Discovered in 1805 by a French priest the ‘grotto’ at Mount St. Mary’s is ground zero for a form of mathematics that holds the power to make something from nothing. What is a grotto-cave but empty space? Confronted by a formless void Father John Dubois reached into the darkness and affixed a cross, that orienting, optical instrument indispensable to seeing things clearly. From the dimness of a yawning chasm emerged something good, true and beautiful-a Catholic university. Today, at the center of the grotto-cave there remains a crucifix. To Catholics this image rivets the attention to the body at its center-Christ crucified. To mathematicians the intersecting lines of the cross create a ‘point’-a seemingly empty space-that is, in fact, the central generative engine for mathematical discovery.
Euclid began his treatise Elements by describing the unembellished ‘point’ as ‘that which has no part’. He does not begin with the isosceles triangle or the square of the hypotenuse but with the geometry’s first essential term. More emphatically, the ‘point’ is defined as having no width, length, or breadth. As the song says ‘Nothing from nothing leaves nothing’. Yet the point does contain one indisputable quality – location. The tiny, valueless speck as described by Euclid, is far from null. It is the starting point whose position is central.
Like the cave the insignificant ‘point’ is a black hole seemingly bereft of value. Yet it is the point, Euclid asserts, that begets the straight line and the profusion of geometric forms that flow from it. A line, postulates Euclid “is that which lies evenly with points on itself”. In nature there are no straight lines. Earth is spherical. Space is curved. Yet the advancements spawned from complex cultures-Egyptian pyramids, Roman columns, American GPS systems-owe their existence to the shortest route from point A to point B.
Describing mathematical rules that govern reality Euclid’s geometry fueled material progress. Squares, triangles and rectangles proved indispensable to Greek civilization building. Yet, theoretical difficulty continued to arise, around shapes that had no sharp corners. Curvature emerged as a nemesis to the straight line. Are there straight lines in a circle? From the solitary ‘point’ the four cardinal directions can be drawn-north, south, east and west. From there an infinite number of ancillary lines can be added, with all points equidistant from the center, until the perimeter of a circle appears, strangely reflecting the contours of the prismatic eye that beholds it.
Invented in Greece, the wagon wheel, is widely touted as the greatest technology ever conceived. But what is a wheel without its axle? How then to steady a spinning disk and capture its furious rotation without the still-point at its center? What is Euclid’s infinitesimal ‘point’ but a circle with zero radius? A building block for flat surface geometry, Euclid’s unyielding ‘point’ would prove to be a foundation stone for higher forms of mathematics to come.
Like Euclid, Rene Descartes obsessed over a ‘point’ when he was a boy. It was a ‘fly’ on the ceiling. To determine the insect’s location mathematically he measured its position by its distance from each wall. By this method of intersecting x/y coordinates to identify a position in space he created the Cartesian plane.
Without knowing it the religiously skeptical Descartes had recapitulated a familiar form. The Cartesian plane is a cross upon which points are plotted and reality is observed. On its shoulders nature is harnessed and science is conscripted to the service of mankind.
Descartes pioneering crosshairs fueled a new era of mathematical advancement providing the first systematic link between the static objects of geometry and the abstract equations of algebra. Calculus, too, arose from this regulating axis accelerating mathematics past the world of objects with its sweeping power to measure invisible forces like the speed of light and the motion of the celestial bodies.
In the beginning there was a ‘point’ of infinite density that existed in a void containing all the matter and energy of the universe. The explosion that rippled outward into all four directions, known as the Big Bang, created time and space. First proposed by Belgian scientist Georges Lemaitre the ‘Primeval Atom Hypothesis’ described an expanding universe, constantly receding with an indisputable beginning-middle-and-end.
The detonation of a single point of immeasurable mass became more problematic when the catalyst for igniting such an explosion was considered. Such a phenomena sounded overtly theological to Lemaitre’s contemporary Albert Einstein. The famous German-born physicist wasn’t going there. But the priest who earned a PHD from MIT had no problem dividing his attention. By reconciling two divergent disciplines – science and religion – Lemaitre demonstrated multiple ways of arriving at a higher truth. As a man of God, Lemaitre knew that the reconciliation of opposites is not a difficulty to be avoided but an invitation to be answered.
When Father John Dubois was beckoned by a point of light he followed it. Stumbling into an arboreal ‘grotto’ he discovered what would become ground zero for a great Catholic university. On many Catholic campuses a relic of Catholic topography remains. A grotto-cave invokes the storytellers of old, who approached the mystery of ’empty space’ as a nascent potentiality not a lifeless nullity. For Christians the cave is the place where Jesus Christ threw off the chains of death-a mathematical ‘singularity’ describing an experience where our equations stop making sense.
Turning dead fiber into living flesh presents a similar singularity. Located at the center of the Catholic Mass the superposition of the Eucharist signifies a sacrifice, a place where an exchange with the unknown is made for the sake of a divine mathematic. The indispensable ‘point’ of Euclid’s geometry, the immoveable axis of Descartes ‘cross’, and the mystery of the Eucharist are not just singularities but are the central generating engines of civilizational advancement and the keepers of reality itself. At the still-point of two clashing and contradictory lines resides a paradoxical location at whose center we are always aiming because that’s where the truth can be found.
When the Fathers of the Church launched the world’s first universities across Catholic Europe they applied the same time-tested mechanism for spiritual inquiry across all disciplines-theology, philosophy, mathematics and the sciences. In a material world where there are no straight lines they must be imposed by a foolproof framework embedded in the navigable periscope of the sea captain, the optical telescope of the physicist, the magnifying microscope of the biologist, and the range-finding cruciate of the theologian. To conceive of a venture is to measure its potential through the intersecting lines of the optical lens. At the center of these reticle crosshairs presides a mysterious, focalizing pupil-a dark cave-that has the capacity to transform chaos into order.
In the Grotto-cave at Mount St. Mary’s the center holds confirming that all of creation hangs on a ‘singularity’-an inconsistency of logic. The mathematics of the Grotto embraces the potential of empty space, making room for the impossible to be done at the locus of causality where a negative becomes a positive.