Poincaré was born on April 29,1854 in Nancy and died on July 17, 1912
in Paris. Poincaré's family was influential. His cousin Raymond was
the President and the Prime Minister of France, and his father Leon
was a professor of medicine at the University of Nancy. His sister
Aline married the spiritualist philosopher Emile Boutroux.
Poincaré studied mining engineering, mathematics and physics in Paris.
Beginning in 1881, he taught at the University of Paris. There he held
the chairs of Physical and Experimental Mechanics, Mathematical
Physics and Theory of Probability, and Celestial Mechanics and
Astronomy.
At the beginning of his scientific career, in his doctoral
dissertation of1879, Poincaré devised a new way of studying the
properties of functions defined by differential equations. He not only
faced the question of determining the integral of such equations, but
also was the first person to study the general geometric properties of
these functions. He clearly saw that this method was useful in the
solution of problems such as the stability of the solar system, in
which the question is about the qualitative properties of planetary
orbits (for example, are orbits regular or chaotic?) and not about the
numerical solution of gravitational equations. During his studies on
differential equations, Poincaré made use of Lobachevsky's
non-Euclidean geometry. Later, Poincaré applied to celestial mechanics
the methods he had introduced in his doctoral dissertation. His
research on the stability of the solar system opened the door to the
study of chaotic deterministic systems; and the methods he used gave
rise to algebraic topology.
Poincaré sketched a preliminary version of the special theory of
relativity and stated that the velocity of light is a limit velocity
and that mass depends on speed. He formulated the principle of
relativity, according to which no mechanical or electromagnetic
experiment can discriminate between a state of uniform motion and a
state of rest, and he derived the Lorentz transformation. His
fundamental theorem that every isolated mechanical system returns
after a finite time [the Poincaré Recurrence Time] to its initial
state is the source of many philosophical and scientific analyses on
entropy. Finally, he clearly understood how radical is quantum
theory's departure from classical physics.
Poincaré was deeply interested in the philosophy of science and the
foundations of mathematics. He argued for conventionalism and against
both formalism and logicism. Cantor's set theory was also an object of
his criticism. He wrote several articles on the philosophical
interpretation of mathematical logic. During his life, he published
three books on the philosophy of science and mathematics. A fourth
book was published posthumously in 1913.
2. Chaos and the Solar System
In his research on the three-body problem, Poincaré became the first
person to discover a chaotic deterministic system. Given the law of
gravity and the initial positions and velocities of the only three
bodies in all of space, the subsequent positions and velocities are
fixed–so the three-body system is deterministic. However, Poincaré
found that the evolution of such a system is often chaotic in the
sense that a small perturbation in the initial state such as a slight
change in one body's initial position might lead to a radically
different later state than would be produced by the unperturbed
system. If the slight change isn't detectable by our measuring
instruments, then we won't be able to predict which final state will
occur. So, Poincaré's research proved that the problem of determinism
and the problem of predictability are distinct problems.
From a philosophical point of view, Poincaré's results did not receive
the attention that they deserved. Also the scientific line of research
that Poincaré opened was neglected until meteorologist Edward Lorenz,
in 1963, rediscovered a chaotic deterministic system while he was
studying the evolution of a simple model of the atmosphere. Earlier,
Poincaré had suggested that the difficulties of reliable weather
predicting are due to the intrinsic chaotic behavior of the
atmosphere. Another interesting aspect of Poincaré's study is the real
nature of the distribution in phase space of stable and unstable
points, which are so mixed that he did not try to make a picture of
their arrangement. Now we know that the shape of such distribution is
fractal-like. However, the scientific study of fractals did not begin
until Benoit Mandelbrot's work in 1975, a century after Poincaré's
first insight.
Why was Poincaré's research neglected and underestimated? The problem
is interesting because Poincaré was awarded an important scientific
prize for his research; and his research in celestial mechanics was
recognized to be of fundamental importance. Probably there were two
causes. Scientists and philosophers were primarily interested in the
revolutionary new physics of relativity and quantum mechanics, but
Poincaré worked with classical mechanics. Also, the behavior of a
chaotic deterministic system can be described only by means of a
numerical solution whose complexity is staggering. Without the help of
a computer the task is almost hopeless.
3. Arithmetic, Intuition and Logic
Logicists such as Bertrand Russell and Gottlob Frege believed that
mathematics is basically a branch of symbolic logic, because they
supposed that mathematical terminology can be defined using only the
terminology of logic and because, after this translation of terms, any
mathematical theorem can be shown to be a restatement of a theorem of
logic. Poincaré objected to this logicist program. He was an
intuitionist who stressed the essential role of human intuition in the
foundations of mathematics. According to Poincaré, a definition of a
mathematical entity is not the exposition of the essential properties
of the entity, but it is the construction of the entity itself; in
other words, a legitimate mathematical definition creates and
justifies its object. For Poincaré, arithmetic is a synthetic science
whose objects are not independent from human thought.
Poincaré made this point in his investigation of Peano's
axiomatization of arithmetic. Italian mathematician Giuseppe Peano
(1858-1932) axiomatized the mathematical theory of natural numbers.
This is the arithmetic of the nonnegative integers. Apart from some
purely logical principles, Peano employed five mathematical axioms.
Informally, these axioms are:
1. Zero is a natural number.
2. Zero is not the successor of any natural number.
3. Every natural number has a successor, which is a natural number.
4. If the successor of natural number a is equal to the successor
of natural number b, then a and b are equal.
5. Suppose:
(i) zero has a property P;
(ii) if every natural number less than a has the property P then
a also has the property P.
Then every natural number has the property P. (This is the
principle of complete induction.)
Bertrand Russell said Peano's axioms constitute an implicit definition
of natural numbers, but Poincaré said they do only if they can be
demonstrated to be consistent. They can be shown consistent only by
showing there is some object satisfying these axioms. From a general
point of view, an axiom system can be conceived of as an implicit
definition only if it is possible to prove the existence of at least
one object that satisfies all the axioms. Proving this is not an easy
task, for the number of consequences of Peano axioms is infinite and
so a direct inspection of each consequence is not possible. Only one
way seems adequate: we must verify that if the premises of an
inference in the system are consistent with the axioms of logic, then
so is the conclusion. Therefore, if after n inferences no
contradiction is produced, then after n+1 inferences no contradiction
will be either. Poincaré argues that this reasoning is a vicious
circle, for it relies upon the principle of complete induction, whose
consistency we have to prove. (In 1936, Gerhard Gentzen proved the
consistency of Peano axioms, but his proof required the use of a
limited form of transfinite induction whose own consistency is in
doubt.) As a consequence, Poincaré asserts that if we can't
noncircularly establish the consistency of Peano's axioms, then the
principle of complete induction is surely not provable by means of
general logical laws; thus it is not analytic, but it is a synthetic
judgment, and logicism is refuted. It is evident that Poincaré
supports Kant's epistemological viewpoint on arithmetic. For Poincaré,
the principle of complete induction, which is not provable via
analytical inferences, is a genuine synthetic a priori judgment. Hence
arithmetic cannot be reduced to logic; the latter is analytic, while
arithmetic is synthetic.
The synthetic character of arithmetic is also evident if we consider
the nature of mathematical reasoning. Poincaré suggests a distinction
between two different kinds of mathematical inference: verification
and proof. Verification or proof-check is a sort of mechanical
reasoning, while proof-creation is a fecund inference. For example,
the statement '2+2 = 4′ is verifiable because it is possible to
demonstrate its truth with the help of logical laws and the definition
of sum; it is an analytical statement that admits a straightforward
verification. On the contrary, the general statement (the commutative
law of addition)
For any x and any y, x + y = y + x
is not directly verifiable. We can choose an arbitrary pair of natural
numbers a and b, and we can verify that a+b = b+a; but there is an
infinite number of admissible choices of pairs, so the verification is
always incomplete. In other words, the verification of the commutative
law is an analytical method by means of which we can verify every
particular instance of a general theorem, while the proof of the
theorem itself is synthetic reasoning which really extends our
knowledge, Poincaré believed.
Another aspect of mathematical thinking that Poincaré analyzes is the
different roles played by intuition and logic. Methods of formal logic
are elementary and certain, and we can surely rely on them. However,
logic does not teach us how to build a proof. It is intuition that
helps mathematicians find the correct way of to assemble basic
inferences into a useful proof. Poincaré offers the following example.
An unskilled chess player who watches a game can verify whether a move
is legal, but he does not understand why players move certain pieces,
for he does not see the plan which guides players' choices. In a
similar way, a mathematician who uses only logical methods can verify
every inference in a given proof, but he cannot find an original
proof. In other words, every elementary inference in a proof is easily
verifiable through formal logic, but the invention of a proof requires
the understanding — grasped by intuition — of the general scheme,
which directs mathematician's efforts towards the final goal.
Logic is — according to Poincaré — the study of properties which are
common to all classifications. There are two different kinds of
classifications: predicative classifications, which are not modified
by the introduction of new elements; and impredicative
classifications, which are modified by new elements. Definitions as
well as classifications are divided into predicative and
impredicative. A set is defined by a law according to which every
element is generated. In the case of an infinite set, the process of
generating elements is unfinished; thus there are always new elements.
If their introduction changes the classification of already generated
objects, then the definition is impredicative. For example, look at
phrases containing a finite number of words and defining a point of
space. These phrases are arranged in alphabetical order and each of
them is associated with a natural number: the first is associated with
number 1, the second with 2, etc. Hence every point defined by such
phrases is associated with a natural number. Now suppose that a new
point is defined by a new phrase. To determine the corresponding
number it is necessary to insert this phrase in alphabetical order;
but such an operation modifies the number associated with the already
classified points whose defining phrase follows, in alphabetical
order, the new phrase. Thus this new definition is impredicative.
For Poincaré, impredicative definitions are the source of antinomies
in set theory, and the prohibition of impredicative definitions will
remove such antinomies. To this end, Poincaré enunciates the vicious
circle principle: a thing cannot be defined with respect to a
collection that presupposes the thing itself. In other words, in a
definition of an object, one cannot use a set to which the object
belongs, because doing so produces an impredicative definition.
Poincaré attributes the vicious circle principle to French
mathematician J. Richard. In 1905, Richard discovered a new paradox in
set theory, and he offered a tentative solution based on the vicious
circle principle.
Poincaré's prohibition of impredicative definitions is also connected
with his point of view on infinity. According to Poincaré, there are
two different schools of thought about infinite sets; he called these
schools Cantorian and Pragmatist. Cantorians are realists with respect
to mathematical entities; these entities have a reality that is
independent of human conceptions. The mathematician discovers them but
does not create them. Pragmatists believe that a thing exists only
when it is the object of an act of thinking, and infinity is nothing
but the possibility of the mind's generating an endless series of
finite objects. Practicing mathematicians tend to be realists, not
pragmatists or intuitionists. This dispute is not about the role of
impredicative definitions in producing antinomies, but about the
independence of mathematical entities from human thinking.
4. Conventionalism and the Philosophy of Geometry
The discovery of non-Euclidean geometries upset the commonly accepted
Kantian viewpoint that the true structure of space can be known
apriori. To understand Poincaré's point of view on the foundation of
geometry, it helps to remember that, during his research on functions
defined by differential equations, he actually used non-Euclidean
geometry. He found that several geometric properties are easily
provable by means of Lobachevsky geometry, while their proof is not
straightforward in Euclidean geometry. Also, Poincaré knew Beltrami's
research on Lobachevsky's geometry. Beltrami (Italian mathematician,
1835-1899) proved the consistency of Lobachevsky geometry with respect
to Euclidean geometry, by means of a translation of every term of
Lobachevsky geometry into a term of Euclidean geometry. The
translation is carefully chosen so that every axiom of non-Euclidean
geometry is translated into a theorem of Euclidean geometry.
Beltrami's translation and Poincaré's study of functions led Poincaré
to assert that:
* Non-Euclidean geometries have the same logical and mathematical
legitimacy as Euclidean geometry.
* All geometric systems are equivalent and thus no system of
axioms may claim that it is the true geometry.
* Axioms of geometry are neither synthetic a priori judgments nor
analytic ones; they are conventions or 'disguised' definitions.
According to Poincaré, all geometric systems deal with the same
properties of space, although each of them employs its own language,
whose syntax is defined by the set of axioms. In other words,
geometries differ in their language, but they are concerned with the
same reality, for a geometry can be translated into another geometry.
There is only one criterion according to which we can select a
geometry, namely a criterion of economy and simplicity. This is the
very reason why we commonly use Euclidean geometry: it is the
simplest. However, with respect to a specific problem, non-Euclidean
geometry may give us the result with less effort. In 1915, Albert
Einstein found it more convenient, the conventionalist would say, to
develop his theory of general relativity using non-Euclidean rather
than Euclidean geometry. Poincaré's realist opponent would disagree
and say that Einstein discovered space to be non-Euclidean.
Poincaré's treatment of geometry is applicable also to the general
analysis of scientific theories. Every scientific theory has its own
language, which is chosen by convention. However, in spite of this
freedom, the agreement or disagreement between predictions and facts
is not conventional but is substantial and objective. Science has an
objective validity. It is not due to chance or to freedom of choice
that scientific predictions are often accurate.
These considerations clarify Poincaré's conventionalism. There is an
objective criterion, independent of the scientist's will, according to
which it is possible to judge the soundness of the scientific theory,
namely the accuracy of its predictions. Thus the principles of science
are not set by an arbitrary convention. In so far as scientific
predictions are true, science gives us objective, although incomplete,
knowledge. The freedom of a scientist takes place in the choice of
language, axioms, and the facts that deserve attention.
However, according to Poincaré, every scientific law can be analyzed
into two parts, namely a principle, that is a conventional truth, and
an empirical law. The following example is due to Poincaré. The law:
Celestial bodies obey Newton's law of gravitation
The law consists of two elements:
1. Gravitation follows Newton law.
2. Gravitation is the only force that acts on celestial bodies.
We can regard the first statement as a principle, as a convention;
thus it becomes the definition of gravitation. But then the second
statement is an empirical law.
Poincaré's attitude towards conventionalism is illustrated by the
following statement, which concluded his analysis on classical
mechanics in Science and Hypothesis:
Are the laws of acceleration and composition of forces nothing but
arbitrary conventions? Conventions, yes; arbitrary, no; they would
seem arbitrary if we forgot the experiences which guided the founders
of science to their adoption and which are, although imperfect,
sufficient to justify them. Sometimes it is useful to turn our
attention to the experimental origin of these conventions.
5. Science and Hypothesis
According to Poincaré, although scientific theories originate from
experience, they are neither verifiable nor falsifiable by means of
the experience alone. For example, look at the problem of finding a
mathematical law that describes a given series of observations. In
this case, representative points are plotted in a graph, and then a
simple curve is interpolated. The curve chosen will depend both on the
experience which determines the representative points and on the
desired smoothness of the curve even though the smoother the curve the
more that some points will miss the curve. Therefore, the interpolated
curve — and thus the tentative law — is not a direct generalization of
the experience, for it 'corrects' the experience. The discrepancy
between observed and calculated values is thus not regarded as a
falsification of the law, but as a correction that the law imposes on
our observations. In this sense, there is always a necessary
difference between facts and theories, and therefore a scientific
theory is not directly falsifiable by the experience.
For Poincaré, the aim of the science is to prediction. To accomplish
this task, science makes use of generalizations that go beyond the
experience. In fact, scientific theories are hypotheses. But every
hypothesis has to be continually tested. And when it fails in an
empirical test, it must be given up. According to Poincaré, a
scientific hypothesis which was proved untenable can still be very
useful. If a hypothesis does not pass an empirical test, then this
fact means that we have neglected some important and meaningful
element; thus the hypothesis gives us the opportunity to discover the
existence of an unforeseen aspect of reality. As a consequence of this
point of view about the nature of scientific theories, Poincaré
suggests that a scientist must utilize few hypotheses, for it is very
difficult to find the wrong hypothesis in a theory which makes use of
many hypotheses.
For Poincaré, there are many kinds of hypotheses:
* Hypotheses which have the maximum scope, and which are common to
all scientific theories (for example, the hypothesis according to
which the influence of remote bodies is negligible). Such hypotheses
are the last to be changed.
* Indifferent hypotheses that, in spite of their auxiliary role in
scientific theories, have no objective content (for example, the
hypothesis that unseen atoms exist).
* Generalizations, which are subjected to empirical control; they
are the true scientific hypotheses.
Regarding Poincaré's point of view about scientific theories, the
following have the most lasting value:
* Every scientific theory is a hypothesis that had to be tested.
* Experience suggests scientific theories; but experience does not
justify them.
* Experience alone is unable to falsify a theory, for the theory
often corrects the experience.
* A central aim of science is prediction.
* The role of a falsified hypothesis is very important, for it
throws light on unforeseen conditions.
* Experience is judged according to a theory.
6. Bibliography
COLLECTED SCIENTIFIC WORKS (in French).
* Oeuvres, 11 volumes, Paris : Gauthier-Villars, 1916-1956
PHILOSOPHICAL WORKS.
* 1902 La science et l'hypothèse, Paris : Flammarion (Science and
hypothesis, 1905)
* 1905 La valeur de la science, Paris : Flammarion (The value of
science, 1907)
* 1908 Science and méthode, Paris : Flammarion (Science and method, 1914)
* 1913 Dernières pensées, Paris : Flammarion (Mathematics and
science: last essays, 1963)
* The first three works are translated in The foundations of
science, Washington, D.C. : University Press of America, 1982 (first
edition 1946).
MAIN SCIENTIFIC WORKS.
* Les méthods nouvelles de la mécanique céleste, Paris :
Gauthier-Villars, 1892 vol. I , 1893 vol. II, 1899 vol. III (New
methods of celestial mechanics, American Institute of Physics, 1993)
* Lecons de mécanique céleste, Paris : Gauthier-Villars, 1905 vol.
I, 1907 vol. II part I, 1909 vol. II part II, 1911 vol. III
WORKS ABOUT POINCARE'.
* Le livre du centenaire de la naissance de Henri Poincaré, Paris
: Gauthier-Villars, 1955
* The mathematical heritage of Henri Poincaré, (edited by Felix E.
Browder) Providence, R.I. : American Mathematical Society, 1983
[Symposium on the Mathematical Heritage of Henri Poincaré (1980 :
Indiana University, Bloomington)]
* Henri Poincaré: Science et philosophie. Congrès international :
Nancy, France, 1994, edited by Jean-Louis Greffe, Gerhard Heinzmann,
Kuno Lorenz, Berlin : Akademie Verlag, 1996 ; Paris : A. Blanchard,
1996
* Appel, Paul, Henri Poincaré, Paris : Plon, 1925
* Bartocci, Claudio, "Equazioni e orbite celesti: gli albori della
dinamica topologica" in Henri Poincaré. Geometria e caso, Torino :
Bollati Boringhieri, 1995
* Barrow-Green, June, Poincaré and the three body problem,
Providence, RI : American Mathematical Society ; London : London
Mathematical Society, 1997
* Dantzig, Tobias, Henri Poincaré. Critic of crisis: reflections
on his universe of discourse, New York : Scriber, 1954
* Folina, Janet, Poincaré and the philosophy of mathematics,
London : Macmillan, 1992 ; New York : St. Martin's Press, 1992
* Giedymin, Jerzy, Science and convention. Essay on Henri
Poincaré's philosophy of science and the conventionalist tradition,
Oxford : Pergamon Press, 1982
* Heinzmann, Gerhard, Entre intuition et analyse : Poincaré et le
concept de prédicativité, Paris : A. Blanchard, 1985
* Heinzmann, Gerhard, Zwischen Objektkonstruktion und
Strukturanalyse. Zur Philosophie der Mathematik bei Poincaré,
Vandenhoek & Ruprecht, 1995
* de Lorenzo, Javier, La filosofia de la matematica de Jules Henri
Poincaré, Madrid : Editorial Tecnos, 1974
* Mette, Corinna, Invariantentheorie als Grundlage des
Konventionalismus : Uberlegungen zur Wissenschaftstheorie von Poincaré
, Essen : Die Blaue Eule, 1986, Essen, 1986
* Mooij, Jan, La philosophie des mathématiques de Henri Poincaré,
Paris : Gauthier-Villars, 1966
* Parrini, Paolo, Empirismo logico e convenzionalismo, Milano :
Franco Angeli, 1983
* Rougier, Luis, La philosophie géométrique de Henri Poincaré,
Paris : Alcan, 1920
* Schmid, Anne-Francoise, Une philosophie de savant : Henri
Poincaré et la logique mathématique, Paris : F. Maspero, 1978.
* Torretti, Roberto, Philosophy of geometry from Riemann to
Poincaré, Dordrecth : D. Reidel Pub. Co., 1978
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