Hilbert's Proof Theory and its modern Development

Ralf Ille

ISBN 10: 3346592898 / ISBN 13: 9783346592897

Published by GRIN Verlag, 2022

About this Item

Druck auf Anfrage Neuware - Printed after ordering - Seminar paper from the year 2021 in the subject Mathematics - Miscellaneous, grade: 1,0, University of Hagen, course: Philosophy of Mathematics, language: English, abstract: David Hilbert first dealt with proofs as independent mathematical objects during the foundational crisis in mathematics at the beginning of the 20th century. Hilbert wanted to dispel all doubts about classical mathematical reasoning by a theory that makes mathematical proofs themselves to its objects (Hilbert, 1923). We examine the reasons and aims of Hilbert's proof theory and show how it came to a surprisingly sudden end. Gerhard Gentzen continued proof theory in the spirit of Hilbert. We will see that Gentzen's system is more closely related to mathematical practice and get an outline how he succeeds in proving the consistency of number theory by means of new methods. Attempts to grasp the real essence of proofs started afterwards. First we show how the important question of proof identity evolved in General Proof Theory. Second, how formal proofs can be represented in a new language by mathematical category theory and the lambda calculus to derive new identity criteria.

Seller Inventory # 9783346592897

Bibliographic Details

• Title: Hilbert's Proof Theory and its modern Development
• Publisher: GRIN Verlag
• Publication Date: 2022
• Language: English
• ISBN 10: 3346592898
• ISBN 13: 9783346592897
• Binding: Taschenbuch
• Condition: Neu
• Dimensions: 210W x 148H x 3D millimeters 210 millimeters width by 148 millimeters height by 3 millimeters depth
• Weight: 56 grams

About this title

Synopsis

Seminar paper from the year 2021 in the subject Mathematics - Miscellaneous, grade: 1,0, University of Hagen, course: Philosophy of Mathematics, language: English, abstract: Imagine a world where the very bedrock of mathematical truth crumbles beneath your feet-this was the reality facing mathematicians in the early 20th century, a period of intense scrutiny and doubt known as the foundational crisis. This book delves into the fascinating story of how David Hilbert, a towering figure in mathematics, sought to rebuild this foundation through his ambitious program of proof theory. Explore the contrasting philosophies of classical versus intuitionistic mathematics, the revolutionary impact of Cantor's set theory, and the ensuing objections that threatened to unravel the entire mathematical edifice. Witness the rise of mathematical formalism, as signs are elevated to objects of study, and the birth of metamathematics, a new discipline dedicated to proving the consistency of mathematical systems. Unravel the intricacies of Hilbert's program, his attempt to secure mathematical certainty through finite means, and the devastating blow dealt by Gödel's incompleteness theorems, which revealed the inherent limitations of formal systems. Discover how Gentzen's groundbreaking work on natural deduction and transfinite induction offered a new path forward, transforming proofs into objects of mathematical inquiry in their own right. Finally, journey into the realm of modern proof theory, where category theory and lambda calculus provide powerful new frameworks for understanding the essence of mathematical truth and developing novel identity criteria for proofs. This book offers a comprehensive exploration of Hilbert's program, its challenges, and its enduring legacy, making it an essential read for anyone interested in the foundations of mathematics, mathematical logic, and the quest for absolute certainty. Keywords: Hilbert's program, proof theory, foundational crisis, classical m