(09-29-2014 03:55 AM)Namir Wrote:  My question to all the math-minded folks out there is, Are there any special or interesting uses for the Collatz sequences?

Charles Krauthammer Wrote:It is the pride of political thought that ideas have consequences. Mathematics, to its glory, is ideas without consequences. “A mathematician,” says Paul Erdös, one of its greatest living practitioners and one of the most eccentric, “is a machine for turning coffee into theorems.” Mathematicians do not like to admit that, because when they do, their grant money dries up — it is hard to export theorems — and they are suspected of just playing around, which of course they are.

Politicians and journalists need to believe that everything ultimately has a use and an application. So when a solution for something like Fermat’s last theorem is announced, one hears that the proof may have some benefit in the fields of, say, cryptography and computers. Mathematicians and their sympathizers, at a loss to justify their existence, will be heard to say, as a last resort, that doing mathematics is useful because “it sharpens the mind.”

Sharpens the mind? For what? For figuring polling results or fathoming Fellini movies or fixing shuttle boosters? We have our means and ends reversed. What could be more important than divining the Absolute? “God made the integers,” said a 19th century mathematician. “All the rest is the work of man.” That work is mathematics, and that it should have to justify itself by its applications, as a tool for making the mundane or improving the ephemeral, is an affront not just to mathematics but to the creature that invented it.

What higher calling can there be than searching for useless and beautiful truths? Number theory is as beautiful and no more useless than mastery of the balance beam or the well-thrown forward pass. And our culture expends enormous sums on those exercises without asking what higher end they serve.

Moreover, of all such exercises, mathematics is the most sublime. It is the metaphysics of modern man. It operates very close to religion, which is why numerology is important to so many faiths and why a sense of the transcendent is so keenly developed in many mathematicians, even the most irreligious. Erdös, an agnostic, likes to speak of God’s having a Book that contains the most elegant, most perfect mathematical proofs. Erdös’ highest compliment, reports Paul Hoffman in the Atlantic, is that a proof is “straight from the Book.” Says Erdös: “You don’t have to believe in God, but you should believe in the Book.”

Charles Krauthammer Wrote:Erdös, an agnostic, likes to speak of God’s having a Book that contains the most elegant, most perfect mathematical proofs. Erdös’ highest compliment, reports Paul Hoffman in the Atlantic, is that a proof is “straight from the Book.” Says Erdös: “You don’t have to believe in God, but you should believe in the Book.”