Why did I study Mathematics?
Of all the branches of human inquiry, I presume that mathematics must remain the most obscure and the one which is, ultimately, most remote from reality... it is probably that branch of knowledge that almost everyone fears (well, at least 99.9% of us).Why?
There are many reasons for this...One explanation is that it requires a special, almost new, way of thinking, thought patterns which are incongruous with our daily experiences. Would you be thinking about the proof of the Fundamental Theorem of Calculus while you're shopping for groceries, or wonder whether or not Mercury will ever crash into the sun while whitewater rafting, how about whether or not the sun will actually rise tomorrow morning while on a date? Probably not... unless you're on your way to being, or are, a mathematician. Thoughts not unlike these are common to the inquiring minds of the men and women dubbed mathematicians . Our minds are ruled by logic and, sometimes, this logic permeates our daily existence to the point where this ultimate mathematical reality takes over and we find ourselves talking sotto voce about things we don't even know exist for sure, except, in the world of mathematics. Although you may find some comfort into my apparently denouncing mathematics as an abstract dreamer's paradise, one ruled by that same queen which governed the intellect of Science Officer, Dr. Spock, on a once popular science fiction television show, that is not why I wound up studying it !
In order to answer this question I'll have to fill you in on some autobiographical information. All I seem to remember about my early years in school is that I was in the top three of my class in grades 1and 2 but thereafter my schooling was completely average. Sure, I went through elementary school without failing a year, but my math. subjects, just as all my other subjects, didn't inspire any of my teachers to encourage me to go to High School. That was then...I always loved science, I adored building things out of wood gathered in the alleys of Montreal's Little Italy where I grew up. I remember building a batmobile complete with cannons armed by firecrackers... I delved into the world of Marvel comics (Thor, Dr. Strange, The Avengers and others) and I read almost anything having to do with Norse gods, Greek gods, legendary figures; these, in turn, fired my imagination even more. I wound up in a high school where most of the Italians from my school went, just so we could be close. There I was admitted into the so-called science program but, since my grades were not that remarkable, only with the understanding that I was to be on trial , i.e., I could stay in the program if I managed to secure the grades required to do so by the end of the year (that was grade 8). My math. was pretty bad then...I was too interested in nonacademic pursuits to do anything meaningful in school; my mother usually signed my report cards, I was terrified of showing them to my dad. So there went another average year. To give you an idea of how average I, and others in my class were, students were assigned classes ranging from A to L, with A containing the smartest of the lot for that year while L ... well, I'm sure you can guess. So, 1A had the brightest of all the grade 8's, while 1L had the not-so-intellectually-gifted; I was in 1I. For grade 9 (i.e., 2K), I had this teacher for Algebra and Geometry who was really tough and condescending (after all, we were only 1 class away from the "last" category) and, unfortunately for us, he also taught the A classes, so there were plenty of comparisons between us and them. O.K., so we weren't all that smart but who cared? That year was Canada's Centennial year and I spent most of my time either at the Dow Planetarium or at Terre des Hommes where expo67 was being held. I also secured my first summer job delivering groceries for a local grocery store. It was truly an exciting time in my life, but, that summer would be short-lived... I failed grade 9 Algebra miserably and, unless I took a summer course in that subject and passed it, I would fail my year. No way! All I wanted to do was to get through this schooling ordeal and get to work, out there, in the real world. I remember being taught by a female teacher that summer, something I enjoyed thoroughly. I barely passed that summer course, and I do mean barely, with 51 per cent. Onwards to grade 10, (i.e., 3G, this time) I had this incredibly bright teacher for Geometry; his problem was that he was just too good for our class. This same year I met an old classmate of mine, Mario D'Angelo; he was always one of those in the A grades. Needless to say, there wasn't much interaction between the A grades and any other grade beyond C! So, I started hanging around those guys in the A grades, simply because I knew how to play chess, and because I knew Mario and he was one of the best in the lot...our High School Graduation Book even had a photograph of Albert Einstein next to his (Mario D'Angelo's) graduation photo. For some reasons which I'll allude to below my grades improved slightly that year and were even better during my final year, the year of my graduation (well, this is nothing to marvel at since they were in the neighnorhood of around 70 per cent in Algebra and 80 per cent in Geometry). I do remember trying out an angle trisection during my grade 11 class ( that would be 4K), a trisection which made the class wonder and the teacher not care, either way. This was Mario's influence for sure ... we spent most of the summer of 1969 playing speed chess and doing Geometry in his back yard, creating all sorts of crazy ruler and compass constructions about triangles... we wanted to add some new result to our High School geometry book; it was a book which consisted essentially of Euclid's Elements.
This was really nuts! All these huge sheets of paper with triangles and arcs of circles and stuff were rolled up and put to the side until the next weekend. Eventually, he filled the gaps in my knowledge of Algebra; I would need this in order to understand ratio and proportion in geometry, and I learned Geometry by actually doing it. That summer we got the thrill of our lives when we discovered a new construction dealing with triangles. We sent this construction to Professor H.S.M. Coxeter at the Uiversity of Toronto, who was then, and still is, The Master of Geometry on this planet. He replied in a wonderful letter mentioning that this should be published as a problem in the Problems for Solution section appended to a Canadian mathematical publication entitled the Canadian Mathematical Bulletin. The problem appeared (sometime in 1970) and remained unanswered for many months until Professor Coxeter himself provided the only solution, a most elegant solution, a solution which we did not understand for many, many years. I guess I got hooked at this point!
I attended Loyola College in Montreal during the Fall of 1969, once again, on trial, because my high school average was too low to be admitted as a regular student into the Faculty of Science there. A low B average in all my subjects (72 per cent) during my first year secured my place in the Faculty of Science as a full time regular student. This was very exciting for me; I was finally in the Faculty of Science! Ever since I had blown the cork off an Erlenmeyer flask during a chemistry experiment in my room (I was making Hydrogen gas and a flame got too close to the actual gas...) sometime in 1968-69, I decided that chemistry was going to be my field, assuming, of course, that I ever got to College in the first place. These two years, 1968-69 were very difficult for me because a guidance counsellor did me a great disservice by telling me that I would never amount up to anything, and would most surely never get to College, so I should take courses in Bookkeeping and Typing and leave the sciences alone. Lucky for me my brother, Marco, five years older and wiser than I, came to the rescue and got the cousellor to authorize me to take one Chemistry course in grade 10... that was good because I scored an 87 per cent in the final examination that year! I developed a lot of confidence in Chemistry as a result. During my second year I followed courses in Thermodynamics, Statistical Mechanics, Organic Chemistry, Inorganic Chemistry, and the usual course in Advanced Calculus among many others. I was torn between three areas which I really enjoyed...Chemistry, Physics, and Mathematics, in that order. I realized quickly that Chemistry was not as challenging as Physics and if I knew Physics it should be possible to understand Chemistry. On the other hand, both of these used Mathematics as their language, as their tool for making predictions. So, I figured, if I understand the math. I should be able to understand the basic elements of ALL the sciences, because all of them use math.. This was 1971. I had gained much confidence in my mathematical abilities because of the summer of 1969, so, I thought that I would major in Mathematics for 1972. I did this and graduated in 1974 with an honors degree in mathematics, went to the University of Toronto on scholarships, even followed two courses with Professor Coxeter himself as my Instructor (he remembered me) and went on to do a Ph.D. specializing in differential equations. That's how it went...