Samuel Damren
By the mid-1500s, the Catholic church was in profound crisis. The Protestant Reformation created chaos throughout the religious and political order of Europe.
The chaos even extended to the calendar.
The Julian calendar was badly out of synch with the solar year. In 1572, newly elected Pope Gregory XIII assembled a commission to reform the calendar. As an adviser, the Pope selected Christopher Clavius, a Jesuit professor of astronomy and mathematics at Collegio Romano.
When the Gregorian calendar was later instituted in 1582, 10 days were erased from that year’s calendar. The underlying problem was solved by designating future years divisible by 100 as leap years except those divisible by 400.
Clavius was the Pope’s spokesperson. Mixing astronomical calculations, mathematics, and scripture, he published a 600-page explanation of the new calendar which convinced a significant number of critics across religious and political borders to accept it.
It was a victory for the Catholic church and taken by supporters as proof that the church could restore truth, order, and regularity to an unruly world.
It was also a victory for the Jesuits, popularly known as the Black Robes, and their commitment to disciplined religious education based on adherence to classical teachings, including Aristotle and Euclidean geometry.
However, alongside geometry lurked an ancient dilemma posed by the smallest of the smallest of mathematical abstractions – the infinitesimals – which by the 1620s had become of interest to Italian mathematicians in Galileo’s circle.
The Black Robes disagreed. They viewed the infinitesimals as a corruption. As a result, in 1632 the Jesuits issued an edict under papal authority banning all teachings based on the infinitesimals.
The history of an ensuing century long conflict that began in mathematics, but would extend well beyond academic and religious circles, is described by Amir Alexander in his 2014 Book, “Infinitesimal – How a Dangerous Mathematical Theory Shaped the Modern World.”
The infinitesimals were first introduced to Western thought in the 5th century B.C. through Zeno’s paradox.
Zeno posited that if you use a system of mathematics analogous to a runner’s “legs,” whereby the runner can only halve the distance to the finish line with each “step” then the runner can never cross the finish line, e.g. first step = 1/2 the distance; second = 1/4; third = 1/8 … eighth = 1/256; ninth = 1/512 …
With each succeeding step, the runner gets closer to zero, but only closer and closer. Those infinitely small steps are the infinitesimals.
Zero is also a mathematical abstraction. It was introduced to Western mathematics in the 1200s from the Islamic world. By the 1500s, its utility as a placeholder to better perform complex multiplication and division than the existing system of Roman numerals was widely recognized. Zero itself, however, is of no utility in multiplication, e.g. 7 x 0 = 0 just as it does for any number multiplied by zero.
Building on his law of falling bodies in the late 1620s, Galileo’s followers were on the cusp of conceiving the infinitesimals as a remedy, in today’s parlance, to the “blank screen” that would appear whenever zero was used in multiplication.
The Jesuit edict of 1632 prevented further exploration.
Calculating uniform speed is straightforward: Distance/Time, e.g. 300 meters/1 minute = an average speed of 5 meters per second. But calculating the speed of an object moving at varying speeds in a “freeze frame” moment – as with planetary motion – is problematic because in that moment and in that position Distance and Time will be zero.
This difficulty prevented Galileo from presenting mathematical proof that the Sun, not the earth, was the center of the solar system. For his continuing insistence on this unsubstantiated assertion, he was condemned by papal authorities to permanent house arrest in 1633.
The “freeze frame” problem was ultimately solved by including infinitesimals as part of the factors for Distance and Time in the equation so that they could “hold” the factors in “place” during reduction but then be discarded from the actual product as “too small to matter.” Instead of being birthed in Italy as it very well could have been, the method of calculus in varying formats was invented by Isaac Newton in England and separately in Saxony Germany by Gottfried Leibnitz.
When Newton combined the mathematics of calculus with his theory of gravity in “Philosophiae Naturalis Principia Mathematica” in 1687, the result could not be overstated.
The instrument of mathematics the Jesuits relied upon to impose religious truth, order, and regularity through the Gregorian calendar turned against the Holy See.
Newton’s “system of the world” also relied on mathematics, but through the method of calculus announced a true explanation of the motion of the planets and other objects based on empirical evidence alone, not faith. The prior “system of the world” - professed by the Catholic church and supposedly mandated in Scripture – was thus proven wrong, undeniably wrong.
In his 1951 book, “The Philosophy of the Enlightenment,” Ernst Cassirer described the momentous effect of Newton’s discovery on the Western world.
“This truth is revealed not in God’s word but in his work; it is not based on the testimony of scripture or tradition but is visible to us at all times …
The truth of nature cannot be expressed in mere words; the only suitable expression lies in mathematical constructions [through which] nature presents itself in perfect form and clarity …”
“In nature, the whole plan of the universe lies before us … waiting for the human mind to recognize and express it … and such a mind had meantime appeared. What Galileo had called for became reality in Newton.”
Through “The Principia” as the work is now known, Newton shattered the infallibility of religious authority. In so doing, through the discipline later called science and then known as Natural Philosophy, he presented the world with a different source of truth, order, and regularity.
Understanding nature would become the scripture of the modern age.
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Samuel Damren is an attorney and author in Ann Arbor.
