Wow, thanks for making me aware of this. I had no idea.

I didnāt look into math because I thought āmath is a specific case of information transformation, a specific case where every symbol of information represents a quantity. But information doesnāt have to be made up of symbols that represent quantities like numbers, information is just symbols and their patterns.ā

So I thought it would be best to go as general as possible and not ānarrow the scopeā as you say, into a particular subdomain; a special case. Is my intuition wrong about that? Iāll admit it seems like my intuition about these kinds of things really doesnāt seem to match the normal intuition, but I donāt yet see why, or if thatās a bad thing.

Anyway, So when I found Lambda Calculus I realized you can reduce them - I think itās called beta reduction or something. Every function can be reduced to itās simplest form. I thought this was important because there are an infinite amount of functions, but if you define a function as āa particular expression of a more general transformationā you have a much larger infinite. Iām a lowly coder so allow me to illustrate my point in pseudo code:

```
def func_a(arg):
return arg
def func_b(arg):
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 Iām familiar with) thereās no way to automatically reduce `func_b`

to `func_a`

but in lambda calculus, there are algorithms for doing so.

Thus I thought, āok, first of all, you want to isolate every possible transformation on information (every possible function) in its simplified expression.ā Though, now that I talk it over, Iām not actually sure that beta reduction is doing exactly what I wanted: by āsimplestā 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. Thereās lots of ways we can transform information; there are lots of functions. Weāve only named a few of these functions, and those names are arbitrary and human-readable (such as āy-combinatorā or āthe identity functionā). We should instead, name every function. But in order to do that we have to know the tree, the shape, the graph of how theyāre 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. Thatās 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 - 0ās and 1ās that are fed to a modern day computing architecture. Every function is a different combination of 0ās and 1ās. Every function has a name, furthermore, similar functions share somewhat similar names. Thatās cool, but itās arbitrary in that instead of english; human language, itās 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 itās 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, āwell you really canāt do that,ā then shouldnāt I be looking into a more generalized domain of like information theory or something? (I mean āinformation theoryā literally, Iām not using the term as it has become to be known as just a pseudonym for āentropy theoryā).

I know that math is the most abstract of all disciplines so for that reason I think, perhaps math is exactly what Iām looking for, but I really donāt 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?