The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue that nevertheless (...) intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (shrink)

According to Husserl’s so-called Exhibition Principle, the propositions “x exists” and “The exhibition of x’s existence is possible” are equivalent. The overall aim of this paper is to debate EP. First, I raise the question whether EP can properly be said to be a principle. Second, I give a general formulation of EP. Third, I examine specific formulations of EP, namely those regarding eidetic and individual objects. Fourth, I identify the readings of EP I hold to be exegetically plausible, that (...) is the transcendental reading, the metaphysical reading, and the hybrid reading. Fifth, I present Husserl’s refutation of Berkeleyan idealism developed in 1902/03 lectures, and I argue that under certain assumptions, both esse est percipi and the metaphysical EP engender an infinite regress. In this regard, I claim there are two options for avoiding such a regress: either to commit oneself to reflexive exhibition of the ego’s actuality or to deny the universality of EP. I show that Husserl has a good argument for rejecting the first option, and I conclude that if the Husserlian idealist chooses the second option while affirming the ego’s actuality, he turns out to be as “dogmatic” as the realist. (shrink)

This paper explicates Husserl’s usage of what he calls “radical Besinnung” in Formale und transzendentale Logik. Husserl introduces radical Besinnung as his method in the introduction to FTL. Radical Besinnung aims at criticizing the practice of formal sciences by means of transcendental phenomenological clarification of its aims and presuppositions. By showing how Husserl applies this method to the history of formal sciences down to mathematicians’ work in his time, the paper explains in detail the relationship between historical critical Besinnung and (...) transcendental phenomenology. Ultimately the paper suggests that radical Besinnung should be viewed as a general methodological framework within which transcendental phenomenological descriptions are used to criticize historically given goal-directed practices. (shrink)

A. Kenny, Frege, an introduction to the founder of modern analytic philosophy. London:Penguin, 1995. viii-h223pp. £7.99 T. Willamson, Vagueness. London:Routledge, 1994. xiii-f-325 pp. £35.00 TOM BU...

What does it mean to become older? This question cannot be answered by biology or sociology alone but must be addressed by studying how aging appears to aging persons and to others who encounter them. The paper presents the outline of such a phenomenological analysis of aging. In contrast to a problematic tendency to reduce phenomenology to the study of first-person experience, it is suggested that aging should be understood as a complex process comprising of both subjective and objective (mental, (...) social and biological) elements, and as partly transcendent to experience, that is, not completely transparent to the aging person. In this it resembles physical and other “transcendent” objects as described by early phenomenologists like Husserl. It is further shown how Marcel Proust’s elaborate description of aging in the last volume of his Remembrance of Time Past can support and contribute to such an analysis, and how it is supported or at least compatible with findings from contemporary gerontology and cognitive psychology. (shrink)

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)

This paper argues that Weyl took formalism to prevail over intuitionism with respect to supporting scientific objectivity, rather than grounding classical mathematics, and that this was what he thought was enough for rejecting pure phenomenology as well.

A new form of skepticism is described, which holds that objectivity and understanding are incompossible ideals of modern science. This is attributed to Weyl, hence its name: Weylean skepticism. Two general defeat strategies are then proposed, one of which is rejected.

The period from 1900 to 1935 was particularly fruitful and important for the development of logic and logical metatheory. This survey is organized along eight "itineraries" concentrating on historically and conceptually linked strands in this development. Itinerary I deals with the evolution of conceptions of axiomatics. Itinerary II centers on the logical work of Bertrand Russell. Itinerary III presents the development of set theory from Zermelo onward. Itinerary IV discusses the contributions of the algebra of logic tradition, in particular, Löwenheim (...) and Skolem. Itinerary V surveys the work in logic connected to the Hilbert school, and itinerary V deals specifically with consistency proofs and metamathematics, including the incompleteness theorems. Itinerary VII traces the development of intuitionistic and many-valued logics. Itinerary VIII surveys the development of semantical notions from the early work on axiomatics up to Tarski's work on truth. (shrink)

A Christian approach to scholarship, directed by the central biblical motive of creation, fall and redemption and guided by the theoretical idea that God subjected all of creation to His Law-Word, delimiting and determining the cohering diversity we experience within reality, in principle safe-guards those in the grip of this ultimate commitment and theoretical orientation from absolutizing or deifying anything within creation. In this article my over-all approach is focused on the one-sided legacy of mathematics, starting with Pythagorean arithmeticism (“everything (...) is number”), continuing with the geometrization of mathematics after the discovery of irrational numbers and once again, during the nineteenth century returning to an arithmeticistic position. The third option, never explored during the history of mathematics, guides our analysis: instead of reducing space to number or number to space it is argued that both the uniqueness of these two aspects and their mutual coherence ought to direct mathematics. The presence of different schools of thought is highlighted and then the argument proceeds by distinguishing numerical and spatial facts, while accounting for the strict correlation of operations on the law side of the numerical aspect and their correlated numerical subjects (numbers). Discussing the examples of 2 + 2 = 4 and the definition of a straight line as the shortest distance between two points provide the background for a brief sketch of the third alternative proposed (inter alia against the background of an assessment of infinity and continuity and the vicious circles present in contemporary mathematical arithmeticistic claims). (shrink)

We argue that no attempt of reducing meaning to a systematic set of rules, according to which the role of linguistic expressions is to be normatively defined, can be abstracted from an irreducibly decisional compound. By comparing Lorenzen’s project of building an Ortho-languageand Brandom’s inferentialist take on meaning, we distinguish two ways of acknowledging this fact, while claiming that Lorenzen’s take is more genuinely constructive, insofar as choices be thought of as genuine features of constructions. It brings into a new (...) perspective the relation between dialogical constructivism and Brouwer’s intuitionism. Finally we bring up a philosophical argument for the claim that interaction rules should be indexed on players and on their choices, when providing deontic bases to semantics. (shrink)

The History of Philosophy as a Problem: Four Fundamental Views in the German Debate and Heidegger’s Hermeneutical Dissolution Resumen: El artículo pretende introducir el problema que supone el estudio de la historia de la filosofía, especialmente hoy cuando esta práctica se ha establecido como elemento indisociable de la formación en la materia. Con el fin de plantear de manera más concreta dicho problema, el artículo trata la legitimidad del enfoque hermenéutico, pues este parece consistir en identificar el estudio histórico de (...) la filosofía con el propio filosofar. Para lograr una evaluación precisa de este enfoque, se lleva a cabo un recorrido por las cuatro posiciones cardinales del debate alemán en torno al sentido, el método y la función de la historia de la filosofía. Mediante la contraposición de la propuesta de Heidegger frente a estas cuatro posiciones cardinales, se espera aclarar cuáles son los fundamentos del pensamiento hermenéutico y sacar a la luz las razones por las que el estudio del pasado filosófico podría ser considerado esencial para la formación en filosofía.: The article aims to introduce the problem that represents the study of the history of philosophy, especially today when this practice has established as the model in the academic formation of the field. In order to raise this problem more specifically, the article focuses on the legitimacy of the hermeneutic approach, as it consists in the affirmation of the necessity of the study of the past for philosophy. The evaluation of this approach is carried out in the light of a description of the four cardinal positions of the German debate about the meaning, method and function of the history of philosophy. By contrasting Heidegger’s proposal with these four cardinal positions, it is hoped to clarify the foundations of hermeneutic thought and to bring to light the reasons because the study of philosophical past can be considered essential for philosophy. Palabras clave: hermenéutica, función de la historia, métodos historiográficos, comprensión, interpretación. Keywords: hermeneutics, function of history, historiographical methods, understanding, interpretation. (shrink)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.

The anti-metaphysical attitude of the neo-positivist movement is notorious. It is an essential mark of what its members regarded as the scientific world view. The paper focuses on a metaphysical variation of the scientific world view as proposed by Heinrich Scholz and his Münster group, who can be regarded as a peripheral part of the movement. They used formal ontology for legitimizing the use of logical calculi. Scholz's relation to the neo-positivist movement and his contributions to logic and foundations are (...) discussed. His heuristic background can be drawn from a set of six methodological ‘articles of faith’, formulated in 1942 and published here for the first time. I would like to thank Gudrun Mikus (Paderborn) for her assistance in collecting the material, Neil Tennant (Ohio State University, Columbus) for his efforts to improve the paper not only in lingual aspects, and Christian Thiel (Erlangen) and two anonymous referees for their helpful comments. CiteULike Connotea Del.icio.us What's this? (shrink)

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...) that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense. (shrink)

This chapter contains sections titled: Connecting Phenomenology and Mathematics Transcendental Phenomenology as a Foundation of Mathematics Examples.

Michael Dummett has interpreted and expounded upon intuitionism under the influence of Wittgensteinian views on language, meaning and cognition. I argue against the application of some of these views to intuitionism and point to shortcomings in Dummett's approach. The alternative I propose makes use of recent, post-Wittgensteinian views in the philosophy of mind, meaning and language. These views are associated with the claim that human cognition exhibits intentionality and with related ideas in philosophical psychology. Intuitionism holds that mathematical constructions are (...) mental processes or objects. Constructions are, in the first instance, forms of consciousness or possible experience of a particular type. As such, they must be understood in terms of the concept of intentionality. This view has a historical basis in the literature on intuitionism. In a famous 1931 lecture Heyting in fact identifies constructions with fulfilled or fulfillable mathematical intentions. I consider some of the consequences of this identification and contrast them with Dummett's views on intuitionism. (shrink)

A. Kenny, Frege, an introduction to the founder of modern analytic philosophy. London:Penguin, 1995. viii-h223pp. £7.99 T. Willamson, Vagueness. London:Routledge, 1994. xiii-f-325 pp. £35.00 TOM BU...

The paper is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker andGödel will be discussed and analysed. The aim of the paper is to show the influence of phenomenological ideas on the philosophical conceptions concerning mathematics. We shall start by indicating the attachment of Edmund Husserl to mathematics and by presenting the main points of his philosophy of mathematics. Next, works of two philosophers who attempted to apply Husserl’s phenomenological ideas to the (...) philosophy of mathematics, namely Hermann Weyl and Oskar Becker, will be briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel will be studied. (shrink)