Novalis. Mathematical Fragments (1798-1800) (original) (raw)
The 'Mathematical' Wissenschaftslehre: On a Late Fichtean Reflection of Novalis
Oxford University Press, 2014
Chapter "The 'Mathematical' Wissenschaftslehre: On a Late Fichtean Reflection of Novalis" in the volume: The Relevance of Romanticism: Essays on German Romantic Philosophy, ed. Dalia Nassar (Oxford: Oxford University Press), pp. 258-272, March 2014: https://academic.oup.com/book/25939 This chapter argues that in his late writings Novalis (1772–1801) was one of the first thinkers to positively grasp the underlying mathematicity of Fichte’s Wissenschaftslehre. In a neo-Platonistic sense, both Novalis and Fichte acknowledged that mathematical and geometrical methods should form an ideal for all scientific philosophy, and that a proper philosophy of mathematics must take into account the intellectual activity of the mathematician. These elements of Fichte’s system only became more widely recognized in the twentieth century by philosophers of mathematics such as Hermann Weyl, Andreas Speiser, and Jules Vuillemin. This may be considered as a belated but independent confirmation of Novalis’s original insights on the relationship between mathematics and philosophy in Fichte’s Wissenschaftslehre. Keywords: Fichte, Novalis, mathematics, geometry, philosophical romanticism, German idealism, Wissenschaftslehre
Recent scholarship has helped to demythologise the life and work of Georg Philipp Friedrich von Hardenberg who, as the poet “Novalis”, had come to instantiate the nineteenth-century’s stereotype of the romantic poet. Among Hardenberg’s interests that seem to sit uneasily with this literary persona were his interests in science and mathematics, and especially in the idea, traceable back to Leibniz, of a mathematically based computational approach to language. Hardenberg’s approach to language, and his attempts to bring mathematics to bear on poetry, is examined in relation to debates that developed late in the eighteenth century over the relation of language to thought—debates which share many features with contemporary ones in this area.
"Le complément supérieur": On the Poetics of Mathematics
A. M. Uluda˘g, A. Zeytin (eds.), Essays on Geometry, nger Nature Switzerland AG 2025, 2025
This chapter considers mathematics, with a special emphasis on geometry, as combining mathematical and artistic thinking. By artistic thinking I do not refer to aesthetic aspects of mathematics or mathematical aspects of art or aesthetics, which subjects are only marginally addressed in the article. Instead, I focus on creative aspects of mathematics as the invention of new concepts, theories, or fields. As my subtitle indicates, this argument, beginning with the term "poetics," follows that of Aristotle's Poetics [Peri poietikẽs], a treatise on ancient Greek poetry, the title of which is derived from the ancient Greek word poeien meaning "making," putting something together. Aristotle's Poetics is about how literature is made or composed. Aristotle did not apply the term poetics to mathematics and did not consider mathematics in this way, focusing instead, in his other works, on logical aspects of mathematics. By contrast, I argue, under the heading of the composition principle, that, as a creative endeavor, mathematics is primarily defined by its compositional nature, rather than by its logical or calculational aspects, essential as the latter are. Of course, while compositional, mathematics is not the same as literature and art. In particular, the poetics of mathematics is the poetics of concepts, which play a more limited role in literature and art, and ally mathematics or mathematical sciences, such as physics, more with philosophy. Two other principles define mathematics, as well as science, in the present view: the continuity principle (found in literature and art as well) and the unambiguity principle (not necessary in literature and art). The chapter introduces yet another principle in considering the nature of reality, the "reality without realism" (RWR) principle, which in the case of mathematics, where the primary reality considered in mental, becomes the "ideality without idealism" (IWI) principle.
Messengers of Mathematics: European Mathematical Journals (1800–1946). Edited by Elena Ausejo and Mariano Hormigón. Madrid (Siglo XXI de España Editores, S. A.). 1993, 1998
La Ciencia es una creación humana de tan variada multiplicidad que ocupa cada vez un mayor y más privilegiado espacio en el pensamiento y en la vida actuales. El protagonismo de todas y cada una de las disciplinas que forman el tronco del pensamiento científico es producto de un largo proceso evolutivo que surge de la mera curiosidad y que se ha ido desarrollando por la vía de la necesidad. La ciencia es una actividad viva porque sus teorías nacen, crecen, se reproducen y mueren dando lugar a cuerpos de doctrina más ambiciosos y veraces. Por eso la ciencia más que ninguna otra actividad intelectual humana es una inevitable confrontación de pasado y futuro. Elena Ausejo y Mariano Hormigón recogen en este libro los trabajos que se presentaron en septiembre de 1991 en el Simposio Internacional sobre Periodismo Matemático. Esta reunión conmemoraba la aparición de El Progreso Matemático, primera revista dedicada a las matemáticas que se publicó en España, y se celebró como homenaje al que fuera su director, Zoel García Galdeano (1846-1924), el matemático español más importante de la época contemporánea. En Messengers of Mathematics aparecen por tanto estudios sobre varias de las revistas matemáticas más destacadas de los dos últimos siglos, a cargo de firmas bien conocidas y prestigiosas en el mundo de la historia de las matemáticas como las de Serguei Demidov, Jean Dhombres, Ivor Grattan-Guinnes, Lubos Novy y otros. Messengers of Mathematics es la primera aproximación seria y rigurosa al análisis de algunas publicaciones periódicas que como los Anales de Mathématiques Pures et Appliquées de Gergonne, los Rendiconti del Circo lo Matematico di Palermo o los Matematischeski Sbornik de Moscú han jugado tan importante papel en el desarrollo de las matemáticas contemporáneas.
Newsletter of the HPM Group, No. 64, pp. 17-19, 2007
In my book Filosofia e matematica [Philosophy and Mathematics, Laterza, Rome 2003], on pp. 38-43, I discussed the reason of the change from the concrete view of the axiomatic method to the abstract one at the end of the nineteenth century-a change beautifully illustrated by the Frege-Hilbert controversy. I summarize the main points here, omitting textual evidence and references to save space. I am grateful to Andrea Reichenberger for valuable comments on an earlier version of this text.
A protagonist of the 18th-century mathematics: Maria Gaetana Agnesi
Lettera Matematica, 2018
We revisit the life and work of Maria Gaetana Agnesi, Milanese mathematician, whose 300th birthday is being celebrated this year. In particular, we describe the main features of her books Propositiones philosophicae and Instituzioni analitiche ad uso della gioventù italiana. A third appendix is devoted to an algebraic curve that has taken its name from Agnesi: the versiera, or witch of Agnesi.
Lakatos' Philosophy of Mathematics: A Historical Approach
Historia Mathematica, 1995
All books, monographs, journal articles, and other publications (including films and other multisensory materials) relating to the history of mathematics are abstracted in the Abstracts Department. The Reviews Department prints extended reviews of selected publications.
The Irony of Romantic Mathematics: Bridging the Historiographies of Literature and Mathematics
Configurations, 2016
This essay juxtaposes a particular a set of novel mathematical ideas from the early nineteenth century with a synchronous development in literary criticism. Important mathematical discoveries of the 1820s, such as non-Euclidean geometries, new impossibility results, and new proof ideals, exhibit structural similarities with notions of Romantic irony expressed by literary critic and philosopher Friedrich Schlegel (1772-1829). The essay shows how both fields-mathematics and literary criticism-were at the time attempting to come to terms with the Kantian divide between the Ding an sich and Ding für uns, and thereby both fields reacted to profound shifts in perspective. By analyzing Schlegel's notion of Romantic irony in relation to these key mathematical examples the essay presents new insights into the historiographical fields regarding literature and mathematics. This contribution to historiography is twofold: the essay sheds new light on a set of important mathematical developments, while at the same time bringing the historiography of mathematics and literature to bear on each other on a deeper, structural level. Like any other creative human activity mathematics is practiced by individuals embedded in rich cultural and cognitive contexts open to historical inquiry.
Massimo Mazzotti, Reactionary Mathematics: A Genealogy of Purity
Review: Reactionary Mathematics by Massimo Mazzotti, 2023
How did mathematics come to be seen as a value-neutral yet essential feature of modern political life? Massimo Mazzotti's Reactionary Mathematics: A Genealogy of Purity traces the emergence of the 'distinctly modernist perception that mathematical knowledgeand the technologies it legitimatesare neutral tools that can be used unthinkingly in the manipulation of both natural and social realities' (p. 2). Mazzotti uses close historical analysis of a mathematical controversy to investigate purity as an epistemic virtue in mathematics in an episode of resistance to modern (that is, French) mathematics and governance in the Kingdom of Naples around 1800. At stake in the controversy, Mazzotti argues, was more than just different mathematical problem-solving methods that divided two mathematical cultures but competing (and incompatible) images of mathematics, politics, history, modernity and reason. Reactionary Mathematics contributes to the ongoing project to radically historize mathematics, particularly the rise of rigorous analysis. Mazzotti also contributes to Neapolitan historiography by making explicit the dynamic intimacy of mathematics and politics in this period. Rejecting the traditional separation between mathematics and culture, Mazzotti goes further still; in Reactionary Mathematics, mathematics is social order, and the 'politics of mathematical modernity' (p. 238) that Mazzotti explores are anything but neutral. Reactionary Mathematics is organized as 'a scaling exercise articulated in two movements' (p. 11). The first movement (Chapters 1-4) zooms in on the local Neapolitan context until we arrive at the heart of the controversy in the 'Intermezzo' chapter separating the movements. Chapter 1 focuses on the development of 'analytic reason' through the reconstruction of Enlightenment ideas about analysis and sentimentalism and their reception in the Neapolitan Enlightenment. In Naples, philosophical and reformist discourses mixed in planning the country's future, and analysis became a critical and soon extractive tool. Chapter 2 traces how analysis in the hands of Neapolitan Jacobins became revolutionary, as politics and mathematics enmeshed in their attempts to revolutionize Neapolitan culture and society, culminating in the short-lived Neapolitan Republic. Mazzotti argues that 'Jacobin science' was not a contradiction in terms but rather an essential part of their project in which analysis was a tool of mathematical and political liberation. Counterrevolutionary and anti-French sentiments ended the republic, the lives of many of the mathematician-reformers involved and the local mathematical culture of analysis. Chapter 3 details how analysis became a tool for control rather than liberation after the French (and analysis) returned to Naples in the Napoleonic age and Restoration. In this new analytic era, proponents insisted upon a clear distinction between analytic
Michel Serres and the Crises of the Contemporary, edited by Rick Dolphijn, Bloomsbury Academic., 2019
If we awaited perfection, we would never move forward. It is everywhere like here: supposing the Greeks had waited for the complete demonstration of their axioms, for their reduction to identical, geometry would still to be done. Mathematics presupposes axiomatics, but not perfection; we can develop the consequences of the former, downstream [en aval], for example by inventing the differential calculus with a lot of pragmatism and little rigour, while simultaneously, going back upstream [à l'amont] of the axioms and definitions in order to logicize them. So is the method of Establishments: once this is agreed upon, this is beyond of dispute. Likewise, without awaiting the perfection of philosophy -in this regard "we are in a certain infancy of the world" [nous sommes dans une certaine enfance du Monde], we are at an antepythagorician stage -we have to work on its elements and to build a first alphabet of human thoughts, but we must also go to the other extremity of the positive and order the proliferation of written or spoken tongues: study Chinese, hieroglyphe, cuneiform, the "Indo-European" languages, demonstrate their harmony. Advancing one enterprise cannot not have effect on the process of the other. By multiplying "samples" set beyond dispute, by pluralising the results of this molecular and regional method, we can hope, bit by bit, to cover unknown territories. Hence achievement [l'achèvement] and perfection are in the end understood as goals, as horizon, not as prior conditions. 1 3 Leibniz, Serres argued, devised a double method of universality: the first one starts from an elemental consistency (the monad, the singular), on which he reads the universal law, while the second starts from the mathesis (the monadology, the system) or abstract generality, from which he proposes to derive the particular. In Leibniz's system, these universalities form a cycle, meaning that these two methods are fundamentally co-conditional and cannot be dissociated. 4 Marcel Hénaff, 'Of Stones, Angels and Humans' in Niran Abbas, Mapping Michel Serres (Ann Arbor: University of Michigan Press, 2005), 183 5 These texts are: 'Ce que Thalès a vu au pied des pyramides', first published Hermès II, L'Interférence (1972) and translated into English as 'Mathematics and Philosophy: What Thalès saw... ' in Hermès: Literature, Science, Philosophy (1982), 84-97 ; and 'Origine de la géométrie' 3, 4 and 5 in Hermès V, Le passage du Nord-Ouest. Only 'Origine de la géométrie 5' has been translated into English as 'The
The Richness of the History of Mathematics
Archimedes Series 66, 2023
This colective book, edited by Karine Chemla, J Ferreirós, Lizhen Ji, Erhard Scholz & Chang Wang, is a unique introduction to historiographical questions concerning the history of mathematics, with essays by many leading scholars, aimed at guiding newcomers to the field. It provides multiple perspectives on mathematics, its role in culture, development, connections with other sciences, with philosophy, etc.