The Case of
|focuses on questions of convenience by describing different numbering systems, which are all different models for describing the same thing: the profound notion of number itself.|
What is Number such that Man can understand it?
What is Man such that he can understand Number?
(introducing his life's quest to his undergraduate mentor,
the Quaker philosopher Rufus Jones)
Then thou shalt be busy as long as thou live!
Rufus Jones' reply
Numbering systems are an interesting special case of models because they are all describing --- in some sense --- the same thing. This makes them good examples for looking at convenience in models, since that is most of what changes between different numerical description schemes. In particular, we'll use numbering systems as an opportunity to introduce two important ways in which models provide convenience.
Symbolic description uses single tokens to describe complex many-parted systems in the world. By moving a single symbol, the user of a model can describe changes in the nature or context of a complex component.
Structural connotation arranges tokens together in configurations with different meanings. The power of structural connotation is that a finite set of basic symbols and a finite set of structural combinations can yield an infinite diversity of structures. This is one of the places where models get so much of their power.
Before we begin our discussion of symbols and structure, we'll look at a brief description of the concept of number. Doing justice to the concept of number could easily take many chapters (we'll be looking at some variations on numbers in the chapters to come), but for purposes of discussing convenience in models, we'll be able to do with a simpler explanation.
The notion of number is fascinating and profound. Number involves a sort of comparison which unites (for instance) these four different figures with one another:
and also with these figures:
The basis of our everday notion of number is comparison by pairing. We can see that two figures have the same number of elements by pairing their elements:
The psychologist Jean Piaget and his successors have discovered that the ability to do this kind of comparison starts when children are four years old. Anthropologists have found that even cultures which don't have names for large numbers still routinely handle those magnitudes in counting by pairing.
The concept of number is so useful because so many actions don't affect number at all. If we rearrange a group of objects or make one-for-one substitutions, the number of objects does not change. If we put a series of objects in a box or bag and later remove them, the number of objects does not change. This gives us something to literally count on in our planning, commerce, and everyday activity.
The notion of number itself presumes a certain model with certain assumptions: we are deciding what we will count. For instance, these two figures:
are not numerically identical if we count line segments (20 and 15) rather than closed shapes (5 and 5).
The notion of number is not limited to human beings. Crows, among other non-humans, can count up to seven. A crow in hiding (or, obviously, which just believes it is in hiding) can see six hunters enter a hunting blind and not leave its own concealment until all six leave the blind. If five leave, it stays in its place. Note that the approximation of number used by crows to count people (and distinct from that used by people to count crow) requires something like the notion of organism which we discussed in the last chapter. The crow must group hunters as "objects" distinct from their arms, legs, rifles, or six-packs.
In this chapter, we take for granted the notion of number based on counting by pairing and the models which enable it. We will look at models of number and the ways in which they simplify operations of comparison, categorization, and arithmetic.
The simplest model of numbers is the analog model which we used briefly in our discussion of convenience. In an analog model, a number is described by a sequence of marks whose "number" is the number being described. The number five would be represented by five marks:
This model uses the "pairing" definition of number and constructs artificial collections (of marks) for pairing with other collections. This works because Pairings can be chained to one another, as below:
so a set of marks can connect collections of objects which have the same number of items. Chaining of mappings in this way is a sort of "model by example" using a particular stylized example --- just marks --- to represent an important commonality between distinct instances.
The chief problem with this analog model is that nearly all operations on it get increasingly awkward as the numbers being described increase. Even if we could copy the number above in one second, if we need to copy the number 186,000 (as we represent it in the digital system), we would need about an hour. To deal with larger numbers, something must be changed and the kinds of changes we can make are the topic of this chapter.
The numbering system which I grew up with (which most of us grew up with) was first developed in India in the seventh century. It went from there to the near east and from there, it travelled with Islamic expansion to Spain, where it was adopted by the European culture which grew into my own. This chain of adoptions was based (at least in part) on a number of striking features:
each of which simplifies the operations of comparison, categorization, and arithmetic on numbers.
Every numbering system besides analog tallies uses symbols to describe particular numbers. For instance, in the digital system, the symbol "5" represents the number which unites the figures we listed above:
Symbols have many advantages, one of which is that we can compare or move symbols in a single operation even if the symbol describes something quite complex with many parts. We can see that "5" is the same as "5" more easily than we can tell that "|||||" is the same as "|||||." This alone increases the convenience of the model.
The ease of recognition extends from the symbol to systems of symbols. Using the digits "0" through "9" to represent the numbers between "|" and "|||||||||," we can make a table:
+ 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 14 15 16 17 18
that allows the addition of any pair of digits in just three operations:
These three operations will usually be much faster than the one to eighteen operations which analog addition would take. However, they rely on the ability of the human user of the model to quickly find symbols in a set. If the human had to look at each symbol individually to find it (checking "is 1 a 5," "is 2 a 5," etc) using the table would be slower than analog addition.
The value of symbols lies not just in the fact that a single token
represents something complex (reference) but in the ability of the
user of the symbol to quickly find, identify, and disambiguate the
symbols (convenience). To see this, consider adding the analog models
||||| and ||||||| in this table instead:
+ | || ||| |||| ||||| |||||| ||||||| |||||||| ||||||||| | 2 3 4 5 6 7 8 9 10 || 3 4 5 6 7 8 9 10 11 ||| 4 5 6 7 8 9 10 11 12 |||| 5 6 7 8 9 10 11 12 13 ||||| 6 7 8 9 10 11 12 13 14 |||||| 7 8 9 10 11 12 13 14 15 ||||||| 8 9 10 11 12 13 14 15 16 |||||||| 9 10 11 12 13 14 15 16 17 ||||||||| 10 11 12 13 14 15 16 17 18
which confers little advantage because we have to count the tallies to find the individual symbols on the top and left. (Note that we had to use symbolic numbers inside the table to make it fit on the page). Symbols take advantage of our perceptual abilities in order to replace long series of operations (like recognizing analog numbers) with single steps. If the symbols don't fit our perceptual abilities, they are much less useful.
In fact, when humans add numbers in the digital notation, they typically have the entire table "in their heads" and recall small sums almost instantly. This might be impossible if we had to think of a sum like "9+9" as a figure like:
||||||||| + ||||||||
with eighteen lines. So symbols simplify operations within our minds as much as the simplify operations "on paper" in tables like the one above.
The use of symbols is likewise limited by the same natural capacities which confer their advantage. When there are too many different kinds of symbols, it can become complicated to identify them or tell them apart. This is why, as we will discuss later, we combine digits into numbers rather than using a huge number of digit symbols.
Every non-analog numbering system we know of uses symbolic numbers
in one way or another, but the different ways in which they combine
the symbols is revealing. Ancient Egyptian hieroglyphic numbers, for
instance, defined symbols for 1 (), 10 (), 100 (), and 1000 (). These symbols were repeated to indicate
The way in which these numbers were written is interesting. The number above could also be written as:
but convention grouped together similar symbols into a certain kind of positional notation. If we were to construct a story of how positional notation was invented, we could imagine it growing from exactly this sort of convention, where having certain magnitudes in the same position makes arithmetic easier.
Egyptian hieroglyphic and Hebrew numbering systems replaced the repetition of symbols with special symbols for multiples of orders of magnitude. Letters of the Hebrew alphabet were used to represent magnitudes like 80 or 90 as well as 2 or 3, as shown on these two fragments of table:
The hieroglyphic and Hebrew systems are both additive systems because the symbols have direct interpretations as numbers which are added together. The problem with simple additive symbols is that representing some numbers will require either more actual symbols (in the hieroglyphic case, representing 444 requires twelve symbols) or more kinds of symbols (in the Hebrew case, representing numbers between 500 and 1000 would require an additional five symbols for 500, 600, etc.).
Hybrid systems address this problem by introducing a set of symbols for basic numbers (e.g. zero to nine) and a separate set of symbols for orders of magnitude (e.g. ones, tens, hundreds, etc). Combining a symbol from the first set with a symbol from the second set produces a combination which describes multiples of a magnitude, for which the hieroglyphic system used repeated symbols and for which the Hebrew system used different symbols.
The spoken form of numbers in English (and many other languages) is a hybrid numerical model, as in "one thousand nine hundred ninety eight" (1998). (The combination "ninety" becomes a single word in English). In a system like this, descriptions stay relatively small (four hundred four-ty four requires only 5 symbols rather than twelve required in the hieroglyphic system above) and increasing the range of the model by a factor of ten requires adding only a single symbol. The spoken English system actually combines the first two magnitude symbols to create sequences ("hundred thousand") which permit the addition of a new symbol to expand the model's range by a factor of a thousand.
But the hybrid system still has the problem that symbols need to be added to increase the range of the model. The positional notation which we use today uses sequential position to indicate magnitude, using physical space --- which is practically unbounded --- to represent magnitude. This allows us to represent any natural number if we have enough space to write it down.
In addition to the use of digits as symbols for numbers, the digital numbering system uses position to uniformly change the meaning of these symbols. In the one digit sequence "5," the digit "5" indicates the number 5 or "|||||" while in the three digit sequence "358" the digit 5 indicates the number 50 (which I won't describe as tallies to keep my typesetter happy). Digits in different positions have their different interpretations which are added together, so "358" is 300 plus 50 plus 8.
In Chapter 4, we described how this model confers a
logarthmic advantage to addition on this encoding of numbers.
This means that the number of digits we need to describe a number
rises as the logarithm of the number's magnitude:
|Numbers less than:||Digits required|
Positional notation makes arithmetic operations like addition more complicated (the rules are more complex) but practically easier (the number of rules we apply for a given addition may be drastically smaller). We can add two "analog numbers" by just putting them next to each other:
which we can do in a single motion. Of course, this gets more complicated if, for instance, we are adding 186,000 and 234,541 (we might need a crane!) but for small numbers, this is very easy. Our use of the positional notation fails to to take advantage of this natural simplicity in order to make operations on it scale readily to large numbers. This is the opposite of our decision to use symbols (which we could quickly and naturally find and recognize) rather than signs (which we may need to examine closely to distinguish and compare). Sometimes, taking advantage of natural structures makes sense but at other time we want to avoid such structure for other advantages. As ever, it depends on our purposes.
Positional notation is significant because it uses structure to assign meaning. Structure comes from the combination of the same components in different ways to mean different things. The string "358" means something different from the string "583" even though they are composed of the same digits. Structured combination allows a model to describe infinitely many things with finite symbols and finite ways of combining them.
Structure can take many forms. The hybrid system used for spoken numbers relies on structure based on adjacency (in "three thousand," "three" is next to "thousand") while the positional system relies on relative positions of digits within the number.
There are many other ways of using structure for description. For example, fractions describe numbers by using vertical space (e.g. ) to distinguish a number which is used to multiply (on the top) from a number which is used to divide (on the bottom). In an example which we will see in the next chapter, the model of numbers on a graph can use vertical space to indicate one quantity and horizontal space to indicate another.
Some human languages rely much more on the adjacency of sounds than on position. In Latin or Finnish, for instance, the ending of a word tells us whether the word is being used as subject or object in a sentence. For example, an English sentence like "Terry kissed Sean" might be rendered thus:
Sean(was done to) kissed(is action) Terry(did it)
where each word is immediately associated with a role (tokens like "was done to") rather than getting the role from its position, as is the case in English.
Historians know of three times when positional notation for numbers has been developed by human beings. In each of these, position was used to indicate orders of magnitude but a different "base" was used to define those orders of magnitude. The Indian system, which we inherit, used a base of ten, so that position indicated multiplication by powers of ten (100=1, 101=10, 1002=100, etc). The Sumerians developed a system of base 60 (e.g. position indicates multiplication by 1, 60, 3600, etc.) while the Mayans used a base of 20 (e.g. 1, 20, 400, etc.)
Encoding numbers with higher bases does not produce quite as marked advantages as the general shift to a positional notation. The number 186,000 in base 60 takes three digits rather than six, which will not simplify addition nearly as much as the initial shift from analog models to digital models (where we went to six digits from 186,000 tally marks).
Higher bases, however, involve another form of complication because the number of symbols increases. The Indian system is unique in assigning single symbols to each of the "basic" numbers 1 through 9. In the Mayan and Sumerian systems, the basic numbers (of which there were twenty or sixty, respectively) were represented as combinations of other still more basic symbols. In the Mayan system, for instance, a dot represented one and a line represented 5, so that was the Mayan "digit" for 13. The Mayan expression of 186,000 would be:
representing the Mayan digits 1, 3, 5, and 0. (Because the Mayan base 20 is a multiple of the Indian base 10, 186,000 is also a relatively "round" number in the Mayan system.)
The Mayan and Sumerian systems used "models within a model" which may complicate some operations. For example, in the Mayan "186,000" above, it is neccessary to figure out that the four dots on the left are actually two distinct digits (1 and 3) rather than one single digit (4). In addition, even if position is made completely clear, there are more "basic symbols" (20) to learn than in the Indian system. But the human perceptual system can, with practice, learn to identify the Mayan digits as single entities, providing that position is noted beforehand and spacing or other cues clearly indicate the boundaries between positions.
The use of different encoding at different levels is also common in natural languages. Most natural languages (including English) have a phonological level and a grammatical level. At the phonological level, walk+ed indicates "walked in the past" and "tree+s" indicates "more than one tree" while on a grammatical level, these "phonological tokens" are then combined (mostly using position) into sentences of a grammatical model. Interestingly, the two models help each other out in tasks like resolving reference. Thus "tree+s" might indicate either a particular set of trees or the set of things called trees. This ambiguity may be resolved at the level of grammar or meaning in sentences like "the trees are overdue" (from the nursery) or "the trees are overdue" (in a recovering ecology).
Though the digital model makes it much easier to deal with large numbers than an analog model, some numbers are still so big that they are awkward to manipulate even in the digital model. For example, the number of water molecules in a kilogram of water is about:
which is a lot of digits to keep track of or manipulate. The digital model is still buying us a lot: if we were using an analog model and a distinct "mark" was half a centimeter wide and took half a second to draw, the "analog" model of this number would reach around the solar system while taking over the life of the Earth to write. Above, in the digital notation, it fits on a single line. However, it is still awkward to handle.
In order to make such numbers easier to deal with, scientists use exponential notation which applies another "logarithmic" transformation to further reduce numbers like these. Exponential notation represents numbers by a regular number (called the mantissa) and an exponent. The exponent specifies how many times to multiply one number (the base) by itself; the mantissa is then multipled by this huge number. In exponential notation, the number above could be written 3.324x1025 or:
but the compact version (3.324x1025) is much easier to deal with. When the exponent is negative, the mantissa is divided by the product instead of multiplied, describing a very small number (like the weight of a single molecule of water).
Exponential notation only works because we can describe the number using repeated multiplications without losing any information. If the number of water molecules in a kilogram were actually:
we would need to describe it precisely as:
which doesn't really help. But if the number can be described by repetitions of the same multiplication (it has a lot of zeros), exponential notation helps. In fact, scientists and engineers sometimes use exponential notation without knowing exact values by thinking about the precision of their models.
A given kilogram of water might indeed contain:
But to tell whether or not this was the case, the kilogram would have to be measured on an exquisitely sensitive balance (more sensitive than anything one can currently construct). Because of this uncertainty in weighing the kilogram of water, scientists and engineers will use the exponential notation because there is no way that they could tell that they're wrong.
Given the uncertainty in these measurements and the regular use of exponential notation, scientists and enginners take advantage of a set of techniques called numerical analysis to reason about the precision of their calculations. When two numbers are multiplied, the leftmost digits have much more of an effect on the final answer than the rightmost digits. This is what makes it reasonable to multiply approximations (like 3.324x1025) and assume that the result is still valid. Other arithmetic operations (like addition and division) have similar properties and numerical analysis determines the precision of an arithmetical result based on the precision of its inputs.
The use of exponential notation also includes assumptions about the physical world itself. Not only does it assume that arithmetic is less sensitive to the unmeasurable details, it assumes that nature itself is less sensitive to them. If some chemical process, for instance, depended on whether a container contained an odd or even number of molecules, exponential notation wouldn't be useful for predicting the outcome of the process. Fortunately, there don't seem to be any "odd" or "even" rules of this sort, but recent work in chaos and complexity (which we won't go into) suggests that some phenomena may indeed depend on such tiny, unmeasurable, details. The question is still open.
One final property of the Indian numbering system is a consequence of the use of positional structure. Each of the positional systems also had to introduce some token to represent what we today call the "zero."
The use of a linear positional notation requires the presence of a placeholder to indicate when a given position is unfilled. So in the digit sequence "305," the digit "0" indicates that there is no digit to be multiplied by ten in determining the number which it represents. We could just have left that space blank, as in "3 5" but that has several problems. First, if several spaces in a row are blank, as in "300005," it may be difficult to tell exactly how many spaces are in "3 5." Second, spaces may already have a meaning in the numbering system, being used to separate different numbers in a sequence of numbers (as opposed to a sequence of digits).
The zero does not correspond to any number we would naturally count, but is neccessary because of the way we are describing numbers. Of course, the notion of zero, while developed because of the description itself, is useful for many things. Indeed, much of the formal structure of mathematics might be thought of as starting with the need to give meaning to the zero developed by arithmetic.
Numerical models provide an interesting context for discussing convenience because they are all describing identical or similar things: the powerful abstract notion of number. Convenience often involves the use of symbols combined into and manipulated as composite structures. The advantage of symbols is that their arbitrariness allows them to be chosen for ease of manipulation. The advantage of structural combination is that it allows a finite collection of symbols to describe an infinite range of objects.
Structure introduces elements which are solely syntactic and these elements are involved in the logic of the description. It is from this property that we proceed to inference in models, where such imaginary constructions begin to take on meaning.