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A History of Analysis
Analysis as an independent subject was created as part of the scientific revolution in the seventeenth century. Kepler, Galileo, Descartes, Fermat, Huygens, Newton, and Leibniz, to name but a few, contributed to its genesis. Since the end of the seventeenth century, the historical progress of mathematical analysis has displayed unique vitality and momentum. No other mathematical field has so profoundly influenced the development of modern scientific thinking. Describing this multidimensional historical development requires an in-depth discussion which includes a reconstruction of general trends and an examination of the specific problems. This volume is designed as a collective work of autho...
The Unattainable Attempt to Avoid the Casus Irreducibilis for Cubic Equations
Sara Confalonieri presents an overview of Cardano’s mathematical treatises and, in particular, discusses the writings that deal with cubic equations. The author gives an insight into the latest of Cardano’s algebraic works, the De Regula Aliza (1570), which displays the attempts to overcome the difficulties entailed by the casus irreducibilis. Notably some of Cardano's strategies in this treatise are thoroughly analyzed. Far from offering an ultimate account of De Regula Aliza, by one of the most outstanding scholars of the 16th century, the present work is a first step towards a better understanding.
Systems, Machines, and Problem-Solving
The notion of problem solving has become central to science education and the cognitive sciences, but it is still peripheral to many philosophies of knowledge and science. In fact, the term only became popular in the course of the twentieth century, as humanity’s ability to solve theoretical and practical problems grew at a seemingly exponential rate. This book questions both the nature of problem solving and its effectiveness in transforming our human practices. We argue that this is linked to the idea that some of our enquiries can be summarized in systematic procedures. Examples are the proof of a theorem within an axiomatic theory, a production line within an industrial factory, or an administrative procedure within a bureaucratic system. Although such a form has been common in mathematics since antiquity, it was only in modern times that the possibility of being systematic in the natural sciences and technical disciplines was discovered. The emergence of the modern concepts of system and machine was key to this expansion and to the scientific, industrial and digital revolutions. Problem solving thus appears as the fundamental form of the modern concept of knowledge.
Contradictions, from Consistency to Inconsistency
This volume investigates what is beyond the Principle of Non-Contradiction. It features 14 papers on the foundations of reasoning, including logical systems and philosophical considerations. Coverage brings together a cluster of issues centered upon the variety of meanings of consistency, contradiction, and related notions. Most of the papers, but not all, are developed around the subtle distinctions between consistency and non-contradiction, as well as among contradiction, inconsistency, and triviality, and concern one of the above mentioned threads of the broadly understood non-contradiction principle and the related principle of explosion. Some others take a perspective that is not too far away from such themes, but with the freedom to tread new paths. Readers should understand the title of this book in a broad way,because it is not so obvious to deal with notions like contradictions, consistency, inconsistency, and triviality. The papers collected here present groundbreaking ideas related to consistency and inconsistency.
A Historical Dictionary of Mathematical Terms
This unique reference work systematically presents and studies mathematical terms, sourcing literature from antiquity until the 20th century. The book provides detailed and reliable information on the origins and developments of more than 1,000 mathematical terms from all branches of mathematics. The dictionary is based on extensive research of the primary literature, cites from original sources, provides historical illustrations, discusses synonyms and competing terms, and quotes critical or approving comments on newly created terms. Each entry is self-contained and includes a bibliography. All major mathematical fields are represented, ranging from geometry and arithmetic to, for example, calculus, topology, and category theory. This is the first of two volumes, covering terms that begin with the letters A through I.
Objectivity, Realism, and Proof
This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematic...
Peirce on Perception and Reasoning
In this book, scholars examine the nature and significance of Peirce’s work on perception, iconicity, and diagrammatic thinking. Abjuring any strict dichotomy between presentational and representational mental activity, Peirce’s theories transform the Aristotelian, Humean, and Kantian paradigms that continue to hold sway today and forge a new path for understanding the centrality of visual thinking in science, education, art, and communication. This book is a key resource for scholars interested in Perice’s philosophy and its relation to contemporary issues in mathematics, philosophy of mind, philosophy of perception, semiotics, logic, visual thinking, and cognitive science.
Salomon Maimon’s Theory of Invention
How can we invent new certain knowledge in a methodical manner? This question stands at the heart of Salomon Maimon's theory of invention. Chikurel argues that Maimon's contribution to the ars inveniendi tradition lies in the methods of invention which he prescribes for mathematics. Influenced by Proclus' commentary on Elements, these methods are applied on examples taken from Euclid's Elements and Data. Centering around methodical invention and scientific genius, Maimon's philosophy is unique in an era glorifying the artistic genius, known as Geniezeit. Invention, primarily defined as constructing syllogisms, has implications on the notion of being given in intuition as well as in symbolic ...
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