Blaise Pascal’s methods

Blaise Pascal (1623-1662) is not a philosopher: he is a scholar and an apologist for the Catholic religion. A scholar, he is in the tradition of mathematical and experimental physics which led from Galileo to Newton. An apologist, he is not one of those who prelude, in his response to the libertines, by demonstrating through reason all those truths of faith that can be demonstrated: it is in history, it is in the human nature taken as a whole that he seeks its evidence, just as he seeks in experience and not in reasoning the proof of a physical truth. Descartes was also a scholar and, to some extent, an apologist: but his genius forbade him to be both, unless he was at the same time a philosopher, unless he brought science and apology into the “chain reasons” outside of which he left the truths of faith. Pascal’s genius, on the contrary, forbids him to bring science and apologia into the fabric of a philosophy, and to leave the truths of faith out of consideration: There is between these men, almost contemporaries, an opposition so profound, so moving too, that no doubt in history can shed more light on the nature of the human spirit.

From his Essay touching conics, which Pascal barely wrote after childhood (1639), a characteristic witticism is revealed: to a precise problem (to seek the principle from which all the properties of conics), he responds by the invention of a precise method, capable of solving this problem, and this problem only. Pascal discovered that all the properties of conics depended on the invention of a certain hexagon, which he called the mystical hexagram. Each problem thus requires an effort of invention each time renewed, where the mathematician has the talent to discover precisely the notions and principles which are useful to him. This is how, later, Pascal will show that, to find the center of gravity of the roulette wheel and the surfaces or volumes which depend on this curve, it is necessary to consider the properties of so-called triangular numbers. As he says in the Thoughts, those who are not geometers will be put off by these definitions and these principles which seem sterile to them, by these propositions incomprehensible to them; they cannot see at a glance and intuitively the slightest relationship between the mystical hexagram and the properties of conics, between the triangular number and the question of centers of gravity. The discovery of these relationships does not depend on a method communicable to all, but on a certain spirit, the geometric spirit, of which very few are gifted; “the few people with whom we can communicate,” said Pascal later, speaking of abstract sciences, “had put me off.” In Pascal, the word method is used in the plural; there are as many methods, that is to say, processes to be invented as there are problems to be solved. The geometer separates objects from one another, and, in turn, the geometric mind separates the geometer from other men.

The geometric mind is not even the whole scientific mind. The Pascal who approaches hydrostatics does not use the same intellectual gifts as the one who invents the mystical hexagram. “Some,” he says, “understand well the effects of water, in which there are few principles; but the consequences are so slight that only extreme righteousness can go there. And these would perhaps not be great geometers, because geometry includes a large number of principles, and a nature of mind can be such that it can penetrate a few principles until basically, and that it cannot penetrate in the least the things where there are many principles.” The search for the effects of water therefore serves the “spirit of accuracy” (since the principle of hydrostatics is unique) which, with its “strength and righteousness”, can nevertheless be narrow: while the surveyor, who must be able to grasp, without confusing them, a large number of principles, has a broad mind, but may have it weak. (1)

Pascal, a scholar, also applies himself to other studies, where knowledge of principles is of no use, or rather, where principles are sought in vain. A Cartesian will claim to establish fullness by principle: “Because, it is said, you believed from childhood that a chest was empty, when you saw nothing there, you believed emptiness possible: it is a illusion of your senses, fortified by custom, which science must correct. And the others say: Because we told you in school that there is no void, we have corrupted your common sense, which understood it so clearly before this bad impression, that it is necessary correct by resorting to your first nature” (no. 82). No recourse to principles is possible, therefore, in the question of the void. On the other hand, experience establishes with certainty that, in the barometric tube, the enclosure placed above the quicksilver is empty; it is a fact no less certain that the weight of the column of quicksilver must be balanced by a pressure acting on the free surface; and the famous Puy-de-Dôme experiment, which shows the height of this column decreasing with the altitude of the device, proves that this pressure is that of the atmosphere. It is not in principle that we can affirm or deny the existence of a vacuum or the gravity of air.

(1) Thoughts, 2 (we refer to the number given to the thoughts in the Brunschvicg edition).