expand_less I've been rocking and rolling with friends in social media about monads. All the possible meanings as possibly related, but coming from an information systems background, my primary model comes from theoretical and applied mathematics. Category Theory (CT) in general and Type Theory (TT) in formal systems and computational theory are already well established as the rising trend. Those doing it and learning is know that this what functional programming (FP) is all about and why its currency is also on the rise. I want to start from something a little bit formal, but quickly branch and twist towards essoteric grounds in Leibntz and his monad and other metaphoric and ancient sources on the monad conceptually.
First, the entirety of the formalism. Though the expression and diagrams are the same for CT, TT and metaphorically, we find the language is all changed. But the whole point of this theory is that it isn't the names but the structure, therefore when you translate the names, the same work becomes meaningful in another domain. Here it is:
All M : M(A) -> A
Reads more or less: For all categories (types) M as in method in TT, functor in CT, of the type (category) Method for any type A to itself.
Then if you just notice that this is essentially recursive because M is a type (category) and can be also in the place of A. In other words, methods (functors) of methods to methods of the same type signature.
The Monad:
{{Monad Image| type: Image}}
This is metaphorically The One of Greek thought, the generalization and sum of all mathematical/logicial possibility spaces. In Pierce, this is the general rule that is necessarily of a class of incomplete infinities like the ones we find in Godel, Church and Turing. Projecting Leibniz through a little Kantian language, the thing in itself is monadic. Existential monads come and go in our consensual realities, but the miriads of relationships and synchronicities that make monads significant just are.
Not to sidetrack this topic, but I want to be clear what this doesn't entail. This isn't existance from the infinities of Platonic forms in their eternal completeness. We want to consider and be consistent with putting the real of consensual realities we can affirm before any eternal abstractions. Abstractions are not necessary to Be, that comes later in time. What is marvelous is that it does turn out to be as comprehensible as it is. You can't postulate a gounding in laws of logic and math for existential foundations. You'll come up with less than air, nothing.
That said, we now have the emerging spaces created by monadic coverage even if completeness is a fool's errand. In other words, when I write code (FP) and it works with other monads, I can then compose my monad with other monads and get more monads. This is so straightforward that the bots can do it. They can also write the formal proofs the security and financial policies require for critical infrastructure. With all the right wiz-bang tooling on the front and the back, this become drag and drop high level programming. Domain experts draw the relationships with a tool, and presto, code.
Turtles and Cows
I have no idea who started either of these metaphors, I got Turtles all the Way Down from Feinman and Sperical Cows from Sean Carroll. The first is simply the idea of founding all theory on a recursive model the doesn't need grounding. The why is same for the cows metaphor, the model is an abstraction and being abstract is how the model does its work. We model Cows and gravitational bodies without any "hair" with no detail. In Newton and Einstein, the object is modelled as being all at the center of gravity, which would make it Black Hole world. Every mass would be one at some scale small enough.
Recursion is grounded by just defining the ground floor. Like any recursive algorithm will have a condition where it just knows or computes the ground value(s) on something concrete typically a constant. We are talking turtles because turtle is the spherical cow from the "The universe rests on a Turtle" theory. The Monad in this essay. The Turtle was the wise man's way of saying stop asking stupid questions. The seeker keeps going, the little man is satisfied and goes about his life.
The answer we find in The Monad is more like the Ouroboros eating its tails and becoming a representation of the infinite. What we can't forget about is the fact that we are always using a model. It is necessary as we cannot understand or compute outcomes without them. We don't need perception to be a reality scope, but if it does not make real some vitally related other, this perceptual entity will cease to be.