The logicist project launched by Frege as a continuation of the arithmetization and rigorization process of analysis during the 19th century, not only served to start the modern field of the philosophy of mathematics but, also, to focus the research in a concrete direction. Frege's last aim was to justify the absolute certitude of the mathematical knowledge through setting secure indubitable foundations for mathematics. This has been the main task of the philosophy of mathematics since then, to such an extent that philosophy of mathematics has often been identified with the search for foundations. Apparently opposite methodologies have been placed in the service of the same final aim by several approaches during these years. I call foundationism to this mainstream in the philosophy of mathematics. In the 1960s, and starting with Lakatos, critical voices emerged opposing the direction adopted in the field since Frege. They considered the question of the foundations as an outdated and irrelevant question from the viewpoint of the mathematics of their time, and committed to the task of articulating the propper methodology of «informal» mathematics, as the primary task for a more relevant philosophy of mathematics. I call non-foundationism to this heterodox viewpoint. In this paper I review some main examples of each of both branches to emphasize their differences in order to obtain a broader panoramic of the field.