This work presents the results of the first empirical research protocol based on the connection between phenomenology, cognitive science and pedagogy, conducted in the Laboratory of Epistemology and Practices of Education (LEPE) in the last three years and aimed at the pedagogical training of psychologists and teachers. The LEPE follows a different path than the more commonly used laboratory strategies because it does not depart from a biographical analysis of the experience but from a preliminary systematization of knowledge about the educational process by means of a comparative-synchronic analysis of theoretical models of education. This analysis is preliminary to the choice and explanation of a model of action, in which formal, non-formal and informal knowledge is interconnected. The training is aimed at acquiring meta-cognitive and self-reflection strategies in order to produce an “autonomous and responsible action” that is the result of conscious and critical “practical” choices, based on understanding, monitoring and management of the largest number of possible alternatives. Relevance: This paper is a synthesis of the author’s book “Il Laboratorio di Epistemologia e Pratiche dell’educazione. Un approccio neurofenomenologico alla formazione dei formatori.” Starting from the nexus between phenomenology and cognitive sciences, it explores intentionality in an educational setting and offers neurophenomenological strategies aimed at the training of the trainers.

Context: Strict finitism is usually not taken seriously as a possible view on what mathematics is and how it functions. This is due mainly to unfamiliarity with the topic. Problem: First, it is necessary to present a “decent” history of strict finitism (which is now lacking) and, secondly, to show that common counterarguments against strict finitism can be properly addressed and refuted. Method: For the historical part, the historical material is situated in a broader context, and for the argumentative part, an evaluation of arguments and counterarguments is presented. Results: The main result is that strict finitism is indeed a viable option, next to other constructive approaches, in (the foundations of) mathematics. Implications: Arguing for strict finitism is more complex than is usually thought. For future research, strict finitist mathematics itself needs to be written out in more detail to increase its credibility. In as far as strict finitism is a viable option, it will change our views on such “classics” as the platonist-constructivist discussion, the discovery-construction debate and the mysterious applicability problem (why is mathematics so successful in its applications?). Constructivist content: Strict finitism starts from the idea that counting is an act of labeling, hence the mathematician is an active subject right from the start. It differs from other constructivist views in that the finite limitations of the human subject are taken into account.

Key words: mathematics, finite, largest number, infinite, limits, budget constraints