L’Ipotesi del Continuo, formulata da Cantor nel 1878, è una delle congetture più note della teoria degli insiemi. Il Problema del Continuo, che ad essa è collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dell’Ipotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema è controverso. In anni più recenti, la ricerca di una soluzione del Problema del Continuo è stata anche una delle ragioni (...) fondamentali per la ricerca di nuovi assiomi in matematica. L’articolo fornisce un quadro generale dei risultati matematici fondamentali, e un’analisi di alcune delle questioni filosofiche connesse al Problema del Continuo. (shrink)

Kurt Gödel made many affirmations of robust realism but also showed serious engagement with the idealist tradition, especially with Leibniz, Kant, and Husserl. The root of this apparently paradoxical attitude is his conviction of the power of reason. The paper explores the question of how Gödel read Kant. His argument that relativity theory supports the idea of the ideality of time is discussed critically, in particular attempting to explain the assertion that science can go beyond the appearances and ‘approach the (...) things’. Leibniz and post-Kantian idealism are discussed more briefly, the latter as documented in the correspondence with Gotthard Günther. (shrink)

Nosso objetivo neste artigo é o de lançar luz sobre alguns aspectos das concepções de Gödel acerca da percepção de conceitos. Começamos investigando a natureza e o papel da analogia entre percepção sensível e percepção de conceitos. A seguir, examinamos as conexões entre percepção de conceitos, razão e compreensão, tentando mostrar que a percepção de conceitos é compreensão de conceitos. Por fim, examinamos aqueles aspectos da concepção de Gödel em que a percepção de conceitos de fato se aproxima perigosamente da (...) ideia de um "olho da mente". Contudo, argumentamos que esses aspectos problemáticos não são intrínsecos à noção de percepção de conceitos, mas sim resultam de uma má compreensão da distinção entre conceito e objeto. Our goal in this article is to illuminate some aspects of Gödel's conception of the perception of concepts. We begin by investigating the nature and function of Gödel's analogy between sense perception and perception of concepts. After that, we examine the connections between perception of concepts, reason, and understanding, trying to show that the perception of concepts consists in the understanding of concepts. Finally, we examine those aspects of Gödel's conception where the perception of concepts actually comes dangerously close to the idea of a "mind's eye". However, we argue that these problematic aspects are not intrinsic to the notion of perception of concepts, but result from a misunderstanding of the distinction between concept and object. (shrink)

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which leads to (...) a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity. The whole undertaking, which takes into account major stages of the research in large cardinals theory, tries to present a defensible argumentation against an ontology of infinity actually rooted in the notion of subjectivity within the world. This means that rather than talking of a general ontology of infinity in the ideal platonic or in the aristotelian sense of potentiality, even in the alternative sense of an ontology of the event in A. Badiou’s sense, one can argue from a subjective point of view about the impossibility of defining cardinalities greater than the first uncountable one \ that would correspond to a distinct existence in real world terms or would be supported by a mathematical intuition in terms of reciprocity with experience. The argumentation from the particular standpoint includes also certain comments on the delimitative character of Gödel’s constructive universe L and the influence of the constructive approach in narrowing the breadth of an ‘ontology’ of infinity. (shrink)

It is well known that in his later years Gödel turned to a systematic reading of phenomenology, whose founder, Edmund Husserl, was highly esteem as a philosopher who sought to elevate philosophy to the standards of a rigorous science. For reasons purportedly related to his earlier attraction to Leibnizian monadology, Gödel was particularly interested in Husserl's transcendental phenomenology and the way it may shape the discussion on the nature of mathematical‐logical objects and the meaning and internal coherence of primitive terms (...) in mathematics. This article, which is less interested in historical facts that are amply presented in other scholars’ accounts, rather tries to point to the mostly indirect influence that the ideas of transcendental phenomenology, including Gödel's reported reference to the notion of (inner) time, had on his philosophy of mathematics, especially with a view to key independent questions of mathematical foundations. A main source for Gödel's philosophical‐epistemological views (of course in addition to his own published material) will be his recorded discussions with H. Wang and S. Toledo, as well as various articles and drafts found in his Nachlass. (shrink)

This article deals with a question of a most general, comprehensive and profound content as it is the nature of mathematical–logical objects insofar as these are considered objects of knowledge and more specifically objects of formal mathematical theories. As objects of formal theories they are dealt with in the sense they have acquired primarily from the beginnings of the systematic study of mathematical foundations in connection with logic dating from the works of G. Cantor and G. Frege in the last (...) decades of the nineteenth century. Largely motivated by a phenomenologically founded view of mathematical objects/states-of-affairs, I try to consistently argue for their character as objects shaped to a certain extent by intentional forms exhibited by consciousness and the modes of constitution of inner temporality and at the same time as constrained in the form of immanent ‘appearances’ by what stands as their non-eliminable reference, that is, the world as primitive soil of experience and the mathematical intuitions developed in relations of reciprocity within-the-world. In this perspective and relative to my intentions I enter, in the last section, into a brief review of certain positions of G. Sher’s foundational holism and R. Tieszen’s constituted platonism, among others, respectively presented in Sher and Tieszen. (shrink)

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...) way of recognizing objects of any kind. (shrink)

This essay will be divided into three parts. In the first part, we discuss the case of infintesimals seen as a bridge between the discrete and the continuous. This leads in the second part to a discussion of the nature of numbers. In the last part, we follow up with some observations on the obvious applicability of mathematics.