Wow, thanks for making me aware of this. I had no idea.
I didn’t look into math because I thought “math is a specific case of information transformation, a specific case where every symbol of information represents a quantity. But information doesn’t have to be made up of symbols that represent quantities like numbers, information is just symbols and their patterns.”
So I thought it would be best to go as general as possible and not “narrow the scope” as you say, into a particular subdomain; a special case. Is my intuition wrong about that? I’ll admit it seems like my intuition about these kinds of things really doesn’t seem to match the normal intuition, but I don’t yet see why, or if that’s a bad thing.
Anyway, So when I found Lambda Calculus I realized you can reduce them - I think it’s called beta reduction or something. Every function can be reduced to it’s simplest form. I thought this was important because there are an infinite amount of functions, but if you define a function as ‘a particular expression of a more general transformation’ you have a much larger infinite. I’m a lowly coder so allow me to illustrate my point in pseudo code:
arg = arg + 1
return arg - 1
func_b is merely a convoluted expression of the transformation in
func_a (the identity function). In regular old imperative programming languages (the kind I’m familiar with) there’s no way to automatically reduce
func_a but in lambda calculus, there are algorithms for doing so.
Thus I thought, “ok, first of all, you want to isolate every possible transformation on information (every possible function) in its simplified expression.” Though, now that I talk it over, I’m not actually sure that beta reduction is doing exactly what I wanted: by “simplest” I mean the fewest number of symbol changes during the transformation.
I mean if you think about it, the problem is really easy to articulate. There’s lots of ways we can transform information; there are lots of functions. We’ve only named a few of these functions, and those names are arbitrary and human-readable (such as ‘y-combinator’ or ‘the identity function’). We should instead, name every function. But in order to do that we have to know the tree, the shape, the graph of how they’re related to each other. Which means we need to know the transformations on data that can be combined to make all other transformations on data. That’s what inspired my question about the prime functions. However, we can go a little further. Every function actually does have a name in the form of byte code - 0’s and 1’s that are fed to a modern day computing architecture. Every function is a different combination of 0’s and 1’s. Every function has a name, furthermore, similar functions share somewhat similar names. That’s cool, but it’s arbitrary in that instead of english; human language, it’s byte code; machine language (the name, the representation of that function, depends on the context of our modern day hardware and software and arbitrary informational structures such as ascii). What we really want is a non-arbitrary way to name every possible transformation on data. Enter lambda calculus. It seemed that this might provide a non-arbitrary mapping of all possible functions, a graph of their relationships with each other; and by so doing; provide the perfect name for every function; the name of it’s relative location in the graph.
Anyway, regardless of my intuition regarding the usefulness of lambda calculus for this endeavor, you suggest taking a mathematical approach with functional analysis. But my question about that is: is functional analysis the most general case to answer this question, or is it, as I suspect, a specific case? In other words, how do I write a mathematical function that transforms symbols into other symbols not tied to quantities (such as transforming language and textual data)? If the answer is, “well you really can’t do that,” then shouldn’t I be looking into a more generalized domain of like information theory or something? (I mean “information theory” literally, I’m not using the term as it has become to be known as just a pseudonym for “entropy theory”).
I know that math is the most abstract of all disciplines so for that reason I think, perhaps math is exactly what I’m looking for, but I really don’t know. Perhaps the context of symbols that represent quantities is important, and general because symbols always represent relativities - that is they only mean something relative to what other symbols mean.
Anyway, I know this is more philosophical, but I feel like I have to get my mind right on this point before I can jump headlong into trying to find the answer via functional analysis; a branch of mathematics. What am I missing?