While solving problems and constructing proofs, mathematicians use many different approaches. A common technique for proving a statement is by contradiction. In this approach, it is supposed that the converse of the statement, or its opposite, is true. If this supposition leads to an absurd result, or contradiction, then it can be said the original statement is true. Hence, exploring incorrect answers and assumptions can often lead to new correct results.
Euclidean and Non-Euclidean Geometry
In 300 b.c.e., Euclid of Alexandria put forward a logical construction of a geometry, which has come to be known as Euclidean geometry. Until the middle of the nineteenth century mathematicians believed that Euclid's geometry was the only type of geometry possible. Euclidean geometry is based on a number of fundamental statements called postulates, or axioms.
In his book Elements, Euclid based his geometry on five axioms. The fifth axiom, also known as the parallel axiom, states the following: Given a line m and a point P not on m, there is only one line through P which is parallel to m.
In mathematics, a set of axioms has to fulfill two conditions: consistency and independence. A set of axioms is consistent if its use does not produce an absurd result that contradicts a statement derived from the axioms. A set of axioms is independent if none of the axioms can be logically deduced from the others.
Since Euclid, a number of mathematicians have thought that the parallel axiom was not independent and could be logically derived from the rest of the axioms. In 1763, the German mathematician Georg Klügel noted nearly thirty attempts to prove the dependence of the parallel postulate. But all attempts failed.
In 1733, a noteworthy attempt was made by Giovanni Girolamo Saccheri. After failing to show the dependence of the parallel axiom, Saccheri declared that Euclid's five axioms are indeed independent. But Saccheri's approach contained all the clues to invent or discover a new type of geometry. However, he failed to see the consequence of his own work because he thought that Euclid's geometry was the only geometry possible.
After Sacherri, a few more mathematicians continued to work on the parallel axiom problem. What Saccheri failed to discover, the young Hungarian mathematician Jénos Bolyai discovered by making a bold declaration. He proposed the first non-Euclidean geometry by replacing the parallel axiom with its "opposite" or negation. In Euclid's fifth axiom, instead of limiting to one parallel line, Bolyai's geometry stated that there is more than one parallel line.
By keeping Euclid's first four axioms the same and combining them with the modified fifth axiom, Bolyai discovered a new consistent geometry. This geometry is known as hyperbolic geometry. Bolyai's bold idea expanded the narrow world of Euclidean geometry. This important discovery occurred partly because of the negative attention that the parallel axiom received for several centuries!
New Negative Solutions
The process of eliminating the wrong answers occasionally results in the discovery of new mathematics. A famous problem, referred to as the four-color hypothesis, proposed in 1852, was partly solved by eliminating incorrect solutions. The four-color hypothesis states that a map drawn on a plane can be colored with four (or fewer) colors so that no two regions that share a common boundary have the same color.
To prove the four-color hypothesis, mathematicians tried to construct maps in which two or three colors were not sufficient. Also, a proof was constructed that showed that any map in the plane can be colored with five colors, so that neighboring regions do not have the same color.
Essentially, this approach showed that three colors are too few and five are more than enough. Therefore, the question that remained was whether four is just enough. In 1976, with the help of a computer, Kenneth Appel and Wolfgang Haken of the University of Illinois presented a proof of the four-color theorem.
see also Consistency; Euclid and His Contributions; Proof.
Dunham, William. The Mathematical Universe. New York: John Wiley & Sons Inc., 1994.