Conjectures and Refutations

To the outside observer, mathematics may appear as the definite system of truth. From this vantage point conjectures may seem a fringe phenomenon to be shunned; the poor man's theorem and an admittance of one's unsuccessful attempts of proving it. I will argue that aforementioned conjecture is false. I will argue for Karl Popper's view that "there is no more rational procedure than the method of trial and error--of conjecture and refutation: of boldly proposing theories; of trying our best to show that these are erroneous; and of accepting them tentatively if our critical efforts are unsuccessful." We will see that conjectures are an integral and essential part of research in the mathematical sciences. This…