Something But Not Too Much: Ontological Considerations in Intuitionistic Mathematics

In the first decade of the twentieth century intuitionist mathematics was born by the Dutchman Luitzen Egbertus Jan Brouwer and was popularized by the patient work of his pupil Arend Heyting. Whereas “classical” mathematics, following a Pythagorean-Platonic heritage, presupposed an ontology whose objects existed independently of human beings, who could only describe them and then infer their properties, in intuitionist mathematics it is the mathematician himself who gives rise to the mathematical entities and their properties, because it is the subject who constructs them in his mind, constrained only by his own mind. In this essay, I will take a close look at the positions of the first two intuitionist mathematicians in order to bring out, in the diversity of their approaches, the richness and the critical aspects of their common enterprise of re-edifying mathematics.