What would be the philosophical implications of a proof of the Riemann hypothesis?

Question

In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the nontrivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.

The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics.

What would be the philosophical implications of a proof of the Riemann hypothesis?

From http://www.ptp-security.com/rsastory.asp

Over two thousand years ago the Greeks proved that every number can be written as a product of prime numbers. A fast and efficient way to find which prime numbers have been used to build up other numbers has eluded mathematicians ever since. What we are missing is a mathematical counterpart of chemical spectroscopy, which tells chemists which elements of the Periodic Table make up a chemical compound. A discovery of a mathematical analogue that would crack numbers into their constituent primes would earn its creator more than just academic acclaim.

In 1903, Cole's calculation was regarded as an interesting mathematical curiosity - the standing ovation he received was in recognition of his extraordinary hard labour rather than any intrinsic importance the problem had. Such number-cracking is no longer a Sunday afternoon pastime but lies at the heart of modern code-breaking. Mathematicians have devised a way to wire this difficult problem of cracking numbers into the codes that protect the world's finances on the Internet. This innocent sounding task is sufficiently tough for numbers with 100 digits that banks and e-commerce are prepared to stake the security of their financial transactions on the impossible long time it takes - at present - to find the prime factors. At the same time, these new mathematical codes have been used to solve a problem that dogged the world of cryptography.

Public-key cryptography

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example 512 bit primes are frequently used for RSA and 1024 bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Wouldn't the ability to crack the codes that protect the information of Governments, Banks and Corporations, and every individuals personal information necessarily have Ethical implications as a possible weapon? (theoretically a person could drastically alter the course of the world using a smartphone on their coffee break) And also wouldn't the ability to predict the distribution of the primes necessarily have philosophical implications regarding pure mathematics and philosophy of mathematics?

Answer 1

One of the most important implications of the Riemannn hypothesis would be an explicit upper bound on the order of $Log(x)\sqrt{x}$ for the approximation of $\pi(x)$ by $Li(x)$.

If you consider that a "philosophical" implication, it's an answer to your question. If you don't consider that "philosophical" implication then surely there are no philisophical implications of the Riemann hypothesis.

Answer 2

Your question appears to assume that there are philosophical implications; since the evidence is heavily weighted towards a positive outcome to the problem (for example there is a positive solution over finite fields) some mathematicians have assumed the result in generating new ideas about number theory & algebraic geometry; in the same way there is nothing to stop philosophers attempting the same gambit or its contrapositive; that they haven't done implies there is very little or no philosophical implications of either a positive or negative solution.

Hardy in his Mathematicians Apology wrote that one couldn't weaponise number theory - and that gave him a great deal of satisfaction; this turned out to be wrong as the development of cryptography showed; still its not quite the equivalent of actually dropping a bomb.

The philosophical issues that abut cryptoraphy deal with privacy, transparency & security; and these are important and vital issues given the prevalence of social meta data, computational power and the penetration of the web.

But the actual implementation details is not important to the discussion; any difficult problem will do - it need not be the factoring of large numbers - for example there is such a thing as elliptic curve cryptography which as you noted relies on the difficulty of calculating the discrete logarithm; thus the Riemann problem is not the philosophical crux of the problem here.

One could ask does the solution of the Riemann problem have consequences for efficient factoring - and this kind of question is best asked at Math.Overflow; for example I do know that the implementation of a Quantum computer would allow for polynomial time factoring given Shors Algorithm.

Answer 3

We don't know whether the Riemann hypothesis is true or false, and if it gets proved true we don't know what the proof will look like. There is no reason to think that a proof will allow numbers to be factorised into primes any faster than they are at the moment, and even if there were, or even if a much faster factorisation method is discovered which has little to do with the Riemann hypothesis, million-digit primes could be used instead of 1024-digit ones - and maybe bankers would have to raise interest rates or accept cuts in their bonuses.