Negative Discoveries
Negative Discoveries
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 NonEuclidean 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 nonEuclidean 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 fourcolor hypothesis, proposed in 1852, was partly solved by eliminating incorrect solutions. The fourcolor 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 fourcolor 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 fourcolor theorem.
see also Consistency; Euclid and His Contributions; Proof.
Rafiq Ladhani
Bibliography
Dunham, William. The Mathematical Universe. New York: John Wiley & Sons Inc., 1994.
Cite this article
Pick a style below, and copy the text for your bibliography.

MLA

Chicago

APA
"Negative Discoveries." Mathematics. . Encyclopedia.com. 18 Aug. 2018 <http://www.encyclopedia.com>.
"Negative Discoveries." Mathematics. . Encyclopedia.com. (August 18, 2018). http://www.encyclopedia.com/education/newswireswhitepapersandbooks/negativediscoveries
"Negative Discoveries." Mathematics. . Retrieved August 18, 2018 from Encyclopedia.com: http://www.encyclopedia.com/education/newswireswhitepapersandbooks/negativediscoveries
Citation styles
Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).
Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.
Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the mostrecent information available at these sites:
Modern Language Association
The Chicago Manual of Style
http://www.chicagomanualofstyle.org/tools_citationguide.html
American Psychological Association
Notes:
 Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
 In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.