⁴I think there is no avoiding the conclusion that, if I was indeed disdainful of Ward, it was for not being very bright. Any amateurloverof mathematics is still doing it to pass the time. As I recall, Ward was calculating Gödel numbers. This was a pointless activity. The usefulness of Gödel numbers is entirely theoretical. Gödel's Incompleteness Theorem can be understood as follows. Every number-theoretic formula φ(x) has a Gödel number, which can be symbolized by ⌜φ(x)⌝. There is a formula θ(x,y) such that, for all formulas φ(x), for all numbers n, the sentence θ(⌜φ(x)⌝,n) is true if and only if the sentence φ(n) has a formal proof (in some predetermined axiomatic system). Let ψ(x) be the negation ¬θ(x,x). Then the sentence ψ(⌜ψ(x)⌝) is true, but has no formal proof. This is the point. Writing down the actual number ⌜ψ(x)⌝ would be impractical. Apparently Ward did not understand this, or did not care.