Essay
Poincaré, Henri (1854-1912)
French mathematician and scientist
Preliminary version of an article
in Noretta Koertge (ed.), New Dictionary of
Scientific Biography, Vol. 6, pp. 121-125, New York, Charles Scribner's Sons, 2007
Scott Walter
Poincaré, Jules Henri (b. Nancy, France, 29 April 1854; d.
Paris, 17 July 1912), mathematics, celestial mechanics,
theoretical physics, philosophy of science. For the original
article on Poincaré see DSB vol. 11.
Historical studies of Henri Poincaré's life and science turned
a corner two years after the publication of Jean Dieudonné's
original DSB article, when Poincaré's papers were microfilmed
and made available to scholars. This and other primary sources
engaged historical interest in Poincaré's approach to mathematics,
his contributions to pure and applied physics, his philosophy,
and his influence on scientific institutions and policy. The result of
these new sources and updated historiography is a new
Poincaré: if Poincaré's outstanding achievements in mathematics
ensure his prominent position in the history of this subject,
his corpus and intellectual legacy are known to extend well beyond
mathematics proper, touching the core methods of theoretical
physics, and central tenets of the philosophy of science. Increased
recognition of the breadth of Poincaré's activity, from geodesy
and electrical engineering to algebraic topology and the philosophy
of space and time, is itself a prime result of Poincaré studies.
A brief summary follows of the ways these studies have revised
or enriched historical understanding of Poincaré's life and
work.
Early Career.
Consider, to begin with, Jean Dieudonné's evaluation of the young
Poincaré's spotty education in higher mathematics, gleaned
from Poincaré's published correspondence with Lazarus Fuchs
and Felix Klein and since then confirmed,
on one hand, by studies of Poincaré's qualitative theory of
differential equations, and on the other hand, by analyses of
the manuscripts submitted by Poincaré for the Grand prix
des sciences mathématiques in 1880, and discovered by Jeremy
Gray a century later. These manuscripts, written between 28 June
and 20 December 1880, show in detail how Poincaré exploited
a series of insights to arrive at his first major contribution
to mathematics: the discovery of the automorphic functions of
one complex variable (called "Fuchsian" and "Kleinian" functions
by Poincaré). In particular, the manuscripts corroborate Poincaré's
introspective account of this discovery (1908), in which the
real key to his discovery is given to be the recognition that
the transformations he had used to define Fuchsian functions
are identical with those of non-Euclidean geometry.
The manuscripts of 1880 also shed light on the origins of Poincaré's
conventionalist philosophy of geometry. Even at this early point
in his career, Poincaré understands geometry to be the study
of groups of transformations. Such a view recalls Klein's Erlangen
program (1872), but where Klein's approach was projective and
hierarchical, that employed by Poincaré in 1880 was entirely
metrical, and free of hierarchy, rendering unlikely any direct
influence of Klein's Program. The influence of Eugenio Beltrami,
however, is apparent in Poincaré's disk model of
hyperbolic geometry, a model he later employed in
the service of his conventionalist philosophy of geometry.
Foundations of Geometry.
Poincaré was elected to the geometry section of the Paris Academy
of Sciences on 31 January 1887 (at age thirty-two), and that same
year published his first paper on the foundations of geometry.
Impressed with the writings of Sophus Lie and Hermann Helmholtz
on the so-called Riemann-Helmholtz-Lie-Poincaré problem of
space, Poincaré characterized plane geometries by considering
intersections of a plane with a certain quadric, and formulating
a common set of axioms. In this way, "straights" and "circumferences"
of hyperbolic geometry are made to correspond, for example, to straights
and circumferences (circles) of Euclidean geometry, and the available
theorems depend on the chosen quadric. (He later popularized
this insight in terms of a "translation dictionary" for Euclidean
and non-Euclidean geometry). Extending his purview to the geometry
of space, Poincaré held hyperbolic geometry and Euclidean geometry
both to be adequate to the task of describing physical phenomena.
Darwinian evolution had provided humans only with the general
notion of a group; consequently, if humans regularly employed
Euclidean geometry instead of hyperbolic geometry, this was by virtue
of the simplicity and convenience of the former, the motion of
solids corresponding roughly to the Euclidean group. Experience
provided the "occasion" for this choice to be made, in that
employment of Euclidean geometry was an outcome contingent upon
the behavior of light rays and solids on Earth.
A few years later, Poincaré explained further that the choice
of hyperbolic geometry entailed non-standard laws of physics, thereby
rendering his view equivalent in essentials to that of Helmholtz
(1876). For Poincaré, the empirical equivalence of the two
possible points of view-Euclidean geometry plus ordinary physics,
or hyperbolic geometry plus some unspecified, alternative physics-ruled
out an empirical foundation of the geometry of physical space.
He regretted that Helmholtz-the champion of methodological empiricism-had
not made this point clear. Few scientists found compelling Poincaré's
extreme view of the geometry of physical space, but philosophers
(including Paul Natorp, Aloys Müller, Moritz Schlick, and Rudolf
Carnap) were swayed by the clever arguments advanced in its favor,
which shaped later debates on the conventionality of the spacetime
metric in general relativity.
Poincaré's contributions to relativity theory have given rise
to priority claims on his behalf, because he presented a theory mathematically
indistinguishable from that of Albert Einstein, four weeks earlier than
Einstein. In fact, Poincaré made a crucial step in 1900 toward
Einstein's 1905 redefinition of physical time and space, when
he realized that the validity of the principle of relativity
for electromagnetic phenomena depended on a certain definition
of the time coordinate in uniformly-moving systems (H.A. Lorentz's
"local time"), realized by an exchange of light-signals between
co-moving observers relatively at rest. Like Einstein, Poincaré
elevated the principle of relativity to a postulate in 1905;
unlike Einstein, he retained the notion of a luminiferous ether
(thus obviating the need for Einstein's light postulate). In
addition, he characterized the Lie-algebra of the Lorentz group,
and derived the first two Lorentz-covariant laws of gravitation.
Along the way, Poincaré provided the four-vectors for Hermann
Minkowski's four-dimensional spacetime theory (1908), but deplored
the latter's Einsteinian view of space and time coordinates,
advocating in its place an interpretative convention equivalent
to the postulation of Galilean spacetime.
In practice, Poincaré applied the principle of relativity as
one prong of a two-pronged method for the discovery of relativistic
laws: whatever laws happen to describe the behavior of natural
phenomena, these laws are required to be covariant with respect
to a certain group of transformations. The first prong left him
with an infinite number of candidate laws, all covariant with
respect to the Lorentz group. The second prong placed a constraint
on the search domain: for velocities that are small with respect
to that of light, the relativistic law should correspond to that
of classical physics. While Poincaré applied his covariance-plus-correspondence
method only to the case of Lorentz-covariant laws of gravitation,
his approach is applicable to any group of transformations, and
any principle of correspondence. Upon further formal elaborations
from Minkowski and Arnold Sommerfeld, Poincaré's method
became a touchstone of theoretical physics.
Work on Trajectories.
Interest in Poincaré's innovative contributions to the theory of
dynamical systems has expanded, with the rise of chaos theory in the
1970s, and the archival discovery of the first version of
Poincaré's submission for the King Oscar II of Sweden competition
to solve the n-body problem in celestial mechanics (1889).
The published version of Poincaré's paper on the restricted
three-body problem (1890) contains the first mathematical description
of a chaotic trajectory in a dynamical system, or what Poincaré
referred to as a doubly-asymptotic (and later, homoclinic) trajectory,
but this extraordinary result is absent from the original,
prize-winning version of the paper. In its place is a hastily-written
corollary concerning unstable periodic solutions of differential
equations, the flaw in which Poincaré discovered while preparing
his manuscript for publication. In a moment of inattention
Poincaré convinced himself of the convergence of the power series
expansion of certain periodic solutions, which pointed to well-behaved
asymptotic trajectories. In fact, the published version shows that the
series in question belongs to the class of asymptotic series he had
defined in 1886 (known later as a Poincaré expansion), and that in
light of his recurrence theorem, the behavior of the corresponding
trajectories becomes quite complicated (or in modern terms,
chaotic). The significance of these doubly-asymptotic trajectories was
not generally recognized at first, perhaps in part because
Poincaré soft-pedaled his discovery, the child of an oversight in
the original submission that was a huge embarrassment both for him,
and for those who awarded him the prize (Karl Weierstrass, Charles
Hermite, and Gösta Mittag-Leffler).
While the fate of doubly-asymptotic trajectories, neglected for
decades, is extreme, relative neglect befell other results obtained
by Poincaré in algebraic topology and mathematical physics.
There were several reasons for this, including an allusive writing
style, and a lack of follow-through. As his former student Émile
Borel put it in 1909, Poincaré was "more a conqueror than
a colonist". A self-made mathematician in the French style,
Poincaré took on no doctoral students, formed no school, and
delivered lectures incomprehensible to all but a handful of auditors.
With the aid of student note-takers, however, he published a
complete series of lectures on mathematical physics and celestial
mechanics, the technical sophistication and logical coherence
of which were much admired; they effectively disseminated Poincaré's
mathematical methods and problem-solving style.
Physics and Philosophy.
Significantly for the historical development of electrodynamics,
Poincaré's 1890 lectures on James Clerk Maxwell's theory of
electromagnetism were the first to appear in German and the second to
appear in French (other than translations of Maxwell's 1873
Treatise). Poincaré emphasized the Scottish physicist's
abstract and powerful Lagrangian approach and proved his remark that
if a phenomenon admits of one mechanical explanation, it admits of an
infinity of such explanations. He also transformed Maxwell's
single-fluid theory into a two-fluid theory, to the irritation of
certain Maxwellians, and misrepresented its notions of charge and
current. Nonetheless, his streamlined, inductive approach appealed to
physicists in Britain as on the Continent, and furthered the cause of
Maxwell's theory and British abstract dynamics.
For the general reader, Poincaré explained the cosmic consequences
of his discoveries in the dynamics of systems, for the foundations
of the second law of thermodynamics, the question of determinism,
and the stability of the Solar system. Starting in the 1890s,
Poincaré laid out in dozens of popular articles a coherent
scientific worldview informed by contemporary research in mathematics
and physics, and a familiarity with the writings of the leading
physicist-philosophers: Maxwell, Helmholtz, Ernst Mach, and Heinrich
Hertz. To Poincaré's surprise, his epistemological reflections
were enrolled by Catholic intellectuals in an effort to undermine
scientific authority. This placed him in the delicate position
of having to explain the value of a science whose pretension
to absolute truth he had systematically destroyed.
Technological advances issuing from fundamental discoveries in
physics were readily apparent in Poincaré's time, some of which
he celebrated (including wireless telegraphy, and the use of
x-ray images in medical diagnostics). Nonetheless, he located
the value of science not in its utility, or even in its power
to relieve human suffering, but in its capacity to awaken the
intellect: "to see," or at least, someday, "to let others see."
The unity and progress of science were linked for Poincaré;
the unity derived not from any shared method of investigation,
a wholly illusory notion, but from the shared mathematical structures
of the physical world. Progress, on the other hand, was realized
by overcoming the obstacles inherent in the methodological disunity
of science. In practice, overcoming such obstacles meant adopting
the most general point of view possible of the problem at hand,
an approach he employed regularly himself.
While scientific progress in Poincaré's sense is possible,
it is by no means inevitable, nor does it lead to absolute truth
in the long run. Objectivity itself is a social construct for
Poincaré: without discourse, he wrote, there can be no objectivity.
All science is not discourse, although some of Poincaré's contemporaries
understood him to be a nominalist. Laws describing natural phenomena
are elaborated pragmatically, in the sense that scientists' choices
remain free, but are guided by experience. Mathematics is a science
apart for Poincaré because the structure of the natural numbers
is intuitively given (in a non-Kantian sense), while the natural
sciences depend on a theory of measurement requiring the real
numbers and arithmetical operations. Progress in mathematics
is itself possible due to the principle of induction, which Poincaré
understood to be a synthetic a priori judgment. Attempts to provide
a logical foundation for mathematics by Bertrand Russell and
Alfred North Whitehead were doomed to failure, Poincaré argued,
because their constructions necessarily involved circular reasoning.
Reputation.
The brilliance of Poincaré's contributions to mathematics,
celestial mechanics, and mathematical physics was recognized
by scientific academies across Europe and in the United States,
most of which counted Poincaré as a foreign member by the end
of the nineteenth century. The mathematical community was unanimous
in its recognition of his accomplishments, the combined depth
and breadth of which none could reasonably hope to emulate. This
intellectual ascendancy had no counterpart in the French institutional
domain, where Poincaré was visibly less skilled than some of
his peers in securing his objectives. At the height of his scientific
authority in 1907, he stood for the position of Perpetual Secretary
for the Physical Sciences at the Paris Academy of Sciences, with
the backing of members of the physics section. Faced with opposition
from the chemistry and mineralogy sections, he withdrew his candidacy,
in order to avoid the humiliation of a loss, or a narrow win.
In return for his retreat, Poincaré's opponents endorsed his
candidacy to join the Académie Française, where he was elected
in 1908.
A second setback occurred in 1910. Following a highly-concerted
campaign, Poincaré garnered a record number of nominations
for the Nobel prize in physics, for the most part in view of
his advances in partial differential equations of mathematical
physics. Support for his candidacy issued largely from France
and Italy, with the British holding back, partly out of concern
over a second de facto extension of the prize domain, this
time to cover work in mathematical physics, only one year after
it had been awarded for applied physics (wireless telegraphy).
In the end, the Royal Swedish Academy chose to recognize the
work of another theorist, Johannes Diderik van der Waals.
While he did not win the Nobel prize, Poincaré cut an authoritative
figure among physicists, even in areas of physics he had ignored,
such as the theory of black-body radiation. In the fall of 1911,
he took an active part in the First Solvay Council, dedicated
to the discussion of problems in molecular and kinetic theory,
along with twenty of Europe's leading experimental and theoretical
physicists. A few weeks after the meeting, inspired by what he
had learned, he proved the quantum hypothesis to be a sufficient
and necessary condition for Planck's law.
Bibliography
For a complete bibliography of Poincaré's writings, a calendar
of his correspondence (with annotated transcriptions and digitized
images), and a list of secondary sources, consult Archives Henri
Poincaré, Laboratoire de Philosophie et d'Histoire des Sciences,
Centre National de la Recherche Scientifique, Nancy-Université,
available from www.univ-nancy2.fr/poincare/).
Works by Poincaré
Papers on Fuchsian Functions. Translated by John
Stillwell. Berlin: Springer, 1985.
Poincaré, Russell, Zermelo et Peano; Textes de la
discussion (1906-1912) sur les fondements des mathématiques:
Des antinomies à la prédicativité. Edited by Gerhard
Heinzmann. Paris: Blanchard, 1986.
L'Analyse et la recherche. Edited by Girolamo Ramunni. Paris:
Hermann, 1991.
New Methods of Celestial Mechanics. Edited by Daniel Goroff. Woodbury,
NY: American Institute of Physics, 1993.
Trois suppléments sur la découverte des fonctions
fuchsiennes. Edited by Jeremy Gray and Scott Walter. Berlin: Akademie Verlag, 1997.
La Correspondance entre Henri Poincaré et Gösta
Mittag-Leffler. Edited by Philippe Nabonnand. Basel:
Birkhäuser, 1999.
The Value of Science: Essential Writings of Henri Poincaré.
New York: Random House, 2001.
L'Opportunisme scientifique. Edited by Laurent Rollet. Basel:
Birkhäuser, 2002.
La Correspondance entre Henri Poincaré et les physiciens,
chimistes, et ingénieurs. Edited by Scott Walter, Etienne
Bolmont and André Coret. Basel: Birkhäuser, 2007.
Other Sources
Barrow-Green, June. Poincaré and the Three Body Problem.
Providence: American Mathematical Society, 1997.
Ben-Menahem, Yemima. "Convention: Poincaré and Some of His
Critics." British Journal for the Philosophy of Science
52 (2001): 471-513.
Darrigol, Olivier. "Henri Poincaré's Criticism of Fin de Siècle
Electrodynamics." Studies in History and Philosophy of Modern
Physics 26 (1995): 1-44.
Folina, Janet. Poincaré and the Philosophy of Mathematics.
London: Macmillan, 1992.
Galison, Peter. Einstein's Clocks and Poincaré's Maps:
Empires of Time. New York: Norton, 2003.
Giedymin, Jerzy. Science and Convention: Essays on Henri
Poincaré's Philosophy of Science and The Conventionalist Tradition.
Oxford: Pergamon, 1982.
Gilain, Christian. "La théorie qualitative de Poincaré et
le problème de l'intégration des équations difféntielles."
Cahiers d'histoire et de philosophie des sciences 34 (1991):
215-242.
Gray, Jeremy. Linear Differential Equations and Group Theory
from Riemann to Poincaré. 2d edn. Boston: Birkhäuser, 2000.
Greffe, Jean-Louis, Gerhard Heinzmann, and Kuno Lorenz, eds.
Henri Poincaré : Science et philosophie: Congrès
international, Nancy, France, 1994. Berlin: Akademie
Verlag, 1996.
Heinzmann, Gerhard. Zwischen Objektkonstruktion und Strukturanalyse:
Zur Philosophie der Mathematik bei Jules Henri Poincaré. Göttingen:
Vandenhoeck & Ruprecht, 1995.
Paty, Michel. Einstein philosophe: La physique comme pratique
philosophique. Paris: Presses Universitaires de France, 1993.
Despite its title, this volume contains a perceptive discussion
of Poincaré's view of relativity theory.
Prentis, Jeffrey J. "Poincaré's Proof of the Quantum
Discontinuity of Nature." American Journal of Physics 63
(1995): 339-350.
Rollet, Laurent. Henri Poincaré: Des mathématiques à
la philosophie. Lille: Éditions du Septentrion, 2001.
Sheynin, Oscar B. "H. Poincaré's Work on Probability."
Archive for History of Exact Sciences 42 (1991): 137-171.
Torretti, Roberto. Philosophy of Geometry from Riemann to
Poincaré. 2d edn. Dordrecht: Reidel, 1984.
Volkert, Klaus. Das Homöomorphismusproblem insbesondere
der 3-Mannigfaltigkeiten in der Topologie 1892-1935. Paris:
Kimé, 2002.
Walter, Scott. "Breaking in the 4-vectors: The Four-Dimensional
Movement in Gravitation, 1905-1910." In The Genesis of
General Relativity, vol. 3, edited by Jürgen Renn and Matthias
Schemmel. Berlin: Springer, 2007.
Zahar, Elie. Poincaré's Philosophy from Conventionalism
to Phenomenology. Chicago: Open Court, 2001.
Footnotes:
Henri-Poincaré Archives (CNRS), and
Department of Philosophy, University of Nancy.