Declarative vs imperative: Math, Algoritms, the universe and everything

 The question of imperative versus pure declarative coding has brought to my mind some may be off-topic speculations.  (so please don´t read it if you have no time to waste):  I´m interested in the misterious relation between mathematics, algoritms and reality (see this for example).  Functional programming is declarative, you can model the entire world functionally with no concern for the order of calculations. The world is mathematical. The laws of physics have no concern for sequencing.

But CPUs and communications are basically sequential.  Life is sequential, and programs run along the time coordinate. Whenever you have to run a program,  you or your compiler must sequence it.  The sequentiation  must be done by you or your compiler or both. The functional declarative code can be sequenced on-the-run by the compiler in the CPU by the runtime. but IO is different. 

You have to create, explicitly or implicitly the sequence of IO actions because the external events in the external world  are not controlled by you or the compiler. So you, the programmer are the responsible of sequencing effects in coordination with the external world. so every language must give you ways to express  sequencing of actions. that is why, to interact with the external world you must think in terms of algorithms, that is , imperatively, no matter if you make the imperative-sequence  (relatively) explicit with monads or if you make it trough pairs (state, value) or unsafePerformIO or whatever. You have to think imperatively either way, because yo HAVE TO construct a sequence. I think that the elegant option is to recognize this different algorithmic nature of IO by using the IO monad.

In other terms, the appearance of input-output in a problem means that you modelize just a part of the whole system. the interaction with the part of the system that the program do not control appears as input output. if the program includes the model of the environment that give the input output (including perhaps a model of yourself), then, all the code may be side effects free and unobservable. Thus, input output is a measure of the lack of  a priori information.  Because this information is given and demanded at precise points in the time dimension with a quantitative (real time) or ordered (sequential) measure, then these impure considerations must be taken into account in the program. However, the above is nonsensical, because if you know everithing a priory, then you don´t have a problem, so you don´t need a program. Because problem solving is to cope with unknow data that appears AFTER the program has been created, in oder to produce output at the corrrect time, then the program must have timing on it. has an algoritmic nature, is not pure mathemathical. This applies indeed to all living beings, that respond to the environment, and this creates the notion of time.

Concerning monadic code with no side effects, In mathematical terms,  sequenced (monadic) code are mathematically different from declarative code: A  function describes what in mathematics is called a  "manifold" with a number of dimensions equal to the number of parameters.  In the other side, a sequence describe a particular  trayectory in a manifold, a ordered set of points in a wider manyfold surface. For this reason the latter must be described algorithmically. The former can be said that include all possible trajectories, and can be expressed declaratively. The latter is a sequence  . You can use the former to construct the later, but you must   express the sequence because you are defining the concrete trajectory in the general manifold that solve your concrete problem, not other infinite sets of related problems. This essentially applies also to IO.

 Well this does not imply that you must use monads for it. For example, a way to express a sequence is a list where each element is a function of the previous.  The complier is forced to sequence it in the way you want, but this happens also with monad evaluation.  

This can be exemplified with the laws of Newton: they are declarative. Like any phisical formula. has no concern for sequencing. But when you need to simulate the behaviour of a ballistic weapon, you must use a sequence of instructions( that include the newton laws). (well, in this case the trajectory is continuous integrable and can be expressed in a single function. In this case, the manifold includes a unique trajectory, but  this is not the case in ordinary discrete problems,) . So any language need declarative as well as imperative elements to program mathematical models as well as algorithms.