The End of Infinity: Where Mathematics and Philosophy Meet

Front Cover
Algora Publishing, Jul 1, 2018 - Philosophy - 198 pages

The idea of infinity stands at the intersection of mathematics and philosophy. As da Vinci said, "Arithmetic is a computational science in its calculations, but it is of no avail in dealing with continuous quantity."

The End of Infinity reviews the philosophical history of infinity, mathematics, numbers, and logic to demonstrate that the modern conception of infinity involves a logical and metaphysical contradiction that argues for a return to Aristotle's conception of potential infinity.

The way we think about infinity shapes the way we think about other subjects. The author says, "My heart fluttered the first time I heard the paradoxical claim of a 19th-century mathematician who said infinity was real, that there was more than one type of infinity, and that not all infinities were the same size! It was either one of the most absurd or one of the most important theories in intellectual history."

The chapters that follow investigate the issue from a variety of perspectives. This book focuses mainly on the philosophy of infinity; it does not require an extensive understanding of mathematics, although relevant terms and concepts are introduced and explained.

From the late 19th to the early 20th century, great thinkers like Cantor, Frege, and Russell attempted to provide a new foundation for mathematics by replacing the intuitions of Kant with logic, rigorous definitions, and axiomatic set theory. One of the results of their work was a new conception of infinity, which included the surprising claims that infinite sets exist, that there are different types of infinite sets, and that not all infinite sets are the same size.

However, the great logician Gödel proved that all axiomatic systems, like set theory, are necessarily incomplete and therefore cannot not provide a solid foundation for mathematics, evidenced by the fact that there are known mathematical truths that cannot be proven or disproven with set theory. Despite this setback, set theory continues to dominate modern mathematics, with limited attempts to reconsider its validity - until now.

 

Contents

Cast of Characters
1
Introduction
3
Time
7
History
13
Aristotle
33
Language Logic
47
The Numbers
69
Static Infinity
87
Dynamic Infinity
115
Counting the Real Numbers
143
_GoBack
156
The End of Infinity
161
Simplified Proofs
177
Index
187
Copyright

Common terms and phrases

About the author (2018)

The way we think about infinity shapes the way we think about other subjects. The author says, "My heart fluttered the first time I heard the paradoxical claim of a 19th-century mathematician who said infinity was real, that there was more than one type of infinity, and that not all infinities were the same size! It was either one of the most absurd or one of the most important theories in intellectual history." The chapters that follow investigate the issue from a variety of perspectives. This book focuses mainly on the philosophy of infinity; it does not require an extensive understanding of mathematics, although relevant terms and concepts are introduced and explained. From the late 19th to the early 20th century, great thinkers like Cantor, Frege, and Russell attempted to provide a new foundation for mathematics by replacing the intuitions of Kant with logic, rigorous definitions, and axiomatic set theory. One of the results of their work was a new conception of infinity, which included the surprising claims that infinite sets exist, that there are different types of infinite sets, and that not all infinite sets are the same size. However, the great logician Gödel proved that all axiomatic systems, like set theory, are necessarily incomplete and therefore cannot not provide a solid foundation for mathematics, evidenced by the fact that there are known mathematical truths that cannot be proven or disproven with set theory. Despite this setback, set theory continues to dominate modern mathematics, with limited attempts to reconsider its validity - until now.

Bibliographic information