account and counting part 3
The practical or theoretical difference between a Monary System and Plain Counting
There is a theoretical and practical difference between a unary system and plain counting, though they are related.
1. Unary System
A unary system refers to a base-1 numeral system, where each unit or increment is represented by a single tally mark or symbol. It’s essentially counting one-by-one, using a single digit (often the number “1”) to represent each unit. Every addition is marked, without grouping into higher denominations like in base-10 (decimal) or base-12 systems.
For example:
- In a unary system, counting to 5 might look like: “1, 1, 1, 1, 1”, “o,0,0,O,o” (dots, handmarks, finger-prints or as tally marks like “|||||”), rather than using a digit notation like “5”, “101”, etc.
In practice, a unary system resembles the use of tally marks, where each mark represents one unit, and there are no shortcuts or grouping into tens, dozens, or other numerically structured groups.
Practical Aspects:
- Monary systems are useful for simple, straightforward tracking of items without needing complex numeral representations.
- They are commonly used in counting things in small quantities or in situations where simplicity is essential (e.g., keeping score in games).
- However, as numbers get large, unary systems become inefficient, as the sheer number of marks or “1”s required to represent large numbers would be cumbersome.
2. Plain Counting
Plain counting generally refers to the act of sequentially enumerating items using any numeral system—whether decimal (base-10), duodecimal, dozenal (base-12), binary (base-2), unary or others. In plain counting, numbers are represented usually using multiple symbols (digits) and grouped according to the base of the system being used.
For example:
- In numerical counting, we move from “0, 1, 2, 3, 4, 5…” all the way to “10” and so on, grouping numbers into tens.
- In a duodecimal system, you would count up to 12 before transitioning to the next unit.
Practical Aspects:
- Plain counting allows for more efficient representation of large numbers by grouping numbers into larger units (such as tens, hundreds, thousands).
- It makes arithmetic operations like addition, multiplication, and division easier due to the grouping of numbers.
- Systems like decimal or duodecimal also have established rules for notation and arithmetic operations, making them highly structured and practical for various applications (trade, science, engineering, etc.).
Theoretical Differences
- Grouping and Efficiency:
- The unary system does not group numbers into larger units, meaning it can be less efficient for large quantities. It relies on pure enumeration, where every single unit is individually marked.
- Plain counting in other bases groups numbers into higher orders (like tens in decimal), making it more compact and efficient for larger quantities.
- Representation:
- In a unary system, the only symbol needed is “1” (or tally marks), which is repeated for each count. There is no symbol for “0,” “2,” or any other numeral. It’s essentially an ungrouped system.
- In plain counting, you use multiple numerals (e.g., 0-9 in decimal), allowing for grouped representations of numbers (e.g., “100” for a hundred).
Practical Applications of Monary Systems:
- Tally marks for keeping score or tracking inventory.
- Primitive accounting systems in early societies, where groups of items were counted and marked without formal numeral systems.
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Conclusion
The unary system is a basic form of counting that lacks grouping and efficiency for large numbers, making it suitable only for simple, small-scale applications. Plain counting, on the other hand, involves structured grouping of numbers, making it more flexible and efficient for larger quantities and mathematical operations. The key difference lies in the grouping of units and the efficiency of number representation.–>
The unary system and plain counting share a common foundation in the act of enumeration, but they diverge in both practice and theory. At its core, the monary system is simple: each unit is represented by a single tally, a mark that stands alone. Imagine someone counting stones by placing one pebble at a time in a row; there are no shortcuts, no grouping, just the steady, deliberate marking of each individual unit. This system, sometimes called a base-1 numeral system, is essentially counting one-by-one with no place for larger groupings or numerical abbreviations. Every count is its own symbol, and if you need to represent a large number, you simply add more marks or symbols—an inherently cumbersome approach as the numbers increase.
In practical terms, this system resembles the familiar use of tally marks, where each mark is simply a line added to the previous ones. For small numbers or in contexts where the amount being counted is manageable, this approach works. But when numbers grow larger, the unary system’s inefficiency becomes obvious. There are no groupings, no tens, dozens, or hundreds to break up the steady stream of individual marks. It’s a method that works in simplicity but falls apart when complexity or volume enters the picture.
In contrast, plain counting involves a more structured approach to numbers, one that introduces grouping. When we count in the decimal system, for example, we don’t continue endlessly in ones. Instead, we reach a certain point—10—and move to a new symbol that represents a higher order, like “10” for ten, “100” for a hundred, and so on. This type of counting introduces efficiency by creating structured groupings that allow for large numbers to be represented with fewer symbols. While the monary system continues to mark off individual units, plain counting creates a way to leap ahead, moving in steps that group numbers together.
The theoretical difference between these systems lies in this concept of grouping. The unary system is essentially ungrouped, relying on pure, sequential enumeration. Each mark stands on its own, and there is no larger structure into which the marks are organized. Plain counting, on the other hand, uses the power of grouping to move numbers into more manageable categories. This makes it possible to perform operations like addition, subtraction, multiplication, and division with relative ease. In practice, the monary system can be useful in contexts like tallying small quantities or simple scorekeeping, but its limitations become clear when faced with the need for larger-scale counting or complex calculations.
The simplicity of the unary system makes it a primitive form of counting, one that may have been used by early human societies before the invention of formal numeral systems. It’s easy to imagine a shepherd or trader marking off goods with notches on a stick, a practice that works well for keeping track of small numbers of items but becomes impractical when trying to count more significant quantities. Meanwhile, plain counting, with its grouping into tens, hundreds, and beyond, reflects the evolution of mathematical thinking, where the need for efficiency and structure became essential for trade, science, and everyday life.
In essence, while both monary, unary and plain counting systems serve the same fundamental purpose—enumeration—the difference between them lies in their efficiency and structure. The unary system offers simplicity but at the cost of practicality for large numbers, whereas plain counting introduces structure and grouping, allowing for the kind of numerical shorthand that makes complex operations and large-scale counting possible.
Shifting the focus from plain counting to finger counting
If plain counting refers to finger counting, the narrative shifts from one of mathematical abstraction to one rooted in a more physical and intuitive practice. Finger counting, after all, is one of the oldest and most natural forms of counting, a tactile system that predates written numbers and even language. This method grounds the act of counting in the body, where each finger becomes a symbol, a unit, representing the numbers being counted. The monary system, too, can be seen as a kind of extension of this approach—each tally mark is akin to lifting a finger, sequential and straightforward.
When we count on our fingers, we typically work in sets of ten—five fingers on each hand. In this sense, finger counting can be viewed as an elementary base-10 system. Each finger is raised individually, forming a unit of counting, but when all fingers are extended, the next level is reached, much like moving from 9 to 10 in our decimal system. However, unlike formal numeral systems, finger counting is limited by the number of fingers you have. It is deeply tied to the body, a reminder that counting originated as a practical tool, something as physical as it is abstract.
The transition from unary (or tally-like) counting to finger counting introduces a subtle but important difference: grouping becomes inherent. While monary counting would continue indefinitely with marks for every single unit, finger counting naturally groups in sets of ten or smaller subdivisions. Even when we extend it through mental or imaginative strategies—using a finger to represent a group of tens or multiples—the physical act of grouping through the fingers remains at the heart of it.
In finger counting, each set of raised fingers is a kind of embodied grouping. For instance, when you reach five, there is a pause, a transition, before you move to the second hand, creating a physical sense of completion before the next sequence begins. It’s a structured yet fluid system, rooted in the human body’s natural limits, yet capable of being extended through mental counting systems.
Unlike written numerical systems, which abstract numbers into symbols and notations, finger counting ties each number to a sensory experience, making it tactile and visual. It’s a practice that brings counting into the realm of mind-pictures, where each finger raised represents not just a number but a memory of counting, a shape or symbol formed in the mind. The transition between the body and mind in finger counting reflects the essence of the relationship between painting and writing—both forms of communication that move between the physical and the mental, creating visual and sensory experiences that are interpreted and extended in thought.
In finger counting, as in both painting and writing, there’s a blending of the physical act with the abstract idea. Each finger raised or lowered is a gesture, much like the stroke of a brush or the movement of a pen. The transition between counting and thinking parallels the way a painter or writer moves between physical action and mental visualization, between the external world and the internal world of mind-pictures. In this sense, finger counting can be seen as a primitive form of both accounting and recounting—a system where each number has its own physical presence, and yet the mind moves beyond these physical limits to imagine greater quantities and more complex numbers.
In sum, plain counting as finger counting introduces an embodied aspect to the process, where the physical act of using one’s fingers parallels the visual and sensory evocation found in both painting and writing. It’s a method rooted in human history, connecting our abstract understanding of numbers with the tactile, bodily experience of counting. Finger counting, like painting and writing, blurs the lines between the physical and the abstract, between the body and the mind, offering a practical yet symbolic way of engaging with the world of numbers.
If one counts on the left hand from 0 to 5, one can count the full 6 on the right hand.
If the sixes are counted binary this enables anyone to count up to 192,
the “double zero” or two fists.
So with ten fingers a human being can count in multiples of sixes(right thumb), twelves.
This way of finger counting is a mix of unary left hand, binary right hand and
overall duodecimal counting system.
The aspect of clustering, aligning, grouping tally-mark-likes presents the shift from simple counting to accounting.
A turn from scratch marking presents and self awareness to writing or painting, because this is a way of composing.