the great natural compressors

the great natural compressors are mathematical formalisms that serve to describe a generalized physical perspective or theory.

For example, there is the group theoretic description of different particles of physics that encapsulates and catalogs their different properties. It is not the only possible way to describe and classify the fundamental particles or fields of nature, but one that happens to be for the moment convenient and popular.

There is also the Hamiltonian formalism to describe the motion of a rigid body , or a chain of them. A system of differential equations can be used to describe completely the thermodynamic behavior and state of an arbitrary physical system. And so on.

The phrase "mathematical formalism" thus refers not merely to a bunch of theorems, but to a distinct coherent scheme of description or model; often one among other possible ones. They encapsulate a great deal of information in terms of two important things. Namely:

(A) the laws and theorems themselves that go into the description or the model.
The formalism usually boils down to (ie, is expressed as / takes the form of a formula of) a single quantity or object
( eg,
the Hamiltonian is the value of the total energy of a system;
the Lagrangian is the action of some segment of a time dependent path (of motion / or in configuration space) defined in terms of some local property of the path ;
the group description is a "representation" (think of it as a set function) of groups of invariant/symmetric and skew/hermitean transformations on observables
).
Despite this, the laws of physics that enter into play in the system being modeled are all derivable from the formalism. For instance, the Lorenz-Maxwell equations of electro-magnetism are all derivable from the Lagrangian formulation. (see [Shankar] ch. 2 for an explicit example of how ; I don't yet know how particle field equations are derivable from group representations).

(B) The formalism readily gives "solutions" that are empirically verifiable when input data is plugged into it.

This is why mathematical formalisms in physics (as in other fields) appear like compression dictionaries (or they are compression schemes). Both functions served by the formalism seem like obvious acts of compression, giving rather fine-detailed and formal descriptions of nature (or whatever the domain of study which is also ultimately a natural subset). These formalisms are a number of conceptual layers up from the underlying "physical theory" (ie the set of proven laws).

They represent a creative act in physical conceptualization / or theoretical thinking, not merely a deductive act. For useful though they are they are interchangeable , not unique. One gaining favor over another by accomplishing a better feat of compression (what is often refered as simplicity, brevity, conciseness, and gaining more generality by separating the field of application from the formal description and manipulation - very much the same way we like to separate content and presentation on computers).

mathematics - ؟ -

These formalisms are even more conceptual layers up from the basic alphabets of literals and idioms and formal patterns we call mathematics. For this seems to be if not the most immediate , than a reasonably good definition of mathematics. It is a set of conventions of linguistic manipulations. Again the spectre of an isomorhpism can be raised between the process of formal reasoning and mathematical expression.

That those conventions have as their by-product the ability to compute, to induct, deduce, the ability to calculate - ie the aspect of calcul or calculus or حساب in mathematics - is the reason we employ them. It's what's great about them.

The quaint thing is, formal though they are, they must be considered part of natural language.


natural vs. formal languages

I do not know why this notion is often ignored (or maybe it isn't) but whatever formal dialects and languages we concoct that are utterly human-readable are to me reasonably considered a subset of natural languages. For two reasons:
1. "we concoct"ed them
2. they are utterly "human-readable".

The arbitrary languages we figuratively feed into state machines in automata theory classes are reasonably formal. Those are the sets of strings acceptable to or generated by FSA ; such as for instance AB*A.

Likewise the programming languages that people develop which are utterly defined , geared to the constraints of a computer architecture are reasonably formal - but frankly the higher-level they are , the more human-readable they become.

But again computer architectures are natural systems.

nature

When we look at what differences there are between the myriad ecological systems that have developed through this planet's history, and the systems that we humans have built ("artificially"), one is tempted to view as a major difference the notion that our artificial systems were consciously developed - rather than emerged.

But nature does not care for conscious acts. although we can study it, bend it to our will, damage it , and hopefully fix it, nature is oblivious to our "will" and our "conscious" efforts.

Our artificial systems are artificial only to ourselves. To any reasonably distant perspective they remain natural systems, as natural as the next quark pair or the next asteroid.

Even within the bowels of human design and architecture, emergence is a common feature and patterns and architectures grow and develop in spite of us.

References

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