An unpublished letter to the editor of The New Yorker, February 2015.

My letter

Alec Wilkinson says in his absorbing profile of the quiet genius Yitang Zhang ("The pursuit of beauty", February 2) that pure mathematics is done "with no practical purposes in mind". I do hope mathematicians will forever be guided by aesthetics more than economics, but nevertheless, pure maths has become a cornerstone of the Information Age, just as physics was of the Industrial Revolution. For centuries, prime numbers might have been intellectual curios but in the 1970s they were beaten into modern cryptography. The security codes that scaffold almost all e-commerce are built from primes. Any advances in understanding these abstract materials impacts the Internet itself, for better or for worse. So when Zhang demurs that his result is "useless for industry", he's mispeaking.

The online version of the article is subtitled "Solving an Unsolvable Problem". The apparent oxymoron belies a wondrous pattern we see in mathematical discovery. Conundrums widely accepted to be impossible are in fact solved quite often, and then frenetic periods of innovation usually follow. The surprise breakthrough is typically inefficient (or, far worse in a mathematician's mind, ugly) but it can inspire fresh thinking and lead to polished methods. We are in one of these intense creative periods right now. Until 2008, it was widely thought that true electronic cash was impossible, but then the mystery figure Satoshi Nakamoto created Bitcoin. While it overturned the conventional wisdom, Bitcoin is slow and anarchic, and problematic as mainstream money. But it has triggered a remarkable explosion of digital currency innovation.

A published letter

Another letter writer made a similar point:

As Alec Wilkinson points out in his Profile of the math genius Yitang Zhang, results in pure mathematics can be sources of wonder and delight, regardless of their applications. Yet applications do crop up. Nineteenth-century mathematicians showed that there are geometries as logical and complete as Euclidean geometry, but which are utterly distinct from it. This seemed of no practical use at the time, but Albert Einstein used non-Euclidean geometry to make the most successful model that we have of the behavior of the universe on large scales of distance and time. Abstract results in number theory, Zhang's field, underlie cryptography used to protect communication on devices that many of us use every day. Abstract mathematics, beautiful in itself, continually results in helpful applications, and that's pretty wonderful and delightful, too.

David Lee
Sandy Spring, Md.

On innovation

My favorite example of mathematical innovation concerns public key cryptography (and I ignore here the credible reports that PKC was invented by the Brits decades before but kept secret). For centuries, there was essentially one family of cryptographic algorithms, in which a secret key shared by sender and recipient is used to both encrypt and decrypt the protected communication. Key distribution is the central problem in so-called "Symmetric" Cryptography: how does the sender get the secret key to the recipient some time before sending the message? The dream was for the two parties to be able to establish a secret key without ever having to meet or using any secret channel. It was thought to be an unsolvable problem ... until it was solved by Ralph Merkle in 1974. His solution, dubbed "Merkle's Puzzles" was almost hypothetical; the details don't matter here but they were going to be awkward to put it mildly, involving millions of small messages. But the impact on cryptography was near instantaneous. The fact that, in theory, two parties really could establish a shared secret via public messages triggered a burst of development of practical public key cryptography, first of the Diffie-Hellman algorithm, and then RSA by Ron Rivest, Adi Shamir and Leonard Adleman. We probably wouldn't have e-commerce if it wasn't for Merkle's crazy curious maths.