4 Dad's Evening Class God made the natural numbers, all the rest is the work of Man.
-leopold kronecker
These were the first words that my father wrote on the board-a bit alarmingly, in German-on the evening of my first class at the Cork Institute of Technology, where he lectures in mathematics. He explained that Kronecker was a German mathematician who lived in the nineteenth century. Little did I know the role that the natural numbers (1, 2, 3, and so on . . .) would play in my life over the next two years.
Although, as I have already said, my father had often taught or helped with my school mathematics at home, this was the first time I had ever attended a formal lecture of his and seen him present work in an organized way to a group of people. This unusual evening class had come about as a result of a conversation he and I had the previous summer, which started something like, "Now that you have decided to do transition year, I must do some math with you." He continued, "I'd like to show you how some beautiful but reasonably elementary mathematics is applied, stuff that you wouldn't ordinarily come across in school."
Not knowing exactly what he had in mind, but not wishing to be collared and dragged to the kitchen blackboard at random times, I replied, "Dad, whatever you do, do something structured!" He promised to think carefully about the most effective but gentle way to give me a realistic glimpse into the mathematical world. Conscious of how much he would have appreciated it had someone done this for him thirty years ago, he felt obliged now to offer me the benefit of his
mathematical knowledge. "Of course, only if you are genuinely interested-I wouldn't force it on you."
Of course he wouldn't! But I was interested. So this is how the evening class was conceived, and this is how I was to be introduced to many wonderful mathematical ideas-accessible ideas I might never otherwise have encountered unless I chose to pursue mathematics as a career.
The title for the intended series of lectures, "Mathematical Excursions," was suggested by one of Dad's favorite books, Excursions in Calculus by Robert M. Young. Dad wanted not to present mathematics in a very rigorous fashion, but rather to explore "elementary" aspects of "higher" mathematics, and learn how these ideas find applications in different fields of study outside mathematics. The only prerequisite for these excursions-besides a thorough grounding in ordinary arithmetic and basic algebra-was interest. The primary aim of the course was that everybody take pleasure in sharing in the spirit of discovery.
The class, which met Tuesday evenings from seven to ten, consisted of eight people and ran for twenty-five nights. At fifteen and a half I was the youngest, though there were a couple others nearly as young. The rest were adults of various backgrounds: one was a secondary school mathematics teacher, one a chemistry graduate working in a medical laboratory, and others were computer scientists. All were there either because they had loved math at school but hadn't continued with it, or because they felt they hadn't properly appreciated the subject the first time around. This noncredit class, which was not part of any major, led to no award or certification at its completion and demanded no homework or study, may have seemed an ideal way of "getting back into math" in an easygoing, informal manner. Dad told us at the outset that there would be no pressure to perform, and that we need not fear being asked questions but we could ask them at any time. We were to feel free to say "any crazy thing" we liked, and not to be in the least afraid of appearing to make fools of ourselves. In fact, he told us, the more you are prepared to make a fool of yourself the more you'll learn. These reassurances convinced me that I was going to enjoy the next few Tuesday nights, even if I was not completely comfortable with the fact that the lecturer was my father. (Students always pity those who have to endure classes given by their parents, though I'm sure the situation is worse for the parent who is the teacher.)
The first excursions were to be in elementary number theory, with some aspects of cryptography as their final destination. Along the way we would see interesting sights, rich in their own right regardless of whether or not they had immediate application. Dad told us that when he was a student, number theory was regarded as the sole province of mathematicians, and those who devoted their lives to its study were considered the purest of the pure, so few were its perceived applications. It wasn't until the 1960s that engineers and scientists began to find applications of number theory, and by the end of the 1970s the discipline came of age, in the applied sense, with the advent of public key cryptography (very briefly, a modern form of cryptography which makes it possible to make public the method by which messages are enciphered, but-amazingly-without revealing how these enciphered messages are deciphered). Now the U.S. government was putting millions of dollars into the study of cryptography, a subject full of wonderful ideas about which, Dad said, he could hardly wait to tell us. Whereas many great number theorists had barely made a living in the past, their modern counterparts were now being actively sought out, so great was the demand for expertise in this emerging science of secrecy which relied so fundamentally on the branch of mathematics that the great mathematician Karl Friedrich Gauss (of whom more later) revered above
all else. It was he who said:
Mathematics is the queen of the sciences and number theory is the queen of mathematics.
As you might imagine, hearing all this filled me with curiosity and interest. Mom said it was like the Sleeping Beauty fairy tale, and she was prompted to write:
A princess was the Theory of Number,
whom no practical use did encumber,
'til Cryptography the prince
('tis all secrecy since),
did kiss her awake from her slumber.
I thought about those who had toiled away through the centuries at unraveling the mysteries of this subject, motivated by nothing more than a passionate desire to know. They could have hardly dreamt of the applications that some of their results would one day find. I wondered what it was they had discovered, and what they would think if they could see how some of these discoveries are now being used. I was eager to learn the subject and surmise for myself whether they would be surprised or not.
We were at once tantalized and put on our guard by the warning that number theory is notoriously deceptive. We were told that very soon we'd find ourselves asking simple questions, many of which others have asked before us through the ages, and some of which have still not been answered to this day.
