What kind of conversation might that be, mathematics and faith, and how could it possibly be of interest to a group of ordinary, non-mathematical, urban churchgoers in the
Greater Toronto Area? So we wondered when Mississauga Mennonite Fellowship announced that it would host Prof. Tim Rogalsky of the Mathematics Department of
Canadian Mennonite University in Winnipeg to bring a lecture series with that title for our annual seminar weekend this year (2011) on March 4, 5, and 6. The series included four sessions, the first on Friday evening entitled Pascal’s Wager, the second Saturday afternoon on Chaos Theory, the third on Saturday evening called Truth and Liars, and the final on the Sunday morning service with the title Fibonacci’s Flowers. All but the
Sunday service were held at Erin Mills United Church. The Saturday event included a meal and group singing between sessions, and gave us the chance to enjoy Tim’s proficiency with the violin and the fiddle, depending on the mood of the song. The sessions were well attended, including guests from the wider Mennonite community and the local non-Mennonite community as well. All agreed that the event was stimulating to the mind (independent of the mathematical background), the heart (that is, our faith quest), and our esthetic senses (with Tim’s musical and photographic contributions.) A true feast for the spirit for all who participated!
Friday Evening: Pascal’s Wager
Blaise Pascal was an early seventeenth century mathematician who founded probability theory and contributed to the beginnings of several practical sciences, like hydrostatics, as well. He is well known for his famous wager that says, “If God does not exist and you choose to worship him nothing is lost in your life, but if he does exist the prize of everlasting bliss is yours. How could you not choose God?” Tim pointed out that this view has often been criticized for leading to a very shallow Christianity, but Pascal did not stop there. Along with a side discussion on probability and the wisdom of buying lottery tickets, Tim explored the biography of Pascal to find that he underwent two religious conversions, the latter one being Christian mysticism. Pascal’s wager, in this context, is his way of saying that reason can lead one to the choice for God, but to enter relationship with God is not of the head, but the heart, faith rather than reason. When one finally perceives his need for God by reason, he must think of himself as a lost traveler, who must ask directions from others with more experience in relating to God to learn from them. For Pascal this meant joining the Catholic church with obedience to its ways with full dedication. In spite of his tendency toward mysticism, Pascal in his lifetime was known as a philanthropist who carried his works of charity to the poor in the streets. We can remember him as one who expressed in his words and his life the importance of both reason and faith in our search for relationship with God.
Saturday afternoon: Glimpses of Chaos Theory
The session began with an 1814 quotation from the mathematician Laplace that expressed the belief that if we could but know enough of the present we could predict perfectly the state of the future in every detail, that the universe is completely deterministic based on the immutable laws of physics. Ironically, Tim then presented us
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with another quote, this time from a Professor Lighthill after 1960, which apologized for science’s previous arrogance and expressed that determinism is no longer a viable worldview. This retraction of determinism was the result of our recent encounter with
Chaos Theory, a study made possible by the computer’s ability to calculate millions of steps in random processes.
Tim began with the idea of the simple attractor, an equation into which you put a random number and then repeat with the previous result as input. A simple attractor takes different forms, but one that he showed us after many tries kept repeating the sequence 4
2 1 4 2 1 4 2 1 … A geometric example was the Sierpinsky triangle, which operates on an equilateral triangle in a fixed way using random numbers; after thousands of random starts a beautiful pattern emerges showing arrays of similar triangles, an example of which can be found in some seashells. He showed us how a simple set of obvious rules fed by random numbers can lead to the simulation of a flock of birds in flight, and when he compared the simulation with a movie sequence of real birds in his garden, it was hard to tell which was real! Tim introduced us to fractals, mathematical functions that use random numbers in repetitive processes. When the results are plotted in two dimensions consistent patterns emerge at different magnifications, as in the Mandelbrot and Giulia sets he showed us. The idea of similarities in nature at different degrees of magnification is expressed in tree branches and shorelines, both of which can be interpreted in terms of fractals.
Perhaps the most profound result of these random processes was discovered by the meteorologist Lorenz who used an early computer and a simple meteorological model to show how weather systems would develop over the course of several weeks. He found that when he interrupted the program and then restarted using what he thought were the accurate representations of the preceding results, the simulated weather systems followed the pattern normally given by the uninterrupted program for the first few days, but then began to diverge rapidly. He attributed the divergence to minute round-off errors he had introduced in the interruption. This result is sometimes expressed by saying when a butterfly flaps its wings in China, it may affect the weather in New York. We say that weather development is non-linear, and Lorenz’s result now limits our reliable weather forecasts to about five days. This leads us to the question of whether the future is knowable, even to God. Tim talked about an Open Theism in which God is engaged in continuous creation, always bringing order out of chaos. He pointed out that this need not weaken our belief in God’s omniscience, and he gave examples of God’s from the Old
Testament, where His actions are determined by the level of obedience of his people.
Open theism seemed to resonate with the audience, giving hope for a future we help God to shape.
Saturday Evening: Truth and Liars
The topic of the evening was a brief review of a class that Tim teaches at CMU on classical logic Logic is the basis of mathematics, where it works perfectly, but Tim showed that when we expect real world experience to follow classical logic, we invite difficulty. He began with the definition that in classical, or bivalent, logic a statement must be either True or False. There are three basic laws of classical logic, each of which he showed to be dubious in many cases. The first is the Law of Non-Contradiction, that a logical argument can allow no contradictions. Yet even his textbook on logic
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acknowledges possible errors in its argument that show it, too, is vulnerable to selfcontradiction. The second law is called in Latin Ex Falso Quodlibet , meaning that from a false premise, any conclusion can be deduced. If taken literally, this law can be shown to be absurd even in ordinary conversation, because some things cannot be logically connected. The third law is that of the Excluded Middle: Either the statement A OR its opposite must be true. Thus if a surface is not white it must be black, and all shades of grey are not recognized. A listener pointed out that a recent US President used this law when declaring any country that cooperated a friend and any who did not could only be an enemy. Tim gave an example of how in church discussion sometimes ideas are treated bivalently when there are in fact alternatives to bivalent logic. One alternative is
Dialetheism, which allows a statement to be True or False or Neither or Both. Another alternative prominent in software design today is so-called “fuzzy logic,” in which a numerical rating is assigned to a statement to estimate its probability of being True.
Tim gave an excellent illustration of a biblical passage that does not fit classical bivalent logic. It came from Galatians, but it appears elsewhere in Paul’s writings, too.
Paul writes the Greek phrase pisteos Christos , where pisteos is agreed by all scholars to mean faith and Christos is of course Christ. There is no word connecting these two in
Greek, but in our languages (at least English and German) we need something. So the possibilities of translating “faith Christ” are at least the following three:
Faith that Christ [died for our sins] as in Martin Luther
Faith in Christ (trusting in his protective presence) as in the theologian Dunn
Faith of Christ (the example for us to follow.)
The difficulty is compounded by the dative and the genitive cases having an identical form in this instance. One can easily see that someone who grew up with a certain translation may try to impose bivalent logic, arguing that his understanding is the only one possible until shown that the passage itself is ambivalent! We seek our guidance from the Holy Spirit through the interpreting community and not by invoking outdated Greek logic.
Some of us came away remembering Tim’s earlier quotation from St. Augustine,
”When you think that you finally understand God, be sure to realize that what you understand is not God.” There is always much mystery for us to ponder with awe.
Sunday Morning: Fibonacci’s Flowers
Our Sunday morning service took up the theme of mathematics, mystery, and awe, beginning with a reading of Psalm 19, “The heavens declare the glory of God.” The mathematical base was the set of numbers known to the Western world as Fibonacci numbers. The sequence is determined starting with zero and one, then making the following number the sum of the previous two. So 0 +1 = 1, 1+1 = 2, 2+1 = 3, 3 + 2 = 5,
5 + 3 = 8 and so on so that the sequence is 0 1 1 2 3 5 8 13 21 … Fibonacci presented this in a puzzle about the number of rabbit pairs you would have at the end of a month under fixed conditions. The mystery and awe became more apparent in Tim’s slide show of beautiful flowers, surprisingly with the number of petals on each blossom given by a
Fibonacci number. Thus blossoms with three or five or eight petals are far more common than those with four, six, or seven petals. A geometric construction using squares with sides given by the Fibonacci numbers generates a Fibonacci spiral, and these spirals were evident in Tim’s photographs of the seed distributions in an artichoke, a pine cone, and a
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sunflower, among others. But how do you place the beginning of the spirals so that the seeds are packed with maximum density> A few obvious choices (like every tenth of the circumference) showed that most choices fill the outer reaches of a seed pad more efficiently than the center. The most uniform distribution uses the Golden Ratio, 0.618, as the spacing choice. The Golden Ratio is the ratio of one very large Fibonacci number to its next larger Fibonacci number. Any large pair will do, because the ratio always tends toward the same Golden Ratio, but the larger the pair the better. So by choosing the start of the Fibonacci spirals of seeds to be 0.618 of a circumference apart, you get a distribution similar to the seed distribution in the sunflower or in the daisy. Tim then showed the use of the Golden Ratio in examples of architecture from classical Greece and
Aztec Mexico, based on rectangular plans the sides of which are in the ratio of 0.618.
Thus the ratio is known to other cultures as well as ours and found to give a pleasing effect. Even the lengths of our limbs are related by the Golden Ratio! What mystery this sequence of Fibonacci numbers holds that transcends so dramatically its simple numerical base! And isn’t this just one of the many natural mysteries in mathematics and nature that reflect the mystery and beauty of the vision that is faith?
As his final tribute to the intersection of mathematics and faith, Tim introduced us to the representation of a four-dimensional cube, called a hypercube, on a twodimensional sheet of paper. In one mode of representation the hypercube takes the form of a three-dimensional cross. This symbol was knowingly employed by Salvador Dali as the cross in his famous painting, The Crucifixion. Tim pointed out the interpretation of the Resurrection as the three-dimensional human Jesus transcending into the fourdimensional world of the Father. This is brought out in the painting by the threedimensional disciples looking up to the figure of the three-dimensional Christ nailed to the four-dimensional cross, while on the floor below is the two-dimensional shadow that we are allowed at this stage to perceive. Thus Dali connected the mystery of faith that is the Resurrection with the mystery of a fourth dimension that mathematics provides. What a fascinating reminder of the mystery that we can only begin to perceive!
And so with the mystery and sense of awe expressed by Psalm 19 at the beginning of the service, we closed the morning and the weekend spiritually refreshed with a renewed vision of the wonder and glory of God that just keeps unfolding before us as we walk in the light of faith with open and inquisitive minds and hearts.
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