These last few days I started to write some more about Zen, but the ideas were ill-conceived and turned to mush. Maybe someday they can be resurrected though I doubt it. To clear my brain I started to write something just for fun. Perhaps it is worth posting.
In the series 1,2,3,… what is the ultimate number? Not the largest number, there being no such thing, but the ultimate number. Answer: The last yianh. As 10 or 11 year olds, my brother and I, playing in our upstairs bedroom, speculated about numbers that were so large that it would be pointless to consider anything larger among the infinity of further numbers. There was an entire series of these, starting with the first yianh, a number so large that a googolplex would fade into an insignificant blob near zero. Of course we had never heard of a googol or a googolplex nor did we know anything about exponential notation, but such specifics are pointless when imagining REALLY large numbers. After a huge series of further numbers one would come to the second yianh. And so on. Finally, after many, many more yianh‘s we would reach the last yianh, a super fabulous number that ended our imaginings. I think now that we had some inchoate idea that the entire tail of the infinite integers would somehow be condensed into this ultimate number. Perhaps this is how mathematics gets developed. One plays around and then some idea like making an infinite series finite by coalescing numbers comes up. Then one sees this as a problem. Can the idea be made logical and coherent? If one succeeds, one has created some new math. If the effort fails, maybe the idea can be deployed somewhere else.
In the nineteenth century there lived a very great mathematician named Bernhard Riemann. Actually there were many very great mathematicians in the nineteenth century, but Riemann was one of the immortals, like Beethoven in music or Shakespeare in drama. (Well maybe not quite as uniquely great as Shakespeare.) Anyway Riemann studied a function, now called the Riemann Zeta function. A function is like a meat grinder. With a meat grinder, one feeds in meat and hamburger comes out. With a function one feeds a number in and another number comes out. The number coming out depends on the one fed in so that one always gets the same second number from the same first. Of course, when talking about the Zeta function there is a little complication that is likely to scare you away if you are a math phobic reader. But be reassured. I will explain this complication with the utmost lucidity and make it completely clear or at least translucent by telling a story about numbers.
Since Greek times there have been many developments about what numbers are. Numbers started as the 1,2,3’s. Then someone discovered a big shortcut, called multiplication, when the same number was added up many times. Immediately, problems arose where one needed to go backwards, so division and with it fractions came into being. The integers and fractions together were called rationals. But horror upon horror, the Greeks found that there were crazy numbers, like the square root of 2, that couldn’t be expressed as fractions. These were called irrationals, disturbing because the Greeks were wedded to being rational. Then when considering such things as debts, people realized that negative numbers were useful, and, more important, could be introduced in such a way that no logical contradictions arose. New math had been created. Later the idea arose that the gaps that still existed among the rational and irrational numbers could be filled to make a continuous stream of numbers with again no contradictions. But here trouble arose. One theme of this blog will be the difficulties we get into by misunderstanding language. The new numbers were called “real” numbers although they were the product of human imagination and could easily have been called “numbers of the imagination” or imaginary numbers for short. Of course, when the numbers we now call imaginary were introduced, this introduction caused all sorts of trouble, not only at the time, but subsequently to generations of math students. I certainly was very dubious about imaginary numbers. If they weren’t real, how could they even exist? Well how can any number exist? It exists because it can be used to calculate things without contradictions arising. So, it turns out we can have what are called complex numbers, consisting of a pair of real numbers, though the second of the pair is called imaginary. Consider imaginary as simply a label used to designate the second of the pair. (It is true, however, that squaring this second number results in a negative “real” number, but that’s really no big deal.) These complex numbers can be used without logical contradictions; but raise the question: What about a triple of numbers? Can a triple be made to act like other numbers? The answer is no. The number pairs are as far as numbers can go. (There are more complicated sets of numbers called vectors, but they work differently from numbers.)
Of course, the reason I’ve gone through this song and dance about numbers is that the numbers fed into the Zeta function are complex numbers, as are the numbers coming out. The Zeta function is called a function of a complex variable and such are studied in graduate level courses in math. (When I took the course at Stanford, I flunked, but later picked up some of the subject on my own.) Since complex numbers are pairs of numbers they can be plotted on a 2 dimensional sheet. We plot the first number on a horizontal x axis and the second, so-called imaginary, number on the vertical y axis. An important situation arises when we feed the first number into the Riemann meat grinder and get 0 out of the function. Such numbers are called “zeros of the Zeta function” although they themselves are not in fact zero, but produce zero when fed into the function. They should have been called zero producers, but that is too long-winded for mathematicians. Meat goes in the grinder, but nothing comes out. So call the meat a zero. Anyway, the Zeta function has many zeros some of whose location turns out to be connected to the distribution of prime numbers. Mathematicians call the others “trivial zeros” and study the ones that matter. Riemann calculated a few of the non-trivial ones (turning the crank of the Zeta function is not easy) and found that they lay on a line with the real part = ½ and the imaginary part on a vertical line rising up from ½ on the horizontal axis. Riemann speculated that all of the important zeros would lie on that vertical line with real part ½. He couldn’t prove it. Nor has anyone proved it in the 156 years or so since, though not for want of trying. (Whether true or false the hypothesis has Yuuge consequences.) Zeros in the billions have been shown to lie on that imaginary line, but billions aren’t equal infinity and a proof would show that all of the infinite number of zeros lay there.
I have a whimsical notion that if Riemann’s hypothesis is incorrect, somewhere up in the far reaches of the imaginary line there is a zero whose real part is not ½. I’ll call this zero the first yianh; other violators the second, third, etc. yianh. One wonders if there is a last yianh or does the sequence of violations never end? Note: I strongly suspect that Riemann’s hypothesis is true, in which case my last remark is even more whimsical.
Prime Obsession: Bernard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire is a fascinating book. A history of the times and a biography of Riemann alternate with chapters that go into the math seldom going beyond high school level. Or go to Wikipedia.
A final note: There are hints that the distribution of the primes for large numbers (primes up there with the yianh’s) has a connection with certain physical properties of the universe. It is difficult to keep “pure” mathematics pure.