Blaise Pascal’s Mathematical Milieu
[The following appeared in Reconsiderations 7:2 (December 2007).]
Blaise Pascal’s Mathematical Milieu
Daniel Julich
Historical study is, from my perspective, about the search for wisdom. It is about listening to the voices and events of the past and allowing them to bring new light and understanding to the way in which we think and behave. But looking into the past must not be oversimplified as merely piecing together big ideas independent of important bits of context that shape the perceived outcome. As one looks at Blaise Pascal, his philosophical and religious ideas cannot be separated from the life with which they are interwoven. This is precisely the holistic, incarnational view of life upon which the Christian tradition is based.
In this essay, I want to lay out some biographical background on Pascal, and then I will describe how his thinking was shaped in no small way by his training in mathematics. Indeed, it is my view that Pascal’s early apprenticeship in mathematics shows up in his later religious and philosophical reflections. While I will only touch on his religious and philosophical ideas here, it will be clear that mathematics provides a prominent backdrop for Pascal’s life in general, and an obvious context for his thinking in particular.
Blaise Pascal was born in 1623 in Clermont, France, a town 260 miles south of Paris and he died in 1662 in Paris at the age of 39. He spent the greatest portion of his adult life in the French capital, but also spent some years in Rouen, a city just a little bit north of Paris. His years include part of the reign of Louis XIII and the commencement of Louis XIV, although the so-called Sun King’s personal reign did not begin until a year before Pascal’s death. Pascal was born and died Catholic, like most of his countrymen, but he embraced a particularly Augustinian version of Catholicism during the latter part of his life. During his lifetime, Pascal became known especially for three endeavors. First was a mechanical calculating machine that he invented in his twenties and continued to develop into his thirties (incidentally, this was the key reason for naming a computer language for him). Second, he was known in his life for the publication of his thoughts and experiments concerning the creation of a void and the demonstration of the weight of the air. And third, he was recognized as the pseudonymous author of the witty and ironic Provincial Letters.
Although well-known by some through these three accomplishments during his lifetime, the publication of some scientific, mathematical, and religious/philosophical works posthumously, made him even more famous. Some of the most well-known aspects of his work, especially for those of faith, are contained in his Pensées, literally Thoughts. These are simply a collection of scribbled fragments ostensibly intended to be a part of a larger religious work but never completed. The order of these fragments is a matter of interpretation, and their place within the grand scheme of the proposed apologetic work is unclear. For anyone who spends much time with Pascal, this disjointedness can be extremely frustrating. Pascal does not give us a coherent articulation of his philosophical and religious views. Despite being unfinished, the Pensées have attracted the attention of thinkers since their first publication. The very famous Wager argument is contained in these Pensées, as well as Pascal’s reflections on the greatness and misery of human beings. Many have also heard of Pascal as having spoken of a “God-shaped void” in the heart of human beings, a paraphrasing of one of the Pensées.[1] Additionally, Pascal’s statement “the heart has reasons which reason does not know” shows up frequently in common parlance.
Pascal’s earliest biographies, written by his family not long after his death, extolled both his precocious genius in mathematics and his exemplary piety. Others followed suit in this evaluation. Mathematically, Leibniz praised Pascal’s first work on conic sections, completed while he was still in his late teens. He has also been hailed as the “founder of modern probability theory” together with Pierre de Fermat. His name has become associated with the arithmetic triangle, sometimes called Pascal’s triangle. Although he did not originate the triangle, he developed many applications of it. The work in physics that came out of the experiments on the void led to what some have seen as significant advances in the field of hydrostatics. And finally, his last mathematical venture dealt with a particular mathematical curve.
One of the especially puzzling and yet fascinating aspects of Pascal’s biography is his supposed renunciation of mathematics and science in order to devote himself fully to religious reflection. His sister, his first biographer, set the stage when she stated that from the time he was twenty-four years old, Pascal “renounced all other knowledge in order to apply himself uniquely to the one thing that Jesus Christ calls necessary.”[2] And Pascal’s writings do indeed suggest a deep unease about the ultimate usefulness and value of geometrical studies. Yet, in his religious writings I found Pascal making use of mathematical metaphors and mathematical ways of thinking that seemed to suggest that his mathematical reflections actually helped him to understand the problems of being human and being a man of faith.
These seemingly unrelated fields of knowledge, mathematics and theology, are in fact part of the philosophical makeup of an informal society of mathematicians by which the young Pascal was shaped. But before describing that group’s ideas and personalities, it would be helpful to understand Pascal’s own family influences that led him to that group. When Blaise was born, his father Étienne was a significant political official in Clermont. Étienne was 35 years old when he became the father of Blaise, his only son. There was an older sister (three years older) and a younger sister (two years his junior), and his mother died when Blaise was just three years old. Étienne, for his part, never remarried during the 25 years of life that remained to him. Instead, he invested his energy in his children, and in 1631 he left his post in Clermont in order to move to Paris and to devote himself to the education of his children and especially his son. This was a move that was not all that unusual at the time. Existing educational institutions were being challenged in their methods and in their outcomes. The Renaissance rediscovery of those ancient Greek and Latin writings hitherto unknown either completely or only imperfectly, had already created an atmosphere that would particularly value the questioning of authority received from the Middle Ages and the application to ancient languages in early life. Furthermore, there was, in the 17th century, a growing recognition of the difference between childhood and adulthood. This, in turn, fostered an emphasis on ensuring that children were not pushed beyond what was prudent for their age. Following these trends, Étienne Pascal first taught Blaise languages and sought to keep him from distractions of other subjects.
Besides Blaise’s education, Pascal’s father also pursued his interest in mathematics. When Étienne arrived in Paris, he began to meet regularly with a group of individuals who were skilled at and interested in mathematics. It is a group that gained notoriety for its specifically mathematical focus. The group is often called the Mersenne Group or the Mersenne Academy, being named for its organizer Marin Mersenne. Mersenne was a monk who is known for his wide correspondence network that included mathematicians, natural philosophers, theologians, historians, and musicians. Mersenne would often bring problems to the group that had been revealed to him by his correspondents and the group would seek solutions for them. These sorts of informal groups were common in the Paris of the time, but this one is specifically noteworthy because of its specific focus on the mathematical disciplines and the mathematical approach to physics and other sciences. These meetings were not simply professional gatherings. In fact, there was probably only one of its members that could be called a professional mathematician.[3] The group was made up of friends with a common interest, and the meeting place would often rotate between the homes of its members. Pascal the son was undoubtedly privy to many of the meetings of the Mersenne group during his late childhood.
His family recounts a story that marks his unofficial entry into mathematical endeavors, and has been one of the major factors for Pascal’s consideration as a precocious “genius.” According to this story, although Pascal wanted to know more about geometry, his father restricted it because according to his understanding, mathematics had the potential to be so alluring that it would monopolize the child’s time and draw him away from his other studies. But when Blaise was twelve or thirteen, as his older sister recounts in a biography, his father walked in on him as he was making figures on the floor with chalk. These figures amounted to the working out of a proof that the sum of the interior angles of a triangle equal two right angles; furthermore, when asked, Blaise explained what he was doing, using the appropriate geometrical reasoning from first principles. Too proud to be angry, Étienne shared his discovery and Blaise was allowed to be a junior attendee of the mathematical conferences in which his father took part.
Pascal was nurtured intellectually then, not only by his father, but also by this group of mathematicians. It was within this context that Pascal did his first work on conic sections, completed when he was only 17. This work dealt with a number of results that he had discovered that were universally true of conic sections – that is, circles, parabolas, ellipses, and hyperbolas. The Mersenne group and especially Mersenne himself, promoted this early work of Pascal’s. By this means, Blaise Pascal’s skill became known to numerous mathematicians in France and the rest of Europe. Importantly, one of the reasons that he was touted was that his mathematical skill was evident at such a young age.
This “mathematical academy” of Marin Mersenne constituted a philosophical and religious approach to the world. The sixteenth and seventeenth centuries saw an increasing attempt to restore orderliness and rationality to the universe. The mathematical approach to the world was a part of this drive. This mathematical group of which Pascal became a part, moved in the same direction as had Galileo, seeking to quantify phenomena in the world and to discover the proportionality that underlay its processes.
The justification of this deep investigation of the mathematical properties of music and of the universe in general had, for these mathematicians, both a religious and a philosophical motivation. One of the most popular quotations for 17th-century mathematicians, given as a religious motivation of their study comes from the Apocryphal Book of Wisdom. This quotation reads: “thou hast ordered all things in measure, and number, and weight.”[4] Mathematicians justified their pursuit of these patterns and proportions, then, in the name of uncovering the thoughts of God the geometer. In the case of music, the harmonious structure represented not just a mathematical, but an aesthetic exploration of God’s creation. The book of nature, Galileo had stated, was written in the language of mathematics.[5] But in Mersenne’s writings we catch an even more in-depth understanding of the type of atmosphere in which Pascal was raised. Mersenne writes of the godlike task of discovering the multitude of consequences that can be produced from a single mathematical principle. At the most basic arithmetic level, he considers the way in which unity (what we call the number 1) can produce an infinity of numbers through addition and multiplication. He compares this to the basic unity of the Godhead which produced the full spectrum of creation in all of its variety. On a more geometrical level, Mersenne says of the figure of the circle: “if one knew all its properties, and its uses, and all that one could draw and conclude from it, one would know more than all that has ever been written in matters of the sciences.”[6] As one seeks out these connections within geometry or the possible permutations of musical compositions, the state of divine knowledge is approximated. So that, when speaking of the remaining mysteries of geometry, Mersenne writes that such discoveries approximate the state of heavenly beatitude. He writes this: “The Angels know all these difficulties perfectly, and we will likewise know them when it pleases God to take us into the ranks of the blessed.” Mathematical knowledge, for Mersenne, is a part of heavenly knowledge.
The interaction between Mersenne’s expressed hopes for mathematics and his description of Pascal place the young man in a category that may perhaps be thought of as a “prophet” of mathematics. In a work written in 1625 (two years after Pascal’s birth and long before Mersenne knew the family), Mersenne expresses the hope for a timely nativity:
May it please God to cause to be reborn in this century some new Archimedeses, who will lead Mathematics to their last perfection…[7]
During the 17th century, the figure of the third-century BC mathematician-engineer named Archimedes came to stand for greatness in mathematics and inventiveness both practical and theoretical. Mersenne’s invocation of the rebirth of Archimedes has indeed the flavor of a prophetic hope. In the 1640s, Mersenne would label each of [ital]two[end ital] outstanding young mathematicians as “Archimedes.” One was the Dutchman Christiaan Huygens (six years Pascal’s junior), later one of the most important of the founding members of the French Academy of Sciences. The other was Blaise Pascal. There is obviously a bit of literary flair to comparing these young men to a great Greek mathematician. But it is also an extension of the attitude that emerged from the Renaissance that knowledge was growing and increasing to such a degree that the progress of ancient thinkers would be taken up and finally come to completion or perfection. A number of utopias appeared during the sixteenth and seventeenth centuries in which the research of the sciences and mathematics by those specifically trained for them were central to the ushering in of the perfect society. This golden age of knowledge was very much akin to Christian hopes of God’s reign on earth. For Mersenne and for others in his circle, mathematics was emerging as the central key to this perfection of knowledge. Hence, the label of Archimedes carried with it this prophetic vision.
This kind of optimism about knowledge was, however, not universally shared. Instead of a coming consummation, many saw the undoing of the possibility of any knowledge. The discovery of new worlds and the questioning of traditional interpretations of the world encouraged many to claim that no knowledge whatsoever was possible. For Mersenne and others like him, mathematics was one of the key weapons against such pessimism about knowledge and truth. Not only was the performing of mathematics an imitation of the mind of God, its methods and structure provided it as the paradigmatic discipline of certainty. In 1625, Mersenne wrote a book entitled The Truth of the Sciences Against the Skeptics. In it he calls mathematics “the sciences very certain and very true in which suspension [of judgment] does not find a place.” The geometrical method of proof based on distinct definitions and clearly accepted axioms was considered by them to show that mathematics, at least, could attain to some certainty. This is not to say, however, that all knowledge could be discovered through mathematics. Besides the proof structure, what gave appeal to those looking for certainty was also that mathematics allowed one to isolate quantitative and numerical aspects of nature. To quantify the world was to make possible exact comparisons. It made for measurability of the world, a true sense of comprehension of it.
The capacities of mathematics to approximate to God’s infinite productivity and attain certainty were extremely attractive. For the group in which Pascal grew up, full of men who were mostly amateurs rather than professionals, proficiency in the mathematical disciplines was a mark of pride. And Pascal was shaped by these values. Growing up among mathematicians who were his father’s friends and some years his senior, he was mentored and promoted by those who were consciously or unconsciously seeking to perpetuate and further their study of geometry. In a later summary of some of his mathematical works Pascal himself would write of “the benevolence which has sustained me since my first years in this learned School.” [8]
Not merely technically but in a profoundly personal way, Mersenne’s mathematical group served as Pascal’s place of initial apprenticeship, and there was a profound sense of responsibility that went along with this. It would be of little surprise if he wrote about religious matters in the Pensées using the metaphors of mathematics. But what I want to suggest is that the connections between Pascal’s mathematical-scientific works and his theological reflections are perhaps even more than metaphor. When he writes his religious thoughts in terms that are similar to those used in mathematics, he is drawing on a kind of structural similarity that reflects theological truth. I believe that the similarity has to do with the relationship between what may be known with certainty and that which transcends the abilities of human knowledge. This relationship is undoubtedly bridged by Pascal’s rich training in mathematics by way of the Mersenne group. It would be the thing that certain people would argue Pascal abandoned for theology, and it would also be the thing which others, including myself, believe enabled him to offer important theological insight. In other words, for Pascal mathematics becomes not a competitor of theology but, rather, a lens through which he would be able to see and articulate an understanding of life that becomes an important thread of the Christian tradition.
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Daniel Julich is a doctoral candidate in history at the University of Florida where he is currently writing his dissertation on Blaise Pascal.
[1] “What else does this craving or powerlessness proclaim to us but that there was previously in mankind a true happiness, of which there remains to him now only the entirely void mark and vestige, and that he tries uselessly to fill with all that surrounds him, seeking in absent things the help that he does not obtain from those present, but which are all incapable of it because this infinite pit/abyss cannot be filled but by an infinite and immutable object, that is to say by God himself,” Le Guern OC 2:591 (Pensée 138).
[2] Mesnard OC 1:578.
[3] Gilles Personne de Roberval, who was a professor of mathematics.
[4] Book of Wisdom, 11:21, Douay-Rheims Bible.
[5] The Assayer (1623).
[6] Mersenne, Vérité des sciences contre les sceptiques, 764.
[7] La vérité des sciences, 750.
[8] Mesnard OC, 2:1032.

