2020/02/27 by Vannieuwenhoven, Nick · 1 citation
Computer Science · Mathematics · #Commutative Algebra and Its Applications #Polynomial and algebraic computation #Tensor decomposition and applications
paper · doi:10.2307/j.ctvw1d7dg.9
openalex created_date 2016/06/24 · openalex publication_date 2020/02/27 · openalex updated_date 2026/07/28
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 will be illustrated with some surprising examples of conjectures and their refutations concerning the complexity of solving linear systems, the dimensions of algebraic varieties, and the equality of symmetric and nonsymmetric rank of symmetric tensors.