In April of this year a relatively obscure mathematician named Yitang Zhang set the world of mathematics on it’s ear when he proved that no gap between two prime numbers will ever exceed 70,000,000. That’s a pretty audacious claim and it must be wrong. But mathematically he seems to have proven something that many others have tried...and failed to do. And he appears to have the support of many an esteemed mathematician. If prime numbers continually get larger, but the gap between them ceases to increase for very large values...and I mean very large values (the largest prime number currently known is 2^57,885,161 − 1, to put that into your calculator you would have to punch 2 x 10 to 57,885,161 and then subtract 1)...will prime numbers eventually run out? Yitang says no...but since prime numbers are the building blocks of all other numbers, it stands to reason that once you run out of gap, you eventually run out of prime numbers between the gap, and then ultimately you run out of numbers. Does this mean we can stop counting? Can you imagine the end of the number line? Kind of like the end of the Earth. There’s a giant waterfall and over you go. Or, perhaps like the Mayan calendar, maybe we just start counting all over again...it actually isn’t the apocalypse, it’s just a Y2K scare.
If it’s true, God will have to go back to the drawing board because it seems, he meant to refer to infinity as simply a concept since personally he was never actually able to count that high. Since Yitang’s proved we still have infinite pairs of prime numbers, even though the gap between them grows not larger then 70 million, his math must be wrong. Regardless he has created quite a stir since many are believing his math to be correct. In situations such as these, I tend to cast a skeptic's eye on the situation. If he’s reached an upper bound, no matter how good his math might be, if it means the end of the number line, either there is no such thing as the infinite or he has made a error. Personally, I hope there is an end to the number line. That will end our search for things that don’t matter. Professionally, however, I sense there must be some mistake. Kind of like neutrinos traveling faster than the speed of light. Go back and check the math there is an error in the assumption.
What’s left to do, however, is to prove, or disprove, that the number of twin pairs of prime numbers, prime numbers separated by a gap of 2, could be infinite. The two seem quite different yet are clearly tied together. If it turns out that prime numbers separated by a gap of 2, are infinite, then again Yitang’s math has to be wrong. If it turns out that they are not infinite, then his math is correct but his assumption that prime pairs are infinite has to be incorrect, and therefore we eventually run out of numbers. Either way it has to be wrong because we can’t run out of numbers...they are an artificial abstraction that we can always increase by 1.
Therefore the error here could be the limit theorem. If you chose a number such as “infinity” to approach, you have chosen a bound. Therefore you can find another bound, inside that bound, if you look far enough. What this means, simply, is that it’s time to expand our definition of infinity... Instead of the world getting smaller, since we reached the end of the number line under our current definition of infinity, the world just got bigger. The point of know return, therefore, just got a whole lot stranger.
I love you, Mooch, but I can’t let this one go…
ReplyDeletePutting forward a post on the subject of Zhang’s bounded gaps result and drawing conclusions about its impact on mathematics is bound (pun intended) to arouse comment on a forum frequented by mathematicians. I haven’t finished Zhang’s paper published in the Annals of Mathematics (and probably still won’t understand his proof when I do) and all this is off the top of my head, but I’ll try to be gentle.
First, you've misstated Zhang’s result. He proved there are infinitely many pairs of primes with gaps no more than 70 million. This in no way implies that successive primes may not have a gap greater than that stated bound; essentially, it looks like Zhang has simply proven a less restrictive version of the Twin Prime Conjecture. Paraphrasing the Prime Number Theorem, only the average gap between primes among the first N integers is proportional to ln(N) and therefore increasing (on average). You know as well as I do how the average works. If I tell you the average is increasing AND the number of “small” gaps is “large” then the number of “large” gaps must be “larger” to balance the average. (BTW, I hate trying to talk about math in English. I can get precise if you need/want it.)
Second, even if Zhang’s result were as you stated (with an absolute bound on the difference between successive primes) it does not “stand to reason” that running out of gap would necessarily imply running out of primes. It is not necessary to have larger gaps between primes for primes to comprise composite numbers (though there would be a contradiction implied with the Prime Number Theorem if the gaps were not, on average, growing without bound).
In any case, Zhang’s result does no more and no less than move the mathematical community closer to an understanding of the primes, a proof of the twin primes conjecture, etc. This has some serious implications for mathematics (and all of its roots and branches), but it certainly doesn't have the philosophical implications you posit. I've concentrated on the mathematical aspects of your argument here and left the philosophical implications aside. Once we get our premises straight, all else is fair game, and you know I’m always up for that discussion.
Merf
There is a place where the number line ends
ReplyDeleteAnd before the street begins,
And there the grass grows soft and white,
And there the sun burns crimson bright,
And there the moon-bird rests from his flight
To cool in the peppermint wind.
Let us leave this place where the smoke blows black
And the dark street winds and bends.
Past the pits where the asphalt flowers grow
We shall walk with a walk that is measured and slow,
And watch where the chalk-white arrows go
To the place where the number line ends.
Yes we'll walk with a walk that is measured and slow,
And we'll go where the chalk-white arrows go,
For the children, they mark, and the children, they know
The place where the number line ends.
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