The subtle art of the mathematical hypothesis

This is not proof, but a conjecture backed by knowledge. But a good hypothesis leads mathematics forward, pointing the way to mathematical obscurity.

The author of the article is Robert Dijkgraaf , a theoretical physicist, a specialist in string theory, director of the Institute for Advanced Study at Princeton, and a professor at the University of Amsterdam.

Mountain climbing is a popular metaphor for mathematical research. Such a comparison is almost impossible to avoid: a frozen world, rarefied cold air, the harsh rigidity of mountaineering resembles an inexorable landscape of numbers, formulas and theorems. Just as a climber contrasts his capabilities with an unyielding object - in his case, a stone wall - so a mathematician often fights in the battle of the human mind against rigid logic.

In mathematics, the role of mountain peaks is played by the great hypotheses - sharply formulated statements, most likely true, but without convincing evidence. These hypotheses have deep roots and wide consequences. The search for their solutions is a large part of mathematics. Eternal glory awaits their first conqueror.

Interestingly, mathematicians have raised the formulation of hypotheses to the level of high art. The most rigorous science loves the mildest forms. A well-chosen, but not proven statement can make its author famous around the world, perhaps even more than the person who offers the final proof. The Poincaré conjecture remains the Poincaré conjecture, even after it was proved by Grigory Yakovlevich Perelman. And after all, the British man George Everest, the chief surveyor of India in the first half of the 19th century, never climbed the mountain that bears his name.

As in any art form, a great hypothesis must meet several mandatory criteria. First of all, it should be non-trivial - difficult to prove. Mathematicians sometimes say, “The task is worth the work only if it resists”, or “If the task does not annoy you, it is probably too easy for you.” If a hypothesis is proved within a few months, its creator might have thought a little longer before opening it to the world.

The first attempt to assemble a comprehensive collection of the greatest mathematical problems was made at the beginning of the last century by David Hilbertwho is called the last universal mathematician. Although his list of 23 issues turned out to be quite influential, looking back, he seems to us rather mixed.

It includes long-standing universal favorites, such as the Riemann hypothesis - often considered the greatest of the greats, remaining Everest for mathematicians for more than a hundred years. When asked Hilbert what he would like to know first, waking up after a 500-year dream, he immediately remembered this hypothesis. It describes the basic intuitive notion of the distribution of primes — atoms of arithmetic — and its proof will have vast implications for many branches of mathematics.

But Hilbert listed much more vague and non-rigorous goals, such as "a mathematical study of the axioms of physics" or "the development of methods of calculus of variations ". One of the hypotheses regarding the equal composition of equal-sized polyhedrons was decided by his student Max Dan in the same year that the list was published. Many of the peaks described by Hilbert turned out to be more like foothills.

The highest peaks do not submit with one attempt. Expeditions carefully set up base camps and stretch ropes, and then slowly climb to the peak. In mathematics, attacking a serious problem often also requires building complex structures. A direct attack is considered stupid and naive. The construction of these auxiliary mathematical constructions sometimes takes centuries, and as a result, they sometimes turn out to be more valuable than the conquered theorem. Then these forests become a permanent addition to the architecture of mathematics.

A great example of this phenomenon is the proof of Fermat's great theorem., which was received in 1994 by Andrew John Wiles. It is known that Fermat wrote his hypothesis on the sidelines of Diophantus's “Arithmetic” in 1639. But its proof required more than three hundred years to develop mathematical tools. In particular, mathematicians had to create a very advanced combination of number theory and geometry. This new area, arithmetic geometry , is now one of the deepest and most far-reaching mathematical theories. It goes far beyond the Fermat hypothesis, and has been used to solve many outstanding issues.

The great hypothesis must also be deep and be in the very middle of mathematics. In fact, the metaphor of conquering the peak does not reflect all the consequences of obtaining evidence. Getting it is not the ultimate goal of a difficult journey, but the starting point of an even greater adventure. A more suitable way would be a mountain pass, a saddle, allowing the traveler to move from one valley to another. This is what makes the Riemann hypothesis so powerful and popular. It reveals many other theorems and ideas, and extensive generalizations follow from it. Mathematicians study the rich valley to which it gives access, despite the fact that it remains purely hypothetical.

Moreover, strong enough evidence should support the hypothesis. Well-known saying by Niels Bohr: “The opposite of a correct statement is a false statement. But the opposite of deep truth can be another deep truth. ” However, for the great hypothesis this is clearly not the case. Since extensive indirect evidence usually speaks in her favor, her denial seems unlikely. For example, the first 10 trillion cases of the Riemann hypothesis were checked numerically on a computer. Who can still doubt her loyalty? However, such supporting material does not satisfy mathematicians. They demand absolute certainty and want to know why the hypothesis is true. Only convincing evidence can give such an answer. Experience shows that it is easy to deceive a person. Counterexamples can hide quite far, like, for example, that Noam Elkis, a Harvard mathematician who refuted Euler’s hypothesis, found a variation of Fermat’s hypothesis, which said that a fourth-degree number cannot be written in the form of three other numbers in the fourth degree. Who would have guessed that in the first counterexample there would be a number of 30 digits?

20 615 673 ⁴ = 2 682 440 ⁴ + 15 365 639 ⁴ + 18 796 760 ⁴

The best hypotheses usually have rather modest roots, like Fermat’s fleeting remark in the margins of the book, but their consequences grow over the years. It is also useful if the hypothesis can be expressed briefly, preferably through a formula with a small number of characters. A good hypothesis should fit on a t-shirt. For example, the Goldbach hypothesisreads: "Any even number starting with 2 can be represented as the sum of two prime numbers." This hypothesis, formulated in 1742, has not yet been proven. She became famous thanks to the story “Uncle Petros and the Goldbach Problem” by the Greek author Apostolos Doksiadis, not least because the publisher offered an advertising trick of $ 1 million to anyone who could prove it within two years after the book was published. The conciseness of the hypothesis develops with its external beauty. You can even define mathematical aesthetics as "the amount of influence per character." However, such elegant beauty can be deceiving. The most concise formulations may require the longest evidence, which again demonstrates Fermat's deceptively simple observation.

To this list of criteria, one can perhaps add the answer of the famous mathematician John Conway to the question of what makes the hypothesis great: "It must be egregious." An attractive hypothesis is also somewhat ridiculous or fantastic, with an unforeseen area of ​​influence and consequences. Ideally, it combines components from areas that are far from each other, which were not previously found in one statement, as unexpected ingredients in an expressive dish.

Finally, it will be helpful to understand that adventure is not always successful. Just as an insurmountable cleft can arise in front of a climber, so mathematicians can be defeated. And if they lose, then they lose completely. There is no such thing as 99% proof. For two millennia, people have tried to prove the hypothesis that the fifth Euclidean axiom - the infamous axiom of parallelism , which says that parallel lines do not intersect - can be deduced from the four previous axioms of planimetry. And then, at the beginning of the 19th century, mathematicians created concrete examples of non-Euclidean geometry, refuting this hypothesis.

But the geometry did not end there. In a perverted sense, the refutation of the great hypothesis may turn out to be even better news than its proof, since failure indicates that our understanding of the mathematical world is very different from reality. Losing can be productive, somewhat opposed to a pyrrhic victory. Non-Euclidean geometry turned out to be an important predecessor of Einstein's curved space-time, which plays such an important role in the modern understanding of gravity and space.

Similarly, when Kurt Godel published his famous incompleteness theoremin 1931, which showed that in any formal mathematical system there are true statements that cannot be proved, he, in fact, answered negatively to one of Hilbert's problems regarding the consistency of the axioms of arithmetic. However, the incompleteness theorem - which is often considered the greatest achievement of logic since Aristotle - did not proclaim the end of mathematical logic. Instead, it led to a heyday that led to the development of modern computers.

So, in the end, the search for a solution to the great hypotheses has slightly different similarities with mountain expeditions to the highest peaks. Only when everyone returned home, in safety - no matter whether the goal was achieved or not - does the true breadth of adventure become clear. And then the time comes for heroic ascension stories.