Experimental prehistory & various digressions: November 2023

Some civilizations are said to not have names for quantities/numbers greater than one and two.

You can find interesting considerations about that on "The Number Warrior" blog article: Is one, two, many a myth (see link at the end of this article if you are interested).

note, 2024/3/3 : i just read some articles about "toki pona", an interesting conlang containing (originally) only 120 words, and only 3 words for numbers (wan=1, tu=2, mute=many); and about some efforts by the speakers to try to complete that set with other names, even trying to promote non-base 10 systems in order to keep close to the original set of 3 names for numbers. (If interested, try to read from here: https://sona.pona.la/wiki/Recommended_learning_resources).

Not having names for numbers greater than two doesn't prevent you from having an accurate representation for quantities. For example, you can still express exact quantities like "two and one", "two for each of your fingers", "two groups of two for each of the joints of your body", "two groups of two and one group of two and one", "two and one for each of the fingers of the members of our tribe" (and that's an accurate quantity, at least for an instant, even if you don't know how many members are in your tribe, as long as none of them looses a finger before you finish your sentence).

And not having names for numbers doesn't even prevent you from performing calculations. We are going to show that, with the example of multiplications performed with the Yupana, the (mythical or real) counting board of the Inca civilization. (Ok, the Inca civilization had names for numbers, but it's irrelevant here... as we plan to badly exaggerate with this concept).

note, 2024/3/3 : the previous remarks would not be totally true for "toki pona": it has no word for "fingers", "joints"... So... you can still have accurate mental representations for numbers by mentally using those equivalences; you would just find it hard to express them verbally, but that would not prevent you from expressing them graphically, or physically (with a counting board, or a khipu, or gestures, etc.).

First, there have been various hypotheses and representations for the Yupana and for the way it was used. But the most general representation is a plane counting board with rows and cells permitting to describe numbers in base ten. (All of that resembling a lot to the north European counting boards still in use in the XVII th century... hmm, does that seem suspect ? or is it just obvious ? because there may not be a lot of ways to design and use a counting board with numbers expressed in base 10).

Modern Yupana

Example below: image of a Yupana promoted under the name "Yupana Inka". It is generally used with a common method where the position of a token in one cell has no particular meaning, the number of dots/spots in a cell representing the value of the each of the tokens that will be put in the cell... knowing that each horizontal row represents a power of 10.

(image extracted from researgate.net - see link to the original at the end of this article)

Let's try to use only "one, two" to describe the standard Yupana, as "one, two" is enough to describe any size of "many" (ok, "one" alone is already enough for that, but well, it's easier with "one, two") :

- in each row, cells with "one", "two", "one and two", "one and a pair of twos" dots/spots.

- each token that will be put on the board will have the value of the cell in which it is put.

- value of one cell in the first (bottom one) row = quantity of dots/spots in it.

- from the second row upward, value of one cell = as many "pairs of the quantity represented in the leftmost cell of the row underneath" as there are dots/spots in the cell... or ..."twice the value of the leftmost cell in the row underneath" for each dot/spot in the cell... or accumulation, for "each finger in a primate" of the "value of the cell underneath";

- a column can also be attributed a value: number of dots/spots in any one of its cells

(ok, yes, we could orientate the counting board in various ways without really changing its interpretation and workings, we would just need to modify the spatial words in the description).

note, 2024/3/3 : and, ok, using "toki pona", we would not easily be able to formulate verbally those explanations... due to the lack of words, we would probably need a bunch of the existing "toki pona" words, with their caracteristically hyper-broad meaning, to try to describe a "row", a "column", a "spot", a "dot", etc... I don't think i'm going to try that, even if i learn a bit more of "toki pona" :-) . The other solution would be, as for every specialized human activity (think medecine, computers, linguistic, philosophy, etc.) to create a specially adapted set of technical words, designed to describe the matter, and only understood and used by the specialists.

Historical Yupana

You can see here a copy of the mythical/historical only known drawing of a Yupana, represented near an Inca accountant holding a quipu (a link to the original image is present at the end of this article).

We can notice that for some cells, some of the dots in the cell are colored differently, which may represent the presence of a token at a particular spot in the cell, with a particular value depending on this spot in this cell.

Following this principle, we can use a "fractal" interpretation of the Yupana that will allow big multiplications (and, obviously, also additions and subtractions, but that is already easy with the common use of the Yupana for which, instead, big multiplications may be cumbersome, unless we use a diversity of tokens having different values: one, two, three, five, for example). Only difference with the standard interpretation of the Yupana: the value of a token in a cell depends on its position in the cell.

Fractal Yupana

In this fractal representation of a Yupana, we have horizontal rows of rectangular cells that contain elliptic spots that contain rounder dots (that contain nothing).
And, yes, someone may already have described that clearly, but they may have named it something else than "fractal Yupana". A quick search on the internet, mid november 2023, didn't show obvious results (i even tried to ask bing's ai and google's ai (bard: seem to give satisfying and precise answers in general - but without references): they didn't find anything). So... i'm trying a description of my own.

The value of a token that will be positioned in a cell on the board will be :
- in/near a spot = quantity of dots in the spot multiplied by value of the cell
- well outside of any spot = value of the cell (= value of its first/lowest spot)

And as in the standard interpretation of the Yupana:

- value of a column: quantity of spots in any one of its cell (or value of its first (bottom one) cell);

- value of a cell in the first row (bottom one): quantity of spots in it;

- value of a cell in a row that is not the first one: quantity of spots in it multiplied by "twice the value of the leftmost cell of the underneath row" or accumulated "value of the cell just underneath" for "each of the toes in a human" or "value of its column" counted as many times as "twice the value of the leftmost cell of the underneath row";

- value of a row: one for the first row; for every subsequent rows: value of the row just underneath counted as many times as the "quantity of fingers in a marmoset";

And, sure, that could be simpler if it was described in base 10... but we are trying to be "one, two, many - compatible", by not using any name for numbers greater than two. We should be allowed to use the words "multiply by" as it is only "accumulated as many times as", a basic concept already used in the deepest prehistory... just guessing...

"One, Two, Many - compatible" multiplication with a fractal Yupana

Basic moves

First, let's consider a few simple (or less simple) moves illustrating the rules previously stated.

If there is no error here, the quantity represented in this portion of a fractal Yupana does not change (for us it would have a name: "five hundred"), only its expression changes, given as a variety of accumulations of group of quantities... (ok, for us that means "sums of products of numbers").

Neither the meaning of the expression nor the quantity that is expressed need to be known (or named) by the user who can just follow the simple rules to correctly transform at random (or at will) the expression of the quantity. Normally, the user would want do that during a calculation in order to obtain a simpler expression (with fewer tokens, requiring less attention, minimizing the risk of errors).

Basic multiplication moves

Multiplying with a fractal Yupana is just putting elementary products (without even knowing their name or their value) on the board, and simplifying the expression when desired. The only elementary products are those produced from two of the quantities "one", "two", "one and two", "one and a pair of twos" (thethat are structuring the board and with which any number can be expressed).

Placing an elementary product a x b in the right spot of a row is a visual task:

- on that row, find a cell that has a (or b) spots in it [it has a column of value a (or b)] and where one of the spots contains b (or a) dots,

- put the token in that spot.

Illustration :

Naturally, the row on which an elementary product is placed depends on the row of each of the constituents (factors) of the product: we show a simple and "mechanical" way of managing that aspect in the following video:

Here is the same multiplication, but performed with a real Fractal Yupana + a simplified technique using less chips. The handling of the chips on the board can become very natural and quick - with some training - if the chips have a convenient shape, becoming easy to seize, push and put aside:

A curiosity...

There are so many ways you can count from 1 to 100 using a single row of the yupana... see next post, to get a feel of this gigantic quantity... Here is one way to count from 0 to 100. It uses 4 of the ways to count from 0 to 25 on the first 3 columns, and the 3 ways to represent 25 using only the last column.

Experimental prehistory & various digressions

"One, two, many" is compatible with accurate calculations - example with a Fractal Yupana