“Lead us from the unreal to the real,”
the Brihadaranyaka Upanishad
Now let’s get serious for a while. Here I claim that Mother Evolution in the macro-process of Biological Evolution, at the level of Molecular Biology have arrived at and “shown us” the most fundamental, Universal “structures” and process shapes (forms), and the mathematics only captured these underlying Universal structures and shapes (forms) and in some cases have clarified their implicit, immanent properties.
But mathematicians have “polluted” the properly captured abstractions by adding convenient to them (for notational , algebraic and for social reasons) non-existent abstract notions,famously claiming that they in principle do not care about the actual existence of their abstractions, even throwign this stance at (You).
So they happily extended the recurringly emergent (from the laws of Causality) shapes with abstract memes of the time and according to the current social contexts, like, adding extra “arrows” to create what they call a full Category, or arbitrary reversion the arrows to produce what they call Duals, and so on.
The underlying Reality, however, does not permit or actualize any arbitrary abstract arrows, just like it does permit or actualize zero, identity elements or morphisms, higher dimensions, or Irrational Numbers. It does not actualize Monoids of Categoies, it “stops at” Semigroups, Natural Numbers, Multiplication as repeated addition.
The point is that if we strip away the added abstract nonsense we will arrive at what is real, or just back to What Is. Molecular Biology has shown us what actually is – some actual “primitives” and even some local optima.
These are:
- an ordered sequence
- a recursive process over an ordered sequence
- the actual duality of “writing down” and “reading”
- the necessary duality of an Initial Algebra and its Interpreter
- an Interpreter (over an ordered sequence) as a Universal Machine
Here is the first aha moment:
| A | T | C | G
is called a Coproduct, (or a Sum-type), but the abstract duality of a Product and Coproduct is an imaginary one.
They are as different as “joins” and “forks”, and these, in turn, are “dual to one another” if and only if we do not use arrows (the notion of directedness) or are frivolously allowed to reverse the direction of the arrows, which the Causality does not allow and reality does not actualize in principle.
One and only one branch can be taken (one and only one of all potential paths will be actualized), direction of arrows can never be changed in principle.
This is exactly what a physical [ordered] sequence encodes and the one-by-one, one-at-a-time process relies on.
An enzyme “observes” either A or T or C or G, and it never changes “direction” (the Order) of observing (reading). The Order is Causal.
On the contrary, the notion of Product.
A, T, C, G
does not imply the particular Order, only the explicit Oder in which the particular sequence has been written down. All the 24 possible combinations (possible different orderings) contain the same information.
Thus the apparent duality is just an abstract bullshit, and they are only conceptually isomorphic to each other, due to the very definition of what it takes to have an isomorphism.
Why I am saying all this? The Molecular Biology does not know anything about abstract Algebraic Structures, yet some underlying structures are just Out There, but they are different from how ow mathematicians define them.
Again, mathematicians define their abstractions as they like them,and then study the properties of what they have ended up with. Reality, however, does not engage in pure intellectual masturbation, apparently due to the energy concerns.
So, the proper notion of a Product – each [of] is presented (and is observable) in the same physical locality, in exactly one of the n! possible orderings (the distinct ways of being observed).
The proper notion of a Sum – any one [pf], and only one [of] (the inclusive OR is an abstract bullshit), is present (and is being observable) in the particular physical locality, as an exactly one [possible] observation .
And these properly captured notions are, obviously, not duals, by capturing, instead of by [symbolic] construction.
This is the end of your first aha-moment.
The second aha-moment is this:
Not all the “unreal” arrows with can be arbitrarily drawn (to “complete” the abstract structure) are Out There, but the associative composition of those which are, is, indeed, “real” .
And our running example is the List type, which is a canonical or even “the” /Recursive Sum-type.
It captures the notion of being either Empty (zero elements) – does not exist in Reality – or being one or more elements in a particular order.
Notice that molecular structures in Reality “begin with 2” – a single atom is not a proper structure – it has no mutual bonds which hold (and define) the structure together. At least two (and their mutual bonds) are necessary to have an actual molecular structure of any kind.
The distinction is similar to just a single element [of a set] and a set of just one element, but, again, over-generalization and over-abstraction loses meaning. There is no such molecular structure of just one atom. Period.
Now the Rules. The rule of structure-preserving (without any information loss) for a composition of Functors (mapping between Categories) is the “real"and Universal one.
One could even postulate, that the only meaningful and well-defined transformations (composition) are those, which preserve some invariant or the other.
The most famous equation of the “composition aspect” of the Category Theory states that the existent “relations” (arrows) should be always structurally preserved – never teared down, never change the wiring.
If there is an arrow (a particular morphism) in the source [Category] there shall be a corresponding arrow (morphism) in the target category.
This is how when one “maps” a function defined on individual elements over a List it gets a List of exactly the same Order (and length) in return. This is not an arbitrary definition of “map”, it is how the preservation of the existing structure is actually achieved.
This is how DNA is getting translated into a bunch the mRNAs and each mRNA is being translated into a corresponding protein. The Causal ordering is invariant, as well as the lengths of the sequence of amino-acids being produced.
Again, the Mother Molecular Biology knows nothing about the necessity of invariants preservation, it just “discovered” that there is the only way of doing a reliable translation (without either any information loss or adding any arbitrary bullshit).
A structure-preserving (immutable) transformations of physical [linear] sequences, which have a Causal Order as its only invariant (the “length invariant” is implied), is the only way to do it.
That was a Homomorphism.
There is also an Catamorphism – a order-preserving reduction of a sequence into a single value, which is a fold, and Anamorphism, which is an unfold (or a scan).
Each of these are recursive processes, and they are proper, non-bullshit generalizations from What Is, almost without any added abstract bullshit (except the notion of an empty list as an identity element, instead of just relying only on the proper exhaustive pattern-matching on the underlying algebraic sum-type),
There is much more to a List.
ghci> :i []
type List :: * -> *
data List a = [] | a : [a]
-- Defined in ‘GHC.Types’
instance Monoid [a] -- Defined in ‘GHC.Internal.Base’
instance Semigroup [a] -- Defined in ‘GHC.Internal.Base’
instance Foldable [] -- Defined in ‘GHC.Internal.Data.Foldable’
instance Traversable []
-- Defined in ‘GHC.Internal.Data.Traversable’
instance Read a => Read [a] -- Defined in ‘GHC.Internal.Read’
instance Show a => Show [a] -- Defined in ‘GHC.Internal.Show’
instance Applicative [] -- Defined in ‘GHC.Internal.Base’
instance Functor [] -- Defined in ‘GHC.Internal.Base’
instance MonadFail []
-- Defined in ‘GHC.Internal.Control.Monad.Fail’
instance Monad [] -- Defined in ‘GHC.Internal.Base’
instance Eq a => Eq [a] -- Defined in ‘GHC.Classes’
instance Ord a => Ord [a] -- Defined in ‘GHC.Classes’
ghci>
Almost everything has been [over]-generalized from what appears to be a “simple” optimal algebraic recursive polymorphic sum-type.
Notice that Maybe is just a List of the length of exactly one (without a recursion), having just two distinct constructions, and Either is the same shape.
The difference is that with Maybe only one of the two possible data-constructors is polymorphic, while with Either both data constructors are, on an arbitrary type, and therefore neither denotes an Emptiness or an Absence of a value.
Again, the underlying Universal algebraic structure is the Coproduct,which captures the universal notion of “any [one] of”, “either [one] of”, “exactly one of”, and never “both” .
A List has an “added” recursion – the notion of “one-or-more-in-a-particular-order” which is a [representation] /invariant of every particular instance of a List, which “becomes” a representation of [a distinct Universal shape of, not a Platonic “concept” of] an Ordered sequence (fuck Platonism).
Here comes the “Data Dominates” principle – the [algebraic] shape of the data stricture dictates the shape of the process which processes (traverses) it one-by-one, one-at-a-time, and this is the discovered Universal principle, which Molecular Biology has “discovered” long before Rob Pike.