Paradoxes and Inconsistent Mathematics

Logical paradoxes, such as the Liar, Russell's, and Sorites, are only the noisiest of many contradictions that arise in everyday life, from the smallest points to the widest boundaries. Zach Weber's book "Paradoxes and Inconsistent Mathematics" uses "dialetheic paraconsistency" to develop this idea rigorously, addressing a longstanding open question about how much standard mathematics can capture paraconsistency. The guiding focus is on why there are paradoxes, which underscores a simple philosophical claim that paradoxes are found in the ordinary and that is what makes them so extraordinary.

Format: Hardback
Length: 260 pages
Publication date: 21 October 2021
Publisher: Cambridge University Press

Logical paradoxes, such as the Liar, Russell's, and the Sorites, are well-known. However, Paradoxes and Inconsistent Mathematics argues that they are just the loudest of many. Contradictions arise in everyday life, from the smallest details to the broadest horizons. In this book, Zach Weber develops the idea of "dialetheic paraconsistency" – a formal framework that allows some contradictions to be true without absurdity – as the foundation for rigorous development, starting from mathematical foundations. By doing so, Weber directly addresses a long-standing open question: how much standard mathematics can paraconsistency capture? The main focus is on a more fundamental question: why are there paradoxes? The details underscore a simple philosophical claim: that paradoxes are found in the ordinary, which is what makes them so extraordinary.

Introduction:
Logical paradoxes have fascinated philosophers and mathematicians for centuries. These paradoxes present statements or scenarios that seem self-contradictory or logically impossible, yet they appear to be true. While some paradoxes, such as the Liar and Russell's paradox, are well-known, there are many others that are less familiar. In Paradoxes and Inconsistent Mathematics, Zach Weber takes a fresh approach to these paradoxes by arguing that they are not the exception but rather the norm.

Paraconsistency:
Weber introduces the concept of "dialetheic paraconsistency" as a formal framework that allows certain contradictions to be true without absurdity. Dialetheic paraconsistency is based on the idea that there are multiple truth values for certain statements or propositions. This means that a statement can be true in one context and false in another, or it can be true at one time and false at another.

Why Paradoxes Exist:
The central question of the book is why paradoxes exist. Weber argues that paradoxes arise from the ordinary, from the way we think and interpret the world around us. He suggests that paradoxes are not unique to mathematics but are found in various fields, including philosophy, logic, and everyday life.

Examples:
Weber provides numerous examples of paradoxes from different fields. For instance, he discusses the Liar paradox, which states that a statement that is always false is true. He also explores the Sorites paradox, which involves an infinite series of numbers that become smaller and smaller, but never reach zero.

Paraconsistency as a Tool:
Weber uses dialetheic paraconsistency as a tool to develop a rigorous framework for analyzing paradoxes. He shows how certain paradoxes can be resolved by adopting a paraconsistent approach. For example, he proposes a solution to the Liar paradox that involves accepting multiple truth values for statements.

Limitations:
While dialetheic paraconsistency provides a useful framework for analyzing paradoxes, it has its limitations. One of the main criticisms is that it can be difficult to apply in practice, as it requires a complex understanding of truth values and logical operators. Additionally, some paradoxes may be intractable, meaning that they cannot be resolved using any known method.

Conclusion:
In conclusion, Paradoxes and Inconsistent Mathematics offers a unique perspective on logical paradoxes. Weber argues that paradoxes are not rare or exceptional but rather a fundamental aspect of the way we think and interpret the world. By introducing the concept of dialetheic paraconsistency, he provides a rigorous framework for analyzing paradoxes and offers potential solutions to some of the most famous paradoxes. While there are limitations to this approach, it remains an important contribution to the study of logic and mathematics.

Weight: 756g
Dimension: 177 x 251 x 27 (mm)
ISBN-13: 9781108834414