On the intelligibility of the universe and the notions of simplicity, complexity and irreducibility

TL;DR

The paper explores the limits of understanding the universe through the lens of simplicity and complexity, arguing that some mathematical facts are inherently irreducible and that the universe's total complexity may be infinite, challenging the notion of complete comprehension.

Contribution

It introduces a philosophical and mathematical framework linking comprehension to compression, and demonstrates the existence of irreducible mathematical facts using algorithmic information theory.

Findings

Mathematical facts can be irreducible and incompressible.

The world of mathematical ideas has infinite complexity.

The physical universe's complexity may be finite or infinite, with current science leaning towards infinity.

Abstract

We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that comprehension is compression, i.e., explaining many facts using few theoretical assumptions, and that a theory may be viewed as a computer program for calculating observations. This provides motivation for defining the complexity of something to be the size of the simplest theory for it, in other words, the size of the smallest program for calculating it. This is the central idea of algorithmic information theory (AIT), a field of theoretical computer science. Using the mathematical concept of program-size complexity, we exhibit irreducible mathematical facts, mathematical facts that cannot be demonstrated using any mathematical theory simpler than they are. It follows that the world of mathematical ideas has infinite complexity and is therefore not fully comprehensible, at least not in a static fashion. Whether the physical world has finite or infinite complexity remains to be seen. Current science believes that the world contains randomness, and is therefore also infinitely complex, but a deterministic universe that simulates randomness via pseudo-randomness is also a possibility, at least according to recent highly speculative work of S. Wolfram.