The symbols all look funky and hard (but in reality are just ways of stating edges and paths in graphs..) you can even see the ‘primatur’ of the Tunny differencing idea – since paths (as sequences of edges) are characterized using the “notation of partial differentials” between the nodes (at each end of the edge).
Given the background Eisenhart lays down earlier (appealing to Turing’s interest in the original math required for general relativity), its interesting to see semi-direct group actions being discussed in terms of “motions”. The quantum mechanical viewpoint had not truly settled (folks are still looking to uphold classical mechanics); and even the part that had settled had not yet settled into the probablistic model.
Given Turing comes back from Princeton a couple of times (and probably does GCCS work on trips back, pre war), its realistic to place his On Permutations group manuscript in the 1937 period. There are lots of hints that its influenced by the Eisenhart studies – though focused on the permutation groups found in enigma math rather than continuous groups (required in Riemannian spaces).
Its also fun to see the Colossus rectangle and its matrix entries as the infinitesimal – and convergence as averaging over neighborhoods. One sees lots of “heat transfer” ideas. CLearly, the notion that one group (e.g. the cycle group) acts on another (the hypercube) is very much a proto-programming idea, as is the idea of an operator transform an eigenfunction (say)– to create a new function that then gets applies to its arguments. We see clearly the idea that complex spaces exist in which the coordinate system relies on a choice of basis, where the basis coordinates are themselves permuted (by the semi-group of cyclic acting on hyper) in order to induce complexity in the (non-linear) estimation process.
We have to remember in the 1930s that Princeton is the intellectual hart of the “then-NSA” math club – playing the role of Cambridge math colleges in the equivalent world of GCCS /Admiralty.
When we look at the very first line
don’t we immediately see notation for a bit vector whose last n bits are sub-vector against whose space calculating a rate of change (which is a way of course of presenting coordinate of a space in a general way). This immediately reminded me of differential binary/boolean calculus and semi-direct products of groups in which the output is “not dependent: on certain components of the vector.