On Mathematical Proofs,Problems and Impossibility
Geometric constructions captivate one and all and is one of the areas of mathematics that is highly preferred by a high school student. Although easy to comprehend and fun to do, a few geometric constructions defy completion with a ruler and a compass. This article takes a look at the confusion in the mindset of many people regarding these ‘impossible’ constructions.
The following are the most famous of the impossible constructions with a ruler and a compass:
Trisecting an angle
Doubling the volume of a cube
Constructing a square equal to the area of a circle
What is a proof? For those of you who are wondering about that, here is the definition for a proof:
‘A proof is that which has convinced and now convinces the intelligent reader’ Who is an intelligent reader? In this context, those people who are known to be mathematicians by the society are the intelligent people.
Also a major requirement for a proof is that,a proof should be in complete harmony with another proven fact.
The ‘Impossibilty of a Problem’:
One often confuses the impossibility of a problem with an unsolved problem. A few problems are unsolved, that is, they have not been solved as of yet, whereas some other problems are insoluble,that is, they cannot be solved.The problem is proved to be impossible to solve. There is no question of anyone coming up with a construction for trisecting an angle because it has been proved mathematically that no one can trisect an angle.Whereas if someone were to claim that he/she (no discrimination) has found a proof of the Riemann’s hypothesis or the Goldbach Conjecture ( famous problems that have not yet been solved but have not been proved to be impossible either), mathematicians shall look into the claim. However for those of you who are itching for a claim to fame by solving one of the unsolved problems, let me remind you the path to success is not a bed of roses.