We present a new English translation of L.E.J. Brouwer's paper ‘De onbetrouwbaarheid der logische principes’ of 1908, together with a philosophical and historical introduction. In this paper Brouwer for the first time objected to the idea that the Principle of the Excluded Middle is valid. We discuss the circumstances under which the manuscript was submitted and accepted, Brouwer's ideas on the principle of the excluded middle, its consistency and partial validity, and his argument against the possibility of absolutely undecidable propositions. (...) We note that principled objections to the general excluded middle similar to Brouwer's had been advanced in print by Jules Molk two years before. Finally, we discuss the influence on George Griss' negationless mathematics. (shrink)

The main ideas behind Brouwer’s philosophy of Intuitionism are presented. Then some critical remarks against Intuitionism made by William Tait in “Against Intuitionism” [Journal of Philosophical Logic, 12, 173–195] are answered.

According to Brouwer’s ‘theory of the exodus of consciousness’, our experience includes ‘egoicity’, a distinct kind of feeling. In this paper, we describe his phenomenology in order to explore and elaborate on the notion of egoic sensations. In the world of perception formed from sensations, some of them are, Brouwer claims, not completely separated or ‘estranged’ from the subject, which is to say they have a certain degree of egoicity. We claim this phenomenon can be explained in terms of the (...) primordial state of consciousness from where the ‘exodus’ starts. Having undertaken the analysis and interpretation of Brouwer’s descriptions and examples of egoic sensations, we provide a formal account of egoicity based on Brouwer’s definition of its relation to estrangement, desire and fear. We show that the four terms can be modeled by a classical and a graded logical hexagon, giving the corresponding axiomatizations. (shrink)

The concept of God is studied using the ontological argument of Anselm of Canterbury that proves God’s existence using a syllogism based on ontology. Unlike metaphysical arguments that demonstrate the existence of God through the study of being and its attributes, the ontological argument aims to reach this same goal based on a concept of God by means of the idea of an entity “greater than anything that can be conceived”. Descartes’ influence highlighted some of the philosophical difficulties with the (...) inherent dualism implicit in ontology. Logic does not say whether ideas are true or false, as Logic is concerned with right or wrong inference. We take a non-dualist approach in contrast to some of the approaches reviewed. To investigate belief and faith, a modal logical formalization is used, especially modal logic S5, which we find appropriate for presenting arguments. While mathematics and logic can convince, they do not help one to believe in God. This can make us think that reason is not enough to accept God, one must have faith to have access to Him. The relation between reason and faith needs clarification, and the ontological argument requires an unsustainable dualism. (shrink)

Friedrich Nietzsche and Luitzen Egbertus Jan Brouwer had strong personalities and freely expressed unconventional opinions. In particular, they dared to challenge the traditional view that considered Aristotelian logic as being absolute and intrinsic to man. Although they formed this opinion in different ways and in different contexts, they both based it on a view of life that considered it as a struggle for power in which logic was a weapon. Therefore, it is interesting to carry out an in-depth analysis on (...) the origins, the core of their opinions concerning logic and the consequences. As a matter of fact, a detailed analysis and a comparison of their contributions to the philosophy of logic have not yet been carried out. In this paper, I intend to present the results of my research.. (shrink)

Cantor argued that absolute infinity is beyond mathematical comprehension. His arguments imply that the domain of mathematics cannot be grasped by mathematical means. We argue that this inability constitutes a foundational problem. For Cantor, however, the domain of mathematics does not belong to mathematics, but to theology. We thus discuss the theological significance of Cantor’s treatment of absolute infinity and show that it can be interpreted in terms of negative theology. Proceeding from this interpretation, we refer to the recent debate (...) on absolute generality and argue that the method of diagonalization constitutes a modern version of the vianegativa. On our reading, negative theology can evoke an attitude of humility with respect to the boundedness of the human condition. Along these lines, we think that the foundational problem of mathematics concerning its domain can be addressed through a methodological attitude of humility. (shrink)

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (shrink)