A Certain Ambiguity

I just finished one of the books I brought along for pleasure-reading during my vacation: A Certain Ambiguity, by Guarav Suri and Hartosh Singh Bal. The book touches on so many of the themes that weave through my own life and philosophical work, as well as the discussions that keep recurring on this blog, that I would recommend it to anyone who finds the issues on this blog captivating.

It is a mathematical novel—one that weaves the philosophy and history of mathematics into a story about the efforts of Ravi, an Indian college student, to understand a transformative episode in his beloved grandfather’s life (and in the process to chart the course of his own life). The mathematical dimensions of the book are captivating in their own right. Reading the book is like having an impassioned math teacher right there with you, inviting you to be as excited as he or she is about Cantor’s proofs of degrees of infinity, or the struggle to prove Euclid’s “fifth postulate” from the other four (and the subsequent emergence of non-Euclidean geometry).

But on a deeper level the book is about the light that these mathematical insights sheds on the prospects (and elusiveness) of certainty, humanity’s quest for meaning, and the complex relationship between religious faith and the kind of intellectual virtues cultivated in mathematics (and other disciplines).

At the heart of the story is the narrator’s discovery that his grandfather, a mathematician named Vijay Sahni, was arrested while visiting a small New Jersey college town in 1919. The charge against him? Blasphemy. For reasons I won’t get into, the young Vijay is motivated to get up in a public forum and discuss the absurdities of Christianity—an act that so offends the community’s sensibilities (after they had gone to such lengths to be hospitable to this strange foreigner!) that he is arrested the next day.

Of course, the New Jersey blasphemy law under which the visiting “Hindoo” is charged does not quite sit comfortably alongside the Constitutional guarantee of free speech, and there is pressure from more progressive voices for the governor to step in and dismiss the charges. The resulting clash of religious conservatives and more progressive constituencies puts the governor in a difficult political situation—one he seeks to extricate himself from by passing the buck.

He does so by sending a respected, conservative judge—a religious man named John Taylor—to review the case and make a legal recommendation concerning whether the case should proceed to trial. The judge decides he needs to interview the foreign blasphemer—and the transcripts of those interviews, along with other relevant documents and newspaper articles that the grandson Ravi uncovers over the course of the novel, tell a story about two intelligent men, one an atheist mathematician from India, the other a Christian judge from America, who form a bond of mutual respect and friendship as their exchanges become about far more than whether Vijay should be put on trial.

Judge Taylor has his own distinctive notion of how and when the blasphemy law can apply without violating the right of free speech—but to make a judgment, he needs to interview the prisoner to discern his motives. Since Vijay claims him motives are rooted in mathematics—more specifically his conviction that mathematics offers the model for how we should go about forming our beliefs—the judge ends up getting an extended lesson in Euclidean geometry. As Vijay's grandson, Ravi, is working his way through these transcripts, he is taking an introductory mathematics course on infinity—in which he is introduced to set theory, including Cantor’s proofs that while the infinite set of integers is equal in cardinality to the infinite set of rational numbers, the infinite set of real numbers has a higher cardinality. In other words, there are degrees of infinity. In fact, there are infinitely many degrees of infinity.

As the story evolves, the sticking point of Euclidean geometry—the fifth axiom, which is more complex than the first four and doesn’t quite seem as if it should be an axiom at all—becomes connected to the sticking point of Cantor’s mathematics—the so-called Continuum Hypothesis, which hold that there is no (infinite) cardinality greater than that of natural numbers but less than that of real numbers.

In both cases, the contested principle defies proof but seems right to many. And yet, when the contested principle is rejected and its negation is assumed, no contradiction emerges. A whole vista of alternative mathematical systems arises instead, depending on what you take to be axiomatic.

And then, of course, the timing of Vijay’s imprisonment takes on meaning. In one of the final documents Ravi uncovers, Judge Taylor and Vijay come face to face with Einstein’s confirmation, in 1919, that space does not conform to Euclidean geometry—that is, it does not fit with the geometry that arises when we accept Euclid’s apparently self-evident fifth axiom (which can be formulated in numerous ways, but is perhaps most simply expressed with the statement that given a line and an adjacent point, you can only draw one line through the point that is parallel to the adjacent line).

What arises is a story about the quest for certainty—for knowledge that can be founded on indubitable starting points—and the epistemic struggles of human beings to figure out what to do when apparently self-evident starting points turn out not to be self-evident after all. What does it mean for human life, for our struggle to understand our world, that we can construct equally coherent geometries or set theories (or interpretive worldviews) out of alternative starting points?

In a concluding story (told through Judge Taylor’s person journal) about the reunion of Vijay and Judge Taylor in 1930, when the judge travels to India to visit the friend he’d made in a prison cell, the still religious judge at one point asks Vijay whether he now believes in God. Here’s the response:

He smiled and shook his head. “No, Judge, that is not in my nature.” He then paused, looked me in the eye, and said, “But I can understand why someone might.”

In a way, this response captures the essence of what I’m about in my philosophical work. While not every “axiomatic system” is consistent or provides an adequate template for understanding our lives and our world, our epistemic situation does not allow us to settle conclusively on just one. Some of us gravitate towards a system of one kind, while others gravitate towards another. It is often hard, once we are enfolded within a functioning axiomatic system, to see it as anything other than the indubitable truth, and to see those who disagree with it in favor of an alternative system as deeply misguided, as caring more about psychological comfort than about truth, or as indoctrinated by some cultural ideology. But while these psychological forces are often at the root of our outlooks, the reality of our epistemic situation is much more complex than the way it appears to be from the vantage point of one particular axiomatic system.

My hope, in the grand scheme of things, is that we will come to see this complexity, and so, in the process, be able to say of those who see the world differently, “It is not in my nature to look at things that way, but I can understand why someone (a reasonable person) might.”

In any event, I recommend the book to anyone interested in these issues.(My copy is from an Indian friend, but it looks as if an American edition will be released at the end of this month).