Pascal himself said in a letter to Pierre Fermat written in 1660 that he felt that his religious writings had little connection with his scientific and mathematical work. His outlook, however, was deeply influenced by what he conceived to be a new way of looking at the world inspired by geometry, and most commentators would agree that all his writings are impregnated with it. He himself made strictly mathematical contributions to number theory, geometry and probability theory, but he also involved himself in the wider polemic about the status of science in his day.
In his letter to Father Noel, he set out the prerequisites of sound scientific methodology, laid down the rules for making affirmative or negative scientific judgments (through axioms and apodictic demonstration), and for establishing or disproving hypotheses about the physical world, which in his view could never be more than provisional. In the same letter, Pascal referred to the rival claims of authority (in this case, the authority of Aristotle) and scientific demonstration; this topic is more fully developed in the Préface sur le traité du vide (Preface to a treatise on the vacuum) (1651?), in which Pascal shows that experiment and correct reasoning should govern the sciences, and that authority and historical example have no place in them.
His view of science is very much a progressive one; as more and more experiments are undertaken with more sophisticated instruments, previously accepted hypotheses are supplanted by newer ones. Thus the hypothesis of occult qualities or powers, which was postulated to explain what lies beyond sensory perception, and the Aristotelian distinction between act and potency, should be replaced if experimentation can show that they are inadequate according to Pascal’s rules. Natural causation and phenomena are unchanging; human attempts to understand them are relative to the historical moment at which the attempts are made. In a striking image, Pascal refers to the successive generations of scientists as a single person in a perpetual state of existence and development. Thus, when we disagree with scientists of the past, we are not contradicting them, since by applying the principle of charity we would have to agree that we would have understood the world in their way had we lived in their times with their resources; and it follows also that they would have agreed with us today for the same reasons. …
De l’Esprit géométrique et de l’art de persuader (On the spirit of geometry and the art of persuasion) (1657-8?) is a yet more sophisticated presentation of the new scientific outlook. Pascal begins by conceding that definitions in geometry are nominal and not real, and that what are taken for axioms are intuitive perceptions which can neither be demonstrated nor reasonably be doubted. The four terms which he identifies in this way are number, space, movement and time. All share the property of being infinitely divisible and infinitely extensible. This insight is counter-intuitive to those who conceive knowledge as finite but, unless it can be grasped, then the geometric spirit itself cannot be comprehended. Pascal is not claiming that man’s capacity for knowledge is unlimited; merely that the immediate information of his senses and his reason have to be transcended if scientific advances are to be made. Geometry emerges from this as superior to logic, in that it can both provide axioms and engage in demonstration, whereas logic can only do the latter.