Pi Of Life: Zero(First Draft of Chapter 1)
The only wisdom is the knowledge that you know nothing.
It was a beautiful sunny day. That, and it being a Friday, ensured that the students rolling into class after lunch would only be ramping up their Sweathog reputation. Vinnie Barbarino, Horshack, and Epstein would have been deserving monikers for many of the math students. On this particular day, some of the students decided to have a lunch that was more liquid than solid — specifically, 75% beer and 25% nachos. Punctuality for the afternoon math class would be taking a Mike Tyson beating that day. The teacher, however, was a very wise and outgoing soul. His approach to teaching mirrored the general disdain for structure found in many of his students. The teacher was no ordinary teacher. He was professor Dave Alexander, a highly respected math teacher at the Faculty of Education in Toronto. I was merely one of his admiring students — and yes, probably the Freddy Washington of the group. It is April, 1992. I am at Teacher’s College at the University of Toronto. While our class contained some brilliant students, it also created and housed a lot of pranksters and provocateurs. We were the general antithesis of the graduating class of future teachers. Our presentations, while rooted in academic research, tended to be framed by undetected mockery of the institutionalized and factory-feel of teacher education. Teaching is more art than science would be the guiding mantra of our collective rebellion. On this particular day — our last math education class before exams — our rebellion was spiked with the hoppy and fruity notes of our favorite draft beer. Dave was unsurprisingly indifferent to our late and laughing entrance. In fact, I distinctly remember him staring out the window in a quasi-Kafka fashion as we found our seats. Most days this would have meant nothing. Not today. After some light questions and remarks about our understandable tardiness and early celebration of the end of the year, Dave asked us to get out a piece of paper. He asked us to discuss in our table groups the most important qualities of being a great teacher. Once we had compiled our list as a group, we would share their results with the rest of the class. Even though the dwindling hours of Friday classes and mild happiness of drinking a few pints in a blistering sun were influencing our interest level, the question posed by Dave had us all collaborating rather quickly. This was, after all, our chance to punctuate our teacher education with our belief that teaching was indeed an art form of the highest calling. Every group was finished within fifteen to twenty minutes. The energy and volume level of the class suggested that every group was confident that they had written the Ten Commandments of Teaching. The first group went up. I cannot recall their answers, but I am sure they put knowledge in their list. After they were done, Dave simply said “Thank you, Next”. What? No affirming “good job”? No gold stars? Where was the feedback? To make matters more confusing, Dave looked out the window quite a bit during the rest of the presentations. Even when our group offered humor as an important criteria for successful teaching, he was unmoved by our choice. The window seemed to offer more interest than responses to a crucial question from future teachers. We would learn that Dave was not so much disinterested in our answers, he was only confident that no group would actually please him with the… two responses he had in mind. Yes. In spite of giving us a red herring requirement that was unbounded, there were only two qualities that Dave felt were needed for a lifelong career of happy and successful teaching. Honesty and being a mutual learner. Even as Dave started to explain his philosophical simplicity, I soon felt his voice recede into this muddied tone that you might hear in a movie — signaling some pivotal moment of reflection for the protagonist. Teaching was not so much an art form; it was simply about being…human. The week prior, we were sitting in this same class examining how to effectively model mathematics for students. Symbols and formulas were discussed with proper pedagogical language. There was plenty of resonance with our brains. Today, our last day, was all about the heart. So while Dave had given some outstanding strategies for teaching mathematics with engagement and clarity the entire year, he wanted our lasting impression of everything — him, our class, teaching, etc. — ending in the warmth of our own humanity. It was never said, and really it didn’t have to be. Teach mathematics as a human endeavor filled with stories of hope, struggle, pleasure, courage, etc. was the last message that we were all given. I would not realize it at the time, but I was given a gift. A gift to shift my rudder on my novice boat of teaching. Familiar waters of formula memorization, procedural memorization, and problem memorization would not be in the cards for me. I would be sailing in calm but rather lonely ocean for the next twenty years, struggling to not only make sense of mathematics, but just to make sense of it all. Period.
Zero. Zilch.Zip.Nil. Naught. They all generally mean the same thing — nothing. India, which gave the world the universal counting system we know today, also gave us zero. In fact, the first documented evidence of this critical number is considered one of the holy sites of mathematics, and has been visited by some of the world’s greatest mathematicians over the years. A 5 hour train ride south of New Delhi will bring you to the city of Gwalior, one of many cities in India overlooked by impressive forts. If you take the path upwards towards the fort, you will approach a tiny temple. It is here that the pilgrimage of many mathematicians ends. A small 100 sq ft room that requires most people to duck their head slightly upon entering. On one of the walls is an inscription that dates back to the late 9th century. It is filled with numbers. And, nestled among these familiar numbers of 1 to 9 is the new number — zero. It occurs twice on the wall. One of them is in the number “50”, with the familiar circle to represent zero. The number here represents the daily gift of 50 garlands of flowers. Fitting that the earliest recording of this revolutionary number — Zero: The Biography of a Dangerous Idea by Charles Siefe is a marvelous read — would be attached to the beauty of nature. As well as zero, India made another contribution to the world, almost parallel in nature. This would be the idea of nirvana, the transcendent state of “nothingness”, when you are liberated from suffering and desires. In fact, the word used in philosophical texts to mean nothing, or the void, is “shunya”, the same word later used to mean zero.
For George Gheverghese Joseph, one of the leading math historians in the world,, the invention of zero happened when an unknown Indian mathematician about two thousand years realized that “this philosophical and cultural concept would also be useful in a mathematical sense.”
Zero denotes nothing. But in India it was derived from the concept of shunya. Shunya means a sort of salvation. According to Renu Jain, professor of mathematics at Jiwaji University in Gwalior “When all our desires are nullified, then we go to nirvana or shunya or total salvation.” In the modern world it is common to see religion and science as always in conflict. Yet in ancient India, mathematics and spirituality are forever intertwined. It is this idea of zero that frames not only this chapter, but essentially the whole book — to find and discover the simple kernels of truth and joy. And, much like the journey to the remote site in which this mathematical treasure is found, so will be the assent to mathematical happiness — filled with broken paths, hidden trails and secret passages. But before anything of philosophical value can be added by zero, it is important to wade into mathematical waters and see — for yourself — how simple and complicated zero can be! When you add or subtract zero from a number, the result is the number you started with. 12 + 0 and 12-0 both give you 12. I know what you are thinking — that was obvious. Obvious might be the most dangerous word used in teaching and learning mathematics. It can not only create an unhealthy haste in learning math, but it can also truncate potentially rich discussions. Zero is a concept that is ripe with contemplative questions. The ones that cause the most trouble are the one when we have to divide a number by…gulp…zero! If there is one operation that has left the strongest residue of discomfort with math, it must be division. And this is because division led us all into that dangerous lair of fractions — the first f word we really learned! Part of the problem is that division divorced us from our comfort zone of just adding numbers. Math was perfectly fine when we restricted our learning to summing up stuff. Even multiplication seemed understandable— it was just the Ferrari of adding. Instead of adding up, let’s say, the number 4 eight times like 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4, we could simply say 4 x 8. We might still have stalled a little bit in knowing the answer, but when I went to school, knowing your times tables was mandatory common sense like brushing your teeth twice a day and looking both ways when you cross a street. But, even if you got stuck, you at least understood what was implied in multiplying. It all came back to adding. Guess what? So does division! If you buy that subtraction is just negative addition, then you will be more at ease with division housing the concept of addition. 12 divided by 2 is 6. Now just shift our idea of division to subtracting until we get to…zero! How many times can you subtract 2 from 12 until you get to zero? 12–2–2–2–2–2–2 = 0. Six times! What about 57 divided by 9? The answer is not as important as knowing that you will have something leftover. You cannot take away 9 some whole number of times from 57. Now what about something like 12 divided by 0? Is this even possible? Is there an answer that makes sense? Well, start taking away zero! How many times can you subtract it from 12 until you get to zero? Try as you might, but you will never get down to zero! You will be stuck at 12 forever! There is a danger of saying the answer is infinity, but it is actually undefined. There is an excellent explanation for this, but I will share simplest and most whimsical. If you say, for example, 12 divided by zero is infinity and 97 divided by zero is infinity, then 12 divided by zero must equal 97 divided by zero. Which means 12 must equal 97 — since the zero denominators equal each other! Now that would be ridiculous! However, in many ways, infinity is the yang to zero’s yin. To quickly clarify, infinity is not a number. It is merely a symbol to denote something without any limit. There are beastly large numbers like Graham’s Number, which is so large that even if every digit was written the size of a sub-atomic particle and the observable universe was a sheet of paper, you still could not write down all the digits of this unimaginable number. Yet, it is still a finite number. It will come to an end. Infinity just marches to an incomprehensible beat forever. If your head is beginning to throb yet so slightly, you are not alone. Georg Cantor, the father of infinity, lived the major part of his later years in depression due to the hostility he faced because of his ideas about infinity from many of his contemporaries. In the BBC documentary Dangerous Knowledge, Cantor’s path to insanity was perhaps due to what he had seen — infinity. Or, in Cantor’s mind — God. It’s some dark conjecturing in the documentary, but there is evidence that the idea of infinity consumed Cantor’s mathematical life in ways that were unhealthy. He died in a sanatorium near the end of World War 1. This will not be the first time a tragedy of a key mathematical figure will be conveyed. Untimely deaths and sadness have been woven intentionally into the Pi of Life to humanize the lives of those that usually endured much scorn and ridicule. Yes. They were brilliant mathematicians, but that was a only a subset of their lives. They usually had other creative talents… and vices. Their failures and failings made them real. Which brings them and all of us really to a singularity that is both comforting and awe-inspiring — being human. Our imperfections and our horribly short 2.5 billion seconds on this planet(78 years is an age we should all be thankful to live up to!) set against the vast cosmos shrinks us to…zero. For me, this is hardly daunting or depressing. It’s almost purifying. To live each day with new found wonder. To not take too much pride in accumulating knowledge, as there are still many mountains to climb and terrains to cross. Zero is a reality of inspiration. Now just think about all the uses of zero that we have in our culture. Absolute zero, the temperature at which, theoretically, all motion stops. Getting down to absolute zero has essentially the identical problem with getting up to the speed of light. Getting to the speed of light requires an infinite amount of work, while getting down to absolute zero requires extracting an infinite amount of heat. Just to make it clear, both of these are impossible. Doesn’t this remind you of the yin-yang relationship of zero and infinity? Ground Zero has a similar connotation of intensity — the point closest to the Earth’s surface to a detonation. Zero generally means nothing, but the void that it creates leaves us with a powerful mathematical or philosophical residue. To zero in is a common idiomatic phrase to denote a high level of focus and accuracy. Zero is also commonplace in pop culture in movies and music. The mid-90's alternative rock sensation, The Smashing Pumpkins, had a song called “Zero”, which had lyrics hinting at a nihilistic connection between all of us, including a deity.
Emptiness is loneliness. And loneliness is cleanliness. And cleanliness is godliness. And God is empty just like me.
Regardless of what you feel about those lyrics, the mystery and weight of zero has been given proper homage. My own idea of zero as applied to learning math is kind of a pureed mixture of all the definitions and ideas — seasoned heavily with the ancient wisdom of the Athenian philosopher and a key founder of Western philosophy, Socrates. I am just as interested in my ignorance of mathematics as I am in my knowledge of it; I am just as keen to discuss my misunderstandings and mistakes that I have made as I am with my correct ideas and notions. Nowhere did I ever make that more apparent — or public — was when I gave a keynote address to 200 people back in 2007. These people were students in grades 4 to 6. I had been invited by a local school to speak to and be involved in their “Math Day”. Perhaps a bit earlier in my career I would not have given the talk that I did. It would have been heavily weighted towards my knowledge of mathematics than to my wonderful ignorance. One would never think of prefacing such a negative word with something so buoyant, so joyful, so…happy. Yet, the seeds of change in philosophical navigation were planted that innocuous and slightly inebriated afternoon in my last math education class fifteen years earlier. In fact, even though this was my first keynote, I didn’t really have a speech prepared. I had two ideas prepared that, looking back now, were a testimonial to my last math lesson ever in Teacher’s College — be honest and always learn with the same passion. The night before my speech, I was busy selecting a few songs that I would play over the loudspeaker as kids filed into the gymnasium and were seated by their teachers. I wanted to set a mood. I remember as a high school student always feeling a slight rush in my blood anytime I heard the school band rehearsing for an assembly. Music is such a comforting balm. I figured I could play two songs. I had over 13000 songs on my iTunes at the time. I knew I wasn’t going to be playing Frank Sinatra or Iron Maiden, but I still mulled over selections until well past midnight. In the end, I chose Yellow by Coldplay and This Is The Sea by The Waterboys. Snippets of lyrics from each song will hopefully make you understand how these songs would wash over the keynote with the warm and affirming tone I was seeking.
Look at the stars, look how they shine for you, and everything you do. Yeah they were all yellow.
This Is The Sea
Once you were tethered .Well now you are free That was the river This is the sea!
Even though I played each song many times the night before, hearing them in the makeshift auditorium gave me the required goose bumps I was hoping for. I wasn't nervous. I was just really happy. The second part of my keynote would be talking about my favorite mathematical hero — Sophie German. The story about a 13 year-old girl living in Paris in 1789 — in the throes of the French Revolution. Because the streets are unsafe, Sophie is forced to stay inside her family’s apartment. A curious child, she discovers her father’s library. She stumbled upon the death of Archimedes.
Archimedes had spent his life at Syracuse studying mathematics in relative tranquility, but when he was in his late seventies the peace was shattered by the invading Roman army. Legend has it that during the invasion Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death.
Germain concluded that if somebody could be so consumed by a geometric problem that it could lead to their death, then mathematics must be the most captivating subject in the world. She was moved by this story and decided that she too must become a mathematician. Sophie pursued her studies, teaching herself Latin and Greek. She read Newton and Euler at night while wrapped in blankets as her parents slept — they had taken away her fire, her light and her clothes in an attempt to force her away from her books. Eventually her parents lessened their opposition to her studies. As she grew up, she had to, in a stealth like fashion, obtain lecture notes from the local university. And, she even took on a nom de plume of Monsieur de Blance to hide her female identity, This would be the name she used on all her correspondence with the university — almost a reverse Tootsie! Sophie’s work would eventually get recognized by what many consider the greatest mathematician ever — Carl Frederich Gauss. Unfortunately, not only would her stellar, self-taught mathematics be recognized, so would her real gender. And, to illustrate how the history of mathematics is inextricably tied to general history, Sophie Germain would not have ever corresponded with Gauss had it not been for the fact that Napoleon was invading Germany, and Gauss’s life was in jeopardy — much like Archimedes. Sophie, French and having connections to the French army, asked that his life be spared. Sophie would write a letter profusely apologizing for her deceit to Gauss. Gauss would, in return, write back one of the greatest letters ever written — smashing the stereotypes of the day to smithereens.
But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius.
Of course, I told an abridged and age-appropriate version of this story. But, even still, even from my distance, I could see the light in the eyes of kids beam — especially the girls. Sophie was one of the greatest mathematicians and fighters of all time. Her story should be inside every math textbook. It is a source of inspiration for not only students, but teachers as well. I knew this story would get their attention, but it was important for me to convey honesty first. I got introduced by one of the teachers, and some polite applause with some adorable howling followed. I took the mic, thanked the school for inviting me, and tried to make meaningful eye contact with some of the students as I spoke. After just a minute or so, I turned to my right and started walking away from the centre of the floor. I went right to the wall and turned around with my back firmly pressed against it. Before I started speaking, I could tell this physical change in speaking location was surprising many kids and teachers in the audience. “See that wall on the other side”, I said in a commanding, but soothing voice. “That wall represents all the mathematical knowledge in the universe. The wall that I am standing against represents just knowing that 1 + 1 = 2. I am going to start walking towards that wall. I want people to start yelling “stop” when they think where I should be — how much math do you think Mr. Singh knows!” The instructions were clear and the anticipation high. I started walking. Besides some murmur and natural giggling and talking, there wasn't any indication from the supportive audience to halt my walk. After all, I was supposed to be this mathematical superhero. Someone who knows libraries upon libraries of mathematical knowledge. I got halfway across the gym floor, and I heard a few “stops”, but nothing largely collective for me to do so. However, I did slow the pace of my gait. And somewhere about 75% of the length of the floor, I heard enough of these instructive stops to end my journey across this figurative floor of mathematical smarts. I turned around and squarely faced the audience, slightly tilting my head head back to the right to compensate for my off-centre location. I shook my head. “Sorry, this is not where I am supposed to be”. I rotated my body to the right, strongly indicating I will be heading back in the opposite direction. While there was some deflation of enthusiasm in the audience that I would not be this omniscient being of numbers, there was marked curiosity as to where Mr. Singh would end up. I repeated the instructions, and this time there was an overwhelming response to stop in the middle of the gym floor. This is where I started the presentation, and perhaps it made compromising sense for me to end up there again. I did take a quick glance over my left shoulder as I passed this logical halfway point. Many kids were wearing faces of mild shock to confusion to “who the hell is this mathematical impostor!” I shuffled my feet to exaggerate the notion I was perhaps shameful of where I was going to end up — which was the wall that I had started at. I put my back against the wall and took the piece of paper that was purposefully in my hands the whole time. I bent down and put the paper between the heels of my shoes and the one plus one wall. “This is how much mathematics I know”, I trumpeted with what was surely confusing enthusiasm! I said some more words, but the expressions on the kids faces — and the teachers — alerted me that I had made my point passionately clear. All of us, more or less, are at the beginning of mathematical understanding. And the beginning is where we will always be — this large huddle of eager learners not afraid to ask questions, show confusion, make mistakes, stumble and fall. We will help each other. Always. For, inasmuch as mathematics teaches us that, simply being human and vulnerable teaches that even more.
I was a classroom teacher for almost twenty years. Almost everyday I wished a question or an idea would come to the surface where I would have the grateful opportunity to proudly proclaim my minuscule knowledge of mathematics. It seems exhausting that everyday in math classes all around the world that there is unreasonable focus on the knowing. As hinted at in the keynote in 2007, I might possess 0.1% of the mathematical knowledge in the universe — hopefully, even less! To have spent every minute of my teaching life dancing on the head of pin would have been disheartening and disingenuous. Don’t you kids want to know the problems I never solved? The ideas that I am still unclear about? A few years back I came across an article in a prominent mathematical magazine in my home province of Ontario. It was authored by a well known leader in mathematics education. The crux of the article was a public confession to his lack of understanding of high school topic called proof by induction. While he had done these proofs correctly as a student and as a teacher, he admitted it was all based on aping procedures found in books — sadly, the most central idea in how mathematics is communicated in schools. Confusion is beyond okay. It needs a figurative seat in every classroom, to disarm and to motivate. Mathematics is not a sprint to be won. It’s a marathon to simply be entered. And, fyi, nobody finishes.
Chris Brownell, a kindred spirit at Fresno Pacific University, sums it up all too well when teaching his students. “ If we were a bit smarter, we wouldn't need it; a bit less intelligent we wouldn't understand it”. In other words, lets relish being human. Inquisitive and humble.
Zero. The beginning and the end. Get comfortable. It’s going to be a long and strange trip — Jerry Garcia would have been an amazing mathematician!
So, thank you, India. Thank you, Dave Alexander. Thanks for…nothing!
Now let’s get Truckin…