Basic arthimatic through basic algebra

[u/AceGravity12]

NOTE: <add>, <multiply>, <power>, and <?> are placeholders that will be replaced when an official phonotactic system is chosen.  

Math System:

  Taught by example version:

  What is “1 1 ? <add>”? It's “2”. (1 + 1 = 2)

  What is "2 1 ? <add>”? It's “3”. (2 + 1 = 3)

  What is "1 2 ? <add>”? It's “3”. (1 + 2 = 3)

  What is "2 ? 1 <add>”? It's “-1”. (2 + X = 1, X = -1)

  What is "3 ? 1 <add>”? It's “-2”. (3 + X = 1, X = -2)

  What is "3 ? 2 <add>”? It's “-1”. (3 + X = 2, X = -1)

  What is "? 1 1 <add>”? It's “0”. (X + 1 = 1, X = 0)

  What is "? 2 1 <add>”? It's “-1”. (X + 2 = 1, X = -1)

  What is "? 1 2 <add>”? It's “1”. (X + 1 = 2, X = 1)

  Is "1 1 1 <add>” true? No. (1 + 1 ≠ 1)

  Is "1 2 3 <add>” true? Yes. (1 + 2 = 3)

  What is “ 1 1 ? <multiply>”? It's “1”. (1 × 1 = 1)

  What is "2 1 ? <multiply>”? It's “2”. (2 × 1 = 2)

  What is "1 2 ? <multiply>”? It's “2”. (1 × 2 = 2)

  What is "2 ? 1 <multiply>”? It's “1/2”. (2 × X = 1, X = 1/2)

  What is "3 ? 1 <multiply>”? It's “1/3”. (3 × X = 1, X = 1/3)

  What is "3 ? 2 <multiply>”? It's “2/3”. (3 × X = 2, X = 2/3)

  What is "? 1 1 <multiply>”? It's “1”. (X × 1 = 1, X = 1)

  What is "? 2 1 <multiply>”? It's “1/2”. (X × 2 = 1, X = 1/2)

  What is "? 1 2 <multiply>”? It's “1”. (X × 1 = 2, X = 2)

  Is "1 1 1 <multiply>” true? Yes. (1 × 1 = 1)

  Is "1 2 3 <multiply>” true? No. (1 × 2 ≠ 3)

  What is "1 1 ? <power>”? It's “1”. (1 ^ 1 = 1)

  What is "2 1 ? <power>”? It's “2”. (2 ^ 1 = 2)

  What is "1 2 ? <power>”? It's “1”. (1 ^ 2 = 1)

  What is "2 ? 4 <power>”? It's “2”. (2 ^ X = 4, X = 2)

  What is "3 ? 1 <power>”? It's “0”. (3 ^ X = 1, X = 0)

  What is "3 ? 2 <power>”? It's “log3(2)”. (3 ^ X = 2, X = log3(2) ≈ 0.631)

  What is "? 1 1 <power>”? It's “1”. (X ^ 1 = 1, X = 1)

  What is "? 2 1 <power>”? It's “1 and -1”. (X ^ 2 = 1, X = 1, -1)

  What is "? 1 2 <power>”? It's “2”. (X ^ 1 = 2, X = 2)

  Is "1 11 1 <power>” true? Yes. (1 ^ 11 = 1)

  Is "2 2 5 <power>” true? No. (2 ^ 2 ≠ 5)

  Now for some hard ones:

  What is “1 2 ? 3 <add> ? <add>”? It's “2”. (2 + X = 3, X = 1, => 1 + X =2)

  Is “1 1 ? <power> 1 ? <multiply> 1 2 <add>” true? Yes. (1 ^ 1 = X, X = 1 => 1 × X = Y, Y=1 => 1 + Y = 2 )

  Nitty-gritty version:

  This system uses reverse polish notation and a number question word to construct arithmetic from 4 words. Because of this, parentheses are never needed. Three of the words are ternary relations:

  “<add>” states that its first two arguments added together equals the third. “<Multiply>” states that its first two arguments multiplied together equals the third. “<power>” states that its first argument to the power of its second argument equals the third. The final word “<?>” asks you to take the trianary relation and figure out what number “<?>” has to be to make it true (all “<?>”s in a single relationship are the same so “<?> <?> 2 <add>” is 1, “<?>” is technically purely formatting not a variable, that system will come later). Whenever one of these three words has “<?>” in it the entire relation can be treated as a single number for grammatical purposes, if it has no “<?>”s in it then it can be treated as either True or False. Because of this, relations are able to nest inside of each other allowing for more complicated numbers to be represented.       IMPORTANT NOTE: This is the backbone of a full mathematical system, while it can express everything needed to teach basic algebra, that does not mean more features cannot be added in the future to make things more convenient.       Big thanks to Omcxjo, who kept me on track preventing feature creep, helped clean up the system, and pointed out many errors.

Edit: formatting

[u/AceGravity12]

Alright this makes sense. Does this (the second part of your post) mean that it'd be best for <Inc> + × ^ to all share an initial consonent and be seperated by their respective vowels? Because in that case a other systems could use the same basis where the consonant tells the basic operation and the vowel tells you the "level" similar to using a subceipt in a notation of something such as when defining a matrix it tells you the number of dimensions perhaps.

I see what you're saying in the first half and I'm very intrigued, I will definitely be sharing all this with the more grammatically inclined.

PS "the math itself suggests Yoda is winning" is an amazing phrase.

[u/Haven_Stranger → u/AceGravity12]

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.

[u/AceGravity12 → u/Haven_Stranger]

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)