The Jordan Curve Theorem, Formally and Informally

1. INTRODUCTION. The Jordan curve theorem states that every simple closed pla-nar curve separates the plane into a bounded interior region and an unbounded exterior. One hundred years ago, Oswald Veblen declared that this theorem is " justly regarded as a most important step in the direction of a perfectly rigorous mathematics " [13, p. 83]. Its position as a benchmark of mathematical rigor has continued to our day. Many vastly underestimate the logical gulf that separates a typical published proof from a fully formal mathematical proof, in which every single logical inference has been generated and checked by computer. A striking example of this logical gulf can be found in Bourbaki's Elements of Sets. How many primitive symbols of logic does it require to represent the number " 1 " in fully expanded form? Bourbaki estimates that the fully expanded representation may require " several tens of thousands of symbols. " The story has it that when A. R. Mathias learned of Bourbaki's estimate, he thought " that must be false, surely only a couple of hundred " are required. This led him to make a careful calculation and to uncover the astounding fact that Bourbaki's own estimate is wrong by several orders of magnitude. Over four trillion symbols are needed to express the number " 1 " [7]. Such is the logical gulf. It is fortunate that not all systems suffer from the same inefficiencies. Who hasn't heard the one about the mathematician who wakes up at night from the smoke of a fire in his hotel? Seeing a fire extinguisher, he notes that a solution to the problem exists and falls peacefully back into a deep sleep. Researchers from Frege to Gödel, who solved the problem of rigor in mathematics, found theoretical solutions but did not extinguish the fire, because they omitted the practical implementation. Some, such as Bourbaki, have even gone so far as to claim that " formalized mathematics cannot in practice be written down in full " and call such a project " absolutely unreal-izable " [1, pp. 10, 11]. While it is true that formal proofs may be too long to print, computers—which do not have the same limitations as paper—have become the natural host of formal mathematics. In recent decades, insomniacs at the same hotel have reworked the foundations of mathematics, putting them in an efficient form designed for real use on …