Developing Understanding for Different Roles of Proof in Dynamic Geometry
Michael D. de Villiers · 2002
Introduction In a recent article submitted to Philosophae Mathematicae Yehuda Rav (1999) poses the interesting hypothetical situation of us having access to an all-powerful computer called PYTHIAGORA with which we can quickly check whether any conceivable mathematical conjecture is true or not. Would such a powerful tool spell the end of proof as we know it today? Perhaps surprisingly to the general public, the answer to this question is a resounding NO! As Rav points out, it is quite often irrelevant in mathematics whether a particular conjecture is true or not. He gives the example of the still unproved Goldbach conjecture that has been the fundamental catalyst for the development of major new theories as mathematicians search for a proof: Look at the treasure which attempted proofs of the Goldbach conjecture has produced, and how much less significant by comparison its ultimate 'truth value' might be! ... Now let us suppose that one day somebody comes up with a counter-example to the Goldbach conjecture or with a proof that there exist positive even integers not representable as a sum of two primes. Would that falsify or just tarnish all the magnificent theories, concepts and techniques which were developed in order to prove the now supposed incorrect conjecture? None of that. A disproof of the Goldbach conjecture would just catalyze a host of new developments, without the slightest effect on hitherto developed methods in an attempt to prove the conjecture. For we would immediately ask new questions, such as to the number of 'non-goldbachian' even integers: finitely many? infinitely many? ... New treasures would be accumulated alongside, rather than instead of the old ones thus and so is the path of proofs in mathematics!