Tilly: Pencaré's conventionism

The French mathematician and physicist Henri Penghalet (1854-1912) developed a positivist and conventionalist interpretation of science, very similar to Mach's positivist theory. According to Pengaller, the basic assumptions of science are convenient definitions or conventions, which are neither obtained by a priori methods nor generalized by induction from experience; our choices among the conventions of possibilities, although inspired and guided by experimental facts, are ultimately governed by simple and convenient considerations. Pengall makes a strict distinction between two main scientific hypotheses: (1) the first type of hypotheses are inherently unprovable; they are the product of the free activity of the mind, imposed on the scientific system by the scientific spirit. Although they can neither be confirmed by experience nor refuted by experience, some of these assumptions are indispensable to any scientific theory conceived in a broad sense. (2) The second type of hypothesis is the usual inductive generalization, and they are valuable because they can be confirmed or falsified by experimental procedures. Scientific theories would encompass both types of assumptions. Poincarle paid more attention to the first type, the one that could not be confirmed by itself, because he believed that students who studied scientific methodology would usually ignore them; while the second verifiable type had been extensively studied by empiricist logicians and methodists from Bacon to J.S. Mill. Pengaller made a full and enlightening account of the nature and function of hypotheses that could not be proven in the body of scientific knowledge. He insisted that such a hypothesis, though not empirically confirmed, was inspired by experience and derived its value from the results of the scientific interpretation of experience. Facts derived from experience can be absorbed into one of an infinite number of hypothetical constructs to choose from; each construct is the product of the free activity of the mind, and the choices made between them are made for the sake of convenience. Thus, the unverified hypothesis is indeed conventional, but not arbitrary: "Experience leaves us with the freedom of choice, but it guides us through the most convenient means of assisting us in insight." "Poincaré's conventionism is far less extreme than Feinger's fictionalism. Feiinger's fictitious constructions are self-contradictory and "contradictory" to reality, while Pengalle's hypothesis of convention is inherently consistent and does not contradict facts, because facts can neither refute nor confirm them.

Of the infinite number of alternative hypotheses that the observed facts can incorporate into them, what controls our choice of one? This question is crucial to conventionism, and Poincaré answers unequivocally — simply. "Of all the possible summaries, we must choose, and can only choose the simplest one. Thus, we are led to action, and other conditions are equal, as if the law of simplicity were more probabilistic than the law of complexity. "We choose the simplest law not because nature loves simplicity, and therefore the simplest is correct in the objective sense, but purely out of the consideration of the economy of thought. The perverted preference to put complex hypotheses on top of simple hypotheses can frustrate scientific enterprises. "When we work out universal, simple, and precise laws based on relatively few experiments... We are merely following an inevitability from which the human mind cannot escape. ”

Pencaré applied his conventionism to two sciences he excelled in: mathematics and physics. In explaining the foundations of mathematics, Poincaré was opposed to both empiricism and rationalism. All forms of transcendentalism in history are unsatisfactory: the axioms of geometry are not a priori intuitions, as Descartes claimed; mathematics cannot be deduced from the principle of contradiction alone through analysis, as Leibniz tried; and Kant's efforts to argue mathematics as a transcendental synthetic truth system based on purely space-time intuition were unsuccessful. Poincar found Müller's empiricist account of mathematics equally unacceptable: the axioms of geometry were not inductive generalizations about the properties of perceptual space. To be sure, experience "plays an indispensable role in the origin of geometry; but it is wrong to conclude that geometry, even in part, is an experimental science." "If it is experimental, it can only be approximate and temporary." Poincaré found another real alternative for empiricism and rationalism in conventionism: the axioms of geometry are axiomatoms, that is, these assumptions are accepted not because they are true, but because they are convenient. Experience "does not tell us which is the most genuine geometry; but which is the most convenient"; thus, although geometry is not an experimental science, it is a science that is "born to correspond to experience". It was in the non-Euclidean system of geometry that Pengall found the proof of his postulatory explanation of mathematics: the observed phenomena could be integrated into both Euclidean geometry and non-Euclidean geometry; he said, it was impossible to imagine a real experiment that could only be explained by the Euclidean system, but not in Lobachevsky's system—a non-Euclidean geometric system. No experience contradicts Euclid's postulate; on the other hand, no experience contradicts Lobachevsky's postulate. Any postulate of geometry, whether Euclidean or non-Euclidean, cannot be proved or refuted; they will be understood as unverifiable (nor falsifiable) types, adopted for reasons of simplicity and convenience.

In his remarkable and enlightening paper, The Value of Science, Pengarh writes: "Mathematics has three purposes. It must provide tools for studying nature. But that's not all: it has a philosophical purpose, I dare assert, and an aesthetic purpose ... His scientific function is to provide us with a simple, precise and economical language for expressing natural knowledge; ordinary language is too clumsy and too vague to express such a rich, precise, and delicate relationship. "The philosophical function of mathematics is to facilitate the philosopher's study of numbers, space, time, quantity, and related categories. But most importantly, the aesthetic value inherent in mathematics, which Poincaré highly admired. "The pleasures that math lovers find in it are comparable to the pleasures derived from music and painting. They worship the subtle harmony of numbers and forms; they are amazed by the unexpected perspectives that new discoveries open up; and although the senses are not involved, is the pleasure thus felt unimpressive? In Science and Method, there is a chapter entitled "The Creation of Mathematics" that is so brilliant that Pengaller presents his profound insight into the nature of the scientist's creative process. He describes, with great psychological acumen, a mathematical discovery of his own; the treatise elucidates the long preparatory stage required for mathematical creation, the role of the unconscious process in gaining final insight, the importance of the free play of analogy, intuition, and imagination, and ultimately the aesthetic satisfaction and the almost mystical delight that accompanies the final result.

Physics, and mechanics in particular, is, in Pengaller's view, the second field of science in which the conventionist hypothesis plays an indispensable role. He judged and examined the basic concepts and fundamental assumptions of the classical mechanics of Newton and Galileo and concluded that they were to a considerable extent conventional. Fundamental assumptions such as force, inertia, absolute space, and absolute time are conventional concepts in terms of neither being proven nor falsified. Pengaller's conventionism, in physics as in mathematics, offers a third important possibility beyond the conflict between traditional empiricism and rationalism. Conventionism provides a credible description of the epistemological character of the basic axioms of mathematics and the fundamental assumptions of the natural sciences, and avoids the arrogance of the authoritarian nature of the a priori truth theory and the Gairan theory of the posterior theory. Conventionism aims to combine the precision and rigor of rationalism with the experimental richness of empiricism.

Science and Hypothesis, 1902, English translation of Science and Hypothesis, 1914; The Value of Science, 1905, English translation of The Value of Science, 1907; Science and Method, 1909, English translation of Science and Method, 1914. The English translation of all of the above works was included in a single volume by G.B. Halsted, entitled The Foundations of Science, 1946.