The Four Color Theorem: The Mathematical Puzzle Behind Map Coloring

## 3. The Breakthrough: Computer-Assisted Proof

Advertisement

1976 was a major turning point not only for this specific issue but also for the discipline of mathematics generally when the Four Colour Theorem was proved. Providing a proof of the theorem, University of Illinois mathematicians Kenneth Appel and Wolfgang Haken accomplished what many believed impossible. But since their approach depended so much on computer help, a first in the field of mathematical proofs, it was groundbreaking and, to some, contentious. Based on the ideas of "unavoidable sets" and "reducible configurations," Appel and Haken showed that every conceivable map has to include one of a set of particular configurations and then showed that each of these configurations could be coloured using just four colours. The issue was that around 2,000 combinations needed to be checked—a chore far too time-consuming and prone to mistakes for human inspection. Computers joined the scene here and dramatically changed the field of mathematical proof. Using computer programmes, the mathematicians examined every one of these configurations—a process requiring more than a thousand hours of computer effort. The outcome was evidence spanning hundreds of pages of mathematical ideas enhanced by computer-generated illustrations and statistics. Unprecedented and generated great discussion in the academic world, this hybrid technique combined conventional mathematical logic with computational capability. For various reasons, using computers in this proof was innovative and divisive. Some purists contended that a proof unable of human verification was not a true proof at all. They argued that human knowledge and verification defined mathematical proof fundamentally. Others saw it as the beginning of a new mathematical era whereby computers could be applied to solve problems beyond human capacity, therefore opening new paths for mathematical discovery. Though there remained debate, most mathematicians finally approved of the Appel-Haken evidence. It challenged conventional ideas of what makes a mathematical proof and launched a new field of computer-assisted proofs. The evidence showed that computers might be fundamental to the mathematics discovery and validation process rather than only calculational instruments. Efforts were undertaken to simplify and streamline the computer-assisted approach in the years following the first proof. Neil Robertson, Daniel Sanders, Paul Seymour, and Robin Thomas developed a more effective proof in 1997 that simplified the computer techniques applied and cut the number of combinations to be examined. This new proof addressed some of the issues expressed with the original Appel-Haken proof and was more manageable and simpler to verify, even while one still depends on computer help. A turning point in the junction of mathematics and computer science, the computer-assisted proof of the Four Colour Theorem It opened the path for more developments in computational mathematics and showed how computers may help to solve difficult mathematical issues. This evidence forced mathematicians to rethink what a legitimate mathematical argument is and created fresh avenues for addressing hitherto unsolvable issues. This method's success in addressing the Four Colour Theorem attracted more attention for computer-assisted proofs for other long-standing mathematical challenges. It generated philosophical debates concerning the nature of mathematical truth and understanding as well as conversations about the part computers play in mathematics teaching and research. More complex proof-checking tools and formal verification systems, tools that have grown ever more crucial in both pure mathematics and computer science, sprang from the proof acting as a spur for growth.