One of the greatest joys that I have found in my religious life as a Dominican friar has been the opportunity to use my previous studies in mathematics to talk about matters of the Catholic faith; grace does perfect nature, after all. So, when I was assigned to work at an all-girls’ high school for ministry this year (a task made less daunting by the Dominican Sisters who run the school), I jumped at the chance to give guest lectures in one of the math classes, among other pastoral activities. While explaining to the geometry class one day the differences and similarities between axioms and theorems, I found an opportunity to draw a parallel, as it were, to the logic of belief.
It is often the case that geometry is the first class in which students are introduced to the method of mathematical proof. Beginning from principles (axioms and postulates), the students devise logical arguments to demonstrate that the desired conclusions are true, and the same type of demonstration occurs in theology as well. While a theorem of geometry is proven in this way, an axiom (from the Greek axios, “worthy”) is proposed to us as worthy of belief, without having been proven.
While a high school textbook would include more axioms (also known as postulates), the first systematic textbook on geometry was built on only five axioms. This book is the Elements of Euclid, who lived in Alexandria in the third century before Christ. The first four axioms are facts that appear intuitively obvious, such as “Two points determine a line,” and they show how to use a straightedge and compass, the two tools used by ancient Greek geometers.
The fifth axiom, however, is not so obvious, and it is often expressed in geometry textbooks as the “Parallel Postulate”: Given a line and a point not on it, there exists a unique line through that point parallel to the given line. No one before Euclid had identified this principle, but his whole system of geometry would break down without it. Other famous results, such as the Pythagorean Theorem for right triangles, or the fact that the angles in any triangle add up to 180 degrees, depend on this non-intuitive axiom. For centuries, mathematicians have tried, and failed, to prove this axiom using the other four. Others have devised alternative systems of geometry that neglect or even deny Euclid’s fifth axiom, which lead to radically different results, such as spherical geometry (where even parallel lines could meet, like lines of longitude at the North Pole) or hyperbolic geometry (where lines in a plane that are not parallel could never meet).
A similar phenomenon occurs in the realm of faith. Just as the geometry book gives some statements as postulates when they can in fact be proven (though with difficulty), the Catholic faith proposes some ideas for belief, such as the existence and uniqueness of God, that can also be demonstrated. These proofs rest on principles that are as self-evident as Euclid’s first four axioms; for example, St. Thomas begins his first proof for the existence of God, “It is certain, and evident to our senses, that in the world some things are in motion.” Yet the proof itself requires some knowledge of metaphysics, and it is easy to make mistakes in the argument; therefore, the Church proposes that we take the existence of God on faith, so that anyone can come to believe in Him.
Yet there are many statements of faith that are neither obvious, nor can they be proven. Take, for example, the Trinity, that the one God is three Persons; or the Incarnation, that God took on human nature in the person of Jesus Christ; or that this same Jesus rose from the dead. These articles, or axioms, of faith, can only be believed as true, if one is to study Christian theology, or more generally, to live the Christian life.
Some theologians have tried to prove these articles (a truly good God should become man to show forth His goodness, right?), but like the attempts to prove the Parallel Postulate, they fall short, as they cannot argue with certainty. Countless other thinkers have denied articles of faith because they are not self-evident and not subject to the standards of rational proof– but in doing so, they end up in a world even stranger than that of non-Euclidean geometry. By believing something contrary to the articles of faith, one could end up walking around in circles (like the spherical case) or diverging along any path imaginable (like the hyperbolic case), rather than living in an intellectual relationship with the living God who leads us on the straight-line path towards the infinity of eternal life.
Furthermore, while the last postulate that holds plane geometry together may come from the mind of Euclid, the axioms of faith can only be revealed by the God who loves us to the point that He communicates His inner life to us and calls us into His company. Because they are revealed by the God who loves us into being, these axioms, like the Trinity, Incarnation, and Resurrection, are truly worthy of our belief, and with God’s grace, we can take them on faith as the basis for living each day of our lives.