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ASSOCIATION of CLASSICAL & CHRISTIAN SCHOOLS

The Quest for Mathematical Truth by Brett Edwards, Atlanta Classical Christian Academy

French Sophie history—from Archimedes to unbeliever is confronted with Germain of the late eighteenth Germain down to the present. this dilemma daily as he watches and early nineteenth centuries What I find so fascinating about the rise, leaves fall, and is considered one of the greatest is its ability to so plants grow in a predictable female of history. completely captivate and consume manner. Why does order exist Germain was thirteen years old the human mind. What is so in the world? The Christian sees when the tumult and chaos of the alluring about mathematics? Why the comprehensibility and the French Revolution forced her to remain indoors. This confinement turned her attention to her father’s . . . what makes mathematics particularly library where she found a book scintillating is that, consistent with the nature of of math history describing the apocryphal story of the death God, there is an infinite region of mathematics of Archimedes. It is often told incomprehensible to the human mind. that as Roman besieged the city of Syracuse during the Second Punic War, a Roman were the minds of Archimedes and incomprehensibility of the created soldier approached Archimedes, Germain so intent on finding truth order as a reflection of God Himself. commanding him to surrender. in mathematics even to the point The Christian math class ought Archimedes was so absorbed in of death in the case of Archimedes? to be a sanctuary for honest, a mathematical diagram that he I would argue that one of the open, and engaging discussion responded saying, “Don’t disturb more engrossing characteristics of both of these attributes. Our my circles.” This sentiment can of mathematics is its dual nature students have an innate longing be understood and appreciated by of being both comprehensible for truth and should see the math math teachers around the world. and incomprehensible. Man is class as another area to study the Unfortunately for Archimedes attracted to those things that very nature of God. Nineteenth- and the world, the soldier decided are both knowable and yet not century English mathematician to disturb his circles and killed fully comprehensible. While Hilda Phoebe Hudson argued the greatest mathematician there is great fulfillment when a that “to all of us who hold the of antiquity. Upon reading of particular mathematical idea or Christian belief that God is truth, Archimedes’ demise, Germain was concept is grasped, the student anything that is true is a fact so impressed with his relentless can return again and again to the about God, and mathematics is devotion to mathematics that she inexhaustible field of mathematics. a branch of .”2 Johannes committed to make it the pursuit Albert Einstein remarked said “the chief aim of all of her own life. Her parents that “the most incomprehensible investigations of the external would often find her staying thing about the world is that it world should be to discover the up all night working through is at all comprehensible.”1 This rational order and harmony which calculations by candlelight on understanding is consistent with has been imposed on it by God and her slate. She would go on to Einstein’s agnostic worldview. which He revealed to us in the become one of the pioneers of In a world without a creator, language of mathematics.”3 Such elasticity theory and for this why should one expect the an understanding gives a hallowed would win the grand prize from created order to be harmonious purpose to every calculation and the Paris Academy of . and comprehensible? Einstein’s proof in the math class. What A l t h o u g h m a n y h u m a n refusal to attribute the order of if the students in our classes pursuits come and go, the quest the world to a God of order is the had this view of their scientific for mathematical truth seems dominant view of the intellectual studies? Such an understanding to persist throughout man’s elite in our secular culture. The of mathematics would surely develop a wonder of and passion Brett Edwards is the head of school at Atlanta Classical Christian for truth in their studies. Academy in Smyrna, GA. Visit http://www.accak12.org/. It would be helpful to dig a

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The Quest . . . little deeper into each of these of mathematics. Gödel’s results Euler’s identity provides attributes and their pervasive shouldn’t come as a surprise to another impressive example of presence in mathematics. First, the Christian but are a rather the incomprehensibility within mathematics is an area in which devastating blow to the progressive the created order. The identity God has provided us access to movement’s hope and trust in the incorporates the five most notable knowledge (comprehensibility). abilities of autonomous man. constants of mathematics in a Throughout human existence Although there is an single equation stating that eiπ we have been able to uncover infinite supply of mathematical + 1 = 0. One poll conducted by mathematical truths which conundrums, two particular a mathematical journal named in turn are used for a host of examples are sufficient to Euler’s identity the most beautiful creative human applications and illustrate the incomprehensibility theorem in mathematics. 7 endeavors. Our ability to find and of mathematics. For thousands of Nineteenth-century American comprehend these truths provides years, man has been amazed and philosopher and mathematician a great sense of achievement perplexed by the nature of prime Benjamin Peirce noted that the and fulfillment especially as numbers. These numbers are the identity “is absolutely paradoxical; the become more basic building blocks of all natural we cannot understand it, and we difficult and challenging. Consider numbers (positive whole numbers). don’t know what it means, but the achievement felt by English There are an infinite amount of we have proved it, and therefore mathematician Andrew Wiles upon prime numbers and what is most we know it must be the truth.”8 successfully proving Fermat’s Last fascinating is that they have no The Christian can connect such Theorem in 1995.4 A proof for this discernible pattern. Eighteenth- a statement to their limited theorem had eluded the greatest century Swiss mathematician understanding of the Trinity. minds for more than 350 years. concluded that Although we know the concept to H o w e v e r , w h a t m a k e s “mathematicians have tried in be true, we cannot understand it. mathematics particularly vain to this day to discover some At root, this passion for scintillating is that, consistent order in the sequence of prime mathematical truth is an with the nature of God, there is numbers, and we have to interest in God Himself. God an infinite region of mathematics believe that it is a mystery into is both comprehensible and incomprehensible to the human which the human mind will never incomprehensible. That this mind. Early in the twentieth penetrate.”5 Twentieth-century dual aspect is also revealed in century, mathematician David Hungarian mathematician Paul creation through the language Hilbert and other prominent Erdös agreed saying that “it of mathematics should not be mathematicians of the will be another million years, surprising to those of the Christian embarked on a “program” to at least, before we understand faith. The psalmist declares “the eliminate all paradoxes the primes.” There are various heavens declare the glory of God, and inconsistencies from the conjectures related to prime and the sky above proclaims His foundations of mathematics. In numbers that continue to stump handiwork. Day to day pours out essence, Hilbert’s pride in the the brightest mathematicians speech, and night to night reveals intellectual potential of man in the world. Among them, knowledge” (Psalm 19:1, 2). The propelled him to believe man Goldbach’s conjecture states that search for truth in mathematics could make logical sense of all “every even integer greater than has found complex and beautiful complexities in the natural world. 2 can be expressed as the sum abstract models that explain the This view believed man could of two primes.”6 Computers have nature of the heavens, the sky, rise to the intellectual level of found the conjecture to be true and all of creation. In describing God. German mathematician and up to 4 x 1018 but no one has the nature of this quest for the philosopher Kurt Gödel would been able to provide a rigorous mathematicians of the sixteenth eventually prove Hilbert’s efforts mathematical proof for this simple through eighteenth centuries, to be logically impossible with mathematical assertion. One secular math historian Morris his incompleteness theorems. can assume that prime numbers Kline says, “Indeed, the work of These theorems would be another will continue to challenge 16th-, 17th-, and most of 18th- victory for the incomprehensibility human minds throughout time. century mathematicians was . . . a religious quest. The search for Volume XIx Number 4 7 ASSOCIATION of CLASSICAL & CHRISTIAN SCHOOLS

The Quest . . . mathematical laws of nature was Academy of in collaboration an act of devotion which would Notes: with numerous specialists, 1911-. reveal the glory and grandeur of Originally begun by publisher B. His handiwork.”9 I believe one 1. Antonina Vallentin, Einstein: A G. Teubner, Leipzig and Berlin. of the great opportunities for Biography, (London: Weidenfeld & Birkhäuser, Boston and Basel, has the Christian math teacher is Nicolson, 1954), 24. continued publication. to appropriately frame the work of their class in this . They 2. Hilda P. Hudson, “Mathematics 6. Eric W. Weisstein, “Goldbach are to communicate clearly the and Eternity,” The Mathematical Conjecture,” Wolfram MathWorld, Gazette, Vol. 12, No. 174 (Jan., connection between mathematics http://mathworld.wolfram.com/ 1925), pp. 265-270. Published online GoldbachConjecture.html and the nature of God. When this by the Mathematical Association, connection has been made they http://www.jstor.org/stable/3603647 7. Paul Nahin, Dr. Euler’s Fabulous can then embark on mathematical Formula: Cures Many Mathematical adventures revealing the intricate 3. Johannes Kepler, Defundamentis Ills, (Princeton, NJ: Princeton complexity of God’s world. In Astrologiae Certioribus, Thesis XX, University Press, 2011), 2-3. finding the rational order behind 1601. God’s creation, the student can 8. Edward Kasner and James truly experience and enjoy the 4. Simon Singh, Fermat’s Enigma, Newman, Mathematics and the glory and grandeur of God. (New York: Anchor Books, 1998). Imagination, (Mineola, NY: Dover, 2001), 103-104. 5. Leonhard Euler, Opera Omnia, Series 1, Vol. 2, 241, edited by the 9. , Mathematics: The Euler Commission of the Swiss Loss of Certainty, (New York: , 1982), 34.

2012 Chrysostom Oratory Competition

Champion Runner-Up Phoebe Hu Daniel Dawson Tall Oaks Classical School Rockbridge Academy Congratulations, Phoebe and Daniel!

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