“Hey, I heard somewhere that you wrote a book about math and your faith. Having never understood how a rational person can possibly subscribe to the Christian dogma (except for having some strong, over-riding subconscious need, perhaps), I’m curious about it, although if it all comes down to ‘faith,’ well, I’ve never had any idea what that really means and don’t think I have any in me. (BTW, I’m a happy member of the Skeptics Society out of Altadena.)”
Those were the concluding lines of an email I received one day from a mathematical acquaintance. His comments revealed what seems to be a prevailing attitude in our culture: faith pertains to the religious sphere; logic and deduction lie in the domain of science, and it is only the conclusions of science that give us firm knowledge.
Such an attitude is not new. It certainly existed during the sixteenth and seventeenth centuries, which can be reasonably claimed as the beginning period of modern science. It was such thinking that, in 1734, helped prompt Bishop George Berkeley to publish The Analyst: A Discourse Addressed to an Infidel Mathematician.1 The purpose of his essay can be gleaned from its subtitle: Wherein it is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.
The “modern Analysis” mentioned in the subtitle refers to the field of calculus that had recently been developed by Isaac Newton. Newton, however, was certainly not the “infidel mathematician” to whom Berkeley alludes. Indeed, in his essay Berkeley chastises those who deride Newton for his religion, yet admire him for his science.2
The general consensus is that Berkeley had in mind the atheist Edmond Halley, discoverer of Halley’s comet. There are a variety of reasons for this attribution. According to the biographer Joseph Stock, many years prior to the publication of The Analyst Berkeley sent Joseph Addison, a member of his parish, to visit Samuel Garth, a London physician who was gravely ill. When Addison urged Garth to confess Christ as his Lord and Savior, Garth’s reported reply was, “Surely, Addison, I have good reason not to believe those trifles, since my friend Dr. Halley, who has dealt so much in demonstration, has assured me that the doctrines of Christianity are incomprehensible and the religion itself an imposture.”3 Then, in 1732, Halley evidently made some critical remarks about Alciphron, a tract Berkeley had just published, whose purpose was to combat the so-called freethinkers of the day.
Freethinkers generally saw reason and logic as the means for resolving differing exegetical conclusions, and mathematical reasoning as the supreme model that theologians should follow. They pointed to religious disputes as supporting their claim that religious dogma is—wrongfully—largely based on authority and tradition, and even when not they saw most theologians as engaging in sloppy thinking with unclear non-empirical references. No freethinking person would have room for a belief system that allowed for mystery, paradox, or ambiguity. Their methodological approach dovetailed nicely with John Locke’s theory of language, where words correspond with ideas, and clear ideas in the mind of one person can be successfully communicated to correspondingly clear ideas in the mind of another. Thus, they saw dramatic contrasts when comparing mathematical and theological reasoning: clear versus obscure ideas, rationality versus superstition, and consensus versus disputes. Most of them were either deists or atheists. With their accompanying reliance on empirical verification, they represented an early version of logical positivism, which is the philosophical view holding that, to be meaningful, a statement must either be empirically verifiable or logically deducible from established truths.
Most people today are not aware that the foundations of Newton’s calculus were very shaky, and that Berkeley aptly pointed out many of the flaws. At the conclusion of his essay he listed 67 rhetorical queries. One of my favorites is the 64th: “Whether Mathematicians, who are so delicate in religious Points, are strictly scrupulous in their own Science? Whether they do not submit to Authority, take things upon Trust, and believe Points inconceivable? Whether they have not their Mysteries, and what is more, their Repugnancies and Contradictions?”
In an earlier blog I listed a few contemporary mathematical conclusions that may qualify as mysteries or repugnancies. But what about contradictions? It would take another century before the French mathematician Augustin-Louis Cauchy made reasonable headway in putting calculus on a firmer foundation, and by the time of his death in 1857 efforts were brewing to firm up all of mathematics.
In 1903 the logician Gottlob Frege was about to take a big step in that direction with the publishing of his Grundgesetze der Arithmetik, Band II (the Basic Laws of Arithmetic, Volume 2). It contained five simple axioms that, Frege hoped, would lay the necessary groundwork for all of arithmetic. Unfortunately, just before the book was to be published, Frege received a disturbing letter from Bertrand Russell, who pointed out that Frege’s fifth axiom was in conflict with the other four. In other words, Frege’s system was inconsistent. It was too late to stop production, so Frege desperately tried to patch things up and inserted a last-minute appendix where he modified his fifth axiom. He also openly explained the situation: “Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr. Bertrand Russell, just when the printing of this volume was nearing its completion.”4
It was subsequently shown that Frege’s fix didn’t work, but the effort to ground arithmetic on a rock-solid foundation went on. Towards that end Bertrand Russell and Alfred North Whitehead published a three-volume work, Principia Mathematica (PM), between 1910 and 1913. Then, by 1922, the logicians Ernst Zermelo and Abraham Fraenkel produced a collection of axioms that, together with another axiom called the axiom of choice, serves as the basis for a large body of mathematics as we know it today. The axiom set is referred to as ZFC. Depending on its formulation, ZFC amounts to about ten axioms.
But a lingering question remains: how does one know that, unlike what happened to Frege, there are no contradictions lurking in PM or ZFC? Showing that these systems are consistent was the goal of many.
In 1931, however, the Austrian-born mathematician Kurt Gödel dealt a fatal blow to that goal. He published a paper demonstrating what is now known as his incompleteness theorems. The first one proved that, if PM is consistent, it contains true propositions within the system that cannot be proven to be true, and it contains false propositions that cannot be proven to be false.
This result would have been a blow to the freethinkers of Berkeley’s day, and was a blow to the logical positivists of Gödel’s day. Recall that, for logical positivists, the only meaningful statements were those that could either be empirically verified or logically deduced. Well, Gödel demonstrated the existence of meaningful statements (propositions) that were not only non-empirical, but also provably not logically deducible.
What’s worse, Gödel showed that, if PM is consistent, then it is impossible to prove that it is consistent. Even worse, Gödel showed that his results apply not only to PM, but to any possible axiom scheme that produces an arithmetic capable of doing addition and multiplication. Thus, ZFC falls prey to the same conundrum.
You’d think that such news would have thrown us mathematicians into a tizzy. It didn’t. We go happily on creating beautiful theories and proving theorems. Why? On what basis do we know that the work we are doing is even consistent? Let’s do a thought experiment and pose that question to a hypothetical mathematician. The answer would almost surely go something like this: “Of course I know that I can’t prove PM and ZFC are consistent, but I’ve got plenty of reasons to believe they are: no contradiction has arisen in close to 100 years, the systems produce results that are very satisfying, and the applications of mathematical theories actually work in the real world. Thus, for me, these outcomes provide evidence of the consistency that I hope for. They also provide a conviction of something that I do not, nor cannot see, namely, a proof of consistency.”
The evidence of something hoped for? The conviction of something not seen? Well, (Hebrews 11:1) that’s faith! Yes, today as always, mathematics requires faith. And practically speaking, perhaps more faith than we realize. It is a safe bet that almost all mathematicians—unless they work in the area of logic—would be unable to recite the axioms that compose either PM or ZFC.
How did I respond to my freethinking friend who sent the email mentioned at the beginning of this blog? I suggested that we get together for breakfast. During our conversation I reviewed the story I just shared. He allowed for some points, but remains a skeptic, although, I hope, more aware of faith-commitments he must make, and more respectful of those espousing religious beliefs. In fact, a respectful attitude is almost demanded of every mathematician. As one wit once commented, if science deals with what can be proved, and religion deals with what cannot be proved, then mathematics is the only science that can prove itself to be a religion.
- The Analyst is available at Wikisource. Click here to access it.
- Newton was not Trinitarian, though he believed in the Christian scriptures and studied them thoroughly.
- An Account of the Life of George Berkeley, available here. Stock’s account, published in 1776, is of disputed reliability.
- Basic Laws of Arithmetic, translated and edited with an introduction by Philip A. Ebert and Marcus Rossberg. Oxford: Oxford University Press, 2013.