Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method
Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method

Author: Carlo Cellucci

Publisher: Springer Science & Business Media

Published: 2013-10-09

Total Pages: 391

ISBN-13: 9400760914

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This volume examines the limitations of mathematical logic and proposes a new approach to logic intended to overcome them. To this end, the book compares mathematical logic with earlier views of logic, both in the ancient and in the modern age, including those of Plato, Aristotle, Bacon, Descartes, Leibniz, and Kant. From the comparison it is apparent that a basic limitation of mathematical logic is that it narrows down the scope of logic confining it to the study of deduction, without providing tools for discovering anything new. As a result, mathematical logic has had little impact on scientific practice. Therefore, this volume proposes a view of logic according to which logic is intended, first of all, to provide rules of discovery, that is, non-deductive rules for finding hypotheses to solve problems. This is essential if logic is to play any relevant role in mathematics, science and even philosophy. To comply with this view of logic, this volume formulates several rules of discovery, such as induction, analogy, generalization, specialization, metaphor, metonymy, definition, and diagrams. A logic based on such rules is basically a logic of discovery, and involves a new view of the relation of logic to evolution, language, reason, method and knowledge, particularly mathematical knowledge. It also involves a new view of the relation of philosophy to knowledge. This book puts forward such new views, trying to open again many doors that the founding fathers of mathematical logic had closed historically. trigger

Rethinking Knowledge
Rethinking Knowledge

Author: Carlo Cellucci

Publisher: Springer

Published: 2017-03-29

Total Pages: 432

ISBN-13: 3319532375

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This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality.

The Making of Mathematics
The Making of Mathematics

Author: Carlo Cellucci

Publisher: Springer Nature

Published: 2022-03-07

Total Pages: 457

ISBN-13: 3030897311

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This book offers an alternative to current philosophy of mathematics: heuristic philosophy of mathematics. In accordance with the heuristic approach, the philosophy of mathematics must concern itself with the making of mathematics and in particular with mathematical discovery. In the past century, mainstream philosophy of mathematics has claimed that the philosophy of mathematics cannot concern itself with the making of mathematics but only with finished mathematics, namely mathematics as presented in published works. On this basis, mainstream philosophy of mathematics has maintained that mathematics is theorem proving by the axiomatic method. This view has turned out to be untenable because of Gödel’s incompleteness theorems, which have shown that the view that mathematics is theorem proving by the axiomatic method does not account for a large number of basic features of mathematics. By using the heuristic approach, this book argues that mathematics is not theorem proving by the axiomatic method, but is rather problem solving by the analytic method. The author argues that this view can account for the main items of the mathematical process, those being: mathematical objects, demonstrations, definitions, diagrams, notations, explanations, applicability, beauty, and the role of mathematical knowledge.

Multidisciplinary Approaches to Neural Computing
Multidisciplinary Approaches to Neural Computing

Author: Anna Esposito

Publisher: Springer

Published: 2017-08-28

Total Pages: 380

ISBN-13: 331956904X

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This book presents a collection of contributions in the field of Artificial Neural Networks (ANNs). The themes addressed are multidisciplinary in nature, and closely connected in their ultimate aim to identify features from dynamic realistic signal exchanges and invariant machine representations that can be exploited to improve the quality of life of their end users. Mathematical tools like ANNs are currently exploited in many scientific domains because of their solid theoretical background and effectiveness in providing solutions to many demanding tasks such as appropriately processing (both for extracting features and recognizing) mono- and bi-dimensional dynamic signals, solving strong nonlinearities in the data and providing general solutions for deep and fully connected architectures. Given the multidisciplinary nature of their use and the interdisciplinary characterization of the problems they are applied to – which range from medicine to psychology, industrial and social robotics, computer vision, and signal processing (among many others) – ANNs may provide a basis for redefining the concept of information processing. These reflections are supported by theoretical models and applications presented in the chapters of this book. This book is of primary importance for: (a) the academic research community, (b) the ICT market, (c) PhD students and early-stage researchers, (d) schools, hospitals, rehabilitation and assisted-living centers, and (e) representatives of multimedia industries and standardization bodies.

From a Heuristic Point of View
From a Heuristic Point of View

Author: Cesare Cozzo

Publisher: Cambridge Scholars Publishing

Published: 2014-07-03

Total Pages: 295

ISBN-13: 1443863351

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How do we get new knowledge? Following the maverick tradition in the philosophy of science, Carlo Cellucci gradually came to the conclusion that logic can only fulfill its role in mathematics, science and philosophy if it helps us to answer this question. He argues that mathematical logic is inadequate and that we need a new logic, framed in a naturalistic conception of knowledge and philosophy – the heuristic conception. This path from logic to a naturalistic conception of knowledge and philosophy explains the title, From a Heuristic Point of View, which recalls the celebrated collection of essays, From a Logical Point of View, by Willard Van Orman Quine, the father of modern naturalized epistemology. The word ‘heuristic’ points to Cellucci’s favorite theme and the main difference between him and Quine: the emphasis on discovery and building a ‘logic’ for generating new knowledge. This book is a collection of essays from leading figures in this field who discuss, criticize, or expand on the main topics in Cellucci’s work, dealing with some of the most challenging questions in logic, science and philosophy.

Heuristic Reasoning
Heuristic Reasoning

Author: Emiliano Ippoliti

Publisher: Springer

Published: 2014-09-05

Total Pages: 194

ISBN-13: 331909159X

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How can we advance knowledge? Which methods do we need in order to make new discoveries? How can we rationally evaluate, reconstruct and offer discoveries as a means of improving the ‘method’ of discovery itself? And how can we use findings about scientific discovery to boost funding policies, thus fostering a deeper impact of scientific discovery itself? The respective chapters in this book provide readers with answers to these questions. They focus on a set of issues that are essential to the development of types of reasoning for advancing knowledge, such as models for both revolutionary findings and paradigm shifts; ways of rationally addressing scientific disagreement, e.g. when a revolutionary discovery sparks considerable disagreement inside the scientific community; frameworks for both discovery and inference methods; and heuristics for economics and the social sciences.

Discoverability
Discoverability

Author: Lorenzo Magnani

Publisher: Springer Nature

Published: 2022-05-20

Total Pages: 170

ISBN-13: 3030933296

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The book analyses the concept of discoverability, and some current epistemological problems related to it, with a special attention to science. It shows that discoverability is closely related to the sustainability of human creativity in an "eco-cognitive" perspective. Advocating the need of an integral ecology and leveraging the important concept of abduction, it demonstrates that an ecology of human creativity should have priority over other needs, i.e that the first ecological duty is to protect and sustain discoverability. Enhancing discoverability will protect human creativity, and it is exactly human creativity, a form of innovative abductive cognition, that can promote the implementation of the other kinds of ecology. The author guides readers through a comprehensive discussion on the concept of discoverability, eco-cognitive situatedness, and eco-cognitive openness and closure alike. By describing some key real-world examples, he highlights the main challenges that are currently posed to human creativity and epistemic integrity. He also describes future eco-cognitive settings, discussing the problem of overcomputationalism and suggesting a reinterpretation of the role of human knowledge. Overall, this book fills an important gap in the literature on the nexus abduction – creativity – discovery, offering a source of inspiration to philosophers, epistemologists, and cognitive scientists. Yet, it also addresses researchers in other disciplines interested in the problems of scientific discovery and epistemic integrity of research.

Great Circles
Great Circles

Author: Emily Rolfe Grosholz

Publisher: Springer

Published: 2018-11-13

Total Pages: 275

ISBN-13: 3319982311

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This volume explores the interaction of poetry and mathematics by looking at analogies that link them. The form that distinguishes poetry from prose has mathematical structure (lifting language above the flow of time), as do the thoughtful ways in which poets bring the infinite into relation with the finite. The history of mathematics exhibits a dramatic narrative inspired by a kind of troping, as metaphor opens, metonymy and synecdoche elaborate, and irony closes off or shifts the growth of mathematical knowledge. The first part of the book is autobiographical, following the author through her discovery of these analogies, revealed by music, architecture, science fiction, philosophy, and the study of mathematics and poetry. The second part focuses on geometry, the circle and square, launching us from Shakespeare to Housman, from Euclid to Leibniz. The third part explores the study of dynamics, inertial motion and transcendental functions, from Descartes to Newton, and in 20th c. poetry. The final part contemplates infinity, as it emerges in modern set theory and topology, and in contemporary poems, including narrative poems about modern cosmology.

Dual-Process Theories of Numerical Cognition
Dual-Process Theories of Numerical Cognition

Author: Mario Graziano

Publisher: Springer

Published: 2018-08-12

Total Pages: 151

ISBN-13: 3319967975

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This book presents a philosophical interpretation to numerical cognition based on dual process theories and heuristics. It shows how investigations in cognitive science can shed light on issues traditionally raised by philosophers of mathematics. The analysis will also help readers to better understand the relationship between current neuroscientific research and the philosophical reflection on mathematics. The author seeks to explain the acquisition of mathematical concepts. To accomplish this, he needs to answer two questions. How can the concepts of approximate numerosity become an object of thought that is so accessible to our consciousness? How are these concepts refined and specified in such a way as to become numbers? Unfortunately, there is currently no model that can truly demonstrate the role of language in the development of numerical skills starting from approximate pre-verbal skills. However, the author details a solution to this problem: dual process theories. It is an approach widely used by theorists focusing on reasoning, decision making, social cognition, and consciousness. Here, he applies this approach to the studies on mathematical knowledge. He details the results brought about by psychological and neuroscientific studies conducted on numerical cognition by key neuroscientists. In the process, he develops the foundations of a new, potential philosophical explanation on mathematical knowledge.