The tension between religion and reason was as familiar to the early moderns as it is to us inhabitants of a postmodern era. However, the concerns in the historical period dubbed the Scientific Revolution were very different from — one might even say diametrically opposed to — the issues that vex us in a world worried about global warming, weapons of mass destruction, nanotechnology and stem cell research. In this scholarly tome, Matthew Jones skillfully knits together history, philosophy and mathematics to introduce us to the mindset of Descartes, Pascal and Leibniz. Theirs was an age where philosophers touted the practices of science and mathematics as models to help one become a moral person. Contrast this to the challenges of our times, where the vector connecting the good (moral, virtuous) life and science points in the other direction, namely our need for an infusion of ethics into science.
Jones carefully reconstructs the philosophical context of the seventeenth and eighteenth centuries in which these three great mathematicians worked. Having set the stage, when they walk out to say their lines (there are plentiful quotes from original sources), we are prepared and able to perform the act of creative imagination that allows us to truly comprehend their standards for truth and falsity, proof and evidence, far removed from our own. The so-called “rational man” was just being constructed, and for Descartes, Pascal and Leibniz, rationality was not limited to maximizing one’s own benefit. “In early-modern Europe, such considerations about the justice due to members of a community were often taken as integral to rationality itself and were not understood merely as a secondary result derived from the rational consideration of self-interest” [Jones, p. 147].
Jones’ skill in stimulating our imaginations is in keeping with the tradition of Descartes, Pascal and Leibniz, who also utilized imaginative exercises as tools for self-transformation, a technique they absorbed from the traditions of the Catholic Reformation. Although the practice of mathematics did not guarantee self-mastery, and one might still achieve a good life without the cognitive discipline that mathematics provided, Descartes was convinced of the efficacy of mathematical exercises, in particular work using geometry, to focus the intellect and elevate the soul to pursue the good life. For Descartes, geometry was the exemplary practice that taught good habits of mental hygiene, enabling one to distinguish the “clear and distinct” and when transferred to the moral sphere, right from wrong.
Descartes, often called the father of coordinate geometry, in fact saw algebra as a powerful tool, but one that was temporary, like scaffolding that needed to be removed. He cautioned against the danger of blind symbol manipulation, losing sight of what the symbols represented. At the risk of sounding whiggish, I thought this seemed remarkably similar to a caveat from a reformed mathematics treatise of today. Indeed, Bernard Lamy and many other educators created pedagogical materials — the “reformed mathematics” texts of their day — which drew heavily on Descartes. With even more passion than the master himself, these thinkers gave his brand of mathematics a central place in their attempts to cultivate good citizens and good Christians. The long-lasting impact of this work is evidenced by a quote from Rousseau’s Confessions: “I did not like the taste of Euclid’s [geometry], which seeks a chain of demonstrations rather than the connections of ideas. I preferred the Geometry of Father Lamy, who became, from then on, one of my favorite authors, whose works I reread with pleasure” [Rousseau, quoted in Jones, p. 49].
Unlike Descartes and his followers, for whom mathematics was a mental discipline that enabled one to distinguish the true from the false, Pascal valued mathematics for its ability to reveal the extremes of human potential, our capacity for both wretchedness and greatness. The fact that many of the greatest philosophers and honnêtes hommes could observe the brutality and cruelty of which humans are capable and not “recognize their infirmity” or “even bother to try” [Jones, p. 153] is evidence of human wretchedness. The fact that any human being can, when goaded/guided to self-reflection, plumb the depths (as well as the heights) of human nature, is evidence of human greatness.
Pascal repeatedly invoked mathematical practice to demonstrate both the powers and limits of human reason. In his Pensées, he had his readers imagine possibilities beyond what their senses could perceive. One of his favorite imaginative exercises was his “two infinities” argument. He started from the two obvious statements that you can always increase (“augment”) a number by doubling it, and you can always decrease a number by dividing it in half. From this we gain insight into the infinitely large and the infinitely small. He then noted that “[a]ll these truths cannot be demonstrated, and yet they are the foundations and principles of mathematics” [Pascal, quoted in Jones, p. 121]. Once again, mathematics reveals two poles of human capability: its greatness in producing certain knowledge about undefined and indefinable things (like infinities); its weakness in basing this certain knowledge on truths that are impossible to prove (like a house built on sand).
Pascal drew two very important conclusions from this exercise. First, it is the essence of the human condition that we “must accept the certain existence of things [we] can never hope to comprehend” [Jones, p. 123]. This, of course, is an argument for faith. Second, he cautioned against the generalization of the logic of mathematics to other domains, such as religion. Pascal claimed that the basic entities of mathematics — space, time, and motion — were examples of things that were clear in themselves, evident and obvious, and a firm basis on which to begin reasoning. In fact he ridiculed those who wasted their time trying to define things that were intuitively clear, like “light” and recommended that their time would be better spent “put[ting] forth propositions concerning the phenomena of light, and then they should attempt to prove or disprove whether those propositions in fact describe the characteristics of light” [Jones, p. 108]. In the case of religion, however, humans do not have certain knowledge of the fundamental objects, and therefore no basis on which to begin reasoning. Without first principles on which to build, “a logic mimicking the mathematical style and applied to other domains can only remain unmoored and useless, merely a speculative logic in the worst sense” [Jones, p. 122]. Even though he recognized the temptation to achieve the certainty of mathematics in other domains, he saw it as a grave mistake to generalize and merely mimic the logical forms without the substance on which mathematics was based. Thinking we can apply this limited human ability — mathematical reasoning — to all domains is an example of human hubris. Realizing the blind spot of this kind of thinking is an example of human humility.
Jones does an excellent job of explaining the mathematics of all three of his subjects, complete with reproductions of pages from their original papers. I was particularly impressed with Jones’ exposition of Pascal’s quadrature of the parabola, using a result well known to his contemporaries: that one could readily find the area under any curve of the form y = kxn (n a positive integer) by considering the ratio of said area to the area of the square that circumscribed it. This led to summing a series of the form Σ ik (k a positive integer, i ranging from 1 to n–1), a sum that Pascal cleverly calculated for all relevant values of n using patterns in his eponymous triangle (see Jones pp. 111–116 for the details). We are all familiar with this famous triangle (versions of which also appear in Chinese, Arabic, Hindu and other mathematical traditions), and for many students it is an early introduction to the magic of patterns and interconnections in mathematics. It has always seemed to me to be a kind of “sandbox” — a place to set playful minds loose to discover their own theorems. Pascal saw the fertility of his triangle for proving propositions about numbers as the very embodiment of mathematical practice. For him, “[m]athematical inventiveness meant constantly discerning new interconnections among mathematical objects” [Jones, p. 102]. He surely would have been delighted by the contemporary connections being made between seemingly disjoint fields.
One of my own favorite linkages is the interplay of the continuous with the discrete and Pascal, too, was fascinated by the connection between these two realms, which at first glance could not seem more disparate. For him the actual result (finding the area) was less important than the demonstration of the utility of mathematical practice for finding surprising relations between apparently unrelated objects (a continuous area and a discrete sum). The sense of wonder that such unforeseen connections could inspire, was one way to provoke humans to acknowledge their limits (no pun intended!), to “cease their investigative quest and simply gawk” [Jones, p. 124]. Whereas Pascal welcomed such moments of cognitive wonder, Descartes excluded such experiences (of the infinite, for example) from mathematics, seeking instead a more limited form of appreciation that would lead to further inquiry into the causes of phenomena, rather than paralysis and “simply gawking.”
Leibniz, like Descartes, hoped that cognitive shock would motivate one to quest after knowledge. But he did not share Descartes’ fear that experiences of wonder, such as encounters with the infinite, would lead to “mere astonishment” and fixating on appearances without looking for their causes. Leibniz, as we know, embraced the infinite. His quadrature of the circle resulted in an infinite series, which he offered as a legitimate representation of exact knowledge: “A value can be expressed exactly, either by a quantity or by a progression of quantities whose nature and way of continuing are known” [Leibniz, quoted in Jones, p. 175]. This was a sharp departure from his predecessors, in particular Descartes, for whom the solution to a geometrical problem was a geometrical construction. Recall that for Descartes, algebraic formalisms were useful tools for obtaining these constructions, but a formula did not in itself constitute a solution. For Leibniz, on the other hand, it was sufficient to know the rule (ratio) connecting the terms of the progression, and to be able to compute the answer to any desired degree of accuracy. “For the nature of a series, even an infinite one, can be understood, even only having perceived a few terms, so long as the ratio of the progression has appeared” [Leibniz, quoted in Jones, p. 175] . In one brilliant move, he promoted symbolic expressions (including infinite ones) — formerly the ghosts behind the machine — to bona fide final results.
In contrast to Pascal, who warned against the generalization of mathematical logic to other realms, Leibniz was constantly seeking to apply successful mathematical methods to solve all problems. You may call him a dreamer (indeed he was satirized by Voltaire) for his prescription for conflict resolution: “Let us calculate.” But he devoted his life to a serious quest for a “universal characteristic,” through which he hoped to “transfer the virtues” of his mathematical results and their philosophical underpinnings “to natural philosophy, theology, and statecraft alike” [Jones, p. 223]. All of his work was geared towards “one end, namely, the solving of problems, and of the greatest of problems: obtaining happiness” [Leibniz, quoted in Jones, p. 198].
Two themes dominate Leibniz’s mathematics (as well as his philosophical and ethical work): a search for the hidden unity behind the seeming diversity (and chaos) in nature; and the development of techniques and tools to improve the (woefully deficient) human capability to see many things all at once. He drew on his early work in both optics and perspective to advance both goals. According to Jones, throughout his life Leibniz continued to invent new “marvel-producing machinery” using whatever technologies he mastered (from optics to symbolic and written expression), to induce “moral and religious transformation” in an audience [Jones, p. 198]. His innovations included a theater of shadows, an educational casino, and dialogues.
Exactly what role did optics and perspective play in Leibniz’s thinking? Early in his career Leibniz invented new optical devices, and his quadrature of the circle was a clever application of perspectival intuition. His method differed from previous attempts (most of which divided up the circle using parallel bands) by dividing up the circle into lines emanating from a single point of view. However, he was not content with his “merely” technical and mathematical achievements. Leibniz was equally (if not more) interested in the philosophical lessons to be drawn from the methods used to obtain his results. Both optics and perspective were ways of improving human vision.
Coming up against the limitations imposed by our physical bodies (in particular, our eyes) motivated us to develop the telescope and microscope. By analogy, Leibniz argued, coming up against the limitations of human reason should naturally lead us to develop symbolic and written expressions. In each case we are merely using technologies to perfect and extend our limited human capacities. And just as optical instruments could aid us in making new discoveries, “blind thinking” with symbols without “consideration of the ideas themselves” could be a powerful aid in creating new knowledge. “[I]n this consists the art of thinking with symbols” [Leibniz, quoted in Jones, p. 205]. Note that this is exactly the kind of thinking abhorred and barred by Descartes. The times they were a-changing.
Similarly, each of our individual minds contributes just one (albeit confused, owing to our physical limitations) perspective on the universe. God’s knowledge, on the other hand, is made up of all views simultaneously. “To understand completely the essence of a building — its plan — one needs infinite perspectival drawings, each contributing some certain knowledge” [Jones, p. 207]. One of Leibniz’s favorite mathematical practices was to change “one formula into various equivalent formulas” [Leibniz, quoted in Jones, p. 213], to create many equivalent, but distinct, symbolic expressions for the same relationship. For him, this was a perfect example of how God “could create different instantiations of the same essence” and suggested that “our knowledge of that essence will come through exploring diverse expressions of it” [Jones, p. 213]. By analogous reasoning, Leibniz “translated a central mathematical practice into his account of creation” [Jones, p. 213].
We are all aware of the obfuscating effects of poor notation and, conversely, the power of good notation to clarify thinking, and, as Leibniz was fond of demonstrating, even to discover new relationships. Along with proper symbols, Leibniz relied on tables to uncover rules and principles underneath apparent chaos. “[J]ust as accurately marking out a circle requires an instrument by which the hand is regulated… we are in need of certain sensible Instruments for thinking correctly, of which I refer to the two great sources, Characters and Tables … [Leibniz, quoted in Jones, p. 236].
One can easily picture him hunched over a desk for days and long into the nights filling page after page, looking for patterns. Nowhere is this more evident than in a paper from 1678 where Leibniz introduces good notation to represent a system of simultaneous linear equations in four variables, and thereby basically discovers (with a few small errors) what we now know as Cramer’s rule. Instead of assigning a new letter to each coefficient, he had the insight to index coefficients to connect them to their respective unknowns and then to represent both the coefficients and the unknowns by their indices only (so, for example, what today we would refer to as a12 is simply written 12 and x2 as 2). Using tables, he carefully kept track of each step as he proceeded through the calculations. This allowed him to track the role of each coefficient and discover general rules for computing what we now call determinants of the matrices of coefficients, which are the key to solving for each unknown. His well chosen notation, and meticulous bookkeeping with tables, permitted him to “focus [his] attention and thus discern the rule underlying the equations, rather than to attempt simply to memorize them or to plod through laborious, blind calculation” [Jones, p. 239].
And if you have plodded this far through the review, I hope you have gained an appreciation for the meticulous and careful scholarship Jones has performed, and are inspired to read all or parts of this book. Each member of the triumvirate gets two chapters, and although there are occasional cross-references, each pair of chapters is relatively independent. Many of the connections among these great men are exercises left for the reader, and I have tried to make some of them more explicit. It is not a quick or easy read, but as Jones so nicely summarizes, “Descartes’ mathematical exercises, Pascal’s arithmetical triangle, and Leibniz’s instructive displays and notational innovations all testified that perfecting human ability would require disciplined work on oneself and on others” [Jones, p. 268].
Bonnie Shulman is Associate Professor of Mathematics at Bates College in Lewiston, ME. Her intellectual passions include history and philosophy of science and mathematics, mathematical modeling, and feminist science studies. She is currently investigating connections between mathematics and the moral sciences.