[Bruno, please forward this to FoR as I am only
on e-l at the moment]

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Dear Bruno,
Thank you for citing Hardy, and your other remarks as well.
The anchor which mathematics provides versus physics
is 'consistency' of (to pirate a term from consciousness
research) 'qualia'. Physics at the moment requires two
or more distinct and mutually exclusive types of qualities
or notions in order to descriptively capture all the events,
behaviors, and observations known. Not so with math.
Relational qualia is singularly consistent in mathematics.
And it is this pandemic language, versus physic's plurality
of languages, which makes math reality seem 'realer' than
physics.
I think that is about to change, if it hasn't already.
Mathematics also displays breaks in consistency. More subtle
ones than the dichotomy of quantum vs continuum qualia, but
present none the less.
Consider commutative vs non-commutative operations; Abelian
vs non-Abelian. Each is valid. Each produces valid relationships
in math, many with correlates in physical reality. Each
generates locally consistent patterns and numeration. But each
is only a piece of the coordinated whole.
So in a sense, math has its internal 'language' diversity
as well.
I am exploring an extension of this realization. There
are states and concordances of information where information
- strict mathematical specificity - changes and transforms.
There are equation forms which generate the seemingly
nonsensical 'identity' of 1=0. Which challenge common sense
and prior mathematical specificity. But, this is no
stranger than Boolean or Cantorian rules versus standard
mathematical rules {eg: a+a=a versus n+n=2n}.
Regarding your conjectures, I would point out to you that
you ought to be considering a simpler standard .. the meaning
behind the geometric inability to deal with axiomatic
inclusion and proving of 'parallel' sets; rather than approaching
the issue through symmetry/assymetry, reversible/irerversible.
The issue behind the parallel-sets problem is: how
can co-existence be proven when the condition to be specified
is non-connectivity and non-communication between sets?
or reversed
Can co-existence generate conditions that perfectly
restrict operant functioning between some of its
subsets/members?
or rephrased
what are all the possible relations and causal
strings of included-middle with/and excluded-middle?
James Rose