Prove that you cannot prove it so you have a proof
The Riemann hypothesis is considered one of the most difficult hypotheses to prove in all of mathematics. For centuries, mathematicians worldwide have cut their teeth on this problem. And out of despair arose the thought that perhaps there is no proof for this hypothesis. But here, too, a loophole can be found. One way to prove the correctness of the Riemann hypothesis would be to prove that there is no evidence for this hypothesis at all. And yes, I know that sounds crazy. And because it sounds so crazy, we need to take a proper look at what is meant by it.
Prime Numbers
Prime numbers are considered to be the atoms of numbers. Every natural number between 2 and infinity can be written as a product of prime numbers. The prime numbers themselves cannot be divided by anything except by themselves and, of course, by 1. There are infinitely many prime numbers. This was already proved by Euclid more than 2000 years ago in ancient Greece. And because of their importance as basic building blocks for all other numbers, research into the properties of prime numbers began even then. For what was particularly peculiar about them was the fact that they popped up without any pattern in between the natural numbers. There was no regularity to be discovered in their appearance. They followed no definite pattern. Why should something as important as the prime numbers be distributed so randomly?
Bernhard Riemann had an idea about this. In line with the trend of his time, he was researching functions with complex variables. In one of his publications, the focus lied on the so-called zeta function. He assumed that the non-trivial roots of this function are all located in the complex plane on a perpendicular straight line through the point 1/2.
This hypothesis does not sound so special at first, especially since it is formulated in a very complex way and hard to get. Only by the connection with the prime numbers one recognizes their value. Because Leonhard Euler had already recognized a connection between the prime numbers and the zeta function some decades before. Riemann, however, had extended this research and made a big step forward. For if he were correct in his assumption, all prime numbers would indeed be randomly distributed.
Proving this statement had turned out to be one of the most difficult problems in mathematics. To this day, no one has succeeded in formulating an acceptable proof without gaps in the logical conclusion. The ideas to prove the correctness of this hypothesis became more and more creative during the centuries. And one way would be to simply prove that there is no proof of the Riemann hypothesis at all. In order to understand what is going through the minds of mathematicians creating ideas like this, we have to get down to the basic pillars of mathematics itself: the axioms.
Axioms and Logic
Mathematics is the language of logic. A statement that has been proven true by mathematicians will be true for all time. Until the end of time. It behaves differently with the natural sciences. What was regarded as certainly true in physics at the beginning of the 20th century has been revised numerous times until today. In a way, this sets mathematics apart from all the other sciences. But is mathematics really so perfect?
Proven mathematical statements are valid for all eternity. But are there statements that cannot be proven? This question was already asked by David Hilbert at the beginning of the last century, when he drew up a list of the 23 most important mathematical problems. In one of the problems Hilbert considered whether mathematics with all its theorems and proofs was free of contradictions in itself. Kurt Gödel found the solution to this problem some time later and formulated it in his so-called incompleteness theorem. And here the axioms of mathematics finally come into play.
All areas of mathematics — from calculus to geometry and on to statistics and even more exotic topics — are based on so-called axioms. These are assumptions whose truth is simply assumed without proving this truth. Axioms were already used by the ancient Greeks in their geometrical considerations and at the end of the 19th century Georg Cantor also used them in his construction of set theory.
Set theory can be seen as the foundation of mathematics. Every mathematical object can be interpreted as a set. The set of natural numbers, the set of rational numbers or the set of real numbers are the best known examples. But also functions are sets, equations are sets, matrices are sets and even triangles and platonic solids are sets.
And because sets are so important, Georg Cantor established axioms for them, which were to form the basis for all logical reasoning in all areas of mathematics. This set of axioms was extended a little later by the mathematicians Ernst Zermelo and Abraham Fraenkl. The two established 10 axioms that still form the basis of modern mathematics. Every mathematical statement — from the infinity of prime numbers to the sum of angles in a triangle — can be derived by logical deduction from one or more of the 10 axioms.
But now let us come back to Hilbert’s question. Was mathematics free of contradictions in itself? Using the axioms of Zermelo and Fraenkl, we can rephrase the question: Are there any contradictions in mathematics arising from the Zermelo-Fraenkl axioms?
Is mathematics free of contradictions?
The answer to this question was the incompleteness theorem of Gödel. This consists essentially of 2 statements. The first statement is that the non-contradiction of mathematics cannot be proved. Thus he answered Hilbert’s question with a clear no. The second statement of the incompleteness theorem was the existence of so-called unprovable statements.
Even if mathematics was free of contradictions (which we cannot prove), there would be mathematical statements for which there is no proof. No proof in this case means that there does not exist a sequence of logical conclusions starting from the Zermelo-Fraenkl axioms and ending at the statement. When Gödel published his incompleteness theorem, the thought spread among mathematicians that the Riemann hypothesis might be such an unprovable statement.
The unprovability as a proof
But the mathematician Paul Cohen had an idea how to take advantage of this. Because if the Riemann conjecture is an unprovable statement, we can neither prove that it is false nor that it is true. If the Riemann conjecture were false, roots would be outside the straight line discovered by Riemann. But if we find a zero outside this straight line, we can prove by logical deduction that the hypothesis was false. But this again contradicts the initially assumed unprovability of the hypothesis. So it can only be true.
So, in summary, if the Riemann hypothesis happens to be unprovable, we simply have to prove this unprovability to show the correctness of the hypothesis itself.
Simple, right?. Well, even that has not quite worked out to this day. But still, in my opinion, it is one of the most creative ideas to approach the solution of a problem.
Mathematics may not be perfect and above all empirical sciences, but its methods will never cease to amaze you. A mathematician moves elegantly through the jungle of theorems, using creative methods to help him or her find a way between propositions. Since Paul Cohen died in 2007, it is now up to us to use this extravagant methodology to solve one of the most difficult problems in mathematics. Any idea how to start?
New to trading? Try crypto trading bots or copy trading on best crypto exchanges
Join Coinmonks Telegram Channel and Youtube Channel get daily Crypto News
