Mathematical and Phenomenological Continuity-Indiscernibility

La Nuova Critica, 2009

Manuscrito – Rev. Int. Fil., Campinas, 2009

This article attempts to link the notion of absolute ego as the ultimate subjectivity of consciousness in continental tradition with a phenomenology of Mathematical Continuum (MC) as this term is generally established following Cantor's pioneering ideas on the properties and cardinalities of sets. My motivation stems mainly from the inherent ambiguities underlying the nature and properties of this fundamental mathematical notion which, in my view, cannot be resolved in principle by the analytical means of any formal language not even by the addition of any new axioms to a consistent first-order axiomatical system such as the Zermelo-Fraenkel (ZF) Set Theory. In this phenomenologically motivated approach I deal, to some extent, with the undecidability of a fundamental statement about the cardinality of Continuum inside ZFC theory, namely the Continuum Hypothesis, and also with the underlying roots of Gödel's first incompleteness result. Resumo: Este artigo procura conectar a noção de ego absoluto enquanto subjetividade última da consciência na tradição continental com a fenomenologia da matemática do contínuo (MC) tal como este termo foi estabelecido seguindo as idéias pioneiras de Cantor sobre as propriedades e cardinalidade de conjuntos. Minha motivação deriva basicamente das ambigüidades subjacentes à natureza e propriedades desta noção matemática fundamental as quais, no meu entender, não podem, em princípio, ser resolvidas pelos meios analíticos de qualquer linguagem formal e nem mesmo pela adição de novos axiomas a um sistema axiomático consistente de primeira ordem como a teoria de conjuntos de Zermelo Fraenkel (ZF). Neste tratamento fenomenológicamente motivado eu lido em certa medida com a indecidibilidade de uma tese fundamental sobre a cardinalidade do contínuo em ZFC, a saber, a Hipótese do Contínuo, e também com aquilo que está na base do primeiro resultado de incompletude de Gödel.

It is known that the notion of the absolute ego of consciousness was described by Husserl in terms of the constitution and the continuous flow of internal time. It is remarkable that gradually Husserl tended to think of it not only in terms of temporality but in its generality as the source of all temporality, “reached” through a radical phenomenological reduction. We take into account the stages by which he was led to the impredicativity of the absolute ego of consciousness and try to demonstrate how this is reflected in the axiomatization of continuum in certain non-Cantorian mathematical theories to the extent that undertake a formalisation beyond natural intuition. We also review those theories’ approach to classical mathematical notions as this approach is more close to Husserl’s shift of the horizon idea.

Philosophia Mathematica, 2002

both recognized that there is something called 'the intuitive continuum'. It is the phenomenon of the intuitive continuum that motivates their developments of constructive real analysis on the basis of choice sequences. Brouwer already mentions the intuitive continuum and describes a few of its features in his doctoral thesis of 1907 [5]. Weyl, evidently unaware of Brouwer's early work, discusses the intuitive continuum in Chapter 2, section 6, of Das Kontinuum (DK) [27], which is entitled 'The Intuitive and the Mathematical Continuum'. The view of the intuitive continuum in DK is based largely on Husserlian phenomenological descriptions of the consciousness of internal time, although Weyl also mentions here and elsewhere some other historical views related to the idea of the intuitive continuum. 5 In this section of DK Weyl takes the experience of the flow of internal time as the model of the intuitive continuum. Brouwer and Weyl both distinguish 'internal', intuitive time from 'external' time. Brouwer calls the latter 'scientific' or measurable time [5, p.61]. Brouwer's notion of the primordial intuition of mathematics is concerned only with internal, intuitive time. Weyl, following Husserl, distinguishes 'phenomenal' time from 'objective' time and says that he will, in effect, 'suspend' the latter and focus only on the former [28, p.88]. It is the phenomenon of the intuitive continuum that he feels he has not captured with the mathematical (or 'arithmetic') theory of the continuum developed in DK. It is this phenomenon, in Weyl's eyes, to which Brouwer's development of intuitionistic real analysis does far more justice, and it is for just this reason that Weyl declares his allegiance to Brouwer in a famous paper published in 1921 [28]. We will describe below some of the features of the phenomenon on which Brouwer and Weyl are focused and then consider their mathematical treatments of it. A mathematics of the intuitive continuum should be founded on the formal or structural features of the intuitive continuum and, for Brouwer and Weyl, these will be structural features of the stream of consciousness that are 1 We are grateful to Robert Tragesser for discussion and comments.

2001

this paper, which is the sequel to [10] and which was partly anticipated in [13], we continue such analysis and show that the notion of continuity can be similarly characterized in structural terms. So also the notion of continuity can be treated following the general principles, which have been exposed in [12] for the philosophy of mathematics in general and in [11] for topology in particular. Thus here too, as in [10], points will be considered on equal rights as opens, which is technically realized by a strict preservation of symmetry, and the meaning of topological notions will be brought to its logical structure

INTENTIO, 2024

The intended scope of this article is to review the Continuum Hypothesis, CH, in foundational mathematics from the viewpoint of a phenomenologically influenced philosopher. Given the capital importance of Cantor’s 19th century conjecture about the cardinality of continuum and the relevance it has acquired over the years in matters of mathematical ontology it is natural to motivate a discussion well beyond its place in foundational mathematics proper. This means that except for the continuing research toward its resolution by new and powerful theories following Cohen’s pioneering forcing method in the 60s, one has come to inquire into the metatheoretical or even extra-theoretical nature of the Continuum Hypothesis, something that implies a philosophical inquiry into the ontological status of the question as such and into the ways it may be shaped by our natural intuitions of the continuous and the discrete. On the latter prompt, among others, and discarding platonistic tendencies I set out to provide an interpretation along subjective-constitutional tracks of the ontological status of the CH and ultimately of the ways it may be reducible to a question of the constitution of continuous unity in subjective-temporal terms. In the more formal part, I have tried to argue against a certain operationally motivated attempt at refuting CH as well as against attempts to resolve the issue of its undecidability relying on the more advanced large cardinals plus the inner models theory.

In “Quality of quantity” /1989/, I have established the relevance of Hegel’s category “qualitative quantity” arguing that there is wide underestimation of this fruitful “side effect of dialectics” due to the static patterns of dialectics and the lack of ontological commitment to the concept of qualitative quantity. Based on the exploration of D'Arcy W. Thompson’s “Growth and Form”, /1917/ findings and examples from Hermann Haken , illustrating the qualitative quantity notion in structural stability, in “Quality of quantity”, I have argued that topology is the field of qualitative quantity and topological homeomorphism To my knowledge, in contemporary dialectics and philosophy research the significant notion of Hegel’s category “qualitative quantity” remained inapparent. Reason for that is probably Hegel’s own warning about “the intellectual difficulty” /§ 777 of the “The Science of Logic”, The Greater Logic/ , accusing “the attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of the alteration” as “tedious like any tautology”. Since the “qualitative quantity” in Hegel is closely related with his “qualitative concept of quantity” /the good quantitative infinity/ and the so called “bad infinity”, after re-calling the Hegel’s account on “qualitative quantity”, I will discuss in the second chapter of the study, Alain Badiou’s Mediation Fifteen on Hegel . Alain Badiou recognizes the “qualitative quantity” as the core of the domain of “quantitative infinity”, claiming that “Quantitative infinity is quantity qu a quantity, the prliferator of proliferation, which is to say, quite simply, the quality of quantity, the quantitative such as discerned qualitatively from any other determination.” The subject of my discussion will be Badiou’s dissagriment with Hegel and Badiou’s claim that “Hegel fails to intervene on number.” According to Badiou “He /Hegel/ fails because the nominal equivalence he proposes between the pure presence of passing – beyond in the void / the good qualitative infinity/ and the qualitative concept of quantity /the good quantitative infinity/ is a trich, an illusory scena of the speculative theatre. There is no symetry between the same and the other, between proliferation and identification. However heroic the effort, it is interrupted de facto by the exteriority itself of the pure multiple. Mathematics occurs here as discontinuity whitin the dialectics. It is this lesson that Hegel wishes to mask by suturing under the same term – infinity - two disjoint – discursive others.” In opposite to Badiou’s claim that “the good quantitative infinity is a properly Hegelian hallucination”, I will argue that “the fragile verbal footbridge” /according Badiou/, build by Hegel is the topology of qualitative quantity. According to Badiou, the name “infinity” suits only the qualitative infinity /in Hegel – “the bad infinity”/ and he argue that Hegel cannot properly ground “the bad quantitative infinity”. My thesis is that what Baduio calls the “Hegelian hallucination” is Hegel’s topology of the fold enveloped in the fourfold of infinities - /1/. the bad qualitative infinity; /2/. the good qualitative infinity; /3/. the bad quantitative infinity; /4/. the good quantitative infinity. In the third chapter of my study, I will discusse two “fragile” issues in Hegel with critical importance for the attempt to reveil the nature of “qualitative quantity” and to establishe the relevance of the topological notion of qualitative quantity. First is the “inapparent” nature of “qualitative quantity” and its notion, second - the “tautology” related /according to Hegel/ with “the attempt to explain coming-to-be or ceasing-to-be on the basis of gradualness of the alteration”. These two issues are the reason to see logically the “verbal footbridge” of Hegel’s “qualitative quantity” as “fragile” and so reason Hegel’s category “qualitative quantity” to remain inapparent. In the fourth chapter of this study I will discusse the topology of this “Hegelian hallucination” – the topology of qualitative quantity as re-solution of the paradox, tautology and ambiguity. The last three chapters of the study will focus on the “Hegel’s topology”, “The philosophical topologies of …” and “The Relevance of D'Arcy W. Thompson’s in the new science as background of the “topology of meaning” and transition from typological thinking to topological thinking”.

Universa. Recensioni di filosofia, vol. 1 (2), 2013., 2013

Journal of Symbolic Logic, 1984

Under the assumption that all “rules” are recursive (ECT) the statement Cont(NN, N) that all functions from NN to N are continuous becomes equivalent to a statement KLS in the language of arithmetic about “effective operations”. Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from R to R and from 2N to N. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.

We start from the history of " continuum problem " and study of infinitely small and infinitely large numbers, rediscover the semi-ring of tropical real numbers and also reconfirm the philosophy that functions are generalized numbers. Alongside, we discuss the " number-shape problem " stating that, numbers and geometric shapes have the same essence.