Whys
Let’s ask some questions that no one usually asks.
Why math is the way it is?
Well, there are a couple of universally fundamental notions behind all of mathematics - the notion of an external observer (its inner representation of the environment based on aggregating sensory input) and the notion of a separate pile as an observed basis of any generalized abstract category.
Another universal and fundamental notion is that things (such as atoms or molecules) and processes have their own unique shapes, compounds and components, which give rise to emergent properties of complex entities (basically, everything is a process).
Generalizations (by the mind of an external observer) and abstractions (which put together entities based on similarity by having particular properties in common, ignoring other differences) form the basis of all mathematics.
For each entity (process) there is a universal notion of locality and related (and relative) notion of a distance with respect to it. Locality is the central notion of physics, while distance is central to geometry.
Last but not least, a further step has been taken to study abstract systems which can be built from a finite set of axioms (which are premises) - statements taken for granted (the same as statements of fact - of What Is).
The mind
So, what the mind of an external observer does? It “parses” the input, distinguishes, categorizes (groups up) and labels. The other word for this is “selects”. The mind observes, categorizes, labels (with language “tags”) and selects.
This is the true basis for what we call Logic and Mathematics and. Logic is not a discipline of rigorous argument, prior to that, it is the discipline of the mind of an external observer.
Notice that neither the observed world or an observing mind are arbitrary or unrelated. The mind has been evolved in the very same environment it has been observing and therefore “incorporate” (is subject to) the fundamenal contreaints of this particular environment (such as gravity, temperature, presence of soild and water, day/night changes, etc).
Why to mention all this? Because this is why, in particular, the Set comprehension notation or notations of some logical formalisms are the way they are.
Set comprehensions
The generalized and abstracted out notions of the mind of an external observer are:
- such that (selection)
- and also (a fucking mess)
- or (a partition)
AND ALSO
The notion of and also is the most overloaded one, and has at least three fundamentally distinct meanings.
- more than one (the notion generalized from a pile, usually denoted as
,) - this and that (in the context of attributes or properties, denoted as
&&) - simultaneously (presented at the same locality, which is prior to and implies at the same time)
Once this mess is sorted out adequately (within a proper context), everything becomes straightforward and simple.
OR ELSE
The notion of or else is just a generalization of a distinct partition or selection from a pile.
Comprehensions
\[x | x \in \mathbb{N}, x > 0\] is just a selection based on more than one properties. Notice the comma.
Programming
This ability to correctly trace everything back to the “What Is” is what makes one a good philosopher and a good programmer, which, by the way, implies a good mathematician.
Unlike mathematicians, we as good programmers, are concerned with representations and implementations of our abstractions, not just classifying, naming and studying emergent properties.
This is why we have understand and focus on the structure of values, just like everything in the Universe has its structure (out of which processes emerge).
Tracing everything back to reality allows us to discover and select among the fundamental and optimal structures (a sequence, a tree, a table, a graph), which, in turn, leads optimal code due to the data dominates principle.
The shape of recursive functions tend to mimic the shape of corresponding recursive structures. Nothing is arbitrary or random.
Nothing is arbitrary or random
It is not a random coincidence that well-researched, principle-guided functional languages (and a few other fundamental notations, such as Set Comprehensions, Lambda Calculus, /BNF etc.) are seem to converge to pure high-order functions (the referential transparency property and the property of a lexical closure) and structural pattern-matching. This is, obviously, a universal language derived by billions of external observers, which mimics (matches) “What Is”.
There is nothing arbitrary in Erlang or ML or even Haskell. Just think about
it. There are the whys to justify why, lets say Erlang is the way it is, and
why Math and Logic are the way they are.