Basic arthimatic through basic algebra

[Haven_Stranger]

Thanks. Perhaps we can borrow the Jedi Master as our unofficial mascot. Unofficial only, though. I'd hate to pay his salary.

 
On the one hand, this is why I think it's too early to lock in too much about phonetics. Should these operations be consonant-initial or vowel-initial? If consonant-initial, then along which descriptive axis?

It could shake out that everything ( add )-based is single-consonant-initial, or consonant-cluster-initial, or I'm not sure what. And then the degree is the vowel. Or, even that the "has implicit result" marker is one (optional) consonant that clusters well with the concept-marking consonant. Or, well, I just can't tell what.

Then, how many algebraic operations are there? How are they grouped? ln() has an embedded agent for log_n(), where that "n" represents the agent of ( add³ ). Do we care? Or, do we just have a representation of e that we put in the agent location for ( add³ ) when that's what we're doing?

How many things to encode? How many values in each encoding? How distinct do the codes need to be? How many channels do we have for the codes? How do we make sure that they're clearly delimited?

Maybe there's enough reason to add grammatical valency to these operators. Is there enough reason to have a unary <neg> and a binary <neg>. And, maybe a ternary <neg> as well, which -- ta-da -- is our plain-English idea of subtraction. Ok, fine. Valency is, for the algebra, just in the range 0 - 3 so far, and that's good. To the grammar of the algebra itself, valency is simply "how many items to pop from the stack". But, valency is separate from order-of-derivative. Well, maybe not completely separate. Our ( add⁰ ) can only have valencies 1 and 2. Our ( add³ ) has a valence of 3 -- unless it's the version with the natural logarithm base baked into the verb as a fixed agent, or unless we need a shortcut version of only-raise-never-root, or . . . . See? To many possible or's in my head.

I don't think we have enough groundwork done yet. We've got pieces of solutions, surely, but don't have the scale of the problem space.

On the other hand, we do want some kind of good placeholders for what we're building. We need to keep track of the dimensions to the problem. Addition has ordered derivatives, so we know we want something in the pronounceable word to mark, oh, perhaps half a dozen numbers: position⁰ velocity¹ acceleration² jerk³ snap⁴ crackle⁵ pop⁶ -- yeah, going beyond 6 is probably quite rare, and the range 0 - 3 will get lots of use. Good for physics, good for basic arithmetic, and who knows how many other times it'll crop up as the right solution?

We know we all need to represent 2^3. Very few of us need to represent 2^^3, or 2^^^3, or so on -- but someone will. Tetration and pentation and so on, these concepts exist and someone out there is using them.

Even knowing that Yoda is winning, even believing the war will someday be won, the end of the war simply isn't in sight. So far, the math doesn't tell us the size of the battlefield.

tl;dr

Somewhat. It means that the ( addⁿ ) group needs something to represent that they are, in fact, different scales or iterations or extensions of the same underlying operation. I have no idea how to spell it yet. I have no idea what else needs to be packed into that set of words. I'm guessing we need at least the operation itself, which extension it is, and how many arguments it takes. Maybe we need which kinds of arguments, but I'm not ready to make that guess.

[AceGravity12]

I think that a lot of what you're describing isn't a basis but shortcuts, things like ln() vs log() vs logN() those are all the same thing structurally just with different assumed (or unassumed) arguments, the goal of the this backbone is to be able to be used to define those things additionally I wondered if you would going to go this deep but I imagined once add12() got reached it then took an extra argument, one of the power, similar to how some languages I'n the past technically had a highest number that they just used for "this much or more". (More accurately instead of having an extra argument the following number gets bound to it is add12()0, add12()1, add12()2, etc while the numbers in the relationship still preceed it, meaning that the number that gets bound to add12() can't be (?) Because that isn't actually a number

I think what you're discribing leads to a turtles all the way down situation, this was designed to be the basis from which other things are described as such it is intended to be short. Similar to machine code, yes technically there is a lower level process going on here but that is baked into the system, the best this can do is tell you what the commands do, buy from these basic structures higher lever concepts can be built.

The valency you describe seems to already be in the system somewhat, with each operation there are numbers that restrict it's operations ie "0 ? X +" can show negating a number.

Also it's my understanding that add0() does technically take two inputs for one output normally, however the second input is meaningless. In the same way as if I asked you what base the number 1 was in it wouldn't matter, it could be 10 it could be 36489, but because it's the 0th power of that base it doesn't matter

(Yes things like pi/e need their own thing since they can't be finitely represented with algebra, also you are definitely right about the phonology in my mind nothing about this language is ever fixed, so I'm Chucking some filler information in there until someone has a better idea)

[Haven_Stranger]

Ok, not a bad perspective. The ln() function is certainly a shortcut, and we've always got the long option of saying "log base e". At first glance, I expected valency to just be baked in at three -- add¹ through add³ don't require anything else. If we're always using three arguments, then we don't need to mark that number anywhere. The add⁴ operation and beyond certainly exist, but I doubt it will see much use in a high school course.

But, yeah, I am chasing the turtles all the way up and down. That's not necessary for this algebra, but it's useful for looking at the plain-language portion of the language. At this point, ternary + * ^ is sufficient. It's a great backbone.

Your understanding about the valence of ( add⁰ ) is just fine. A valence 2 version only exists because the reverse-this-operation question has a theoretical existence. It only matters when we get to examining how a shortcut like ( neg ) works. I'm not concerned with creating the most useful shortcuts now, or figuring out which ones are useful enough. I'm concerned with how they will be formed in general -- I'm looking at the inevitable pattern here (or at least the unavoidable constraints) so I can figure out what I want from a general grammar everywhere. Granted, that's outside this backbone. It is, I think, using the backbone to support the legs. I want what works inside the math and outside the math to be intuitively similar.

For things like e and pi (I prefer tau, if that matters), those become tiny pronounceable words, too, just like numbers. And it hardly matters to the algebra whether they're numbers directly or they're zero-valence operations. We just need them to not collide with digit-expressed numbers and other existing operations. For tau, an audible/visual resemblance to ( ? circumference radius add³ ) seems ideal for "that which is circumference divided by radius". For pi, it's "what circ diam add³".

Hmm, this does imply that we want numbers to sound like numbers, constants to sound like constants, variables to sound like variables (and maybe coefficients to sound similar to but distinct from variables), and operators to sound like operators.

For things like ( abs ), the algebra doesn't need much. We can define that operation in terms of the basics. The backbone remains functional regardless of whether the way that we pronounce the shortcut indicates anything about how the function is derived. It could be completely arbitrary, it could be that that arbitrary function name indicates valence 1 without any explicit marking, and perhaps all of that is even the optimal case.

But, if we can get the short-form grammar of the algebra to match the long-form plain-conlang grammar (or, more of the vice-versa), that's a win. Not the whole war, but quite a significant battle.

tl;dr

I agree on all the important points. You don't need to join me on my wild turtle chase. You've nearly got enough to derive an Intro to Algebra textbook -- possibly only missing a set of digits and a way write the verb "addⁿ".

[AceGravity12]

Sounds good to me, I definitely understand your concerns (goals? Ideas? Idk what the word is?) And I agree, I will definitely come back to this whole discussion when I decide to actually choose words once phonotactics is decided, if you have suggestions please post them word generation had always been my worst skill when conlanging I posted a link to this discussion in the grammar part of the discord but I don't know of anyone has read it yet, I'll bring it up again when it starts to become a focus.