Intuition and the Axiomatic Method

Intuition and the Axiomatic Method

By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their ...

Author: Emily Carson

Publisher: Springer Science & Business Media

ISBN: 1402040393

Category: Philosophy

Page: 324

View: 238

Following developments in modern geometry, logic and physics, many scientists and philosophers in the modern era considered Kant’s theory of intuition to be obsolete. But this only represents one side of the story concerning Kant, intuition and twentieth century science. Several prominent mathematicians and physicists were convinced that the formal tools of modern logic, set theory and the axiomatic method are not sufficient for providing mathematics and physics with satisfactory foundations. All of Hilbert, Gödel, Poincaré, Weyl and Bohr thought that intuition was an indispensable element in describing the foundations of science. They had very different reasons for thinking this, and they had very different accounts of what they called intuition. But they had in common that their views of mathematics and physics were significantly influenced by their readings of Kant. In the present volume, various views of intuition and the axiomatic method are explored, beginning with Kant’s own approach. By way of these investigations, we hope to understand better the rationale behind Kant’s theory of intuition, as well as to grasp many facets of the relations between theories of intuition and the axiomatic method, dealing with both their strengths and limitations; in short, the volume covers logical and non-logical, historical and systematic issues in both mathematics and physics.
Categories: Philosophy

Axiomatic Method and Category Theory

Axiomatic Method and Category Theory

They may be or be not supported by some sensual intuitions; the sensual intuition is a separate issue which must not ... quite rightly stresses the fundamental role of this restricted notion of intuition in Hilbert's Axiomatic Method.

Author: Andrei Rodin

Publisher: Springer Science & Business Media

ISBN: 9783319004044

Category: Philosophy

Page: 285

View: 330

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method.
Categories: Philosophy

Mathematics and Mind

Mathematics and Mind

The present collection will be of interest to students of both mathematics and of mind.

Author: Alexander George

Publisher: Oxford University Press on Demand

ISBN: 9780195079296

Category: History

Page: 204

View: 360

The essays in this volume investigate the conceptual foundations of mathematics illuminating the powers of the mind. Contributors include Alexander George, Michael Dummett, George Boolos, W.W. Tait, Wilfried Sieg, Daniel Isaacson, Charles Parsons, and Michael Hallett.
Categories: History

Bourbaki

Bourbaki

He unwittingly used many properties that were suggested by visual intuition rather than included in the axioms or proved rigorously . ... Despite this , what Euclid had developed was an important precursor to the axiomatic method .

Author: Maurice Mashaal

Publisher: American Mathematical Soc.

ISBN: 0821839675

Category: Biography & Autobiography

Page: 168

View: 430

The name Bourbaki is known to every mathematician. This book presents accounts of the origins of Bourbaki, their meetings, their seminars, and the members themselves. It also discusses the lasting influence that Bourbaki has had on mathematics, through both the Elements and the Seminaires.
Categories: Biography & Autobiography

Rethinking Knowledge

Rethinking Knowledge

Intuition. In the analytic-synthetic method, intuition plays an essential role. As already mentioned, in such method principles must be true. ... byproduct of the analytic-synthetic method is the material axiomatic method.

Author: Carlo Cellucci

Publisher: Springer

ISBN: 9783319532370

Category: Philosophy

Page: 427

View: 984

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.
Categories: Philosophy

Rethinking Logic Logic in Relation to Mathematics Evolution and Method

Rethinking Logic  Logic in Relation to Mathematics  Evolution  and Method

2.7 Logic, Intuitive Thinking and Discursive Thinking . ... 31 2.12 Intuitive Thinking, Discursive Thinking and Greek Mythology .. 33 2.13 OriginoftheName'Logic'. ... 43 3.9 Hippocrates of Cos' Criticism of the Axiomatic Method .

Author: Carlo Cellucci

Publisher: Springer Science & Business Media

ISBN: 9789400760912

Category: Philosophy

Page: 389

View: 136

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
Categories: Philosophy

Naturalizing Logico Mathematical Knowledge

Naturalizing Logico Mathematical Knowledge

Let's name this result G. Now, if the method of mathematics really is the axiomatic method, how could we accept that Tholds ... as irrelevant and the consistency of axioms establishable by merely relying on an evolved sort of intuition.

Author: Sorin Bangu

Publisher: Routledge

ISBN: 9781351998444

Category: Mathematics

Page: 306

View: 117

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato’s dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. Naturalizing Logico-Mathematical Knowledge contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers’ willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.
Categories: Mathematics

Rational Reconstructions of Modern Physics

Rational Reconstructions of Modern Physics

Functions of intuition in quantum physics. In E. Carson & R. Huber (Eds.), Intuition and the axiomatic method (pp. 267-292). Dordrecht: Springer. Hentschel, K. (1990). lnterpretationen und Fehlinterpretationen der speziellen und ...

Author: Peter Mittelstaedt

Publisher: Springer Science & Business Media

ISBN: 9789400755932

Category: Science

Page: 141

View: 271

Newton’s classical physics and its underlying ontology are loaded with several metaphysical hypotheses that cannot be justified by rational reasoning nor by experimental evidence. Furthermore, it is well known that some of these hypotheses are not contained in the great theories of Modern Physics, such as the theory of Special Relativity and Quantum Mechanics. This book shows that, on the basis of Newton’s classical physics and by rational reconstruction, the theory of Special Relativity as well as Quantum Mechanics can be obtained by partly eliminating or attenuating the metaphysical hypotheses. Moreover, it is shown that these reconstructions do not require additional hypotheses or new experimental results. In the second edition the rational reconstructions are completed with respect to General Relativity and Cosmology. In addition, the statistics of quantum objects is elaborated in more detail with respect to the rational reconstruction of quantum mechanics. The new material completes the approach of the book as much as it is possible at the present state of knowledge. Presumably, the most important contribution that is added to the second edition refers to the problem of interpretation of the three great theories of Modern Physics. It is shown in detail that in the light of rational reconstructions even realistic interpretations of the three theories of Modern Physics are possible and can easily be achieved.
Categories: Science

Intuition in Science and Mathematics

Intuition in Science and Mathematics

Put in another way, what I am saying is that I consider it just as necessary to train the intuition for finding and writing ... It is also important to note that Suppes does not write on “teaching the axiomatic method but rather on ...

Author: H. Fischbein

Publisher: Springer Science & Business Media

ISBN: 9780306472374

Category: Education

Page: 226

View: 278

In writing the present book I have had in mind the following objectives: - To propose a theoretical, comprehensive view of the domain of intuition. - To identify and organize the experimental findings related to intuition scattered in a wide variety of research contexts. - To reveal the educational implications of the idea, developed for science and mathematics education. Most of the existing monographs in the field of intuition are mainly concerned with theoretical debates - definitions, philosophical attitudes, historical considerations. (See, especially the works of Wild (1938), of Bunge (1 962) and of Noddings and Shore (1 984).) A notable exception is the book by Westcott (1968), which combines theoretical analyses with the author’s own experimental studies. But, so far, no attempt has been made to identify systematically those findings, spread throughout the research literature, which could contribute to the deciphering of the mechanisms of intuition. Very often the relevant studies do not refer explicitly to intuition. Even when this term is used it occurs, usually, as a self-evident, common sense term.
Categories: Education

Great Currents of Mathematical Thought

Great Currents of Mathematical Thought

and the end of the unique property that it enjoys of compelling the reader's assent , since then every mathematical proposition would be as debatable as the intuition of its author ! Axiomatic methods are often criticized for their ...

Author: François Le Lionnais

Publisher: Courier Corporation

ISBN: 0486495795

Category: Mathematics

Page: 266

View: 777

50 essays by eminent scholars include meditations on "Structures," Disciplines," "Space," "Function," "Group," "Probability," and "The Mathematical Epic" (Volume I) and on "Mathematics and the Human Intellect," "Mathematics and Technology," and "Mathematics and Civilization" (Volume II). 1962 edition.
Categories: Mathematics