A Topological Theory of Empirical Simplicity and its Connection to the Truth

Simplicity is analyzed topologically. Ockham’s razor is shown, under very general conditions, to be the unique, convergent strategy that jointly minimizes errors, retractions, and retraction times prior to convergence to the true theory. 1 Empirical Questions and Solutions 1.1 Input Streams Let I be a set of potential inputs. An input stream is an element of Iω. If w is an input stream let w|i = (w(0), . . . , w(i− 1)), so that w|0 = (), where () denotes the empty sequence. An input sequence of length n is a member of In. A finite input sequence is a finite input sequence of some length. If e is a finite input sequence, let |e| denote the length of e, which is the same as the cardinality of e viewed as a set of ordered pairs. If e is a finite input sequence and e′ is an input stream or finite input sequence, define e ≤ e′ if and only if there exists n such that e′|n = e. 1.2 Empirical Problems An empirical problem is a pair (K, Θ) where Θ is a partition of Iω so that ∅ ⊂ K ⊆ ⋃ Θ ⊆ I. Then K is called the empirical presupposition of the problem and Θ is called the question posed by the problem. Let Hw denote the unique cell of Θ that contains (i.e., is true of) w. Define: K̂ = {w|i : w ∈ K and i ∈ ω},