Godel's Mistake The Role Of Meaning In Mathematics [Paperback]

Why Is Mathematics Incomplete?

Godels incompleteness theorem is a foundational result in mathematics that proves that any axiomatic theory of numbers will be either inconsistent or incomplete. Turings Halting problem is a foundational result in computing proving that computers cannot know if a program will halt. Godels Mistake connects these theorems to the question of meaning. The book shows that the proofs arise due to category confusions between names, concepts, things, programs, algorithms, problems, etc. The book argues that these problems can be solved by introducing ordinary language categories in mathematics.

Where the Solution Lies

The solution to the problem, the author argues, requires a new approach to numbers where numbers are treated as types rather than quantities. To view numbers as types requires a foundational shift in which objects are constructed from sets rather than sets from objects. Since sets denote concepts, this shift implies that objects are created from concepts. This also changes our view of space-time from linear and open to hierarchical and closed. In this hierarchical description, objects are symbols of meaning, rather than physical things. The author calls this theory the Type Number Theory (TNT) and shows that the type view of numbers is free of Godels Incompleteness and Turings Halting Problem.

How This Book Is Structured

Chapter 1: Mechanizing Thoughtprovides an overview of mathematical, philosophical, linguistic and logical issues that preceded Godels and Turings results and shows that the problems encountered in mathematics have a wider undercurrent extending into other areas of science.

Chapter 2: Godels Mistrickdiscusses Godels Incompleteness Theorem and Turings Halting problem and shows how their proofs rest on category mistakes. The chapter also connects the theorems to the issues of sentenlS!