In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, ...

Author: Fernando Ferreira

Publisher: Springer

ISBN: 3030776565

Category: Mathematics

Page: 197

View: 614

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

Accordingly, only formulae, not propositions, can ultimately satisfy the requirements of exact specification of an axiomatic thinking (denken). If this is how Hilbert eventually came to understand the requirement that axioms be ...

Author: Reinhard Kahle

Publisher: Springer

ISBN: 9783319101033

Category: Mathematics

Page: 561

View: 989

Gerhard Gentzen has been described as logic’s lost genius, whom Gödel called a better logician than himself. This work comprises articles by leading proof theorists, attesting to Gentzen’s enduring legacy to mathematical logic and beyond. The contributions range from philosophical reflections and re-evaluations of Gentzen’s original consistency proofs to the most recent developments in proof theory. Gentzen founded modern proof theory. His sequent calculus and natural deduction system beautifully explain the deep symmetries of logic. They underlie modern developments in computer science such as automated theorem proving and type theory.

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, ...

Author: Fernando Ferreira

Publisher: Springer

ISBN: 3030777987

Category: Mathematics

Page: 283

View: 843

In this two-volume compilation of articles, leading researchers reevaluate the success of Hilbert's axiomatic method, which not only laid the foundations for our understanding of modern mathematics, but also found applications in physics, computer science and elsewhere. The title takes its name from David Hilbert's seminal talk Axiomatisches Denken, given at a meeting of the Swiss Mathematical Society in Zurich in 1917. This marked the beginning of Hilbert's return to his foundational studies, which ultimately resulted in the establishment of proof theory as a new branch in the emerging field of mathematical logic. Hilbert also used the opportunity to bring Paul Bernays back to Göttingen as his main collaborator in foundational studies in the years to come. The contributions are addressed to mathematical and philosophical logicians, but also to philosophers of science as well as physicists and computer scientists with an interest in foundations.

108 • 4 Discovery of Formal Reasoning The second axiom is an instance of a principle that is crucial whenever a logical ... To give an example of such axiomatic thinking, say axiom 2 has the instance (A ⊃ (A ⊃ B)) ⊃ ((A ⊃ A) ⊃ (A ...

Author: Jan von Plato

Publisher: Princeton University Press

ISBN: 9780691174174

Category: Science

Page: 400

View: 433

The information age owes its existence to a little-known but crucial development, the theoretical study of logic and the foundations of mathematics. The Great Formal Machinery Works draws on original sources and rare archival materials to trace the history of the theories of deduction and computation that laid the logical foundations for the digital revolution. Jan von Plato examines the contributions of figures such as Aristotle; the nineteenth-century German polymath Hermann Grassmann; George Boole, whose Boolean logic would prove essential to programming languages and computing; Ernst Schröder, best known for his work on algebraic logic; and Giuseppe Peano, cofounder of mathematical logic. Von Plato shows how the idea of a formal proof in mathematics emerged gradually in the second half of the nineteenth century, hand in hand with the notion of a formal process of computation. A turning point was reached by 1930, when Kurt Gödel conceived his celebrated incompleteness theorems. They were an enormous boost to the study of formal languages and computability, which were brought to perfection by the end of the 1930s with precise theories of formal languages and formal deduction and parallel theories of algorithmic computability. Von Plato describes how the first theoretical ideas of a computer soon emerged in the work of Alan Turing in 1936 and John von Neumann some years later. Shedding new light on this crucial chapter in the history of science, The Great Formal Machinery Works is essential reading for students and researchers in logic, mathematics, and computer science.

Author: William Maxwell HetheringtonPublish On: 1867

There is at the very least this peculiar value in à priori or axiomatic thinking , that if its primary positions cannot be proved , neither can they be refuted . It is impossible for any man , by prosecuting axiomatic thinking ...

Author: William Maxwell HETHERINGTONPublish On: 1867

There is at the very least this peculiar value in à priori or axiomatic thinking , that if its primary positions cannot be proved , neither can they be refuted . It is impossible for any man , by prosecuting axiomatic thinking ...

While Mises resisted strictly axiomatic thinking, and Hayek begins to introduce axioms with his increased formalization of economic science and search for fixed patterns of causality, both engage with the Pure Logic of Choice insofar as ...

Author: Sonia Marie Scott

Publisher: Routledge

ISBN: 9780415699211

Category: Business & Economics

Page: 302

View: 416

The history of European economic thought has long been written by those seeking to prove or disprove the truth-value of the theories they describe. This work takes a different approach. It explores the philosophical groundwork of the theoretical structure within which economic subjects are presented. Demonstrating how the subjects of economic texts tend to be defined in and through their relationship to knowledge, this study addresses the epistemological constitution of subjectivity in economic thought.

It is a pure thought construction, of which one can no longer say that it is true or false. ... is perfectly clear from his later work (1918), his sustained hymn to the axiomatic method, entitled, incidentally, "Axiomatic thinking.

Author: Alexander George

Publisher: Oxford University Press on Demand

ISBN: 9780195079296

Category: History

Page: 204

View: 697

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.

I think I find axioms to be mere definitions , to be true because the words used express and embody certain relations ... A child is younger than its father ' is an axiom : it is proveable by referring to meaning of the word father .

As well as being a beautiful mathematical result, this is also an interesting illustration of the axiomatic ... Their results show that, if you think a person's preferences satisfy these axioms, you have to think functions u,m and um ...

Author: Edward Cartwright

Publisher: Routledge

ISBN: 9781351600026

Category: Business & Economics

Page: 556

View: 606

Over the last few decades behavioral economics has revolutionized the discipline. It has done so by putting the human back into economics, by recognizing that people sometimes make mistakes, care about others and are generally not as cold and calculating as economists have traditionally assumed. The results have been exciting and fascinating, and have fundamentally changed the way we look at economic behavior. This textbook introduces all the key results and insights of behavioral economics to a student audience. Ideas such as mental accounting, prospect theory, present bias, inequality aversion and learning are explained in detail. These ideas are also applied in diverse settings such as auctions, stock market crashes, charitable donations and health care, to show why behavioral economics is crucial to understanding the world around us. Consideration is also given to what makes people happy, and how we can potentially nudge people to be happier. This new edition contains expanded and updated coverage of contract theory, bargaining in the family, time and risk, and stochastic reference points, among other topics, to ensure that readers are kept up to speed with this fast-paced field. The companion website is also updated with a range of new questions and worked examples. This book remains the ideal introduction to behavioral economics for advanced undergraduate and graduate students.