Poincaré Jules Henri (1854–1912), French mathematician and influential philosopher of science. Born into a prominent family in Nancy, he showed extraordinary talent in mathematics from an early age. He studied at the École des Mines and worked as a mining engineer while completing his doctorate in mathematics (1879). In 1881, he was appointed professor at the University of Paris, where he lectured on mathematics, physics, and astronomy until his death. His original contributions to the theory of differential equations, algebraic topology, and number theory made him the leading mathematician of his day. He published almost five hundred technical papers as well as three widely read books on the philosophy of science: Science and Hypothesis (1902), The Value of Science (1905), and Science and Method (1908).
Poincaré’s philosophy of science was shaped by his approach to mathematics. Geometric axioms are neither synthetic a priori nor empirical; they are more properly understood as definitions. Thus, when one set of axioms is preferred over another for use in physics, the choice is a matter of ‘convention’; it is governed by criteria of simplicity and economy of expression rather than by which geometry is ‘correct.’ Though Euclidean geometry is used to describe the motions of bodies in space, it makes no sense to ask whether physical space ‘really’ is Euclidean. Discovery in mathematics resembles discovery in the physical sciences, but whereas the former is a construction of the human mind, the latter has to be fitted to an order of nature that is ultimately independent of mind. Science provides an economic and fruitful way of expressing the relationships between classes of sensations, enabling reliable predictions to be made. These sensations reflect the world that causes them; the (limited) objectivity of science derives from this fact, but science does not purport to determine the nature of that underlying world. Conventions, choices that are not determinable by rule, enter into the physical sciences at all levels. Such principles as that of the conservation of energy may appear to be empirical, but are in fact postulates that scientists have chosen to treat as implicit definitions. The decision between alternative hypotheses also involves an element of convention: the choice of a particular curve to represent a finite set of data points, e.g., requires a judgment as to which is simpler. Two kinds of hypotheses, in particular, must be distinguished. Inductive generalizations from observation (‘real generalizations’) are hypothetical in the limited sense that they are always capable of further precision. Then there are theories (‘indifferent hypotheses’) that postulate underlying entities or structures. These entities may seem explanatory, but strictly speaking are no more than devices useful in calculation. For atomic theory to explain, atoms would have to exist. But this cannot be established in the only way permissible for a scientific claim, i.e. directly by experiment. Shortly before he died, Poincaré finally allowed that Perrin’s experimental verification of Einstein’s predictions regarding Brownian motion, plus his careful marshaling of twelve other distinct experimental methods of calculating Avogadro’s number, constituted the equivalent of an experimental proof of the existence of atoms: ‘One can say that we see them because we can count them. . . . The atom of the chemist is now a reality.’ See also CONVENTIONALISM , PHILOSOPHY OF MATHEMATIC. E.M. polarity, the relation between distinct phenomena, terms, or concepts such that each inextricably requires, though it is opposed to, the other, as in the relation between the north and south poles of a magnet. In application to terms or concepts, polarity entails that the meaning of one involves the meaning of the other. This is conceptual polarity. Terms are existentially polar provided an instance of one cannot exist unless there exists an instance of the other. The second sense implies the first. Supply and demand and good and evil are instances of conceptual polarity. North and south and buying and selling are instances of existential polarity. Some polar concepts are opposites, such as truth and falsity. Some are correlative, such as question and answer: an answer is always an answer to a question; a question calls for an answer, but a question can be an answer, and an answer can be a question. The concept is not restricted to pairs and can be extended to generate mutual interdependence, multipolarity. See also MEANING, PHILOSOPHY OF LAN- GUAG. M.G.S.