Metaphysics and mathematics: Perspectives on reality

Gideon J Kühn

doi:10.4102/hts.v73i3.4663

Abstract

The essence of number was regarded by the ancient Greeks as the root cause of the existence of the universe, but it was only towards the end of the 19th century that mathematicians initiated an in-depth study of the nature of numbers. The resulting unavoidable actuality of infinities in the number system led mathematicians to rigorously investigate the foundations of mathematics. The formalist approach to establish mathematical proof was found to be inconclusive: Gödel showed that there existed true propositions that could not be proved to be true within the natural number universe. This result weighed heavily on proposals in the mid-20th century for digital models of the universe, inspired by the emergence of the programmable digital computer, giving rise to the branch of philosophy recognised as digital philosophy. In this article, the models of the universe presented by physicists, mathematicians and theoretical computer scientists are reviewed and their relation to the natural numbers is investigated. A quantum theory view that at the deepest level time and space may be discrete suggests a profound relation between natural numbers and reality of the cosmos. The conclusion is that our perception of reality may ultimately be traced to the ontology and epistemology of the natural numbers.

Highlights

When the 19th century mathematician George Cantor developed a set theory in his in-depth study of the natural numbers, a chain of reasoning was initiated that culminated in the incompleteness theorems of the mathematician Kurt Gödel and halting problem of the theoretical computer scientist Alan Turing
The Pythagorean School viewed the essence of number as the cause of the existence of the universe, and so it remained until Cantor in the late 19th century initiated a deep study to understand the fundamental properties of numbers
He showed that the cardinality of the natural numbers are transfinite and have counter-intuitive properties: the elements of an infinite set may be put into a one-to-one correspondence with the elements of a proper subset of the set, which contradicts Euclid’s axiom that the whole is greater than the part

Summary

Introduction

When the 19th century mathematician George Cantor developed a set theory in his in-depth study of the natural numbers, a chain of reasoning was initiated that culminated in the incompleteness theorems of the mathematician Kurt Gödel and halting problem of the theoretical computer scientist Alan Turing. The universe is fundamentally digital in nature, and reality may at the deepest level be a mathematical construct.

Results

Conclusion