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"[T]he untrammeled reflections of a broad and cultivated mind upon the procedure and the postulates of scientific discovery." - Bertrand Russell
Henri Poincaré's Science and Method (1908) was one of the most talked-about books in turn-of-the-century France, and it remains a fine example of the author's philosophical scholarship. Starting from first principles and written in Poincaré's unique and inimitable style, the book encompasses a wide variety of methodological topics in science. It also contains Poincaré's personal views on much-discussed contemporary issues such as the theory of relativity, the applicability of the calculus of probability in physics, and the foundations of mathematics. Few would disagree that it is the clarity of the exposition that makes Poincaré's writing so compelling and impressive, as it conveys even the most complex idea in a down-to-earth and unaffected manner.
About the Author:
Born in 1854 to a distinguished family whose members included the prime minister and president of France, Jules Henri Poincaré excelled in mathematics from early childhood. The young "Monster of Math," as his schoolteacher used to call him, was soon to become one of the greatest minds in turn-of-the-century France. In philosophy he shall be always considered the godfather of conventionalism and logical positivism, but for the general public he shall remain the scholar who turned his gifts to describing the meaning of science and mathematics to the layman.
"Mathematicians are born, not made," said the man who is credited for many of the intellectual discoveries in mathematical physics that continue to astonish us even today, hisown life exemplifying the truth in those words. Born in 1854 to a distinguished family whose members included the prime minister and president of France, Jules Henri Poincaré excelled in mathematics from early childhood. The young "Monster of Math," as his schoolteacher used to call him, was soon to become one of the greatest minds in turn-of-the-century France and, along with the German David Hilbert, the most important mathematician of that era. It is hard to overestimate Poincaré's contributions made during his lifetime to fields so diverse as topology, mathematical physics, and celestial mechanics; his popular essays on the foundations of mathematics and the philosophy of science - the third of which is Science and Method - were equally influential and provocative. For many physicists, Poincaré will be remembered along with Lorentz and Fitzgerald as the father of one of the predecessors of the special theory of relativity; for many mathematicians, as the founder of chaos theory, nonlinear dynamics, and algebraic topology. In philosophy he shall be always considered the godfather of conventionalism and logical positivism, but for the general public he shall remain a famous and opinionated scholar who turned his gifts to describing the meaning and importance of science and mathematics to the layman.
Poincaré's three philosophical works - Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908) - reached a wide public of non-professionals and were immediately translated into German and English. Written in a period of less than eight years, they differ slightly in their aim and scope. The first is mainly a conventionalist manifesto and presents Poincaré's views on the conventionality of geometry and the role played by mathematical concepts and by hypotheses and experiments in science and in physics. The second is a collection of Poincaré's thoughts on mathematical creation and includes an account of one of his greatest mathematical discoveries. Science and Method complements them by responding to the criticism they raised and by presenting an opinionated view on the methodology of science, on the newly discovered special theory of relativity, and on the role that the calculus of probability plays in scientific theories. Apart from their main themes, what unites the three works is that they are all a joy to read. For in addition to his uncanny mathematical gifts, Poincaré had the talent of expressing himself beautifully in writing. Even in translation, his prose has an admirable transparency and grace, and his aphoristic style often makes him highly quotable.
Published in an epoch that saw the overthrow of the Newtonian worldview and the first cracks in the solid foundations of mathematics, Poincaré's philosophical writings capture the flavor of this fascinating period of transition in science, and they are highly informed by contemporary scientific and philosophical discussions on topics such as the existence of the ether; the debate on the deterministic and reversible character of the fundamental laws viz. a viz. the discovery of the subatomic world and the development of thermodynamics and the kinetic theory of gases; and the suspicion towards Bertrand Russell's and Gotlob Frege's ambitious set-theoretic approach in the foundations of mathematics. Since Poincaré was personally involved in these discussions, and apparently was aware of the writings of other key figures such as Maxwell, Lorentz, Fitzgerald, and Russell, who also participated in them, Science and Method is best appreciated in the context of the academic turmoil that these debates generated.
But here a word of caution is merited. Although Poincaré is recognized as a genius in math, some of his views on physics are now considered misguided. The discovery of the general theory of relativity, for example, made his conventionalist views on the nature of the geometry of space unsustainable. But even with respect to the special theory of relativity he advocated what would be considered today a radical view. Like Lorentz and Fitzgerald and contrary to Einstein, he did not actually reject the concept of 'ether,' the physically preferred frame of reference that dominated nineteenth-century physics, and he saw the key relativistic effects (space contraction and time dilation) as physical effects of the motion with respect to it instead of a result of spatio-temporal transformations. Physics cognoscenti will surely notice this when reading Science and Method, but they would also appreciate the remarkable fact that in some of his writings Poincaré anticipated Einstein's work. The case of the special theory of relativity is well known: Poincaré is responsible, along with Einstein, for the introduction of the relativity principle and the relativity of simultaneity, the formulation of the equation of the relativistic mechanics and transformation laws for the electromagnetic field and current, and the establishment of the Lorentz group as a symmetry group of nature. But while these could reflect individual, albeit convergent, thinking, his influence on Einstein is rarely acknowledged, even by Einstein himself, who commented on Poincaré's views in his famous article "Geometry and Experience" (1921). Poincaré's seemingly skeptical equivalence arguments and their geometrical implications (in Science and Method they appear in Book II, Chapter I, Section I) clearly influenced Einstein's formulation of his equivalence principle and were used by Einstein to wrest striking empirical consequences which later led to what he famously called "the most beautiful idea of my life," namely, the conception of the general theory of relativity. Science and Method is where Poincaré turns some of his discoveries into methodological and pedagogical tools - the misguided conclusions he had drawn from them notwithstanding.
Over and above the sections on the relativity of space and the conventionality of geometry, in which Poincaré re-expresses his neo-Kantian views first propounded in Science and Hypothesis (1901), a substantial part of the book is dedicated to the calculus of probability and its application in physics. Toward end of the nineteenth century the deterministic arrogance of Newtonian mechanics was threatened by the kinetic theory of gases and the discovery of the subatomic world. Motivated by the revival of the atomic hypothesis, Maxwell and Boltzmann, Poincaré's contemporaries, were introducing statistical and probabilistic posits into the equations of motion in order to account for thermodynamic phenomena with dynamical models based on classical (Newtonian) mechanics. The challenge Poincaré was responding to in classic chapters such as "Chance" (a reprint of Poincaré's article "Le Hasard" that was published a year earlier) was to square these probabilistic posits that were necessary for the success of the aforementioned models in predicting thermodynamic phenomena such as approach to thermodynamic equilibrium with the deterministic character of the underlying mechanical laws. In his response Poincaré insists on a distinction between determinism and predictability, and with the help of the most prosaic examples (Roulette wheels, weather predictions, asteroids distribution, and the attempt to balance a cone on its apex) demonstrates how the law of large numbers can be applied in physics, and how 'chance' can be defined in an objective manner in terms of dynamical properties.
Many today regard this classic chapter in which Poincaré redefines the notion of chance in physics as containing the first conception of chaos theory, and Poincaré himself as the grandfather of the flourishing science of nonlinear dynamics. While it is true that the chapter includes one of Poincaré's most quotable phrases on the notion of exponential sensitivity to the initial conditions which is the hallmark of chaos (". . .it may happen that small differences in the initial conditions produce very great ones in the final phenomena"), the four examples Poincaré presents, the sensitivity to the initial conditions they involve, and the unpredictability with which they are associated are all practical in character and hence are much less threatening to determinism. In fact, Poincaré sowed the seeds of chaos elsewhere in yet another great discovery of his in the domain of celestial mechanics, i.e., in his solution to the three-body problem.
Poincaré had won a prestigious prize sponsored by King Oscar II of Sweden and Norway for a paper on the three-body problem: the problem of determining the motion of three bodies (idealized as material points) in an otherwise empty space under Newtonian gravitation. Although plain wrong on the critical matter and hastily revised by the author only after publication, Poincaré's solution to the problem appeared in 1890 in the young journal Acta Mathematica and is now understood as the prime example of chaotic behavior. Remarkably, the prize fiasco went unnoticed until the late twentieth century, and Poincaré's results themselves did not receive the attention that they deserved. In fact, the scientific line of research that Poincaré opened was neglected until in late 1963 meteorologist Edward Lorenz rediscovered a chaotic deterministic system while he was studying the evolution of a simple model of the atmosphere. Another interesting aspect of Poincaré's study which concerned the real nature of the mixing distribution in phase space (the abstract space by which physicists describe a dynamical system) of stable and unstable points (known now to be fractal-like) did not begin until Benoit Mandelbrot's work in 1975, a century after Poincaré's initial insight.
Given that Poincaré was awarded the prize for his discovery, the question arises of why his research on chaos was neglected. Two theories pose possible answers. First, scientists and philosophers were primarily interested in the revolutionary new physics of relativity and quantum mechanics, while Poincaré's results belonged to classical Newtonian mechanics. Second, the description of chaotic deterministic behavior requires numerical solutions whose complexity is incredible. Without the help of a computer the task is almost hopeless.
Another substantive part of Science and Method is devoted to Poincaré's attack on logicism in the foundations of mathematics. Around the turn of the century, Poincaré and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts, claiming as they did that the terms point, line, and plane can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist - Russell against Poincaré and Frege against Hilbert - who maintained the dying view that geometry essentially concerns space or spatial intuition. But in the debate on the nature of mathematical reasoning in general Poincaré and Russell switched roles. Siding with Kant, Poincaré argued that mathematical reasoning is characteristically non-logical. Russell advocated the contrary view, maintaining that the plausibility originally enjoyed by Kant's view was due primarily to the underdeveloped state of logic in his (i.e., Kant's) time, and that with the aid of recent developments in logic, it is possible to demonstrate its falsity.
Russell relied on Frege's ambitious attempts to reduce mathematics to logic and to set theory, but the discovery of the famous set theoretic paradoxes (to which Poincaré refers as 'antinomies') such as Russell's paradox or Burali-Forti's paradox demanded a revision of strategy. In Science and Method Poincaré presents a lucid attack on Russell's Principia and on Zermelo's axiomatic system - two famous responses to the paradoxes - and in so doing anticipates the rise of the intuitionist school in mathematics that was created in 1923 by the Dutch mathematician Luitzen Brouwer (who had already published these ideas in his Ph.D. thesis in 1907 but abandoned them for almost two decades).
The basic claim of Poincaré and the intuitionist school that followed him is that mathematical demonstrations and proofs must be constructive. The existence of irrational real numbers, for example, can be proven only by giving a concrete construction of such a number, e.g., Ö2, with the Pythagoras theorem. The alternative, i.e., the proof that such a number exists from elementary premises of set theory, leaves open the question of how to find it, and hence is unwarranted. For intuitionists, mathematicians are architects and engineers rather than explorers; their theorems are of their own making, and the tools they can use are correspondingly limited to those appropriate for construction. Poincaré's first-person experience with mathematical discovery (Book I, Chapter I, Section III), his rejection of the problematic axiom of choice that Zermelo embraced ("There is no actual infinity"), and his famous remark "Logic therefore remains barren, unless it is fertilized by intuition" (Book II, Chapter II, Sections X, XI) epitomize his distaste for Russell's logicist approach. It's no wonder that Russell in his preface to the first English edition of Science and Method dismisses Poincaré's criticisms of mathematical logic and remarks that they "do not appear to me to be among the best parts of his work," and that "he was already an old man when he became aware of this subject…."
Poincaré is famous for saying that "[t]o doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection." Reading him is a fascinating experience, and his reflections, right or wrong, express the fruits of an original and unique mind. For this reason, if no other, Science and Method shall remain an enduring classic in the philosophy of science.
Amit Hagar is a philosopher of physics with a Ph.D. from the University of British Columbia, Vancouver. His area of specialization is the conceptual foundations of modern physics, especially in the domains of statistical and quantum mechanics.