You struck a personal nerve in me with your following remarks:

Bruno Marchal wrote:
> They are degrees. The worst "unreasonableness" of a (platonist or
> classical or even intuitionist) machine is when she believes some plain
> falsity (like p & ~p, or 0 = 1). The false implies all propositions, so
> that such machine believes everything, including everything about their
> maximal consistent extensions or histories (which does not exist).
> Those machines are just inconsistent.

particularly ,
    "some plain falsity (like p & ~p, or 0 = 1)".

Rather than treat these as 'blatantly false' I have been 
exploring the notion for several years .. 'what conditions,
situations, criteria or states would allow such statements 
to be 'true', and what would it mean in how we define and
manipulate and operate the rest of mathematics?'.

I have discovered that an unprecedentedly un-appreciated 
realm of mathematical relations has existed right before
our minds.  The lack, having kept us trying to cope with
'anomalies' and math issues without the full toolkit of
mathematical instruments.

An example at the core of it is a most simplistic  

                1^1 = 1^0

[one to the exponent one  equals  one to the exponent zero]

To all mathematicians, this is a toss-out absurdity, with 
no 'real meaning'.  n^0 is a convenience tool at best ; along
with  'n/0 is 'undefined''.   We note the consistent/valid
notation, but walk away from any active utility or application.

My thesis is that doing so was a missed opportunity.

To be hyper-consistent, the equation set-up   

                1^1 = 1^0

indicates that there -must- be some valid states/conditions
(not just 'interpretation') when 0 and 1 are 'equal' in some
real meaning/use of the word "equal".  If they can be substituted
in the above equation, without changing a resultant of
calculations (they are embedded in), then they must somewhere
somehow in fact be identical in some way or condition.

The entire ediface of physics is hamstrung because of this,
because mathematical definitions and language compounded
the error by applying - actually DIS-applying - a related
concept .. the notion of 'extent' .. also known as 'dimension'.

Physics and mathematics transform and wholly open up when
we throw away the old concept of 'dimensionless' and instead
reformulate -everything- as 'dimensional'.  Including zero;
including numbers unassociated with variables.

As musch as you are brilliant and mathematically inventive,
your statement  "some plain falsity (like p & ~p, or 0 = 1)"
shows you haven't quite awoken to everything yet.  I hope
I'm in the process of stirring you from your slumber.

Jamie Rose

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