Subtle art of the mathematical hypothesis

This is not a proof, but a guess, supported by knowledge. But a good hypothesis leads the mathematics forward, pointing the way into mathematical uncertainty.

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

Mountaineering is a popular metaphor for mathematical research. Such a comparison is almost impossible to avoid: the frozen world, the thin cold air, the harsh rigidity of mountaineering resembles the inexorable landscape of numbers, formulas and theorems. Just as a mountaineer opposes his abilities to an intractable 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 great hypotheses - sharply formulated statements, most likely true, but without convincing evidence. These hypotheses have deep roots and broad implications. The search for their solutions makes up a large part of mathematics. Eternal glory awaits their first conqueror.

Interestingly, mathematicians raised the formulation of hypotheses to the level of high art. The most rigorous science loves the softest forms. A well-chosen, but not proven statement may make its author famous throughout the world, perhaps even more than the person who offers the final proof. The Poincaré conjecture remains a Poincaré conjecture, even after it was proved by Grigori Yakovlevich Perelman . And after all, the Briton George Everest, the chief surveyor of India in the first half of the XIX century, never climbed the mountain bearing his name.

As in any form of art, the great hypothesis must meet several mandatory criteria. First of all, it must 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 the hypothesis is proven for several 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 Gilbert , who is called the last universal mathematician. Although his list of 23 problems turned out to be very influential, looking back, he seems to us rather mixed.

It includes longtime general favorites, such as the Riemann hypothesis - often considered the greatest of all greats, the remaining Everest for mathematicians for more than a hundred years. When Gilbert was asked what he would like to know first, waking up after a 500-year-old sleep, he immediately remembered this hypothesis. It describes the basic intuitive understanding of the distribution of primes — the atoms of arithmetic — and its proof will have broad consequences for many branches of mathematics.

But Hilbert listed some more vague and lax goals like “mathematical study of the axioms of physics” or “development of methods of variational calculus ”. One of the hypotheses concerning the equal composition of equal-sized polyhedra was decided by his student Max Dan in the same year when the list was published. Many of the peaks described by Gilbert turned out to be more like foothills.

Highest peaks do not submit on one try. Expeditions carefully set up the base camps and stretch the ropes, and then slowly climb to the peak. In mathematics, to attack a serious problem, it is often also necessary to build complex structures. Direct attack is considered stupid and naive. The construction of these auxiliary mathematical constructions sometimes takes centuries, and in the end they sometimes turn out to be more valuable than the conquered theorem. Then these forests become a constant addition to the architecture of mathematics.

A great example of this phenomenon will be the proof of the great Fermat theorem , which Andrew John Wiles received in 1994. It is known that Fermat wrote his hypothesis in the fields of Diophantus "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 field, arithmetic geometry , is now one of the most profound and far-reaching mathematical theories. It goes far beyond the Fermat hypothesis, and was used to solve many outstanding issues.

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

Moreover, the hypothesis must be supported by sufficiently strong evidence. Niels Bohr's famous statement: “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 is usually in her favor, her denial seems unlikely. For example, the first 10 trillion cases of the Riemann hypothesis were tested numerically on a computer. Who can still doubt her loyalty? However, such supporting material does not satisfy mathematicians. They require 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, such as the one that Noam Elkis found, a Harvard mathematician who disproved Euler's hypothesis, a variation of Fermat's hypothesis, which said that the number in the fourth degree cannot be written in the form of three other numbers in the fourth degree. Who would have guessed that in the first counter-example 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 on the margins of a 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 symbols. A good hypothesis should fit on a T-shirt. For example, the hypothesis of Goldbach says: "Any even number, starting with 2, can be represented as the sum of two primes." This hypothesis, formulated in 1742, has not yet been proven. She became famous for the story “Uncle Petros and Goldbach's Problem” by the Greek author Apostolos Doxiadis, not least because the publisher offered $ 1 million as an advertising ploy to someone who could prove it within two years after the book was published. The conciseness of the hypothesis is formed with its external beauty. You can even define mathematical aesthetics as "the amount of influence on one symbol." However, such elegant beauty can be deceptive. The most concise formulations may require the longest evidence, which again demonstrates the deceptively simple observation of the Farm.

Perhaps to this list of criteria one can add the answer of the famous mathematician John Conway to the question of what makes the hypothesis great: "It must be glaring." An attractive hypothesis is also somewhat ludicrous or fantastic, with an unanticipated area of ​​influence and consequences. Ideally, it combines ingredients from distant areas that had not previously been encountered in the same statement, as unexpected ingredients in an expressive dish.

Finally, it will be helpful to understand that adventure does not always end in success. As the climber can face an irresistible cleft, so mathematicians can be defeated. And if they lose, they lose completely. There is no such thing as 99% proof. For two millennia, people tried to prove the hypothesis that the fifth Euclidean axiom - the notorious axiom of parallelism , which says that parallel lines do not intersect - can be derived from the four previous axioms of planimetry. And then, at the beginning of the XIX century, mathematicians created specific examples of non-Euclidean geometry, refuting this hypothesis.

But this is not the end of the geometry. In a perverted sense, a refutation of the great hypothesis may even be better news than its proof, since failure means that our understanding of the mathematical world is very different from reality. Losing can be productive, something opposite to the Pyrrhic victory. Non-Euclidean geometry proved to be an important predecessor of Einstein's warped space-time, which plays such an important role in the modern understanding of gravity and space.

Similarly, when Kurt Godel published his famous incompleteness theorem in 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 concerning the consistency of axioms of arithmetic. However, the incompleteness theorem — which is often considered the greatest achievement of logic since the days of Aristotle — did not proclaim the end of mathematical logic. Instead, it led to a flourish that led to the development of modern computers.

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