledge related to the theory without properly checking the possible logical
difficulties that would arise from this addition.61 Also the question of the role of
probability arguments in physics was not a settled one in this context. In Hilbert’s
view, the axiomatic treatment was the proper way to restore order to this whole system
of knowledge, so crucial to the contemporary conception of physical science.
Hilbert accepted without any further qualification the controversial atomistic
assumptions underlying the classical approach to this theory as developed by Ludwig
Boltzmann. What Hilbert saw as especially problematic was the status of the
probabilistic arguments in a physical theory. Even if we know the exact position and
velocities of the particles of a gas—Hilbert explained—it is impossible in practice to
integrate all the differential equations describing the motions of these particles and
their interactions. We know nothing of the motion of individual particles, but rather
consider only the average magnitudes that constitute the subject of the probabilistic,
kinetic theory of gases. In an oblique reference to the arguments that Boltzmann had
devised as replies to the objections raised against his theory, Hilbert stated that the
combined use of probabilities and infinitesimal calculus in this context is a very
original contribution of mathematics, which may lead to deep and interesting
consequences, but which at this stage has in no sense been fully justified. As an
example Hilbert mentioned the equations of the vis viva. In the probabilistic version of
the theory, Hilbert said, the solution of the corresponding differential equation is not
derived using the differential calculus alone and yet it is correctly determined. It could
well be the case, however, that the application of the probability calculus could have
led here to results that contradict well-known results of the theory. Such results would
clearly be considered false. Hilbert explained what he meant with this warning, by
showing how a fallacious probabilistic argument could lead to contradiction in the
theory of numbers.62
Following the detailed discussion of these issues, Hilbert formulated a very
interesting, and surprisingly pragmatist, opinion concerning the role of probabilistic
arguments in mathematical and in physical theories. According to this opinion the
calculus of probability is not an exact mathematical theory, but one that may
appropriately be used as a first approximation, provided we are dealing with
immediately perceivable mathematical facts. Otherwise it may lead to significant
contradictions. The use of the calculus of probabilities is justified—Hilbert
concluded—inasmuch as it leads to results that are correct and in accordance with the
facts of experience or with the accepted mathematical theories (p. 182).
61. Two classical, detailed accounts of the development of the kinetic theory of gases and the
conceptual problems involved in it (particularly during the late nineteenth century) can be
consulted: Brush 1976 and Klein 1970 (esp. 95-140).
62. See Hilbert 1905, 178-180. For more details on these arguments see Corry 1997,
167.
24
Hilbert and Physics:
An in important feature of Hilbert’s discussion of insurance mathematics is the
methodological comparison he drew between this domain and thermodynamics. In the
latter domain, the state of matter had been expressed in terms of a function H ( v , H ) .
A similar thing Hilbert intended to do here for describing the relevant state of a person:
an individual person is characterized, for the purposes of insurance, by means of a
function p(x,y), defined for y > x. This function expresses the probability that a person
of age x will reach age y, and it is required to satisfy the following axiom:
The probabilities p(x,y) p’(x',y') associated with two different individuals are independent for
all pairs x,y x',y' of positive numbers.
A collection of individuals, such for that any two of them p(x,y) = p’(x,y), is called an
equal-risk group. From the point of view of insurance, the individuals of any of these
collections are identical, since the function p wholly characterizes their relevant
behavior. Now, very much like in thermodynamics, where the main accomplishment
achieved in the lectures was the explicit derivation of the form of the function
f (T , H ) , using to this end only the particular axioms postulated, Hilbert declared aim
for this domain was the determination of a certain function of one variable starting
from the axioms. For every equal-risk group associated with a function of probability
p(x,y) Hilbert defined a “virtual mortality-order” (fingierte Absterbeordnung). In this
way, with every such group one can associate a function of the continuous variable x,
l(x), called the “number of living people of age x” or “life function”, satisfying some
simple properties. Without entering here into the details of the axioms postulated by
Hilbert for this function, I will only say that Hilbert did actually not prove any of the
results pertaining to this theory and to the functions p and l. He just stated that such
proofs would involve a kind of deductions similar to those used in the other domains.
He added, however, that also in these deductions, an axiom of continuity of the kind
assumed in the former domains—the particular version of which he would not
formulate explicitly in this case—plays a central role.
The last example I want to mention here concerns Hilbert’s treatment of
psychophysics. Hilbert referred in this section to a recent work on the theory of color
perception published by Egon Ritter von Oppolzer, a psychologist from Innsbruck
(Oppolzer 1902-3). Oppolzer’s article was a classical representative of the German
school of experimental psychology, going back to the work of Gustav Fechner (18011887).63 Its declared aim was to characterize the sensation of light in “total colorblind
systems” by means of a single, purely psychological parameter—the brightness
(Helligkeit)—as opposed to the physically characterizable concept of intensity
(Intensität). The problem addressed by Oppolzer, as Hilbert characterized it in his
lectures, was to express the magnitude of this parameter as a function of the intensity
and wave-length of light.64 The derivation of the analytic expression of this function,
starting from axioms fitted very well with the other domains considered earlier in the
lectures.
63. On Fechner’s contributions see Boring 1929, 265-287. More generally, on the German
school, see there, pp. 237-401. Oppolzer is mentioned neither in Boring’s classical account, nor
in other, standard similar works.
64. Hilbert 1905, 189: “Das Hauptproblem ist, diese Helligkeit x als Funktion der
Bestimmungstücke der das Licht physisch (sic) zusammensetzenden homogenen Lichter (d.i.
Intensität und Wellelänge eines jeder) darzustellen.”
25
Hilbert and Physics:
As in the case of Bohlmann’s work on probabilities, the axioms mentioned by
Hilbert for the case of psychophysics can be only retrospectively found in Oppolzer’s
own article. Oppolzer himself described his basic assumptions discursively, sometimes
loosely, and not only in the opening sections, but rather throughout the article.
Needless to say, he didn’t analyze the independence, consistency or any other property
of his “axioms”. Yet, precisely for the unsystematic way in which Oppolzer discussed
principles and ideas drawn from works as diverse as those of Goethe and the German
physiologists, Newton and Thomas Young, this work seems to have presented Hilbert
with a further, unexplored territory in which the axiomatic approach could usefully be
applied. In fact, Oppolzer’s article was in this sense symptomatic of a more
generalized situation in contemporary research in psychophysics,65 and was therefore
well-suited to exemplify Hilbert’s claims concerning the unmindful introduction of
new assumptions into existing physical theories.
The manuscript of the lectures makes no mention of the differences between
Hilbert’s formulation and Oppolzer’s own one. Hilbert simply put forward his axioms,
which are defined for a collection of “brightnesses” x1, x2, ..... The axioms postulate
the following properties that the brightnesses are demanded to satisfy:
1. To every pair of brightnesses x1, x2, a third one [x1,x2] can be associated, called “the
brightness of the mixed light of x1,x2.” Given a second pair of brightnesses x3,x4, such that
x1 = x3 and x2 = x4, then [x1,x2] = [x3,x4].
2. The “mixing” of various brightnesses is associative and commutative.
3. By mixing various homogeneous lights of equal wave-lengths, the brightness of the mixed
light has the same wave light, while the intensity of the mixed light is the sum of the
intensities.
Experience, said Hilbert, amply confirms these three axioms. The first one contains
what Hilbert called the law of Grassmann, namely, that intensities that are psychically
equal (but which may be physically different), after undergoing the operation of
physical mixing, remain equivalent at the psychical level.
If one calls the uniquely determined number [x1,x2], x12, one can write it as a
function of the two parameters
x(12 )
f ( x1 , x 2 )
.
From the second axiom, one can derive the functional equation:
f ( f ( x1 , x 2 ), f ( x3 , x 4 ))
f ( f ( x1 , x 3 ), f ( x 2 , x 4 ))
f ( f ( x1 , x 4 ), f ( x 2 , x3 )) .
One can then introduce a new function F that satisfies the following relation:
F ( x12 )
F ( f ( x1 , x 2 ))
F ( x1 ) F ( x 2 )
.
65. As Kremer 1993, 257, describes it: “For a variety of philosophical, institutional and
personal reasons, color researchers between 1860 and 1920 simply could not agree on which
color experiences are quintessential or on what criteria are appropriate to evaluate hypothetical
mechanisms for a psychoneurophysiological system of sensation.”
26
Hilbert and Physics:
From axiom 3, and assuming the by now well-known general postulate of continuity, it
follows that the function F , for homogeneous light, is proportional to the intensity.
This function is called the “stimulus value” (Reizwert), and once it is known, then the
whole theory becomes, so claimed Hilbert, well-established. One notices immediately,
Hilbert went on to say, the analogy with the previously studied domains, and especially
with the theorem of existence of a function l(x) in the life-insurance mathematics. This
very analogy could suffice to acknowledge, he concluded, that also in this latter
domain, so far removed from the earlier ones, the approach put forward in the whole
course would become fruitful.66
Hilbert’s treatment of psychophysics, at least as it appears in the manuscript,
was rather sketchy and its motivation was far from obvious, since he did not provide
any background to understand the actual research problems of this domain. Moreover,
as in the case of probabilities, Hilbert did not examine the logical interrelations among
the axioms, beyond the short remarks quoted in the preceding paragraphs. Yet, in the
context of his treatment of other physical domains and of the confused state of affairs
in contemporary psychological research, one can grasp the width of application that
Hilbert envisaged for the axiomatic method in science. Hilbert’s ideas don’t seem to
have influenced in any tangible way the current research of German psychologists, and
one wonders whether or not there was some personal contact between him an his
psychologist colleagues, at least in Göttingen.
There is no direct evidence to judge what was the reaction of the students who
attended these lectures of Hilbert, that were listed among the elementary courses
offered in Göttingen in that semester. Before those students stood the great Hilbert,
quickly overviewing many different physical theories, together with arithmetic,
geometry and even logic, all in the framework of a single course. Hilbert moved form
one theory to the other, and from one discipline to the next, without providing
motivations or explaining the historical background to the specific topics addressed,
without giving explicit references to the sources, without stopping to work out any
particular idea, without proving any assertion in detail, but claiming all the while to
have a unified view of all these matters. The impression must have been thrilling, but
one wonders to what extent his students could really appreciate the ideas presented to
them. In fact, it is hard to determine with exactitude to what extent Hilbert himself
dominated the physical subtleties of the issues he discussed, though there can be no
doubt that his unusual abilities helped him overcoming very easily what perhaps
presented great mathematical difficulties to others. Still, the picture of Hilbert’s
knowledge of physics that arises from these lectures is impressive both in its breadth
and its incisiveness.
In Hilbert’s treatment of physical theories of 1905 we come across diverse
kinds of axioms. In the first place, every theory is assumed to be governed by specific
axioms that characterize it. These axioms usually express mathematical properties
establishing relations among the basic magnitudes involved in the theory. Then, there
are certain general mathematical principles that Hilbert thought should be valid of all
66. Hilbert 1905, 190: “Das mag zur Kennzeichnung genügen, wie auch in diesem von den
früheren so ganz verschiedene Gebiete unsere Gedankengänge fruchtbar werden.”
27
Hilbert and Physics:
physical theories. In the lectures he stressed above all the “continuity axiom”,
providing both a general formulation and more specific ones for each theory. As an
additional general principle of this kind he suggested the assumption that all functions
appearing in the natural sciences should have at least one continuous derivative.
Furthermore, the universal validity of variational principles as the key to deriving the
main equations of physics, especially in mechanics and electrodynamics (which were
not described in detail above) was a central underlying assumption of all of Hilbert’s
work on physics, and they appear throughout these lectures as well. In each of the
theories he considered in his 1905 lectures, Hilbert attempted to show how the exact
analytic expression of a particular function that condenses the contents of the theory in
question could be effectively derived from the specific axioms of the theory, together
with more general principles. On some occasions he elaborated this more thoroughly,
while on others he simply declared that such a derivation should be possible.
There is yet a third type of axiom for physical theories, however, which Hilbert
avoided addressing in his 1905 lectures. They comprise claims about the ultimate
nature of physical phenomena, an issue which was particularly controversial during the
years preceding these lectures. Although Hilbert’s sympathy for the mechanistic
world-view is apparent throughout the manuscript of the lectures, his axiomatic
analysis of physical theories contain no direct reference to it. The logical structure of
the theories should be fully understood independently of any particular position one
would assume in this debate. As will be seen below, this position would change around
1913, when Hilbert wholeheartedly adopted the view that all physical phenomena
should be explained in terms of electrodynamic processes. I discuss this issue below.
After the summer semester of 1905, Hilbert lectured again on mechanics (WS
1905-06)67 and continuum mechanics (SS 1906 and WS 1906-07). Since then and until
1910, he taught no additional courses on physics. But at the same time, from 1907 and
until his death in 1909, Hilbert’s colleague, Minkowski, dedicated much efforts to the
study of electrodynamics and the principle of relativity. It would seem as if Hilbert had
left the stage open to Minkowski alone to work out his ideas on these issues. As a
matter of fact, however, Minkowski’s work can be seen to a large extent as a
realization of Hilbert’s program of axiomatization, in which the specific role of the
newly adopted principle of relativity in various physical theories was thoroughly
investigated for the first time. Moreover, Hilbert’s lectures on physical issues over the
years following Minkowski’s death indicate that the former fully adopted the point of
view, the notations, and the concepts introduced by the latter. Thus a description of
Minkowski’s work on electrodynamics may also serve as a parallel description of how
Hilbert’s conceptions applied to the new situation created in physics by the
introduction of the principle of relativity, and at the same time, as evidence of what
Hilbert’s own ideas on physical issues looked like over that period of time of which we
have no additional, direct evidence.
67. A deatiled analysis of this course, with especial emphasis on the role of variational
principles in Hilbert’s conception of physics appears in (the unpublished) Blum 1994.
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Hilbert and Physics:
Elsewhere I have presented a comprehensive picture of Minkowski’s work on
electrodynamics, in which his motivations and the details of his work are interpreted in
the lines suggested in the preceding paragraph.68 For the purposes of the present article
it may suffice to discuss briefly only one of Minkowski’s articles, dealing with “The
Basic Equations of Electromagnetic Processes in Moving Bodies” (Minkowski 1908).
This was Minkowski’s only publication on this issue that appeared in print during his
lifetime. This was also the article that Hilbert considered as his friend’s most
significant contribution to this field. In his obituary of Minkowski, Hilbert stressed the
importance and the innovative character of the axiomatic analysis presented in that
article, as well as of Minkowski’s derivation of the equations for moving matter
starting from the so-called “World-postulate”, and from three additional axioms. The
correct form of these equations had been theretofore an extremely controversial issue
among physicists, but this situation had totally changed—so Hilbert believed—thanks
to Minkowski’s work (Hilbert 1909, 93-94).
Minkowski’s article opened with an analysis of current developments in the
theories of the electron and of the role played by the principle of relativity in them.
Minkowski distinguished three possible different meanings of this principle. First, one
has the plain mathematical fact that the Maxwell equations, as formulated in Lorentz’s
theory of electrodynamics, are invariant under the Lorentz transformations.
Minkowski called this latter fact the “theorem of relativity.” The “postulate of
relativity” differs from the theorem in that it expresses more of a confidence
(Zuversicht) than an objective assessment concerning some actual state of affairs: a
confidence that the domain of validity of the theorem may be extended to cover all
laws governing ponderable bodies, including laws that are still unknown. He compared
this postulate to the postulation of the validity of the principle of conservation of
energy, which we assume even for forms of energy that are not yet known. Finally,
there is the “principle of relativity” which expresses the assertion that the expected
Lorentz covariance actually holds as a relation between purely observable magnitudes
relating to a moving body (Minkowski 1908, 353).
Minkowski’s declared aim was to deduce the exact expression of the equations
for moving bodies from the principle of relativity. He claimed that his formulation of
the principle had never before been articulated the way he did and that under earlier
formulations the equations were not truly invariant. Minkowski believed that his
axiomatic analysis of the principle of relativity and of the electrodynamics theories of
moving bodies was the best approach for unequivocally obtaining the correct
equations. Minkowski started by presenting the equations for pure ether. Then, in the
second part of his article, he discussed how the equations change when matter is added
to the ether. For the case of a body at rest in the ether, Minkowski relied on Lorentz’s
version of Maxwell’s equations, and analyzed the symmetry properties of the latter. He
formulated the equations as follows:
(I)
(II)
curlm div e
68. See Corry 1997a.
we
wt
U
s
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Hilbert and Physics:
(III)
(IV)
curlE wM
wt
div M
0
0
Here M and e are called the magnetic and electric intensities (Erregung) respectively,
E and m are called the electric and magnetic forces, U is the electric density, s is the
electric current vector (elektrischer Strom). The properties of matter, in the case of
isotropic bodies, are characterized by the following equations:
(V)
e
HE , M
Pm , s
VE
,
where H is the dielectric constant, P is the magnetic permeability, and V is the
conductivity of matter.
From the basic properties of the equations for bodies at rest, Minkowski
deduced the fundamental equations for the case of a body in motion. This deduction is
where the detailed axiomatic derivation is realized: Minkowski assumed the validity of
the previously discussed equations for matter at rest and to them he added three more
axioms. He then sought to derive the equations for matter in motion exclusively from
the axioms, together with the equations for rest. Minkowski’s axioms are:
1. Whenever the velocity v of a particle of matter equals 0 at x, y, z, it in some reference
system, then equations (I)-(V) also represent, in that system, the relations among all the
magnitudes: U, the vectors s, m, e, M, E, and their derivatives with respect to x, y, z, it.
2. Matter always moves with a velocity which is less than the velocity of light in empty space
(i.e., ~v~= v < 1).
3. If a Lorentz transformation acting on the variables x, y, z, it, transforms both m,-ie and M,iE as space-time vectors of type II, and s,iU as a space-time vector of type I, then it transforms
the original equations exactly into the same equations written for the transformed
magnitudes.69
This last axiom is what Minkowski called the principle of relativity.
Minkowski deduced the equations for moving bodies in a section were he
showed in detail how every step of the deduction is allowed by one of the axioms. On
first reading, his straightforward argument may seem somewhat out of place amidst all
the elaborate mathematical and physical arguments displayed throughout his long
article. But when seen in the light of the kind of axiomatic conceptual clarification
promoted by Hilbert in his lectures on physics, it becomes clear that Minkowski was
simply stressing this deduction as a central task of his exposition of the theory.
Moreover, Minkowski went on to check to what extent different existing versions of
the equations satisfy the principle as stated in his axioms. Minkowski’s implicit
assumption was that only such equations can be accepted as correct, which comply to
his own version of the principle. Thus Minkowski showed that the equations for
moving media formulated by Lorentz in his Encyclopädie article (Lorentz 1904) are in
certain cases incompatible with his principle. Minkowski also discussed the equations
69. See Minkowski 1908, 369. For the sake of simplicity, my formulation here is slightly
different yet essentially equivalent to the original one.
30
Hilbert and Physics:
formulated in 1902 by Emil Cohn, pointing out that they agree with his own ones up to
terms of first order in the velocity (Minkowski 1908, 372). After having formulated the
equations and discussed their invariance properties, Minkowski dealt in detail, in three
additional sections, with the properties of electromagnetic processes in the presence of
matter.
Minkowski’s article also contained an appendix discussing the relations
between mechanics and the postulate of relativity. In this appendix the similarity of
Minkowski’s and Hilbert’s treatment of physical theories is more clearly manifest: it
explores the consequences of adding the postulate of relativity to the existing building
of mechanics, and the compatibility of the postulate with the already established
principles of this discipline. The extent to which this addition can be successfully
realized provides, in Minkowski’s view, a standard to assess the status of Lorentz
covariance as a truly universal postulate of all physical science.
Using the four-vector formalism that he had introduced in the earlier sections
of his article, Minkowski showed that the equations of motion of classical mechanics
are invariant under the Lorentz group only under the assumption that c = f. It would
be embarrassing or perplexing (verwirrend), he said, that the laws of transformation of
the basic expression
x 2 y 2 z 2 c 2t 2
into itself would necessitate a certain finite value of c in a certain domain of physics
and a different, infinite one, in a second domain. In view of this situation, the postulate
of relativity (i.e., our confidence in the universal validity of the theorem) compels us to
see Newtonian mechanics only as a tentative approximation initially suggested by
experience, which however must be corrected to make it invariant for a finite value of
c. Minkowski thought that reformulating mechanics in this direction was possible, and
moreover— expressing himself in terms that could have been equally found in
Hilbert’s lecture notes— he asserted that such a reformulation would seem only to
perfect, to a considerable extent, the axiomatic structure of mechanics as currently
conceived.70
Clearly, the universal validity of the postulate of relativity could only be
asserted if one could show that it does not contradict the observable phenomena related
to gravitation. To this effect, in the last section of his article, Minkowski sketched a
proposal for a theory of gravitation that would also be Lorentz covariant. This sketch
was no more than a preliminary attempt in this direction, that neither Minkowski
himself, nor Hilbert went on to pursue.
A main motivation of Minkowski’s work in electrodynamics, then, is a
systematic investigation of the logical consequences of assuming the universal validity
of Lorentz covariance for all physical disciplines. This is exactly the formulation used
by Hilbert in his future lectures to describe the contents of the “new mechanics.” The
70. Minkowski 1908, 393 (Italics in the original.): “Ich möchte ausführen, daß durch eine
Reformierung der Mechanik, wobei an Stelle des Newtonschen Relativitätspostulates mit c = f
ein solches für ein endliches c tritt, sogar der axiomatische Aufbau der Mechanik erheblich an
Vollendung zu gewinnen scheint.”
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postulate of relativity had been strongly suggested by experimental results obtained
during the late nineteenth century, and its theoretical implications had been
investigated from different perspectives in recent works, noticeably those of Lorentz,
Poincaré and Einstein. Yet, in a spirit similar to that underlying Hilbert’s program,
Minkowski believed that the logical structure of the physical theories built on the
principle of relativity had not yet been satisfactorily elucidated. The postulate of
relativity should be taken as a further axiom appearing at the basis of each and every
physical theory, together with the particular axioms of that theory. In his work
Minkowski was able to prove for certain domains of physics that the ensuing theory
indeed produced a consistent logical building. Concerning gravitation, he was less
successful, but always declared his conviction to have indicated in principle how a
consistent, Lorentz covariant theory of gravitation could eventually be elaborated in
detail.
For Minkowski the postulate of relativity was not simply an additional axiom,
with perhaps a wider domain of validity in physics than others. It was an axiom of a
different nature: a principle that should be valid for every conceivable physical theory,
even those theories that were yet to be discovered or formulated. As I mentioned
above, also Hilbert laid a great stress on the importance of this kind of universal
physical principle, and focused especially on the “principle of continuity.”
Minkowski’s comparison in this context, of the status of the postulate of relativity with
that of the principle of conservation of energy, had been drawn earlier by Einstein.71
But Einstein’s and Minkowski’s comparisons were basically different. Einstein spoke
in his article of two “open” principles of physics, with a strong heuristic character.
Unlike Minkowski and Hilbert, Einstein did not see the principle of relativity and the
principle of energy conservation as parts of strictly deductive systems from which the
particular laws of a given domain could be derived.72 More generally, although
Einstein introduced the principle of relativity together with the constancy of light at the
beginning of his 1905 article as “postulates” of the theory (in some sense of the word),
it is necessary to draw a clear difference between what he did and what Minkowski and
Hilbert had in mind when speaking of an axiomatic analysis of the postulate of
relativity. As a matter of fact, one of the main explicit aims of Hilbert’s program was to
address situations similar to the one created here by Einstein, which had the potential
of turning out to be problematic, namely, that faced with conflict between an existing
theory and new empirical findings, it was common for physicists to add new
hypothesis that apparently settle down the disagreement in question but which perhaps
contradict some other consequences of the existing theory. Hilbert thought that an
adequate axiomatic analysis of the principles of a given theory would help clear away
possible contradictions and superfluous additional premises created by the gradual
introduction of new hypotheses into existing theories. This was also what Minkowski
was pursuing with his analysis: to verify that the introduction of the principle of
relativity will not create such a problematic situation.
71. Einstein 1907. Minkowski was very likely aware of this specific article of Einstein, if only
because it appeared in the Annalen der Physik as a reply to an earlier article of Paul Ehrenfest
who at that time was in Göttingen.
72. Cf. Einstein 1907, 411: “Es handelt sich hier also keineswegs um ein ‘System’, in welchem
implizite die einzelnen Gesetze enthalten wären, und nur durch Deduktion daraus gefunden
werden könnten, sondern nur um ein Prinzip, das (ähnlich wie der zweite Hauptsatz der
Wärmetheorie) gewisse Gesetze auf andere zurückzuführen gestattet.”
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Hilbert and Physics:
Minkowski’s analysis of the place of the postulate of relativity in physics
avoided any assumption about, and certainly any commitment to, any particular
conception of the ultimate nature of physical phenomena.73 Perhaps Minkowski had
some kind of clear position of his own on these issues, though we have no direct
evidence of it. Minkowski’s admiration for Hertz and the fact that in 1910 Hilbert
sided with the mechanistic world-view when lecturing on mechanics under the
declared influence of Minkowski’s ideas74 might seem to suggest that this was also the
latter’s view. Hilbert was initially very attracted to the mechanistic reductionism of
Hertz or Boltzmann and around 1913, as will be seen below, he changed his position
diametrically and adopted an electromagnetic, reductionistic world-view based on
Gustav Mie’s theory of matter. Yet, both Minkowski and Hilbert, at least by this time
considered that the much needed task of axiomatically analyzing the logical building
of physical theories should be carried out while leaving aside this kind of unsettled
physical issues.
6. Mechanics, Kinetic Theory, Radiation Theory: 1910-1914
Until 1912, Hilbert’s mathematical efforts concentrated on the theory of linear integral
equations, publishing the successive installments of what finally constituted his
classical treatise on this topic (Hilbert 1912). At the same time, however, after
Minkowski’s death, Hilbert returned to teach courses on physical issues. Hilbert now
moved into disciplines that he had never taught in the past. Thus, after teaching
mechanics and continuum mechanics in 1910 and 1911, Hilbert taught statistical
mechanics for the first time in the winter semester of 1910-11. This course marked the
starting point of Hilbert’s definitive involvement with a wider variety of physical
theories. In December of 1911 he presented to the GMG an overview of his recent
investigations on the kinetic theory of gases, that were soon to be published.75 The
kinetic theory was also the topic of his course during the winter semester of 1911-12,
and in the following semester he taught a course on radiation theory (Hilbert 1911-12).
In 1912 Hilbert enrolled an assistant for physics, who was commissioned with the task
of keeping him abreast of current developments in the various branches of physics.
Paul P. Ewald (1888-1985), who had recently finished his dissertation in Munich, was
the first to hold this position, which Hilbert maintained for many years to come. In
April 1912, Hilbert was among the lecturers that took part in a twelve-day seminar on
the axiomatic foundations of physics, opened for high-school teachers.76
73. This claim, which differs from most existing interpretations of Minkowski’s work, is
argued in detail in Corry 1997.
74. As manifest, i.e., in Hilbert’s lecture notes: see Hilbert 1910-11, 295.
75. See the announcement in the JDMV Vol. 21, p. 58.
76. See the announcement in the JDMV Vol. 21, p. 166. Unfortunately, no recorded evidence
of what Hilbert did in this Ferienkurs seems to have been kept.
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Hilbert’s increasingly deep involvement in physics led him to ponder again, now from
a wider perspective, some basic questions concerning the foundations of this
discipline. By 1910 Hilbert’s approach, as already said, was dominated by the view
that all physical phenomena could be reduced to mechanics. This view was clearly
manifest in the courses he taught. Between 1910 and 1913, however, although his
reductionistic inclinations did not change, he moved from the mechanistic to the
electromagnetic point of view. Electromagnetic reductionism became the basis of all of
Hilbert’s work in physics thereafter.
From the manuscripts of Hilbert’s courses between 1911 and 1913, as well as
from his publications, it becomes evident that Hilbert was very much impressed by
recent developments in quantum theory. The importance of these developments had
particularly been discussed and highlighted during the First Solvay Conference, held in
Brussels in October 1911,77 and whose echoes must have reached Hilbert through his
physicist colleagues. “Never has been a most proper and challenging time than now”—
said Hilbert in the opening lecture of a course taught in 1912—“to undertake the
research of the foundations of physics.” What seems to have impressed him more than
anything else were the recently discovered, deep interconnections, “of which formerly
no one could have even dreamed, namely, that optics is nothing but a chapter of the
theory of electricity, that electrodynamics and thermodynamics are one and the same,
that also energy possesses inertial properties, that physical methods have been
introduced into chemistry as well.”78 And above all, the “atomic theory”, the
“principle of discontinuity”, as Hilbert said, which today is not hypothesis anymore,
but rather, “like Copernicus’s theory, a fact confirmed by experiment.”79 Very much
like the unification of apparently distant mathematical domains, which played a
leading role throughout his career, the unity of physical laws exerted a strong attraction
on Hilbert.
The kind of interests manifest in Hilbert’s lectures of 1911 and 1912 were
directly reflected in his publications of 1912 as well. The latter include articles dealing
with these two issues: the kinetic theory and radiation theory.80 Hilbert’s first major
physical publications bring to the fore an interesting mixture of all the themes formerly
found in his lectures and published works in both mathematics and physics: an attempt
to connect apparently distant issues by uncovering their underlying, structural
similarities; an appeal to the axiomatic analysis of physical theories in order to
redesign their logical building and to clarify the specific roles of their basic principles;
a use of deeply sophisticated mathematical tools; an attempt to clarify the interrelation
between probabilistic and analytic reasoning in physics.
77. See Kormos Barkan 1993.
78. Hilbert 1912c, 2: “Nun kommen wir aber zu eigentlicher Physik, welche sich auf der
Standpunkt der Atomistik stellt und da kann man sagen, dass keine Zeit günstiger ist und keine
mehr dazu herausfordet, die Grundlagen dieser Disziplin zu untersuchen, wie die heutige.
Zunächst wegen der Zusammenhänge, die man heute in der Physik entdeckt hat, wovon man
sich früher nichts hätte träumen lassen, dass die Optik nur ein Kapitel der Elektrizitätslehre ist,
dass Elektrodynamik und Thermodynamik dasselbe sind, dass auch die Energie Trägheit
besitzt, dann dass auch in der Chemie (Metalchemie, Radioaktivität) physikalische Methoden
in der Vordergrund haben.”
79. Hilbert 1912c, 2: “... wie die Lehre des Kopernikus, eine durch das Experimente bewiesene
Tatsache.”
80. Hilbert 1912a, 1912b, 1913,1913a, 1914.
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This is not the place to describe the details of Hilbert’s involvement with kinetic theory
and with raditation theory.81 Still, some aspects of it are worth mentioning here, since
they help understanding properly the actual place of physics in Hilbert’s world. Thus,
for instance, shortly after the publication of his first article on kinetic theory, Hilbert
organized a seminar on this topic, together with his former student Erich Hecke. The
seminar was also attended by the Göttingen docents Max Born, Paul Hertz, Theodor
von Kármán and Erwin Madelung. A contemporary report in a journal published by
German students of mathematics provides an invaluable source of information
concerning the issues discussed in that seminar, which included the following:82 the
ergodic hypothesis and its consequences; on Brownian motion and its theories;
electron theory of metals in analogy to Hilbert’s theory of gases; report on Hilbert’s
theory of gases; on dilute gases; temperature split by the walls; theory of dilute gases
using Hilbert’s theory; on the theory of chemical equilibrium, including a referat on the
related work of Sakur [sic]; diluted solutions. It is not clear whether the term “Hilbert’s
theory of gases” was meant here as what Hilbert published in his article or as the ideas
he had developed in his courses, but in any case the very use of the terms suggests that
Hilbert considered to have in his hands more than a simple local addition to the theory
as then conceived. Moreover, the fact that the seminar discussions involved the
younger colleagues mentioned above indicates that the deep physical issues that were
at stake, and that are listed in detail in the report, could not have easily been ignored or
discussed only superficially. Especially indicative of Hilbert’s surprisingly broad
spectrum of interests is the reference to the work of Otto Sackur (1880-1914). Sackur
was a physical chemist from Breslau, whose work dealt mainly with the laws of
chemical equilibrium in ideal gases, on Nernst law of heat, and who wrote a widely
used textbook on thermochemistry and thermodynamics (Sackur 1912). Sackur work
included also an important experimental side, and in general his work was far from the
typical kind of mathematical physics with which Hilbert and the Göttingen school may
more easily be associated.83
Also significant is the fact that Hilbert’s ideas on kinetic theory seem to have
influenced the work of several of his students. In the first place one can mention two
doctoral dissertations written under his supervision on issues related to the kinetic
theory: one by Hans Bolza (concluded in July of 1913) and one by Bernhard Baule
(concluded in February of 1914).84 Second, other young Göttingen scientists, like Max
Born, Theodor von Kármán and Erich Hecke, who had attended Hilbert’s seminar,
published in this field under its influence.85 But perhaps of a much greater impact on
the development of the theory was the work of the Swedish physicist David Enskog
(1884-1947), who attended Hilbert’s lectures of 1911-12.86 Building on ideas
81. Such a discussion can be found in Corry 1998.
82. Reference to this seminar appears in Lorey 1916, 129. Lorey took this information from
the Semesterberichte des Mathematischen Verereins, but he does not give the exact year of the
seminar.
83. See Sackur’s obituary in Physikalische Zeitschrift Vol. 16, 1915, 113-115. Sackur died in
an explosion at the Kaiser-Wilhelm-Institut in Berlin-Dahlem, while taking part in current
research on explosives, related with the war effort.
84. This was published as Baule 1914.
85. Cf. for instance: Bolza, Born & van Kármán 1913; Hecke 1918; Hecke 1922.
86. See Mehra 1973, 178. I know of no evidence to decide whether Enskog attended the above
mentioned seminar.
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contained in Hilbert’s article, Enskog developed what has come to constitute, together
with the work of Sydney Chapman, the now standard approach to the whole issue of
transport phenomena in gases.87 Although a detailed analysis of Hilbert’s influence on
Enskog is yet to be written, there can be little doubt that such an influence can indeed
be traced down to the 1911-12 lectures. Finally, a further issue that should be
investigated in this context, is the possible influence of Hilbert on the publication of
Paul and Tatyana Ehrenfest’s famous Encyclopädie article on the conceptual
foundations of statistical mechanics (Ehrenfest 1956 [1912]). Ehrenfest studied in
Göttingen between 1901 and 1903, and returned there in 1906 for one year, before
moving with his wife Tatyana to St. Petersburg. Tatyana had studied mathematics in
Göttingen. The idea of writing this Encyclopädie article arised following a seminar
talk in Göttingen, to which Paul Ehrenfest was invited by Felix Klein.88 the
Ehrenfests’s style of theory clarification, as manifest in this article, is strikingly
reminiscent of Hilbert’s lectures in many respects, and strongly suggests a possible
influence of the latter.
Worth of mention are also the critical reactions to Hilbert’s published papers on
radiation theory. Such reactions came especially from Ernst Pringsheim (1859-1917),
and they led to a somewhat heated debate between the two.89 This debate is very
instructive as to the way in which a physicist could have reacted to Hilbert’s approach
to physical issues, and to how Hilbert’s treatment, rather than presenting the systematic
and finished structure characteristic of the Grundlagen, was piecewise, ad-hoc and
sometimes confused or unilluminating.90
A further, compelling testimony of Hilbert’s deep interest in physics over this
period of time is furnished by the proceedings of a meeting held at the Royal Academy
of Sciences in Göttingen in May of 1913. The meeting consisted of a series of lectures
on the current state of research in the kinetic theory, and the lecturers included some of
the finest physicists of the time. Max Planck, whose work on radiation Hilbert had
studied in great detail when writing his own articles, lectured on the significance of the
quantum hypothesis for the kinetic theory. Peter Debye (1884-1966) had become in
1914 professor of physics in Göttingen; his talk dealt with the equation of state, the
quantum hypothesis and heat conduction. Walther Nernst (1864-1941), whose work on
thermodynamics Hilbert had been following with interest,91 spoke about the kinetic
theory of rigid bodies. Marian von Smoluchowski (1872-1917) came from Krakow
and lectured on the limits of validity of the second law of thermodynamics. Arnold
Sommerfeld (1868-1931) came from Munich to talk about problems of free
trajectories. Hendrik A. Lorentz (1853-1928) was invited form Leyden; he spoke on
the applications of the kinetic theory to the study of the motion of the electron. That the
87. See Brush 1976, 449-468.
88. See Klein 1970, 81-83.
89. Pringsheim published his objections in Pringsheim 1913, 1913a.
90. For more details on this debate see Corry 1998.
91. In January 1913, Hilbert had lectured on Nernst’s law of heat at the Göttingen Physical
Society. The manuscript of the lecture is preserved in Hilbert’s Nachlass, Ms Cod 590. See also
a remark added in Hilbert’s handwritting in Hilbet 1905, 167.
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Hilbert and Physics:
meeting was an initiative of Hilbert is clear from the fact that it was sponsored by the
Wolfskehlstiftung, whose chair was Hilbert himself. Hilbert wrote a report on the
lectures delivered in the meeting,92 as well as the introduction to the published
collection of lectures, in which expressed his hope that it would stimulate further
interest, especially among mathematicians, and lead to additional involvement with the
exciting world of ideas created by the new physics of matter.93
7. Electrodynamics and General Relativity: 1913-1915
By the year 1913 Hilbert’s interest in a wide variety of physical disciplines had
become a truly central feature of his current research and teaching concerns. In 1912
Hilbert solved the Boltzmann equation, which was intimately connected to his current
research in the theory of linear integral equations, but which at the same time raised
intriguing physical problems that directly attracted his attention. Hilbert’s student
Lüdwig Föppl completed in March 1912 his dissertation on atomic stability (Föppl
1912), and so did Hans Bolza with his own one dealing with the theory of gases.
Hilbert also gave public lectures on Maxwell’s theory of gases, statistical mechanics
and Nernst’s law of heat. The meetings of the GMG, clearly with Hilbert’s approval, if
not under his direct initiative, discussed Gustav Mie’s recent work on an
electromagnetic theory of matter (in December 1912) as well as Albert Einstein and
Marcel Grossmann’s first attempt to formulate a relativistic theory of gravitation (in
December 1913), the famous Entwurf paper.94
This increased interest in physics is also manifest in Hilbert’s lectures of 1913.
The topics of his lectures on physics had expanded way beyond the more traditional
ones of classical mechanics and continuum mechanics and now covered also statistical
mechanics, radiation theory and the molecular theory of matter. In the summer of 1913
Hilbert returned to an old field of interest that would occupy his thoughts for several
years now: electromagnetism. He taught a course on electron theory and, at the same
time, a second course on the principles of mathematics, very similar to the earlier one
of 1905, where he included again a long section on the axiomatization of physics. Over
the following years he would also lecture on electromagnetic oscillations, statistical
mechanics, the structure of matter, and, in 1916-17, on the general theory of relativity
and the theory of the electron. These years were also characterized by Hilbert’s
unqualified adoption of an electromagnetic, reductionist view of physical phenomena.
More generally, his views on physical issues during this period became more articulate
and, in many respects, much more dogmatic.
Hilbert’s 1913 course on the theory of the electron was conceived as an
axiomatic treatment of electrodynamics. Following Minkowski’s work, Hilbert laid
special stress on the role of the (special) relativity principle, i.e., on the assumption that
all laws of nature are expressible as formulas that are invariant with respect to
92. See JDMV Vol. 22 (1913), pp. 68-69.
93. See Planck et al. 1914.
94. For more details on the relevant activities of the GMG see the appendix to this article.
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Hilbert and Physics:
simultaneous, homogenous (orthogonal) transformations of the four variables x,y,z,t.95
Hilbert dedicated some effort to highlight the importance of the specific contributions
of his Göttingen colleagues: Minkowski’s “World vector analysis”, and Born’s rigid
body. The Maxwell equations and the concept of energy, he explained, do not suffice to
provide a complete foundation of electrodynamic. An additional concept is thus
needed: the concept of rigidity. Electricity must be attached to a steady scaffold. This
scaffold is what we call an electron. The electron, he explained, embodies the concept
of a rigid body of Hertz’s mechanics. All of the laws of mechanics can be derived, in
principle at least, from these three ideas: Maxwell’s equations, the concept of energy,
and rigidity. From them also all the forces of physics can be derived, and in particular
the molecular forces. Only gravitation, he concluded, has evaded until now every
attempt at an electrodynamic explanation.96
Hilbert also stressed very much the mathematical difficulty involved in solving
the n-electron problem. This difficulty, he asserted, provides additional justification for
studying the movements of the electron based on the principles of statistical
mechanics.97 It is interesting to notice that Gustav Mie and his electromagnetic theory
of matter were not mentioned explicitly in the lectures, despite the fact that back in
December 1912 this theory had been discussed in the meeting of the Göttingen
Mathematical Society.
Parallel to this course, Hilbert also lectured in the summer of 1913 on the
“Elements and Principles of Mathematical Thinking”. A glance at the contents of the
course indicates the extent to which Hilbert still considered physics and mathematics
as tightly interconnected at their most fundamental and essential aspects, and how he
thought that the axiomatic method should be similarly applied for the benefit of both
domains. The contents of the course also illustrate, like was the case with the similar
1905 course, how a typical Hilbert lecture could look like: several, not obviously
interconnected, mathematical issues could be discussed in successive lectures, in very
general terms and without entering into any kind of details. The initial program of the
course could always be changed, according to the way in which it developed.
One significant difference between this course and the one taught earlier, in
1905, is the addition of a new section dealing with the axioms of the theory of
radiation, which essentially repeated what had appeared in his published work on this
topic. Like in the 1905 course, a main motto that appears in Hilbert’s treatment of each
95. Hilbert 1913b, 1 (Emphasis in the original): “Der Inhalt des Relativitätsprinzip ist nun die
Behauptung, dass die gesamten Naturgesetze mathematischen Formal ihren Ausdruck finden,
die invarianten gegen einen simultanen homogenen (orthogonale) Transformation der vier
Variablen x,y,z,t sind.”
96. Hilbert 1913b, 61-62 (Emphasis in the original): “Auf die Maxwellschen Gleichungen und
den Energiebegirff allein kann man die Elektrodynamik nicht gründen. Es muss noch der
Begriff der Starrheit hinzukommen; die Elektrizität muss an ein festes Gerüst angeheftet sein.
Dies Gerüst bezeichen wir als Elektron. In ihm ist der Begriff der starrer Verbindung der
Hertzschen Mechanik verwirklicht. Aus den Maxwellschen Gleichungen, dem Energiebegriff
und dem Starrheitsbegriff lassen sich, im Prinzip wenigstens, die vollständigen Sätze der
Mechanik entnehmen, auf sie lassen sich die gesamten Kräfte der Physik, im Besonderen die
Molekularkräfte zurückzuführen. Nur die Gravitation hat sich bisher dem Versuch einer
elektrodynamischen Erklärung widersetzt.”
97. Hilbert 1913b, 83: “Je komplizierte aber das Problem, mit umsomehr Recht wenden wir
später das Grundprinzip der statistischen Mechanik auf die Elektronenbewegung an.”
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Hilbert and Physics:
separate field is the need to determine the exact analytic form of a certain function,
which lies at the heart of the theory, starting from the axioms alone. The main laws that
must be derived here are Kirchhoff´s laws of emission and absorption. We are given an
empty closed sphere at a given temperature. Independently from the material
composition of the sphere, the law establishes that the radiation energy of a welldetermined color depends only on the wave-length and the temperature. The function
whose expression must be deduced in this case is the wave function. Hilbert
formulated axioms from which he believed the precise analytic expression of the
function might be derived as a logical consequence, in a way that in a state of thermal
equilibrium, and independently of the kind of matter involved and of its location, the
equation would hold.98 A second innovation found in these lectures, compared to those
of 1905, is an elaborate treatment of special relativity or, as Hilbert called it, of the new
conceptions of space and time.
In the winter semester of 1913-14 Hilbert taught his last course on a physical
issue before eventually turning to general relativity. It dealt with electromagnetic
oscillations. In the introductory lecture of this course he referred once again to the
example of geometry as a model of an experimental science, our thorough knowledge
of which has transformed into a mathematical, and therefore a “theoretical science”. In
a passage that characterizes very aptly his conception of the relation between physical
disciplines and mathematics, Hilbert said:
From antiquity the discipline of geometry is a part of mathematics. The experimental grounds
necessary to build it are so suggestive and generally acknowledged, that from the outset it has
immediately appeared as a theoretical science. I believe that the highest glory that such a
science can attain is to be assimilated by mathematics, and that theoretical physics is presently
on the verge of attaining this glory. This is valid, in the first place for the relativistic
mechanics, or four dimensional electrodynamics, of whose belonging to mathematics I am
already convinced for a long time.99
But, here, for the first time in Hilbert’s lectures, we come across an explicit suggestion
that electrodynamics is the field that might provide the correct foundation for all of
physics. In fact, Hilbert seems to have conceived a much more comprehensive, unified
picture of science that also covered all of mathematics and physics. In a somewhat
unclear passage, Hilbert claimed:
98. Hilbert 1913c, 107: “Die Axiome müssen die Tatsache als logisch Folgerung ergeben, daß
die thermischen Gleichgewicht, unabhängig von Stoff und Ort, die Beziehung gilt:
m=f(J, z).”
99. Hilbert 1913-14, 1 (Emphasis in the original): “Seit Alters her ist die Geometrie eine
Teildisziplin der Mathematik; die experimentelle Grundlagen, die sie benutzen muss, sind so
naheliegend und allgemein anerkannt, dass sie von vornherein und unmittelbar als theoretische
Wissenschaft auftrat. Nun glaube ich aber, dass es der höchste Ruhm einer jeden Wissenschaft
ist, von der Mathematik assimiliert zu werden, und dass auch die theoretische Physik jetzt im
Begriff steht, sich diesen Ruhm zu erwerben. In erster Linie gilt dies von der
Relativitätsmechanik oder vierdimensionalen Elektrodynamik, von deren Zuhörigkeit zur
Mathematik ich seit langem überzeugt bin.”
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Hilbert and Physics:
It appears, however, as if theoretical physics has finally and totally been absorbed by the
electrodynamics, to the extent that every special question should be solved, in the last
instance, by appealing to electrodynamics. Following the methods prevailingly used in the
individual mathematical disciplines, one could also classify mathematics—more from the
point of view of contents than from a formal one—into one-dimensional mathematics (i.e.,
arithmetic), then function theory (which in essence is limited to two dimensions), geometry,
and finally four-dimensional mechanics.100
This more general, unified conception would remain basic to Hilbert’s subsequent
dealings with physics, and particularly to his involvement with the problems
associated with the general theory of relativity.
While teaching his 1913-14 course, Hilbert’s interest in the work of Gustav
Mie must have already become evident. As already mentioned, Max Born lectured
before the GMG on this theory in December 1912. On October 22, 1913, Mie wrote a
letter to Hilbert expressing his satisfaction for the interest that the latter had manifested
on Mie’s recent work (presumably in an earlier letter which has not been
preserved).101 Then, on December 12, 1913, Born lectured again at the meeting of the
GMG, this time on his own contributions to Mie’s theory. The reformulation of Mie’s
theory at the hands of Born was to become a crucial turning-point in Hilbert’s way to
the problems of general relativity.102 Incidentally, Born’s lecture was communicated to
the society by Hilbert himself,103 At the same time, recent work on gravitation and
relativity began to be discussed more intensely in Göttingen. Einstein and Grossmann's
Entwurf paper, as already mentioned, was discussed in the GMG on December 12,
1913. But also the works of Max Abraham and Gunnar Nordström were studied with
great interest.104
Hilbert’s physical activities in 1914 were less intense than in previous years.
He published the third installment of his work on the foundations of the theory of
radiation, and lectured once again on statistical mechanics. Also, in June 1914, his
student Kurt Schellenberg presented a dissertation dealing with the applications of
integral equations to the theories of electrolysis (Schellenberg 1915). The beginning of
the war obviously altered the normal course of activities in Göttingen, and in particular
the presence of students and young docents there over the next years. Already in
November 3, 1914, Hilbert discussed in the meeting of the GMG the consequences of
war on the society’s activities.105
100. Hilbert 1913-14, 1: “Es scheint indessen, als ob die theoretische Physik schliesslich ganz
und gar in der Elektrodynamik aufgeht, insofern jede einzele noch so spezielle Frage in letzter
Instanz an die Elektrodynamik appellieren muss. Nach den Methoden, die die einzelnen
mathematischen Disziplinen vorwiegend benutzen, könnte man alsdann—mehr inthaltlich als
formel—die Mathematik einteilen in die eindimensionale Mathematik, die Arithmetik, ferner
in die Funktionentheorie, die sich im wesentlichen auf zwei Dimensionen beschränkt, in die
Geometrie, und schliesslich in die vierdimensionale Mechanik.”
101. Mie’s letter is in Hilbert’s Nachlass, NSUB Göttingen - Cod Ms David Hilbert 254 - 1.
102. For more details see Corry 1998a.
103. See the announcement in Jahrb. DMV Vol. 22, p. 207. Born’s work appeared later as
Born 1913.
104. See Born 1922, 593. For an historical account of Nordström’s work in relativity see
Norton 1992.
105. See the announcements of the meetings of the GMG for that year in the JDMV Vol. 23
(1914).
40
Hilbert and Physics:
In the summer semester of 1915 Hilbert lectured on the structure of matter. The notes
of this course illuminate yet another significant aspect of Hilbert’s wide range of
physical knowledge and interests. The course focused, in fact, on Born’s theory of
crystals, which was based on the study of the potential energy of a lattice of particles.
For Hilbert, this theory, in the one hand, and the theory of dilute gases, on the other
hand, were complementary to each other in accounting for the properties of matter
(Hilbert 1915, 1). Hilbert discussed briefly the mathematical aspects of the theory
(crystallographic groups), and also, in much more detail, its physical aspects: wave
displacement inside the lattice, crystal elasticity, specific heat in the lattice, the piezoelectric effect, etc.
The announcement of meetings at the GMG for the summer semester of 1915
record a lecture on structure of crystals by Felix Klein (together with Hilbert and
Mügge) in May, and a lecture by Sommerfeld “On Modern Physics” in June. Then
from June 29 to July 7, 1915, Einstein gave a series of lectures on the current state of
his research on gravitation and relativity. Unfortunately not much is known about the
visit itself, except that, as I mentioned in the introduction to this article, Einstein felt
that his work had been understood to the details.106 Over the months following that
visit, and especially in October and November, Hilbert devoted most of his efforts to
what he later called “The Foundations of Physics”, namely, the formulation of a
unified field theory, based on Mie’s electromagnetic theory of matter.107 He presented
the results of his investigations in the meeting of the Göttingen Scientific Society on
November 20, 1915.108
Hilbert’s work during these months and the publication of his field equations of
gravitation opened a new phase in his career, during which he and his colleagues in
Göttingen would dedicate much efforts to the general theory of relativity.109 From that
time on, Hilbert became increasingly enthusiastic about the significance of general
covariance. In a lecture held in 1921, for instance, he asserted that no other discovery
in history had aroused so much interest and excitement as Einstein’s relativity theory
(“the highest achievement of human spirit”) did. This excitement was indeed justified
in Hilbert’s view since, whereas all former laws of physics were provisory, inexact and
special, the principle of relativity (and Hilbert meant by this the general covariance of
physical laws) signified “for the first time, since the world exists, a definitive, exact
and general expression of the nature laws that hold in reality.”110 At the same time,
Hilbert’s involvement with general relativity provided additional support to his
106. I have made some efforts to gather documents related to this visit, so far without much
success. What I did find in Hilbert’s Nachlass in Göttingen, nevertheless, are the handwritten
notes taken from the first of Einstein’s lectures (Staats- und Universitätsbibliothek Göttingen,
Cod Ms D Hilbert 724). These notes have been published meanwhile in Kox et al (eds.) 1996,
App. B, 586-590.
107. For an account of this period and the details of Hilbert’s theory, see Corry 1998a.
108. They were published in March of the following year as Hilbert 1916.
109. In Corry 1998, 1998a, I give detailed historical accounts of Hilbert’s way of general
relatvity, as well as the more immediate context of his publications. David Rowe’s contribution
to the present volume discusses the developments in Göttingen around general relativity
between 1916 and 1918.
110. Hilbert 1921, 1: “... denn das Relativitätsprinzip bedeutet, wie mir scheint, zum ersten
Mal, seit die Welt steht, eine definitive, genaue und allgemeine Aussage über die in der
Wirklichkeit geltenden Naturgesetze.”
41
Hilbert and Physics:
empiricist conception of geometry. Although a detailed analysis of the place of
geometry in Hilbert’s work on relativity would be much beyond the scope of the
present account, it is pertinent to quote here from the manuscript of his course on this
issue, taught in the winter semester of 1916-17, where Hilbert expressed this
connection as part of a more general, unified view of all branches of human
knowledge. Hilbert thus said:
In the past, physics adopted the conclusions of geometry without further ado. This was
justified insofar as not only the rough, but also the finest physical facts confirmed those
conclusions. This was also the case when Gauss measured the sum of angles in a triangle and
found that it equals two right ones. That is no longer the case for the new physics. Modern
physics must draw geometry into the realm of its investigations. This is logical and natural:
every science grows like a tree, of which not only the branches continually expand, but also
the roots penetrate deeper.
Some decades ago one could observe a similar development in mathematics. A theorem was
considered according to Weierstrass to have been proved if it could be reduced to relations
among integer numbers, whose laws were assumed to be given. Any further dealings with the
latter were laid aside and entrusted to the philosophers. Kronecker said once: ‘The good Lord
created the integer numbers.’ These were at that time a touch-me-not (noli me tangere) of
mathematics. That was the case until the logical foundations of this science began to stagger.
The integer numbers turned then into one of the most fruitful research domains of
mathematics, and especially of set theory (Dedekind). The mathematician was thus compelled
to become a philosopher, for otherwise he ceased to be a mathematician.
The same happens now: the physicist must become a geometer, for otherwise he runs the risk
of ceasing to be a physicist and viceversa. The separation of the sciences into professions and
faculties is an anthropological one, and it is thus foreign to reality as such. For a natural
phenomenon does not ask about itself whether it is the business of a mathematician or of a
physicist. On these grounds we should not be allowed to simply accept the axioms of
geometry. The latter may be the expression of certain facts of experience that further
experiments would contradict.111
111. Hilbert 1916-17, 2-3 (Emphasis in the original): “Früher übernahm die Physik die Lehren
der Geometrie ohne weiteres. Dies war berechtigt, solange nicht nur die groben, sondern auch
die feinsten physikalischen Tatsachen die Lehren der Geometrie bestätigen. Dies war noch der
Fall, als Gauss die Winkelsumme im Dreieck experimentell mass und fand, dass sie zwei
Rechte beträgt. Dies gilt aber nicht mehr von der neuesten Physik. Die heutige Physik muss
vielmehr die Geometrie mit in den Bereich ihrer Untersuchungen ziehen. Das ist logish und
naturgemäss: jede Wissenschaft wächst wie ein Baum, nicht nur die Zweige greifen weiter aus,
sondern auch die Wurzeln dringen teifer.
Vor einigen Jahrzehnten konnte man in der Mathematik eine analoge Entwicklung verfolgen; einen Satz
hielt man damals nach Weierstrass dann für bewiesen, wenn er auf Beziehungen zwischen ganzen
Zahlen zurückführbar war, deren Gesetz man als gegeben hinnahm. Sich mit diesen zu beschäftigen,
wurde abgelehnt und den Philosophen überlassen. Kronecker sagte einmal: ‘Die ganzen Zahlen hat der
liebe Gott geschaffen.’ Diese waren damals noch einen noli me tangere der Mathematik. Das ging so
fort, bis die logischen Fundamente dieser Wissenschaft selbst zu wanken begannen. Nun wurden die
ganzen Zahlen eines der fruchtbarsten Arbeitfelder der Mathematik uns speziell der Mengenlehre
(Dedekind). Der Mathematiker wurde also gezwungen, Philosoph zu werden, weil er sonst aufhörte,
Mathematiker zu sein.
So ist es auch jetzt wieder: der Physiker muss Geometer werden, weil sonst Gefahr läuft, aufzuhören,
Physiker zu sein und umgekehrt. Die Trennung der Wissenschaften in Fächer und Fakultäten ist eben
etwas Antropologisches, und der Wirklichkeit Fremdes; denn eine Naturerscheinung frägt nicht danach,
ob sie es mit einem Physiker oder mit einem Mathematiker zu tun hat. Aus diesem Grunde dürfen wir
die Axiome der Geometrie nicht übernehmen. Darin könnten ja Erfahrungen zum Ausdruck kommen,
die den ferneren Experimenten widersprächen.”
42
Hilbert and Physics:
But as was the case with his earlier contributions to physics, Hilbert’s work in
general relativity did not attain wide recognition. Einstein himself, like many other
physicist, disliked both Hilbert’s derivation of the equations from a variational
principle and his excessive reliance on Mie’s theory of matter. Nevertheless, seen in
the context of his other contributions to physics, there can be no doubt that, like in
other disciplines, Hilbert’s involvement in general relativity was far from being a
short-range, opportunistically motivated endeavor.
8. Concluding Remarks
Hilbert had a sustained interest in physics that can be traced throughout his career. In
the present article I have shown how this interest is manifest from his early dealings
with geometry, around 1894, and up until 1915. This interest, however, continued to
occupy a central place in Hilbert’s overall conception of science until the end of his
career. Although Hilbert’s published work on physical issues covers only a small
portion of his overall output, he actually dedicated a much more significant part of his
efforts to various physical disciplines than the amount of his publications could
indicate. This effort can properly be understood only by examining Hilbert’s docent
and organizational activities in Göttingen, as I have done in the preceding pages.
In order to determine whether Hilbert’s ideas had any actual influence on the
development of twentieth century physics it is necessary to undertake a more detailed
examination of his published and unpublished writings in each individual discipline,
and of the people who were exposed to these ideas. I have suggested above that this
influence may have been instrumental, in one way or another, in shaping the ideas of
several persons who contributed significant work in physics: Max Born, Hermann
Minkowski, Paul Ehrenfest, David Enskog, as well as other, relatively minor figures.
Other names come to mind which were not mentioned in this article but which
certainly were influenced by Hilbert in different domains of physics, such as Hermann
Weyl in relativity and John von Neumann in quantum theory.
Many physicist reacted with lack of enthusiasm, to say the least, to Hilbert’s
incursions into physics. Pringsheim’s reaction to Hilbert’s articles on radiation offers a
good example of this. Also Einstein expressed many reservations concerning Hilbert’s
approach to relativity. In a letter to Hermann Weyl, Einstein criticized Hilbert’s use of
the variational principle in this context and judged his approach to the general theory
of relativity to be “childish ... in the sense of a child that recognizes no malice in the
external world.”112 Weyl himself considered that Hilbert’s work in physics was of
rather limited value, especially when compared to his work in pure mathematics. In
particular Weyl considered that Hilbert’s application of the axiomatic method was of
little significance for physics. Weyl thought, that a valuable contribution to physics
required a different kind of skills than those in which Hilbert excelled. In an obituary
of Hilbert, Weyl wrote:
112. In a letter of November 23, 1916. Quoted in Seelig 1954, 200.
43
Hilbert and Physics:
The maze of experimental facts which the physicist has to take in account is too manifold,
their expansion too fast, and their aspect and relative weight too changeable for the axiomatic
method to find a firm enough foothold, except in the thoroughly consolidated parts of our
physical knowledge. Men like Einstein and Niels Bohr grope their way in the dark toward
their conceptions of general relativity or atomic structure by another type of experience and
imagination than those of the mathematician, although no doubt mathematics is an essential
ingredient.113
Max Born was perhaps the physicist that expressed a more consistent
enthusiasm for Hilbert’s physics. He seems also to have truly appreciated the exact
nature of Hilbert’s program for axiomatizing physical theories and the potential
contribution that the realization of that program could bear. On the occasion of
Hilbert’s sixtieth birthday, the journal Die Naturwissenschaften dedicated one of its
issues to celebrate the achievements of the master. Several of his students were
commissioned with articles summarizing Hilbert’s contributions in different fields.
Born, who as a young student in Göttingen attended many of Hilbert’s courses, and
later on as a colleague continued to participate in his seminars, wrote about Hilbert’s
physics. Besides his extolling summary of Hilbert’s achievements, Born also explained
why in his view Pringsheim had misunderstood Hilbert and why the former’s
reproaches to the latter were unjustified. In doing so he also clarified, in a very
succinct formulation, the nature of Hilbert’s axiomatic treatment and why, in general,
physicist tended not to appreciate it. Born put it in the following words:
The physicist set outs to explore how things are in nature; experiment and theory are thus for
him only a means to attain an aim. Conscious of the infinite complexities of the phenomena
with which he is confronted in every experiment, he resists the idea of considering a theory as
something definitive. He therefore abhors the word “Axiom”, which in its usual usage evokes
the idea of definitive truth. The physicist is thus acting in accordance with his healthy instinct,
that dogmatism is the worst enemy of natural science. The mathematician, on the contrary, has
no business with factual phenomena, but rather with logic interrelations. In Hilbert’s language
the axiomatic treatment of a discipline implies in no sense a definitive formulation of specific
axioms as eternal truths, but rather the following methodological demand: specify the
assumptions at the beginning of your deliberation, stop for a moment and investigate whether
or not these assumptions are partly superfluous or contradict each other. (Born 1922, 591)
In fact Hilbert never performed for a physical theory exactly the same kind of
axiomatic analysis he had done for geometry, though he very often declared this to be
the case. In no case in the framework of his lectures, did Hilbert actually prove the
independence, consistency or completeness of the axiomatic systems he introduced. In
certain cases, like vector addition, he quoted works in which such proofs could be
found. In other cases there were no such works to mention, and Hilbert limited
himself—as in the case of thermodynamics—to state that his axioms are indeed
independent. Still in other cases, he barely mentioned anything about independence or
other properties of his axioms. Also his derivations of the basic laws of the various
disciplines from the axioms are rather sketchy, when they appear at all. Many times
Hilbert simply declared that such a derivation was possible. Among his published
works, his last article on radiation theory contains—perhaps under the pressure of
criticism—his more detailed attempt to prove independence and consistency of a
system of axioms for a physical theory. But what is clear in all cases is that Hilbert
always considered that an axiomatization along the lines he suggested was plausible
113. Quoted in Sigurdsson 1994, 363.
44
Hilbert and Physics:
and could eventually be fully performed following the standards established in the
Grundlagen. Be that as it may, there can be no doubt that the kind of conceptual
clarification attained in Hilbert’s examination of physical theories, as well as in works
of those who followed his lead, provided an important contribution in and of itself.
Whether or not physicists should have looked more closely at Hilbert’s ideas
than they actually did, and whether or not Hilbert’s program for the axiomatization of
physics had any influence on subsequent developments in this discipline, it is
nevertheless important to stress, that a full picture of Hilbert’s own conception of
mathematics cannot be complete without taking into account his views on physical
issues and on the relationship between mathematics and physics. More specifically, a
proper understanding of Hilbert’s conception of the role of the axioms in physical
theories—a conception condensed in the above quoted passage of Born, and illustrated
throughout this article—helps us understanding his conception of the role of axioms in
mathematical theories as well. The picture that arises from such an understanding is
obviously very far away from the somewhat widespread image of Hilbert as the
champion of a formalistic conception of the nature of mathematics.
Acknowledgments
The research and archival work that preceded the writing of this article was conducted
as part of a larger project on the historical context of the rise of the general theory of
relativity at the Max-Planck-Institut für Wissenschaftsgeschichte in Berlin, where I
worked during the academic year 1994-95. I would like to thank the staff for their
warm hospitality and diligent cooperation, and especially to Jürgen Renn for inviting
me to participate in the project and for his constant encouragement. During my stay in
Berlin, I benefited very much from illuminating discussions on the history of modern
physics with Jürgen Renn, Tilman Sauer and John Stachel.
This article was partly written at the Dibner Institute during the academic year 199596. I wish to thank the Directors and staff of the DI, as well as the other fellows with
whom I was fortunate to share my time. Special thanks to Ulrich Majer and David
Rowe for long and interesting discussions on the issues analyzed in this article.
I also thank Jeremy Gray and the participants of the meeting on Geometry and Physics,
1900-1930, at the Open University, Milton Keynes (March 14-19, 1996) for their
illuminating lectures and for their fruitful remarks and criticism following my own
talks.
Arne Schirrmacher read an earlier version of this article and provided helpful
corrections and suggestions for improvement. I thank him very much.
Original manuscripts are quoted in the text by permission of the Staats- und
Universitätsbibliothek Göttingen (Handschriftenabteilung), the library of the
Mathematisches Institut Universität Göttingen. I thank Ralf Haubrich and the Hilbert
Edition staff, for allowing me to quote the motto of this article from their provisory
typescript draft of the Tagebuch. Special thanks I owe to Peter Damerow for allowing
me to read and quote from the manuscript of Hilbert 1913c, belonging to his private
collection.
45
Hilbert and Physics:
Appendix
All the lists included in this appendix have been compiled from various sources.
Among them: the manuscripts of the lecture notes in the Lesezimmer of the
mathematical institute in Göttingen; Hilbert’s Nachlass (Niedersächsiche Staats- und
Universitätsbibliothek Göttingen); announcements in the Jahresbericht der Deutschen
Mathematiker Vereinigung and in the Physikalische Zeitschrift; Wilhelm Lorey’s book
quoted in the bibliography. It is conceivable that additional sources may add further
items to each of these lists.
Table 1:
Hilbert’s Courses on Physics (1898-1927)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
Mechanics
Lectures on Potential Theory
Selected Chapters of Potential Theory
Continuum Mechanics - Part I
Continuum Mechanics - Part II
Exercises on Mechanics Logical Principles of Mathematical Thinking (and of Physics)
Mechanics
Continuum Mechanics
Lectures on Continuum Mechanics
Mechanics
Continuum Mechanics
Statistical Mechanics
Radiation Theory
Molecular Theory of Matter
Foundations of Mathematics (and the axiomatizaion of Physics)
Electron Theory
Electromagnetic Oscillations
Statistical Mechanics
Lectures on the Structure of Matter
The Foundations of Physics I (General Relativity)
The Foundations of Physics II (General Relativity)
Electron Theory
Space and Time
Space and Time
Mechanics and the New Theory of Gravitation
Basic Principles of the Theory of Relativity
Statistical Mechanics
SS 1898
WS 1901/02
SS 1902
WS 1902-03
SS 1903
WS 1904/05
SS 1905
WS 1905/06
SS 1906
WS 1906/07
WS 1910/11
SS 1911
WS 1911/12
SS 1912
WS 1912/13
SS 1913
SS 1913
WS 1913/14
SS 1914
SS 1915
SS 1916
WS 1916/17
SS 1917
WS 1917/18
WS 1918/19
SS 1920
SS 1921
SS 1922
46
Hilbert and Physics:
29.
30.
31.
32.
Mathematical Foundations of Quantum Theory
On the Unity of Science
Mechanics and Relativity Theory
Mathematical Methods of Quantum Theory
WS 1922/23
WS 1923-24
SS 1924
WS 1926/27
Table 2:
Hilbert’s Doctoral Students on Physical Issues
1. Lüdwig Föppl: “Stabile Anordnungen von Elektronen im Atom” (March 1, 1912)
2. Hans Bolza: “Anwendungen der Theorie der Integralrechnungen auf die
Elektronentheorie und die Theorie der verdünnten Gasen.” (July 2, 1913)
3. Bernhard Baule: “Theoretische Behandlung der Erscheinungen in verdünnten
Gasen.” (Feb. 18, 1914)
4. Kurt Schelenberg: “Anwendungen der Integralgleichung auf die Theories der
Elektrolysie.” (June 24, 1914)
Table 3:
Hilbert’s Seminars and Public Lectures on Physics
A. Seminars:
1. Mechanics (together with Klein)
2. Mechanics
3. Electron Theory (together with H. Minkowski et al)
4. The Equations of Electrodynamics (with Minkowski)
5. Hydrodynamics
6. Electrodynamics
7. Kinetic Theory of Gases
B. Public Lectures:
1. Maxwell’s Theory of Gases
2. Statistical Mechanics
3. On Nernst’s Law of Heat
4. Space and Time
5. Nature and Mathematical Knowledge
1896 (?)
WS 1904-05
SS 1905
SS 1907
???
???
1912 (?)
1912 (?)
1912
(G_ttingen) 1913
(Bucharest) 1918
(G_ttingen) 1919-20
47
Hilbert and Physics:
6. On the Laws of Chance
7. Nature and Mathematical Knowledge
8. Science and Mathematical Thinking
1920
(Copenhagen) 1921
(G_ttingen) 1922-23
48
Hilbert and Physics:
Table 4:
Lectures on Physical Issues delivered at the meetings of the
Mathematical Society in Göttingen: 1904-1918
(According to the announcements in the Jahrb. DMV)
A. Hilbert’s Lectures:
1. Continuum Mechanics
Feb. 24, 1903
2. Continuum Mechanics
Aug. 4, 1903
3. The relations between variational principles and the theory of partial
differential equations, with applications to the integral principles of
mechanics.
Jan. 18, 1910
4. Kinetic theory of gases
Dec 19, 1911
5. Axiomatic Foundations of Physics (Ferienkurs for high-school
teachers)
April 15-27, 1912
6. Theory of Radiation
July 30, 1912
7. Theory of Radiation
Jan. 21, 1913
8. Theory of Radiation
July 28, 1914
9. The Fundamental Equations of Physics (General Relativity)
Nov. 16, 1915
10. Theory of Invariants and the Energy Principle
Jan. 25, 1916
11.
The Causality Principle in Physics
Nov. 21 & 28, 1916
12.
Non-Euclidean geometry and the new Gravitation Theory
Jan. 23, 1917
13.
Laue’s Theorem
June 12, 1917
14.
Reply to Klein’s “On Hilbert’s first note on the
Foundations of Physics”
Jan. 29, 1918
15.
The Energy Principle for the Motion of Planets in the New
Theory of Gravitation
June 4, 1918
16.
On Weyl’s Communication to the Berlin Academy (May 2, 1918)
“The Energy Principle in the General Theory of Relativity”
July 15, 1918
B. Lectures by Others:
1. On the Axioms of Vector Addition (R. Schimmack)
2. Molecular Theory of Heat Conduction (G. Prasad)
3. Capillarity (H. Minkowski)
4. Linear Heat Conduction in Surfaces (G. Prasad)
5. Maxwell’s Work on Stress Systems (F. Klein)
6. Euler’s Equations of Hydrodynamics (H. Minkowski)
7. Electromagnetic Quatity of Motion (M. Abraham)
8. Gibb’s Thermodynamical Surfaces (H. Happel)
June 9, 1903
June 9, 1903
June 23, 1903
June 23, 1903
June 23, 1903
June 28, 1903
July 14, 1903
Dec. 8, 1903
49
Hilbert and Physics:
9. Variational Principles in Electrodynamics (K. Schwarzschild)
Jan. 26, 1904
10. Can the Electron Reach the Speed of Light (P. Hertz)
Jan. 26, 1904
11. Overview of a Seminar on Hydrodynamics and Hydraulics (F. Klein) Feb. 9, 1904
12. Motion of a Material Particle on a Uniformly Moving
Plane (P. Ceresole)
May 17, 1904
13. On the Elasticity of the Earth (G. Herglotz)
June 28, 1904
14. On Sommerfeld’s Works on Electron Theory (G. Herglotz)
Dec. 6, 1904
15. Motion of a Fluid with Little Friction (L. Prandtl)
Dec. 13, 1904
16. On a Talk by Poincaré on the Future of Mathematical
Physics (C.H. Müller)
Jan. 24, 1905
17. On Gases ans Vapors (L. Prandtl)
May 23, 1905
18. On Poincarés Published Lectures on Mathematical
Physics (M. Abraham)
Feb. 6, 1906
19. On Poincarés Investigations on Rotating Fluid Masses (H. Müller) Feb. 13, 1906
20. On Gibb’s Book on Statistical Mechanics (E. Zermelo)
Feb. 20, 1906
21. Graphical Methods in Mechanics and Physics (C. Runge)
Feb. 27, 1906
22. On Painlevé’s Work on the Foundations of Mechanics
(C. Carathéodory)
May 28, 1906
23. On W. Nernst’s “On Chemical Equilibrium” (H. Minkowski)
June 26, 1906
24. Problems of Aeronavigation (L. Prandtl & E. Wiechert)
Oct. 30, 1906
25. The Mathematical Theory of Elasticity (C.H. Müller)
Nov. 6, 1906
26. On Botzmann’s H-Theorem (P. Ehrenfest)
Nov. 13, 1906
27. The Evolution of the Theory of Radiation through the Works
of Lorentz, Rayleigh, W. Wien and Planck (H. Minkowski)
Dec 1, 1906
28. On H. Witte’s “On the Possiility of a Mechanical Explanation
of Electromagnetic Phenomena” (M. Abraham)
Dec. 18, 1906
29. On the Application of Probability Calculus to Astronomy
(K. Schwarzschild)
Jan. 8, 1907
30. Theories of the Effects of Air Resistance (L. Prandtl)
Jan. 22, 1907
31. Seismic Waves (E. Wiechert)
Jan. 29, 1907
32. Statistical Stellar Astronomy (K. Schwarzschild)
Feb. 19, 1907
33. Seismic Rays (G. Herglotz)
May 14, 1907
34. Solutions of Differential Equations for Gas Spheres (Gaskügeln)
(K. Schwarzschild)
July 30, 1907
35. On the Equations of Electrodynamics (H. Minkowski)
Nov. 5, 1907
36. Graphical Methods in Fluid Mechanics (C. Runge)
Nov. 26, 1907
37. Applications of Quaternions to Electron Theory (F. Klein)
Dec. 10, 1907
38. A New, Simple General Proof of the Second Law of
Thermodynamics (C. Carathéodory)
Dec. 17, 1907
39. An Overview of Man’s Attempts to Fly (C. Runge)
March 3, 1908
40. Report on a Joint Seminar on Hydrodynamics (F. Klein,
L. Prandtl, C. Runge, E. Wiechert)
May 5, 1908
41. An Experiment on Stabilization of Air Balloons (L. Prandtl)
May 12, 1908
50
Hilbert and Physics:
42. On Lanchester’s Book “Aerodynamics” (C. Runge)
May 12, 1908
43. On the Equations of Electrodynamics (H. Minkowski)
July 28, 1908
44. On Recent French Research on Aviation (C. Runge)
Nov. 3, 1908
45. Recent Works on Earth Pressure (Th. Van Kármán)
Nov. 24, 1908
46. Theory of Earth Pressure (A. Haar & Th. Van Kármán)
Dec. 8, 1908
47. Position Determination from and Air Balloon (C. Runge)
May 11 & 18, 1909
48. Defintion of a Rigid Body on the “Einstein-Minkowski”
System of Electrodynamics (M. Born)
June 8 & 15, 1909
49. Average Motion in the Theory of Perturbations and
Applications of Probability to Astronomy (F. Bernstein)
June 22, 1909
50. On Minkowski’s Nachlass (Electrodynamics) (M. Born)
Feb. 8, 1910
51. On the Definition of a Rigid Body (M. Born)
Juni 21, 1910
52. Stable Orderings of Electrons in the Atom
(L. Föppl - PhD Dissertation supervised by Hilbert)
Nov. 21, 1911
53. On Herglotz Work on Deformable Bodies in the
Theory of Relativity (M. Born)
Dec. 12, 1911
54. The Behavior of Solid Bodies and Hooke’s Law (L. Prandtl)
Jan. 16, 1912
55. A Newly Discovered Relation Between Elasticity of Crystals
and Optical Oscillations (M. Born & Th. van Kármán)
Feb. 13, 1912
56. Mollecular Oscillations and Specific Heat (Born & van Kármán)
May 14, 1912
57. Theory of Disperssion in Crystals (P.P. Ewald - PhD Dissertation)
June 4, 1912
58. New Works of Poincaré end Ehrenfest on the Axiomatic
Foundation of Quantum Theory (Th. van Kármán)
Juli 16, 1912
59. Statistical Mechanics (P. Hertz)
Nov. 26, 1912
60. On Sommerfeld’s Article on the Theory of
Oscillating Equations (H. Weyl)
Dec. 10, 1912
61. Mie´s Theory of Matter (M. Born)
Dec. 17, 1912
62. Motion of Fluids (L. Prandtl)
Feb. 4, 1913
63. Reports on the Solvay Conference, Brussels 1911
(Born & van Kármán)
Feb. 25 & March 4, 1913
64. An Application of Diophantine Approximations to a
Question in Statistical Mechanics (E. Hecke)
May 20, 1913
65. On the Structure of Crystals (M. Born)
June 7, 1913
66. Recent Work of J.J. Thomson on Canal Waves (Kanalstrahlen)
(C. Runge)
Juni 24, 1913
67. An Application of Quantum Theory to Capillarity
(M. Born & R. Courant)
July 1, 1913
68. On a Recent Work of E. Noether on Turbulences in a Fluid
(Th. van Kármán)
July 30, 1913
69. On Poincaré’s Book on Cosmogonic Hypotheses (L. Föppl)
July 30, 1913
70. Propagation of Light in Transparent Media (W. Behrens)
Nov. 4, 1913
71. On Mie’s theory of Matter (M. Born)
Nov. 25, 1913
72. The Solution of an Integral Equation of Spectroscopy (C. Runge)
Dec. 2, 1913
51
Hilbert and Physics:
73. Recent Work of Einstein and Grossmann on Gravitation (F. Böhm) Dec. 9, 1913
74. On Mie’s Theory of Matter (M. Born)
Dec. 16, 1913
75. Theoretical Treatment of Phenomena in Diluted Gases
(B. Baule - PHD Dissertation Supervised by Hilbert)
Feb 24, 1914
76. Review of Recently Published Works by von Smoluchowski
(Brownian Movement), and Einstein (On Light Deflection;
On the Determination of Molecular Dimensions) (P. Hertz)
Feb. 24, 1914
77. Lattice Theory of Diamonds (M. Born)
March 3, 1914
78. Intensity Distribution in Spectral Lines (P. Debye)
Dec. 18, 1914
79. Foundation and Problems of Quantum Theory (P. Debye)
Feb. 23, 1915
80. Dynamics of Crystal Lattices (M. Born)
Feb. 25, 1915
81. Structure of Crystals (F. Klein, with Hilbert and Mügge)
May 18, 1915
82. On Herglotz’s Research on Potentials in the Interior of
Attracting Masses (Wiarda)
June 1, 1915
83. On Modern Physics (A. Sommerfeld)
June 15, 1915
84. On Gravitation (A. Einstein)
June 29, 1915
85. Theory of Distant Forces (Uhlich-Pirna)
July 20, 1915
86. History of Mechanics up until Galileo (C. Müller)
March 2, 1916
87. Four-dimensional Vectorial Analysis (C. Runge)
Dec. 5, 1916
88. Foundations of a Theory of Matter (G. Mie)
June 5-8, 1917
89. On the Riemannian Curvature (F. Klein)
Oct. 30, 1917
90. On the Riemannian Curvature (F. Klein)
Nov. 6, 1917
91. On G. Herglotz’s Paper on Curvature and Gravitation (F. Klein)
Dec. 4, 1917
92. On Liquid Crystals (M. Born)
Dec. 11, 1917
93. On Invariants of Arbitrary Differential Expressions (E. Noether)
Jan. 15, 1918
94. On Hilbert’s First Note on the Foundations of Physics (F. Klein)
Jan. 22, 1918
95. On Einstein’s Cosmological Ideas of 1917 (F. Klein)
May 7, 1918
96. Four Lectures on Quantum Theory (M. Planck)
May 13-17, 1918
97. On Einstein’s “On Gravitational Waves” (C. Runge)
Jun. 31, 1918
98. On Einstein’s Cosmological Ideas of 1917 (F. Klein)
June 11, 1918
99. On the Three-body Problem (C. Carath_odory)
June 24, 1918
100. On Einstein’s “Energy Principle in General Relatitvity” (F. Klein) July 4, 1918
101. Hilbert’s Energy Vector (F. Klein)
July 22, 1918
102. Invariant Variational Problems (E. Noether) July 23, 1918