If numbers cannot have infinite strings of digits, then the future can never be perfectly preordained.
It sure did for Sir Isaac Newton, this from The World of Mathematics: Newton took his degree from Cambridge early in 1665. In the autumn of that year the great plague, which was raging in London, caused the University to close, and Newton went back to live at the isolated little house at Woolsthorpe where…
One of my students pointed out this video showing how it’s done, from Ireland.
I was forced to broaden my horizons in my PhD pursuit. That’s because, although I’ve done coding since I was eighteen, I had to acquire a deeper understanding for two things: linear algebra and numerical methods. It’s no understatement to say that both of these are at the core of the advances wrought by computerisation, whether we’re talking about statistical analysis or (in my case) simulation.
After my initial boffo performance, I turned to my Iranian friends for more help. So they let me use some of the books they found useful for study back in the “old country”. One of those was a sizeable book entitled Applied Numerical Methods by Brice Carnahan, H.A. Luther and James O. Wilkes. As was the case with their wedding video, the heart skipped a beat, because the middle author, Hubert A. Luther, was my Differential Equations teacher at Texas A&M, forty years ago this spring.
Applied Numerical Methods was, AFAIK, the first really comprehensive textbook which combined linear algebra, numerical methods, and coding (in their case, FORTRAN IV) in one text. Although some of the methodologies have been improved since it was published in 1969, and languages have certainly changed, it’s still a very useful book, although a little dense in spots. Many of the books on the subject that have come afterwards have learned from its mistakes, but still refer back to the original.
Dr. Luther taught me the last required math class in my pursuit of an engineering degree at Texas A&M. It wasn’t an easy class, even after three semesters of calculus (which I did reasonably well at). Although he was originally from Pennsylvania, he acclimated himself to the Lone Star State with western shirt, belt and string tie, the only professor I can remember who did so. The start to his course was especially rough; the textbook was terrible, he was a picky grader, the scores I got back were low. I thought I was facing the abyss…until another one of those “aha” moments came along.
We (the engineering students) were standing outside our Modern Physics class, which came before Differential Equations. I found out I wasn’t the only one having this problem. But one of my colleagues, a Nuclear Engineering student who went on to become my class’ wealthiest member, had a simple suggestion. Go visit his office, he said. He’s lonely (he was nearing retirement) and likes the company. Your grade will go up.
I wasn’t much for visiting my professors, but I was desperate enough to try anything. I made a couple of office visits. I’m not sure how helpful his advice was, but his grading became more lenient and I got through the course OK.
Today I’m on the other end of the visitation. I spend a lot of time in the office with no student visits. Part of the problem comes from scheduling, both theirs and mine. But I’ve found out something else about student visits: the students that come to see you really care about what they’re supposed to be doing in your class. Although there are still students who think it their duty to “tough it out” without asking questions, many others just want to get through in the quickest and least time-consuming way they can find.
I’m glad I took my classmate’s advice and made the office visits. But there are two other lessons I have learned since that time.
The first is that I wish I had taken a numerical methods course taught by Dr. Luther, it would have prepared me for what I’ve been doing both before and during the time of my PhD pursuit.
The second is that, when I started my MS degree twenty years later, I took a course over basically the same material taught by a Russian. I found out that there was a great deal I hadn’t learned from Dr. Luther, and that American math education leaves a lot to be desired of. So sometimes making the way easier up front comes back to get you in the end.
Although it’s been out there for a long time, these days the clash between religion and science has been especially heated. The simplest way to solve the problem would be to cancel elections and let the self-proclaimed know-it-alls run the show. In this way they could ignore the religious “masses” and insure the continuity of their funding. Funded scientists are happy scientists…
But what would happen if there was a synergy between the two? Basically the same thing that is happening between religion and science now: an academic slugfest. (Reminds one of Rabbi Jonathan Sacks’ joke about “the tradition”…) That’s pretty much the bottom line of the life of the life of Georg Cantor (1845-1918) and his formulation of set theory.
He was born in St. Petersburg, Russia to parents who originally came from Denmark. When he was eleven, they moved to Frankfurt, in the German Electorate of Hesse (the Hessians were the ones George Washington crossed the Delaware to defeat at Trenton). As Carl Boyer notes in his A History of Mathematics:
His (Cantor’s) parents were Christians of Jewish background–his father had been converted to Protestantism, his mother had been born a Catholic. The son Georg took a strong interest in the finespun arguments of medieval theologians concerning continuity and the infinite, and this militated against his pursuing a mundane career in engineering as suggested by his father. In his studies at Zurich, Göttingen and Berlin the young man consequently concentrated on philosophy, physics and mathematics–a program that seems to have fostered his unprecedented mathematical imagination.
His central “claim to fame” is the elucidation of set theory (or, as the Germans are wont to call it, mengenlehre). It’s not an understatement that set theory has come to dominate the teaching of mathematics and its conceptualisation, as I found out the hard way taking advanced linear algebra a couple of years ago, complete with the bizarre notation that has just about taken over math textbooks. In the 1960’s it was the centrepiece of the “new math” that came into primary and secondary school curricula, and that was controversial, but a great deal more useful than its critics would admit.
The controversy didn’t end with the sets themselves. Cantor realised that set theory forced him to consider something that mathematicians had danced around for almost two centuries: infinite quantities, or more precisely transfinite quantities. Sets can have an infinite number of elements, but just what that means was something Cantor plunged very deeply into.
It’s easy to get lost in Cantor’s reasoning, as the concepts he proposed are very profound. I’ll try to keep things as uncomplicated as I can, taking the risk that I may oversimplify the business.
Let us consider the set of integers. We know instinctively that there are an infinite number of integers. Now let us consider the number of even integers. You’d think that there are half as many even integers as all integers, right? But both quantities are in fact infinite, which means that dividing it by two doesn’t mean much. In fact Cantor proved that, if we considered the set of all integers and the set of even integers, we would have a one-to-one correspondence between each member of each set. So the size of the two sets is equal, even though one set is a subset of the other.
Things get more complicated when we pass from integers and rational numbers to transcendental numbers like e and pi. Cantor proved that the number of transcendental numbers was larger than that of either/or or both/and the integers and rational numbers, even though all of them were infinite. Cantor had shown, in effect, that not all infinities were equal to each other!
One device that Cantor, and just about anyone else who deals with transfinite numbers, uses is the limit. But one major difference between Cantor and many of his contemporaries–and predecessors–is that Cantor showed that infinity was in fact an existing quantity, the problem with the transcendentals not withstanding.
That lit several fuses. Before Cantor’s time the French mathematician Cauchy stated the following:
I protest against the use of an infinite magnitude as something completed, which is never permissible in mathematics. Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close.
The most deadly grenade pin that Cantor pulled, however, was that of Leopold Kronecker (1823-1891), after whom the famous Kronecker Delta is named. Kronecker, like Cantor of Jewish origins but a Christian, famously stated that “God made the integers, and all the rest is the work of man.” Kronecker made a career out of academically trashing Cantor, blocking appointments and delaying publications. Cantor, not the scholarly pugilist the situation called for (he should have read Jerome with the mediaevals) had his first nervous breakdown in 1884. After that time he published little, and died in a psychiatric hospital in 1918, although by then his work was receiving the recognition it deserved. David Hilbert pithily stated that “From the paradise created for us by Cantor, no one will drive us out”.
So how did the mediaevals influence this revolution in mathematics? The problem of infinity wasn’t as far-fetched as you might think. It had sat there since Newton and Leibniz set forth the calculus, which in turn hangs on infinitesimals. Between two finite points there is an infinite number of infinitesimals. Mediaevals have been jeered for wondering how many angels could dance on the head of a pin, but as long as they were infinitesimals, the answer is clear for finite pins. It was only a matter of someone putting infinitesimals and infinities together, and that person (with help from others) was Cantor.
Anyone who has explored the philosophy of the scholastics with a mathematical background sooner or later will consider the relationship between their idea and the mathematics of infinity. Coming off of a master’s degree, I found myself doing that in My Lord and My God. Although I would not dare to rank myself any where near Cantor, I discovered that all infinities were not equal, and, although they could not have a finite ratio with finite quantities, they were not necessarily equal to each other. That in turn helped me to see that subordination in God does not impair the deity of the subordinate persons, which solves many problems. Unfortunately there are those who either can’t or–ahem–won’t see that relationship, and there is always the problem that prelates and seminary academics and are often mathematically challenged.
Today we live in a world where science and religion are forcibly bifurcated. But it was not always so. Cantor–and Kronecker and others for that matter–allowed the two to intermingle, and before that Euler was more religiously conservative than Voltaire. And the Nineteenth Century in Europe was a golden age in mathematics, where advances came one after the other.
But there’s a price. If you want to get into serious trouble, read the mediaevals, and that’s true for mathematicians and theologians alike.
Note: in addition to Boyer’s book, I used Jane Muir’s Of Men and Numbers: The Story of the Great Mathematicians (Dover Books on Mathematics) in writing this piece.