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 1980s.
Does that not sound like exactly the sort of person to whom one might look to solve such a problem?
Yitang Zhang's proof, though, seems to have come from out of nowhere. We can hope that this means that even at these levels of mathematical discourse, traditional distinctions between elite and non-elite are crumbling. The center (consisting of Oxbridge, and US Ivy League schools) cannot hold. Great.
Infinity and Size
The ultimate goal of theorists working in this area seems to be a proof that there are infinitely many pairs of primes that differ by just two. SAs I said yesterday: it is intriguing that from the perspective of infinity, the difference between 2 and 70 million is a matter of detail.
This also reminds me of the contrary point: there is a difference of literally infinite importance between any positive number, however small, and zero.
This comes up in discussions of the economics of energy. There still exists a certain naïve sort of enthusiast who believes that one or another technological breakthrough will make energy so widely available there will be no way to sell it, the "too cheap to meter" goal.
Now: whatever the rumored breakthrough in question, either the enthusiast means that energy will be free, or he means that energy will have a tiny cost, although one still expressible in positive numbers. Such a person should be informed repeatedly (until it sinks in) that the difference is of infinite importance, and that the plausibility of such claims from one generation to the next depends upon obscuring that point.
Okay, it sounds like that last paragraph is a stretch from Zhang's work but ... not really.