Equations from God:
Pure Mathematics and Victorian Faith
by Daniel J. Cohen
Johns Hopkins University Press, 256 pages, $50
It is tempting to treat mathematics as though it existed in a socio-historical vacuum, unaffected by what happens to people and societies. Though, like any other form of human knowledge, mathematics is born from our day-to-day experience, mathematicians carefully try to sandblast away every bit of specificity, leaving behind their abstract formal structure.
In fact, by restricting itself to the study of such structures, mathematics has achieved a unique status among human disciplines. On the one hand, its questions and ideas seem to survive unchanged over the centuries. On the other, it attains an extraordinary level of certainty: While there are factions and opinions among mathematicians, there is hardly ever much discussion about whether something is true or not, whether a proof is correct or fallacious.
Thus, for an historian, mathematics poses interesting challenges. If it is true, as many have assumed, that mathematical truths are transcendent, timeless, and independent of human minds, then how can mathematics even have a history? At best, one could write the history of the discovery of those truths or of our understanding of how those truths relate to our (untranscendent and definitely time-dependent) everyday reality.
Over the years, there has been quite a bit of the former, especially when mathematicians write the history themselves: We get the discovery of facts, often technical and, as Cohen puts it in one of his footnotes, “without addressing the wider cultural and historical issues that are of interest to mainstream historians.”
But, in recent times, we've gotten as well a great deal of the context. Neither mathematicians interested in history nor historians interested in mathematics are likely to argue, today, for an unhistorical view of mathematics. As we read the work of mathematicians of other times, we are usually struck by how different it is from today's mathematics. Yes, one still can, by sandblasting away the particulars, recognize the same underlying formal structure (or, sometimes, a slightly different formal structure that is logically equivalent to the one we see today). But the particulars are just too interesting, and those particulars are what history is about. Mathematics is still a human production, and the marks of the humanity of the mathematicians are all over their works. No serious historian of mathematics would now claim that history can be done entirely in the world of ideas, without reference to its context.
Religion has been an important part of this context. All mathematicians, throughout history, have lived in cultures that were religious. Many of them have been deeply involved with this aspect of their cultures. Most of the mathematicians of medieval Europe were, after all, clerics. Gerbert of Aurillac, one of the first Europeans to investigate the mathematical discoveries of Islamic cultures, was later Pope Sylvester II, for example. Nicole Oresme, probably the most brilliant late-medieval mathematician, eventually became bishop of Lisieux. Many of the mathematicians of medieval Islam seem to have been devout. Blaise Pascal is more famous for his religious and philosophical writing than for his mathematics, though he excelled at both. Isaac Newton's papers contain more pages about biblical interpretation than about mathematics and physics. Leonhard Euler, the greatest mathematician of the eighteenth century, was a devout Christian and wasn't afraid to defend his views in print. And Srinivasa Ramanujan, a brilliant Indian mathematician whose short life has made him a romantic figure, claimed to have received many of his ideas from the goddess Namagiri.
As in any cross-section of intellectuals, one also finds skeptics and atheists. Asked about God's role in his “system of the world,” Laplace famously told Napoleon that he “felt no need of that hypothesis.” G.H. Hardy believed that God either did not exist or was out to get him. Many mathematicians of the late-nineteenth and early-twentieth centuries got caught up with the various intellectual fashions of their times, from positivism and Marxism to the darker obsessions of the 1930s.
Most historians of mathematics, however, feel a little uncomfortable about religion as a part of the sociohistorical context of the discipline. They tend to brush off the religious convictions of mathematicians as either merely conventional or a kind of strange eccentricity that is irrelevant to the main story. While few go as far as E.T. Bell—who described Pascal as “a gifted mathematician who let his masochistic proclivities for self-torturing and profitless speculations on the sectarian controversies of his day degrade him to what would now be called a religious neurotic”—their discomfort is often palpable.
Hence, one can only welcome Equations from God, Daniel J. Cohen's investigation of the influence of religion on mathematics during the Victorian era. Given the book's period and Anglo-American focus, the religion in question is mostly Protestant Christianity. The Catholic Church is a noticeable but not central presence, and Jews are mostly there in the margins, but the big debates are about Unitarianism, various kinds of “higher spirituality,” and liberalism. In the mathematical world, the main action at the beginning of this period was definitely on the European continent. English mathematics was recovering from a period of relative isolation and was institutionally focused on mathematical physics—and, at the beginning of the Victorian age, still required of all fellows of Oxford and Cambridge to take religious orders.
Cohen focuses the major part of his attention on three specific figures: Benjamin Pierce, George Boole, and Augustus De Morgan. Pierce (the father of Charles Saunders) was a professor at Harvard and was probably the most important American mathematician of the time. Boole was English but taught in Ireland for most of his professional life. De Morgan was involved in the founding of the University of London, which was created specifically as a university with no religious requirements for professors. De Morgan was a prolific author and a tireless organizer who played a crucial role in the move toward professionalization, including the founding of the London Mathematical Society. All three held broad religious views, considering religion important but sectarianism harmful, and all three felt that religious doctrine was an impediment to true religion.
In other words, all three were fairly standard-issue nineteenth-century religious liberals. Rather than looking at their actual mathematical work, Cohen's attention is focused mostly on what they said about mathematics and in particular about how mathematics fit into their religious views. He seems to take these often strongly rhetorical statements at face value.
This is in welcome contrast to those historians who would dismiss them as merely conventional, but it may go too far in the other direction. What mathematicians say about mathematics is often said as an afterthought, the real action being located in the mathematical work itself.
Cohen's approach yields a fairly clear narrative. In Pierce we find great respect for the transcendent qualities of mathematics that put it either in the same realm or close to spiritual experience. Mathematics showed, for Pierce, that the human mind could transcend the limitations of time and place and reach for the infinite and unconditioned. As such it served as a model for the sort of platonic speculation Pierce felt could save religion from the sectarianism he despised.
Boole's distaste for sectarianism was even more intense. He seems to have hoped that a mathematical analysis of human thought would allow religious disputes to be settled by calm and dispassionate calculation rather than power games and rhetoric. This hope played a part in Boole's development of mathematical logic, a truly new idea that was to bear very strange fruit in the twentieth century.
De Morgan, while agreeing with the goals of his predecessors, felt that mathematics needed to be made more professional, purer, and more abstract precisely in order to insulate it from the religious sectarianism he despised. The progression, then, is from a platonic view of mathematics as part of the search for the transcendent, through the idea that mathematics might serve as a useful tool in that search, to arrive finally at a separationism that saw a need to purge mathematics of any connection with religious ideas.
It is an interesting story, but one that I find far from persuasive. If you asked a typical historian of mathematics to name the three most important English-speaking mathematicians of the Victorian era, the answer would probably be William Rowan Hamilton, Arthur Cayley, and J.J. Sylvester. Hamilton and Sylvester get brief mentions in the book, but Cayley doesn't appear at all. This is something of a pity, because, while Hamilton might fit the narrative well, Cayley was a devout evangelical and Sylvester was Jewish, so that one might expect their views to add complexity to the overall picture. I doubt, for example, that Cayley would have agreed with even the premises of the discussion of sectarianism in Cohen's book.
Several other aspects of Cohen's approach give the reader pause. He seems, for example, to take the heated rhetoric of nineteenth-century writing about religion and philosophy at face value, though that heat was largely conventional. He takes for granted the accuracy of his principals' perception that sectarianism was a problem in Victorian England. By focusing on three specific individuals, he avoids having to figure out what the mathematical community as a whole was thinking and doing. He neglects to point out that the professionalization of mathematics happened well-nigh simultaneously all over Europe, and that the motivations for the move were many and complex. Most of all, he tends not to look at the mathematics at all, restricting himself almost entirely to what is said about mathematics.
That, I think, is the decisive question for the historian: not how religious ideas affected or were affected by ideas about mathematics, but how those ideas affected or were affected by the mathematical ideas that these men were thinking about. Did Pierce's broad-minded Unitarianism have an influence on his approach to linear associative algebras? I think so, and I wish Cohen had looked into it. Did Cayley's and Sylvester's more conservative points of view influence their invariant theory? Did Boole's conception of thought as something formalizable arise from his mathematics or from other premises? How did non-mathematicians react to it?
Cohen has opened the door a crack and shown us a few interesting things, but there's a lot more to explore here.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and coauthor, with William P. Berlinghoff, of Math through the Ages: A Gentle History for Teachers and Others.