Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide (...) a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth. (shrink)

The novel use of symbolism in early modern mathematics poses both philosophical and historical questions. How can we trace its development and transmission through manuscript sources? Is it intrinsically related to the emergence of symbolic algebra? How does symbolism relate to the use of diagrams? What are the consequences of symbolic reasoning on our understanding of nature? Can a symbolic language enable new forms of reasoning? Does a universal symbolic language exist which enable us to express all knowledge? This book (...) brings together a collection of papers that address all these and related questions which were initially posed at a conference held in Ghent (Belgium) in August 2009. Scholars working on philosophy of science, history of philosophy and history of mathematics provide an insight into the role and function of symbolic representations in the development of early modern mathematics. The papers cover the period from early abbaco arithmetic and algebra (14h century) up to Leibniz (early 18th century). (shrink)

By the end of the twelfth century in the south of Europe, new methods of calculating with Hindu-Arabic numerals developed. This tradition of sub-scientific mathematical practices is known as the abbaco period and flourished during 1280–1500. This paper investigates the methods of justification for the new calculating procedures and algorithms. It addresses in particular graphical schemes for the justification of operations on fractions and the multiplication of binomial structures. It is argued that these schemes provided the validation of mathematical practices (...) necessary for the development towards symbolic reasoning. It is shown how justification schemes compensated for the lack of symbolism in abbaco treatises and at the same time facilitated a process of abstraction. (shrink)

Promises and disappointments of artificial intelligence for scientific discovery Recent successes within Artificial Intelligence with deep learning techniques in board games gave rise to the ambition to apply these learning methods to scientific discovery. This model for discovering new scientific laws is based on data-driven generalization in large databases with observational data using neural networks. In this study we want to review and critical assess an earlier research programme by the name of BACON. Though BACON was based on different AI (...) technology, we can learn from its limitations and thus adjust our expectations. The BACON program had two ambitions, one descriptive and one normative. On the one hand, it wanted to provide an explanation how philosophers of nature in history arrived at general laws from observational data and on the other hand it also aimed to reconstruct historical discoveries and realize new ones. We will assess the claims of the BACON program by means of two historical cases studies: the discovery of the sine law of refraction in optics and the discovery of Kepler’s third law of planetary motion. I will demonstrate that, despite the formulated claims, BACON did not use the same historical data available to its discoverers and, more importantly, that the model of inductive generalization of observational data does not correspond with the historical methods. Finally I will question the value of data-driven induction for scientific discovery in general. (shrink)

This paper critically assesses the claim by Gavin Menzies that Regiomontanus knew about the Chinese Remainder Theorem (CRT) through the Shù shū Jiǔ zhāng (SSJZ) written in 1247. Menzies uses this among many others arguments for his controversial theory that a large fleet of Chinese vessels visited Italy in the first half of the 15th century. We first refute that Regiomontanus used the method from the SSJZ. CRT problems appear in earlier European arithmetic and can be solved by the method (...) of the Sun Zi, as did Fibonacci. Secondly, we pro-vide evidence that remainder problems were treated within the European abbaco tradition independently of the CRT method. Finally, we discuss the role of recreational mathematics for the oral dissemination of sub-scientific knowledge. (shrink)

This introductory paper provides an overview of four contributions on the epistemological functions of mathematical symbolism as it emerged in Arabic and European treatises on algebra. The evolution towards symbolic algebra was a long and difficult process in which many obstacles had to be overcome. Three of these obstacles, related to the circulation and adoption of symbolism, are highlighted in this special volume: 1) the transition of material practices of algebraic calculation to discursive practices and text production, 2) the transition (...) from manuscript production to printed works involving material limitations of typefaces, and 3) the transition of algebraic symbolism from a system of notation and representation to a tool of mathematical analysis. This paper will conclude with the observation that the whole development towards symbolization can be considered an obstacle by itself in the sense of ‘epistemic obstacles’ as used by Brousseau. (shrink)

We introduce the question whether there are specific kinds of writing modalities and practices that facilitated the development of modern science and mathematics. We point out the importance and uniqueness of symbolic writing, which allowed early modern thinkers to formulate a new kind of questions about mathematical structure, rather than to merely exploit this structure for solving particular problems. In a very similar vein, the novel focus on abstract structural relations allowed for creative conceptual extensions in natural philosophy during the (...) scientific revolution. These preliminary reflections are meant to set the stage for the following contributions in this volume. (shrink)

The early calculus is a popular example of an inconsistent but fruitful scientific theory. This paper is concerned with the formalisation of reasoning processes based on this inconsistent theory. First it is shown how a formal reconstruction in terms of a sub-classical negation leads to triviality. This is followed by the evaluation of the chunk and permeate mechanism proposed by Brown and Priest in, 379–388, 2004) to obtain a non-trivial formalisation of the early infinitesimal calculus. Different shortcomings of this application (...) of C&P as an explication of inconsistency tolerant reasoning are pointed out, both conceptual and technical. To remedy these shortcomings, an adaptive logic is proposed that allows for conditional permeations of formulas under the assumption of consistency preservation. First the adaptive logic is defined and explained and thereafter it is demonstrated how this adaptive logic remedies the defects C&P suffered from. (shrink)

Explores the theories of democratic individualism articulated in the works of the American transcendentalist writer Ralph Waldo Emerson, pragmatic philosophers William James and John Dewey, and African-American novelist and essayist Ralph Ellison.

Wirth beschreibt in diesem Buch Volkstum und Weltmacht in der Geschichte, im Original 1904 erschienen.

An account of the conflict between our positive and negative emotional relationships to the Earth and how they will be resolved for the Symbiocene, the next period in the history of the Earth.

Die neu gegründete Reihe Paracelsus Studien bietet ein wissenschaftliches Forum für interdisziplinäre Forschungen, die sich kultur- und wissenschaftshistorisch Modellen eines ganzheitlichen Weltbildes widmen und den Dialog zwischen den Disziplinen anregen. Der frühneuzeitliche Alchemist, Mystiker, Theologe und Philosoph Paracelsus (1493-1541) dient hierfür als Leitfigur und Vorbild. Als eröffnender Sammelband fokussiert Mystik und Natur Epochen und Persönlichkeiten der Geistesgeschichte von der Spätantike bis in die Gegenwart, die sich der Integration von Spiritualität und Wissenschaft widmeten: Hildegard von Bingen, Jakob Böhme, Emanuel Swedenborg, Josef (...) Goerres und Johann Wolfgang von Goethe stehen ebenso im Mittelpunkt der Untersuchungen wie gegenwärtige ganzheitliche Ansätze und ihre historischen Bezüge. (shrink)

This is an anthology of literary, religious, and philosophical texts from the entire Middle Ages and the early modern age that address already quite explicitly religious toleration and even tolerance.

The notions of other peoples, cultures, and natural conditions have always been determined by the epistemology of imagination and fantasy, providing much freedom and creativity, and yet have also created much fear, anxiety, and horror. In this regard, the pre-modern world demonstrates striking parallels with our own insofar as the projections of alterity might be different by degrees, but they are fundamentally the same by content. Dreams, illusions, projections, concepts, hopes, utopias/dystopias, desires, and emotional attachments are as specific and impactful (...) as the physical environment. This volume thus sheds important light on the various lenses used by people in the Middle Ages and the early modern age as to how they came to terms with their perceptions, images, and notions. Previous scholarship focused heavily on the history of mentality and history of emotions, whereas here the history of pre-modern imagination, and fantasy assumes center position. Imaginary things are taken seriously because medieval and early modern writers and artists clearly reveal their great significance in their works and their daily lives. This approach facilitates a new deep-structure analysis of pre-modern culture. (shrink)