Origin Of Geometry Research Articles

Crafts and the Origins of Geometry

In this paper, we explore the possible roles of craftwork at the origins of geometry. Since shapes are the characters of geometry, our focus is on crafting shapes. We review two approaches to the nature of making: hylomorphism and hylonoesis. Related to these, we distinguish between ‘shape’ and ‘shaping’. From studies on prehistoric pottery, we explore a conception of ‘shaping’ as thoroughly implicated by the manifold social, biological, and material life of a community. Then we explore the significance of ’shaping’, which animate shapes, immerse them in the vagaries of materiality, and fill them with secrets longing for their open realization; a realization that is at once — inseparably — material and imaginary. We make shapes as they shape our bodies, and then we become capable of imagining and gesturing them.

Phenomenology Out of Bounds? Jean-Yves Lacoste’s Phenomenology and the Presence of God

This article articulates Jean-Yves Lacoste’s account of phenomenology. It does so by tracing Lacoste’s relation to Husserl. Although the influence of Heidegger on Lacoste’s thinking has been sufficiently studied, his relation to the father of phenomenology perhaps is not. The aim of this essay is to see whether Lacoste’s practice of phenomenology still qualifies as phenomenology proper or whether, as some might be inclined to think, it is an improper venturing into the terrain of theology. For this, we offer an account both of Lacoste’s conception of “theological thinking” and of the phenomenon he describes so beautifully, the presence of God in liturgical and sacramental presence. This article concludes by, perhaps, putting into parentheses some of Lacoste’s findings by pointing to Jacques Derrida’s take on Husserl’s Origin of Geometry, questioning once more the bounds and boundaries of Lacoste’s phenomenology of faith and opening avenues for further research.

HUSSERL AND DERRIDA ON THE PROCESS OF SENSE FORMATION—GAPS AND EXCESSES

This paper deals with the problem of the origin of sense and meaning. For Husserl, the determination of the ideal identity of something new can only take place retroactively in the totality of the preceding series by stepping back towards the original foundation of sense. In this regard, J. Derrida questions the ideality of the same as presence and the possibility of retrieving any arché of sense in his writings Speech and Phenomena and Edmund Husserl’s Origin of Geometry. Phenomenology is not oblivious to these difficulties, as results from a closer reading of the correlation between idealities and sensory experience: It will show that there are at least four interrelated gaps in the Husserlian phenomenology which testify the difficulty in grasping a retraceable arché of sense. First, the gap between the ideality of sense and its representation. Second, the gap between sensory and categorial intuitions, whereby both exceed one another. Third, the gap between the ideality of sense and the sense which exceeds our expectations. Fourth, the gap between the experience of the new that overcomes us and its apprehension by consciousness. Hence, there is a fundamental gap between the conceptual idealities and the essential indeterminacy of our phenomenological sensory experience, which is correlative to an excess of one in respect to the other. In this sense, Derrida’s concepts of « différance » and “invention” allow us to conceive of this self-givenness of sense as the expression of its constant self-renewal. This expression takes the form of a trace, which, neither present nor absent, suspends meaning and full presence, leaving the narrative “open” for the reinvention of sense.

The Origin of Geometry

The Origin of Geometry

Geschichte als Grenzproblem

This paper aims to bring a hidden “metaphysical” horizon in Husserl’s famous text “The Origin of Geometry” to light. This will be achieved in three steps: first, by analyzing the two essential manners in which the problem of history is thematized in this text, namely, both as a “basic problem” and as a “limit problem”; second, by highlighting the direct connection of this twofold problematization of history to Husserl’s own concept of metaphysics; and third, by exploring the relation between the “metaphysical” limit-problem of history implicit in the “Geometry” text and the explicit historical reflections in the Crisis itself. In so doing, Husserl’s concept of metaphysics, which remains largely unknown, and the thematization of history as a “limit problem” in phenomenology, which gives expression to the fundamental engagement of phenomenology in history, can be illuminated through each other.

Interpreting the concept of sedimentation in Husserl’s Origin of Geometry

In the influential text Origin of Geometry, Edmund Husserl argues that even the invariant meaning found in theoretical disciplines like geometry has a historical becoming: through gradual abstraction and stabilization, ending in a completely rational discipline. This is a process which Husserl proposes is due to language and other symbolic systems. In the absence of a system allowing for stable communication of meaning, geometry or any other tradition would constantly have to begin anew. At the same time Husserl also sees the historical process of meaning stabilization in linguistic form as detrimental. It allows for a reception of an established meaning, which simultaneously entails the forgetfulness of the experiential basis and intuitive knowledge that made ideality possible in the first place. Husserl calls this Janus-faced dialectical process between discovery and forgetfulness sedimentation. This paper analyzes this concept in Origin of Geometry and places it in the context of Husserl’s thought more generally. In contrast to Husserl’s negative view of the effects that sedimentation has for an authentic meaning, I discuss four interpretations of sedimentation that provide more constructive perspectives on the concept. These interpretations also differ considerably from one another, a fact which speaks both to the richness and the tensions in Origin of Geometry.

Interpreting the concept of sedimentation in Husserl’s Origin of Geometry

In the influential text Origin of Geometry, Edmund Husserl argues that even the invariant meaning found in theoretical disciplines like geometry has a historical becoming: through gradual abstraction and stabilization, ending in a completely rational discipline. This is a process which Husserl proposes is due to language and other symbolic systems. In the absence of a system allowing for stable communication of meaning, geometry or any other tradition would constantly have to begin anew. At the same time Husserl also sees the historical process of meaning stabilization in linguistic form as detrimental. It allows for a reception of an established meaning, which simultaneously entails the forgetfulness of the experiential basis and intuitive knowledge that made ideality possible in the first place. Husserl calls this Janus-faced dialectical process between discovery and forgetfulness sedimentation. This paper analyzes this concept in Origin of Geometry and places it in the context of Husserl’s thought more generally. In contrast to Husserl’s negative view of the effects that sedimentation has for an authentic meaning, I discuss four interpretations of sedimentation that provide more constructive perspectives on the concept. These interpretations also differ considerably from one another, a fact which speaks both to the richness and the tensions in Origin of Geometry.
 
 Author Biography
 Johan Blomberg has a PhD in General Linguistics from 2014, for the dissertation Motion in Language and Experience. He has since then worked in the Division for Cognitive Semiotics at Lund University, to which he continues to be affiliated. His main interests include motion semantics and the relations between language and thought, on which he has published extensively in journals like Cognitive Linguistics, Frontiers of Psychology and Language and Communication.

The origin of number and the origin of geometry: issues raised and conceptions assumed by Edmund Husserl

The objective of this article is to present the concept of origin, as presented in Husserl’s initial studies, and the same concept as it appears in his final work. The views he assumed in the different phases of his life are addressed: in Halle, when he follows Brentanian psychology to support the origin of number; in Göttingen, where he remains until 1916, when his thinking about reduction matures; and the final stage in Freiburg. The article presents ways through which he understood and explained the origin of number, considered in the first phase and his first studies as central for the clarification of the fundaments of mathematics. It further presents how he explained the origin of geometry, under the dimension of the life-world, as well as his conception of knowledge and reality, already understood in the 1930s and throughout the years nearing his death as life-world. Keywords: Phenomenology; Origin of number; Origin of geometry; Crisis of modern sciences; Historical a priori.

Mathematics, Science, And Cultural Change

According to Roger Bacon (1214-1294), “Mathematics is the gate and key of the sciences. Neglect of Mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or things of this world, and what is worse, men who are thus ignorant are unable to perceive their own ignorance and so do not seek a remedy”. Mathematics enables various branches of science draw implications of their observations and experimental findings. A computer is now an inevitable tool in science; its internal working rests on basic mathematics -arithematic to base2 Mathematics is at the heart of our important scientific theories; Newtonian mechanics, Electromagnetic theory of Maxwell, Einstein’s theory of relativity, quantum theory of Planck etc. Further, vast areas of mathematics grew out of efforts to find solutions to problems in science. We give two examples. The first one is about geometry, also sometimes referred to as the gift of the Nile. The story goes that in the 14th Century B.C., King Sesotris divided the land among Egyptians, each receiving a rectangle of same area, and was taxed accordingly. If anyone lost some of his land during the annual overflow of the Nile, he had to report the loss. An overseer would then be sent to measure the loss and make an abatement of tax. Some sources claim that this is the origin of geometry. The second example is more recent; it is about topology, whose origin is associated with a problem on a bridge. During the 18th century, in the German town of Konigsberg (now the Russian city of Kaliningrad), people enjoyed strolling along the banks of the Preger river, which meandered through the town, and was crossed by seven bridges which ran trom each bank of the river to two islands in the river, with a bridge joining the islands. One question asked is as follows: “How can you take a stroll so that you cross each of the seven bridges exactly once?”. Leonhard Eu!er ( 1707 -1783) converted this problem to one on vertices and networks in a diagram. He made general discoveries about networks, and found that the answer to the Konigsberg bridge problem, is in the negative, namely, that it is impossible to take a stroll so that each of the seven bridges is crossed only once! In the process of obtaining this result, he originated a new kind Of geometry, a geometry which does not depend on the size or shape of the figure, but about places, and how they are connected by arcs. Out of this grew the branch of mathematics called ‘Topology”.

The Generative Dimension of Translation

This article seeks to disclose the sense(s) of translation by attending to it in the phenomenological key with Edmund Husserl’s The Origin of Geometry. I suggest that the text could be approached through the prism of generative phenomenology, and I investigate it from two different perspectives: xenological and Derridean. Both perspectives are employed toward the same objective: to demonstrate the theoretical relevance of Husserl’s text to the understanding of a complex social phenomenon such as translation. The emphasis on sociality, historicity, and tradition, as these themes transpire in The Origin, brings about a new understanding of translation as a phenomenon of the liminal in-between, with its basic structure defined as the encounter with the alien. Derrida gives this emphasis much attention in his commentary on The Origin. In this commentary, he focuses on the constitution of a communal history, which is not possible without the iteration of the origin, the basic means of which is communication on the general level, while on the level of different socio-cultural worlds translation comes up as the most optimal mode for the encounter with the alien. The encounter takes place in the liminal in-between which points to repeatability as the original sense of translation inscribed in the acts of binding and approximation.

Pochwała donkiszoterii, czyli o tym, co strukturalizm wniósł i co wciąż może wnieść do historii

The paper discusses various possibilities of using structuralism in the current historical research. It seems, e.g., that this methodology or mode of reasoning could still be helpful to understand the origins of some historic phenomena. Despite the widely-held belief, the most important thinkers associated with structuralism, such as Jean-Pierre Vernant, Roland Barthes, or Pierre Vidal-Naquet, did not reduce their interests to the study of synchronic relations, but – as Foucault put it – attempted to fashion the instrument for a precise historical analysis. It was also a case of Michel Serres’ reflection on the origins of geometry. This example is closely examined in the paper to indicate some differences between both traditional and structuralist approaches to that question.

Zur geschichtlichen Wende der genetischen Phänomenologie. Eine Interpretation der Beilage III der Krisis

The paper addresses the methodological tensions between Husserl’s phenomenology and history by reinterpreting the Addendum III of the Krisis-work in view of genetic phenomenology. Thus, the paper starts out by retracing the traditional criticism against the unhistorical character of Husserl’s phenomenology as voiced by Heidegger, Adorno and others. Afterwards, it moves on to analyse the troubled relationship between static and genetic phenomenology, on the one hand, and between genetic phenomenology and empirical genesis, on the other hand. Finally, it arrives at a step by step methodological reconstruction of Husserl’s considerations on the “origin of geometry”, which are regarded to be an application of the methods of genetic phenomenology to the field of history.

Husserl and Jacob Klein

The article explores the relationship between the philosopher and historian of mathematics Jacob Klein’s account of the transformation of the concept of number coincident with the invention of algebra , together with Husserl’s early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein’s research is shown to present the historical context for Husserl’s twin failures in the Philosophy of Arithmetic: to provide a psychological foundation for the proper concept of number (Anzahl), and to show how this concept of number functions as the mathematical foundation of universal (symbolic) arithmetic. This context establishes that Husserl’s failures are ultimately rooted in the historical transformation of number documented in Klein’s research, from its premodern meaning as the unity of a multitude of determinate objects to its modern meaning as a symbolic representation with no immediate relation to a concrete multiplicity. The argument is advanced that one significant result of bringing together Klein’s and Husserl’s thought on these issues is the need to fine-tune Husserl’s project in The Crisis of European Sciences and Transcendental Phenomenology of de-sedimenting the mathematization of nature. Specifically, Klein’s research shows that “a ‘sedimented’ understanding of numbers” “is superposed upon the first stratum of ‘sedimented’ geometrical ‘evidences’” uncovered by Husserl’s fragmentary analyses of geometry in the Crisis. In addition then to the task of “the intentional-historical reactivation of the origin of geometry,” recognized by Husserl as intrinsic to the reactivation of the origin of mathematical physics, Klein discloses a second task, that of “the reactivation” of the “complicated network of sedimented significances” that “underlies the ‘arithmetical’ understanding of geometry.”

Husserl’s existentialism: ideality, traditions, and the historical apriori

Husserl’s concept of an “historical apriori” is marked by a tension: It simultaneously departs from, and develops his long-standing commitment to philosophy as transcendental phenomenology. This paper looks at some reasons for this tension in the context of Husserl’s attempt to determine philosophy as a “tradition” in The Origin of Geometry. Husserl is convinced that philosophy is a scientific tradition, and the historical apriori serves in the analysis of the conditions that define a distinctively scientific “handing down.” The key here is a kind of writing—mathematics—thanks to which the participants in the tradition can be certain that they are in essential connection with the “original” idealities of that tradition. But Husserl’s discovery about the individual philosopher’s relation to the philosophical tradition—namely, that it is always mediated by a kind of “poetic invention”—undermines his conviction that philosophy can be a scientific tradition. This is Husserl’s “existentialism.”

From Ideality to Historicity, What Happens?

1. Historicity and Origin of Geometry1.1 as Prototype of Pure HistoricityIn his lecture about Philosophy and Crisis of European Humanity (1935), also known as Vienna Lecture Edmund Husserl describes development of philosophy and science as an irreversible transformation of human civilization. This transformation was due to introduction of a sense of Along with which established a system of ideal objects and deductive reasoning, there arrived possibility of manipulating universal and infinite objects by finite means of human reason.2 Infinite horizons for universal validity in knowledge, ideas and ideal objects that are not subjected to transformation and becoming, that stand eternally and omnitemporally both outside historical becoming and as regulative horizons for historical progression, were made possible for first time. But introduction of infinity not only entailed a shift in science and philosophy. Human values and human art, ethical and political life, also begin to be shaped and guided by infinite horizons. Thus Greek miracle involved a transformation of way in which human beings produce their own civilization. Human culture became a culture of trans-historical or trans-generational objects, and began producing humanity itself according to such infinite objects, indeed as infinite objects.3The introduction of infinity made possible concept of historicity itself. As Derrida puts it in his commentary on Husserl's posthumous essay concerning of published as an appendix to Crisis, the openness of infinite is only . . . openness of history itself, in utmost depths and purity of its essence (Derrida 1962: 143; 1989: 131). Historicity is not senseless mortality or senseless finitude; it is not opposed to infinite realization of human values; it is not realm of delusion and mere appearance against background of absolute being and truth. Historicity is realm in which infinity becomes actual, concrete and true for first time. Historicity is infinite horizon for infinite realization of humanity's true knowledge and absolute value.Husserl's phenomenological investigations into concept of history take as a prototype of science and philosophy in general, and therefore, ultimately, as a prototype of transformation giving rise to new idea of historicity defining European humanity. For that reason, question about origin of geometry, as surprising as it may seem at first sight, must be understood as a question concerning concept of historicity itself. The origin of geometry is origin of historicity, hence origin of any possible understanding of historical origins in general. Geometry will therefore be treated as being much more than, or something totally different from, a simple mathematical discipline or theory. It will not be taken, for instance, as general concept of normal way in which science is practiced, that is, as a 'paradigm' in Thomas Kuhn's sense (Kuhn 1996). It is not even taken as one horizon or one infinite task, among others. Geometry is analysed as infinite horizonality, as ideality as such, as possibility itself of an idea of infinite progression and infinite task, in sum as methodological basis upon which any historicity of truth is thinkable and therefore any concept of science and of history of science is possible, including idea of science as a cultural practice historically articulated through shift of paradigms that are incommensurable to each other.4 is considered a prototype of ultimate horizon for any trans-generational practice of meaning and for any formation of world-views or world-conceptions capable of providing representations of historical progression (Heidegger 2002; Derrida 2013: 196-202).The problem of origin of geometry is crucial for understanding formation and development of Derrida's early conception of historicity (1989). …

The Creative Expression of the Sense of Experience

En este trabajo estudiamos la prioridad que Husserl otorga en sus primeras obras al sentido frente a la expresión, considerada improductiva. En sus últimos escritos, en cambio, la génesis del sentido es correlativa a la de la expresión y la comunicación intersubjetivas. De la mano de Merleau-Ponty, nos centraremos concretamente en el giro que supone El Origen de la Geometría de Husserl. No obstante, contextualizaremos en cada una de sus obras su reiterada interpretación de la intención husserliana de llevar la experiencia a la expresión de su sentido con el propósito de demostrar que nunca dejó de pensar con Husserl, tematizando incluso sus impensados. La concepción merleau-pontiana de la reducción incompleta, la intencionalidad operante y la fenomenología como interrogación así lo pondrá de manifiesto. Con ella, se abre, yendo más allá de Husserl, la posibilidad de una expresión creadora.

Husserl’s Archaeology of Exact Science

Why is nature amenable to mathematical description? This question has received attention in the philosophy of science but rarely from a phenomenological perspective. Nevertheless Husserl’s late essay “The Origin of Geometry,” which has received some critical scholarly attention in recent years, contains the beginning of a striking answer. This answer proceeds from Husserl’s main claim in that essay, which he also makes in the Crisis of the European Sciences, that the original meaning of science has been covered over or “sedimented” by concepts that obscure the true intentional core of scientific meaning. In the first three sections of this paper I develop Husserl’s central insights about mathematics in light of two contemporary critiques of his project of “reactivation” of the original, sedimented meaning of science. In the latter two sections, I argue that accepting Husserl’s account of the original meaning and development of science offers a promising explanation of why nature is amenable to mathematics. This explanation hinges on a conception of the objects and methods of mathematics and the mathematized physical sciences as accomplishments, that is, as constituted contents of consciousness.

Factorizations of Cycles and Multi-Noded Rooted Trees

In this paper, we study factorizations of cycles. The main result is that under certain condition, the number of ways to factor a d-cycle into a product of cycles of prescribed lengths is d r-2. To prove our result, we first define a new class of combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the cardinality of this new class which with proper parameters is exactly d r-2. The main part of this paper is the proof that there is a bijection from factorizations of a d-cycle to multi-noded rooted trees via factorization graphs. This implies the desired formula. The factorization problem we consider has its origin in geometry, and is related to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the standard translation of Hurwitz numbers into group theory, our main result is equivalent to the following: when the genus is 0 and one of the ramification indices is d, the degree of the covers, the pure-cycle Hurwitz number is d r-3, where r is the number of branch points.

Translating Euclid: Designing a Human-Centered Mathematics

Translating Euclid reports on an effort to transform geometry for students from a stylus-and-clay-tablet corpus of historical theorems to a stimulating computer-supported collaborative-learning inquiry experience. The origin of geometry was a turning point in the pre-history of informatics, literacy, and rational thought. Yet, this triumph of human intellect became ossified through historic layers of systematization, beginning with Euclids organization of the Elements of geometry. Often taught by memorization of procedures, theorems, and proofs, geometry in schooling rarely conveys its underlying intellectual excitement. The recent development of dynamic-geometry software offers an opportunity to translate the study of geometry into a contemporary vernacular. However, this involves transformations along multiple dimensions of the conceptual and practical context of learning. Translating Euclid steps through the multiple challenges involved in redesigning geometry education to take advantage of computer support. Networked computers portend an interactive approach to exploring dynamic geometry as well as broadened prospects for collaboration. The proposed conception of geometry emphasizes the central role of the construction of dependencies as a design activity, integrating human creation and mathematical discovery to form a human-centered approach to mathematics. This book chronicles an iterative effort to adapt technology, theory, pedagogy and practice to support this vision of collaborative dynamic geometry and to evolve the approach through on-going cycles of trial with students and refinement of resources. It thereby provides a case study of a design-based research effort in computer-supported collaborative learning from a human-centered informatics perspective. Table of Contents: Acknowledgments / Background / Vision: The Cognitive Potential of Collaborative Dynamic Geometry / History: The Origin of Geometry / Philosophy: The Obfuscation of Geometry / Mathematics: Demythologizing Geometry / Technology: Deconstructing Geometry / Collaboration: Group Geometry / Research: Analyzing Geometry / Theory: Resources for Geometry / Pedagogy: Designing Geometry / Practice: Doing Geometry / Design-Based Research: Human-Centered Geometry / Author Indes / Bibliography / Author's Biography

Reflections on the Future University: Introduction

Click to increase image sizeClick to decrease image size Notes Notes 1. “The Future University,” Centre for Research in the Arts, Social Sciences and Humanities (CRASSH), Cambridge University, UK, 30 June–2 July 2011; (www.crassh.cam.ac.uk/events/1321). 2. Bernard Stiegler, “Our Ailing Educational Institutions,” Culture Machine 5 (2003); (http://www.culturemachine.net/index.php/cm/article/view/258/243); and Stiegler, For a New Critique of Political Economy (Cambridge: Polity Press, 2010). See also Patrick Crogan, “Knowledge, Care, and Trans-Individuation: An Interview with Bernard Stiegler,” Cultural Politics: An International Journal 6.2(July 2010): 157–70. 3. An “infinite we of geometers” alludes at once to the sign over Plato's Academy (No one ignorant of geometry may enter) and to Husserl's Origins of Geometry, thus to the notion of an academic community and the conditions of rational thought. 4. Stiegler developed Gilbert Simondon's concept of individuation with reference to the works of Plato, Husserl, and Derrida, amongst others, in his keynote address “Pharmacology of the University: Towards a Spiritual Peace Treaty” (altered from the title in the printed programme). Citations from this and other papers presented at the conference are from my own notes. 5. The definitions are taken from the glossary of terms of the Ars Industrialis website (http://www.arsindustrialis.org/glossary), “an international association for an industrial politics of the technologies of spirit” founded by Stiegler to disseminate his ideas and those of the group's associates, and to develop dialogues and actions to counter the global process of disindividuation. Translations from this website are my own (accessed 8.10.2012). 6. Alan Liu, “Where Is Cultural Criticism in the Digital Humanities?” in “The Future University,” CRASSH. 7. Cathy Davidson, “Digital Literacies and the Future of the Humanities,” in “The Future University,” CRASSH. 8. Danielle Allen, “Strong Ties: Why Democracies Still Need Universities in the Digital Age,” in “The Future University,” CRASSH. 9. Don Randel, President of the Mellon Foundation, in the panel discussion on “The Fate of the Humanities,” in “The Future University,” CRASSH. 10. Chris Newfield, “Refunding the Humanities: The Case for Succession,” in “The Future University,” CRASSH. 11. Christopher Newfield and the Edu-factory Collective, “The Corporate University and the Financial Crisis: What Is Going On?” in Toward a Global Autonomous University: Cognitive Labor, The Production of Knowledge, and Exodus from the Education Factory (New York: Autonomedia, 2009), 183.