  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.
