Dear John S Schultz,
A brilliant observation that, while mathematical logic severely constrains what can be known using algorithmic mathematical patterns, it is not necessarily relevant for other, non-algorithmic types of patterns.
You say "this appears to open the door to a possibility that if human cerebration is non-algorithmic, perhaps we can know the world more completely than the limitation theorems would seem to apply."
You might appreciate the first paragraph on page 8 of my essay Deciding on the nature of time and space.
The history behind that paragraph is the reason that I deeply appreciate your observation that the limitation [math] theorems,... born from the study of algorithmic patterns... have no necessary pertinence to non-algorithmic patterns - and thus no necessary limitations on knowability.
You are correct: it is a big claim. I have for years been trying to formulate this as clearly as you do, with no success. I knew it, but could not so clearly state it. I wish I had been able to make your argument in my essay The Nature of Mind, which had the highest community rating in the 2016-2017 essay contest.
In short, human cerebration is not algorithmic, although of course it includes 'logic', hence algorithmic patterns. The mass-density-based flows in axons and across synaptic gaps generate gravito magnetic fields, and the [endless possible] flow configurations of local fields are 3-D and dynamic, and as I posit this field to be the 'consciousness field', wherein lies awareness, said awareness is not algorithmically structured. You have provided a major missing argument to me, and for this I sincerely thank you.
The dynamic mass-density flows in the brain's trillions of possible neural net connections enable 3-D fields capable of physically approximating any 3-D 'shape' and the self-aware field hence internally 'models' the 3-D reality we either observe or imagine [we do both!] We do not algorithmically compute 3-D from xs, ys, and zs. It goes without saying that no other consciousness model has any idea how we obtain 3-D awareness of the world, at all scales.
The local field is, of course, just a very dense, very organized field self-identified with the local brain, but contiguous with the self-aware universe that Wheeler and many others propose. This field, in its densest possible form, is the field to which all known forces are assumed to converge at the creation/Big Bang. If this gravitational/consciousness field is the essence of self-awareness, it was here at the beginning, and has self-evolved through to the present, increasingly self-aware.
This lack of inherent limitation of knowability is why I reject Hodge's claim that it is 'hubris' to claim awareness of ontology.
I hope your essay makes as much sense to others as it does to me. I think it is brilliant. It is the missing link argument for me, that puts consciousness in 'mathematical' perspective.
Edwin Eugene Klingman