Jules Henri Poincaré (1854-1912) |
Jules Henri Poincaré was a mathematician, physicist, and philosopher of science. He is best
known to philosophers for his
forceful development of the philosophical doctrine of conventionalism.
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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.
Arithmetic, Intuition and Logic
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 + xis 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.
Conventionalism and the Philosophy of Geometry.
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 gravitationThe law consists of two elements:
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.
For Poincaré, there are many kinds of hypotheses:
Regarding Poincaré's point of view about scientific theories, the following have the most lasting value:
COLLECTED SCIENTIFIC WORKS (in French).
Oeuvres, 11 volumes, Paris : Gauthier-Villars, 1916-1956
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)
Le livre du centenaire de la naissance de Henri Poincaré, Paris : Gauthier-Villars, 1955
Appel, Paul, Henri Poincaré, Paris : Plon, 1925
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
Mauro Murzi
Email: murzim@yahoo.com