Jamesian Philosophy Refreshed

In yesterday's entry , I discussed a recent proof devised by a professor at the University of New Hampshire that there are "bounded gaps" between prime numbers. Specifically, Yitang Zhang has established that there are infinitely many pairs of primes that differ by 70 million or less. There are two fascinating things about this tidbit about which I wish to comment today. First, who the heck is Yitang Zhang? Second, infinity and size. Who the heck? One might naively expect the burning questions of a recondite field to be settled by the elites of the relevant expertise.  Andrew Wiles, the fellow who proved Fermat's Last Theorem correct (though in a way that can't "fit into a margin") was a Royal Society Research Professor at Oxford University, specializing in number theory. Before that, he had been a professor at Princeton University in the early 1980s and a Guggenheim Fellow at the Institut des Hautes Etudes Scientifiques in France in the late 1...

An odd campaign emerged in the 19 th century – the effort to reduce numbers to sets. This brings us back, though in a rather different light, to the question with which we opened: what is a number? The simplest possible kinds of numbers are the counting numbers, those we tick off from one to ten on our fingers. There can be no doubt that they are numbers, they are the paradigm, they are at heart what we mean by “numbers.” Thus, whether anything else is a “number” is, if you will, a question about the family resemblance between the counting numbers and that other sort. Whatever else is a number: three is a number! “Is pi a number?” means, “does pi have a close family relationship to three?” “Is i a number?” means the same.    But in the 19 th century, for the first time, mathematicians and logicians started wondering, not idly but in all workaday seriousness, what it means to call three a number. They decided that numbers were special sorts of set. The...

Leonhard Euler (1707-1783) was surely one of the most prolific of great mathematicians. Among his contributions, we need to mention two, each of which comes down to us as a single letter: the letter e and the letter i .  E uler was born in Basel, Switzerland, so his life and work might fittingly be considered a riposte to the old anti-Swiss jibe (originally from The Third Man ) that Switzerland has produced nothing for all its years of peace and democracy more than the humble cuckoo clock.   Since Euler’s day and because of his work, i stands for the simplest of the numbers that Descartes had called “imaginary.” This i refers to the square root of -1. We don’t need to bother ourselves further with the question “ what is the square root of -1?” It is simply  i , by stipulation. We don't end there, of course, but we can start from there and build something new and important.   Also since Euler’s day and because of his work, e stands for perhaps the m...

The story of another candidate for numberhood, the infinitesimal, is even stranger than the story of the irrationals or the imaginaries. The infinitesimal is the limit of a process, where the method stipulates that the end of that process can never be reached. Consider: can a single mathematical point have a slope? Our intuitive answer, trained by Euclid, is: no. A point is pure position. It isn’t a line, it isn’t even a tiny part of a line, so it can’t have a slope! Can there be a smallest possible line?   How short can a line get and still have a slope! That’s a question that Euclid taught us not to ask. It is akin to suggesting that we can have two adjacent points. If we could have two adjacent points, then they would presumably constitute the smallest possible line and they would have a slope. But we can’t. Between any two points, properly speaking, infinity of other points can fit. In the seventeenth century it was common for the mathematicians at its cuttin...

Continuing... How has Taleb made his name? Look at the above graph. The blue line represents a normal or Gaussian distribution, also known as the "bell curve." Events on the far right or ar left side of that line, where the blue is approaching zero along the X axis, are sometimes call hundred-year storms. If we think of this in a finance/business context, the blue line may represent what a certain business thinks are its profits for the coming year. The tip of the bell represents the most likely result (a modest profit in line with that of most of its competitors, perhaps.) Toward the right end of the curve you get to ever higher but more unlikely profits, to the left you get losses, and then ever larger losses, though here too the fall-off in the line toward the zerobase of the X axis implies that certain disastrous results are very unlikely. But what if probabilities in finance don't have a normal outcome distribution? If you draw a flattened curve with "fa...

I've been writing here of late about the disappointment I felt upon reading Hilton Ratcliffe's book, The Static Universe. Not only does it fail to make much of a case against the Big Bang Theory, but it tries to do so at the expense of all of modern geometry, going back to the great Gauss himself. The key line of argument comes late, in chapter 8. One might say Ratcliffe has buried his lede although, this being his lede, it may be natural to try to bury it. In chapter 8, after praise for Gauss' simple life, teaching skills, and generous spirit, we learn that he had one weakness, an "obsession with the abstract." That would seem to be a job requirement for a geometer, but by calling it an "obsession" Ratcliffe has established to his own satisfaction that it is a weakness. After working on global cartography, Ratcliffe tells us, Gauss succumbed to his eagerness to abstract and "presented his scientific progeny the gift of Differential ...