Here are some comments about the prevailing view that the
concept of the World or the Universe is safe to use in
any kind of arguments related to referential transparency.
I would be quite cautious here. I am not an expert on
these issues in relation to FP, but I have seen enough
tough examples in physics that make me a bit sceptical here.
I am not fighting the concept itself, but I would like
to see it well described first to know exactly the limits
of its applicability.
In my view, once I have introduced the idea of the outside
Universe into my model I am bound to complicate things quite
dramatically, because now I need to consider the interactions
between the two subsystems A and B (Universe). And they are
often not so simple..
To rigorously describe, and -- what is much more difficult --
to solve such a system one has to treat the compound A+B
as a whole. There are many such examples in physics, where
it is absolutely necessary to think globally. We cannot,
for example describe the system of two interacting electrons
via individual (interfering or not) waveforms: psi(x1) and
psi(x2). We have to use a function psi(x1,x2), which is an
amplitude of probability that one (we do not know exactly
which one) electron is at x1 and another at x2.
But physics also teaches us how to simplify things by
isolating the subsystem A from B -- as long as such
simplification makes practical sense. It is not always
possible though, and often not quite acurate. When you
describe the free fall of a mass "m" in the Earth gravitational
field you do, in fact, simplify the rigorous model by replacing
the two-body problem by the one-body problem. Knowing that
the gravitational pull of mass "m" has practically no effect
on the movement of Earth mass "M", you can just ignore
the Earth as such and instead introduce "one-way" gravitational
force into your model. This methodology of removing
constraints and replacing them by reaction forces (and/or
torques) dates back to Newton and is the first thing
to do in solving countless problems of classical mechanics.
You can also apply it to a quantum model of the hydrogen atom,
because the proton is much, much heavier than the electron.
But you cannot exactly apply it to the model of helium, because
now you deal -- in addition to heavy nucleus -- with two
electrons of the same mass.
I think that the monadic IO provides us with such a
simplification. As long as we realize what are its limitations
and as long as we stay within reasonable limits of the concept
we should be fine here. The operative word here is "realize".
Do we really know those limitations?
I have seen engineers nondiscriminantly applying simplified
engineering formulas to problems where such simplifications
were not valid at all. You know, formulas of the sort "beam
bending moment", taken from some engineering handbook. Such
formulas had been, of course, derived from some basic theory
-- but with a lot of simplifying assumptions: linearity, isotropy,
small deflections, limits put on ratio of dimensions, etc.
But, by the time the formula hit the handbook, all those
assumption have been long forgotten.
I am afraid that once we start taking the monadic concept
too far we are bound to find some obstacles (philosophical or
practical) sooner or later. Same applies to the concept of
the Universe, because we often do not know what is exactly our
model of Universe. How do we model interactions of unknown
or imprecise nature?
Jan
