> Certainly things that we can imagine even slightly, like real-valued
> observers, already have a kind of existence, in that they cause us
> to argue about them.


> That's a bit like saying there is some truth to 1+1=3 just because we 
> can argue about it


> 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.  

When I say 1+1=2 isn't doubtful, without redefining "1","+","=","2", I
am assuming the particular traditional rewrite rules used by everybody,
not alternative systems where symbol "2" is replaced by "6", or "+" by
addition modulo group size.

> And you can build universes in such systems, where the
> arithmetic you find so correct never rears it misshapen head.

Algorithmic TOEs are indeed about all possible rewrite systems,
including your nontraditional ones.

Perhaps you would like to argue that our traditional rewrite system is
doubtful as it cannot prove its own consistence? But algorithmic TOEs
include even rewrite systems that are inconsistent, given a particular
interpretation imposed on innocent symbol strings. They are limited to 
all possible ways of manipulating symbols.

>From any description-oriented and communication-oriented viewpoint,
however, this does not seem much of a limitation as we cannot even
describe in principle things outside the range of algorithmic TOEs.

> 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!

But you cannot even unambiguously define in a formal way (using any
describable set of axioms or rewriting rules) what your intuitive notion
of the "continuum of reals" really means. All provable properties of
the reals are expressible as finite symbol strings, and also have an
interpretation in a noncontinuous, countable model.

Maybe there _are_ important things beyond formal description, things we
cannot sensibly talk about. It seems wise though not to talk about them.

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