> [re. the existence of non-computable real-valued observers]
> That's a bit like saying there is some truth to 1+1=3 just because we 
> can argue about it

> [Re. Dubito ergo cogito]
> Many things are doubtful. 2+2=4 isn't.

There you go again.  But being sure isn't the same as being right.

Despite the intuitively compelling nature of arithmetic as we know it,
it is really quite arbitrary.  It is compelling only because we
evolved in a world that provided some survival advantage to brains
that interpreted sense experience that way, by way of major
approximations and conflations.  But its formalizations, like the
Peano axioms and the inference mechanism that produces theorems like
1+1=2 really are just arbitrary system of rewriting rules.

Its perfectly easy to construct equally pretty systems where 1+1 = 3
or 1+1 = 1, starting with different initial strings or using different
rewrite rules.  And you can build universes in such systems, where the
arithmetic you find so correct never rears it misshapen head.

What's more, there are situations in our own neighborhood where
alternate arithmetics are more natural than everyday arithmetic.  For
instance, in a lasing medium, if you introduce one photon in a
particular quantum state, and then add another photon in the same
state, it is likely that you will find three photons in that state
(then more and more - Boson statistics: the probability of a new
recruit to a state occupied by n particles is proportional to
n/(n+1)).  Photons in the same state are in principle
indistinguishable from one another, so occupancy of a quantum state is
a purer model of counting than the everyday one: when you count
pebbles, thay remain quite distinguishable from one another, and it
takes an arbitrary high-handed act of abstraction to say that THIS
pebble, with its unique shape, color and scratch pattern is somehow
the same as this other, completely different pebble.

The quantum world in general, with its superpositions, entanglements
and ephemeral virtual particles is probably poorly served by bimodal
Aristotelian logic, never mind mathematical frameworks idealized from
grouping pebbles.

But because you are so exclusively wedded to these parochial ways of
thinking, you feel you can just reject out of hand the existence
(among many other things) of beings able to store, compute and
communicate reals, even though many of their properties can be puzzled
out.  PAH!

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