This article argues against the concept of a priori and essence as they have been traditionally thematized in the course of the old metaphysical-idealist tradition. Specifically, I argue against the existence of an ontological a priori, often endowed with metaphysical-platonic connotations, by attempting to “relocate” it in the subjective sphere and thus reduce it to the being and modes of being of a transcendental subjectivity. To do so, I will be appealing to a phenomenological, Husserlian approach, while pointing to a (...) possible connection with the Kantian views on the matter and also taking into account certain views in the secondary literature. Since a substantial part of my position is associated with the notion of a constituting, transcendental subjectivity further reducible to the origin of inner temporality, I intend to show that the objectivity constraints put in this way on the conception of the transcendental a priori may ultimately lead to a “destruction” of its traditional ontological sense. Given that in transcendental phenomenology the concept of a priori “appears” both in the constituting and the constituted level, I aim to show that it is precisely in this context that the a priori cannot rid itself of the vestiges of factuality brought in by means of the very constitutive, subjective processes implying ipso facto the question of the role of the constituting origin of temporality. (shrink)

A debated issue in the mathematical foundations in at least the last two decades is whether one can plausibly argue for the merits of treating undecidable questions of mathematics, e.g., the Continuum Hypothesis, by relying on the existence of a plurality of set-theoretical universes except for a single one, i.e., the well-known set-theoretical universe V associated with the cumulative hierarchy of sets. The multiverse approach has some varying versions of the general concept of multiverse yet my intention is to primarily (...) address ontological multiversism as advocated, for instance, by Hamkins or Väätänen, precisely for the reason that they proclaim, to the one or the other extent, ontological preoccupations for the introduction of respective multiverse theories. Taking also into account Woodin’s and Steel’s multiverse versions, I take up an argumentation against multiversism, and in a certain sense against platonism in mathematical foundations, mainly on subjectively founded grounds, while keeping an eye on Clarke-Doane’s concern with Benacerraf’s challenge. I note that even though the paper is rather technically constructed in arguing against multiversism, the non-negligible philosophical part is influenced to a certain extent by a phenomenologically motivated view of the matter. (shrink)

The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed practices. (...) At the same time the view endorses the sense that mathematical reality is given to mathematicians with a sense of independence. As mathematical social constructions are products of highly constrained, intersubjective practices and accord with the phenomenologically clarified experience of mathematicians, positing them is phenomenologically justified. The social ontological approach offers a way to build mathematical ontology out of the practice with no metaphysical magic. (shrink)

Although the notion of the crisis of European sciences has a general meaning, Husserl mainly focuses on this phenomenon in relation to the modern establishment of a mathematical natural science. However, he does not provide a definitive clarification of how its new method is specifically involved in bringing about such a crisis. Without trying to offer a faithful exegetical contribution, this paper further elaborates on Husserl’s analyses in the Krisis to give a possible answer to this question. After defining the (...) concept of crisis in general, I single out why algebraic thinking is the core of the method that essentially characterizes mathematical physics. Then, I compare this method to a case of pre-modern discipline (namely, Euclidean geometry) to show that the sedimentation of meaning at work in the progressive technization of algebra is exceptional and sufficient to explain the specificity of the modern character of the crisis of the exact sciences. The exclusively technical development of formal mathematics, in fact, elicits those shifts of meaning (such as objectivism) that phenomenology is required to amend to overcome the crisis and to establish an authentic form of knowledge. On the basis of these results, I conclude by suggesting a few consequences that are useful for the understanding of the project of phenomenology of the later Husserl. (shrink)