I like this course on topology as i feel like I’m sitting in Turing’s 1920s shoes while still at school – when he got roughly the same lecture from some now infamous Cambridge math lecturers. Our current teacher has a nice, non-formalist manner to his lecturing. It aligns nicely with the approach take in 1900 math – before the very formalism that was studying for machines took over the humans doing math.
Of particular interest is the material on constant functions.
Turing’s language game suggests he is assuming that any reader would be part of an ongoing course and thus would have context that helped support each argument point. One notes that rationale is simply NOT stated, being presumably contextually defined. In particular, his arguments concerning uniformity, continuity and word problems all make a lot more sense (as does the one-dimensional function ‘alpha()’).
Turing’s argument form, in his On Permutations manuscript, is interested in relations between groups and their (distance) invariants, where the groups happen to be those used in enigma machines and the like (doing polymorphic encoding based on the friedman square formed by the rotation and translation of suitable groups)
We are used, in coding theory, of the trivial code being part of the basis set – one of the atomic blocs out of which certain entire families of other codes can be formed. And here we see two further notions, related to this: 1) the mapping of various point in (I,I) to traversal of a single path, in image space, and 2) an indicator function, taking 1 when ones image point is one said line and 0, otherwise.
Back on Turings’ 7 point geometry, in one particular coset the “particles” wandering (in parallel) along the geometry are on fixed circuits. One can look them as 3 independent cycles (each supporting 1 particles motion) or as 1 cycle (in 3 dimensional space) with an single electron that manifest itself in 3 places at once (based on the magic of quantum mechanics)