Students of mathematics are well versed in the knowledge that some problems have no solutions. We learn in grade school that division by zero is simply impossible; the fabric of spacetime may unravel if we brazenly attempt such a feat. But this is more akin to a rule violation than a truly unsolvable problem. Are there legitimate mathematics problems that are so difficult that humans may never solve them?
Mathematicians agreed upon in Paris in the year 2000 as the biggest challenges facing the field. Most of them are far too advanced and esoteric for the average mathematician, let alone the average person. To make things interesting, the Clay Mathematics Institute offered a $1 million prize for each of the seven problems. Since then, only one has been solved – the Poincaré Conjecture, which involves the mathematical description of surfaces and, in Devlin’s words, demonstrate “the deep and fundamental ways in which a doughnut is the same as a coffee cup” – by Russian mathematician Grigoriy Perelman. Then, in stereotypically mysterious Russian fashion, he declined the $1 million prize, refused to give interviews, quit his job, retreated into obscurity, and moved in with his mother. Perelman also declined the Fields Medal, which is the mathematical equivalent of the Nobel Prize.
For laymen, perhaps the most comprehensible of the seven (now six) problems is the Riemann Hypothesis, which deals with a topic that even the Ancient Greeks found fascinating: The prime numbers. Prime numbers are those numbers whose whole number factors include only 1 and itself. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. A number like 6 is composite, not prime, because it can be factored as 2 x 3. (As a side note, 6 is also a “perfect number,” because the whole number factors excluding itself – 1, 2, and 3 – sum to 6.) We know that there are fewer and fewer prime numbers the higher we go up the number line, and using a bit of logic, Euclid proved roughly 2300 years ago that the prime numbers go on infinitely. But modern-day mathematicians are not satisfied. They want to know if there is a pattern to the primes.
In 1859, Bernhard Riemann extended the concept to include imaginary numbers (i.e., those that contain the square root of -1, also known as i). Furthermore, he proposed that when the function equals 0, the solutions provide an insight into the distribution of the prime numbers. Specifically, he claimed that while there is an overall pattern to the prime numbers (namely, that they become less and less common), we cannot predict in advance which numbers will be prime.
This is known as the Riemann Hypothesis, and it is considered the greatest outstanding question in the field of mathematics. While mathematicians assume the hypothesis is true, it has not been proven in 160 years.
Source: [url=https://www.acsh.org/news/2020/02/05/math-politics-some-problems-don’t-have-solutions-part-i-14555]From Math To Politics, Some Problems Don’t Have Solutions (Part I)[/url]
10 Feb 2020, 10:39 am
