The four-color problem and its philosophical significance

TLDR: Albertson as discussed by the authors raised the question of whether the 4CT is really a theorem and showed that four colors are not sufficient to color a map in such a way that neighboring regions are never colored alike.

Abstract: Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact support@jstor.org. T HE old four-color problem was a problem of mathematics for over a century. Mathematicians appear to have solved it to their satisfaction, but their solution raises a problem for philosophy which we might call the new four-color problem. T h e old four-color problem was whether every map on the plane or sphere can be colored with no more than four colors in such a way that neighboring regions are never colored alike. This problem is so simple to state that even a child can understand it. Nevertheless , the four-color problem resisted attempts by mathematicians for more than one hundred years. From very early on it was proved that five colors suffice to color a map, but no map was ever found that required more than four colors. I n fact some mathematicians thought that four colors were not sufficient and were working on methods to produce a counterexample when Kenneth Appel and Wolfgang Haken, assisted by John Koch, published a proof that four colors suffice.? Their proof has been accepted by most mathematicians , and the old four-color problem has given way in mathematics to the new four-color theorem (4CT). T h e purpose of these remarks is to raise the question of whether the 4CT is really a theorem. This investigation should be purely philosophical, since the mathematical question can be regarded as definitively solved. I t is not my aim to interfere with the rights of * I would like to thank Michael Albertson, Joan Hutchinson, and IVilliam Marsh for reading a draft of this paper and …