I will discuss Leibniz's ideas on complexity (Discours de metaphysique, 1686),

leading to modern work on program-size complexity, the halting probability and incompleteness. Leibniz's principle of sufficient reason asserts that if anything is true it is true for a reason. But the bits of the numerical

value of the halting probability are mathematical truths that are true for no reason.

More precisely, as I will explain, they are irreducible mathematical truths, that is, true for no reason simpler than themselves.