An Understanding of our fundamental limitations is among the most important contributions of science and of mathematics. There are quite a few cases where things that seemed possible and had been pursued for centuries in fact turned out to be fundamentally impossible. Ancient geometers thought that any two geometric lengths are commensurable, namely, measurable by the same common unit. However, for a right triangle with equal legs, the leg and the hypotenuse are incommensurable. In modern language (based on the Pythagorean theorem), this is the assertion that the square root of two is not a rational number. This was a big surprise in 600 BCE in ancient Greece (the story is that this discovery, attributed to a Pythagorean named Hippasus, perplexed Pythagoras to such an extent that he let Hippasus drown). Two centuries later, Euclid devoted the tenth book of his work The Elements to irrational quantities. The irrationality of the square root of 2 is an important landmark in mathematics. Similarly, the starting point of modern algebra can be traced back to another impossibility result. Algebraists found formulas for solving equations of degrees two, three, and four. Abel and Galois proved that no such formula is possible for general equations of degree five and above. This theory also led to a solution of a problem that remained unsolved from the ancient Greek era: finding a method of trisecting a general angle with a compass and a ruler. Galois’ theory demonstrated that no such method exists.
There are other important impossibility results. We have already mentioned Gödel’s impossibility result, asserting that it is not possible to prove (nor even to state) the consistency of mathematics from within mathematics. It is believed that there are inherent impossibilities for computers (and for any computational devices). This is the famous conjecture that P is different from NP. An impossibility theorem of Gibbard and Sattherthwaite asserts that for an election with more than two candidates and at least two voters, there does not exist a voting method; that is, there is no method for choosing a winner, based on the voters’ preferences, which is immune to manipulation. Impossibility insights outside mathematics, like the impossibility of building a perpetuum mobile, of turning iron into gold, and (most likely) of traveling in time, are related to profound scientific understanding. And, of course, the impossibility of traveling quicker than the speed of light is one of the famous and mind-boggling insights of science.
Here is a post on related things by Dick Lipton,