Chapter E

Counting and Quantifying

Introduction

The common language can generate misimpressions about numbers. Each integer is an element in a succession. Element x may be separated from zero, for instance, by many or few associations relative to that same kind of separation in the case of another element, y. This is the essential difference between any pair of integers. We often speak of "big numbers" and "bigger numbers" as though big numbers were somehow larger than smaller numbers. Element y is said to be greater than element x whenever element y is next in order from element x, but the common association between greater than and bigger is unfortunate and misleading. It is true that their names get longer as more and more integers are named, but the metaphor is wrong. Some might say that they get farther and farther from zero. The graphical representation of a succession of numbers along a line is widespread, but the concept of distance only applies to such graphical representations. Unlike planets and particles, numbers do not have physical sizes or locations. They are non-set elements associated in ordered sets. They name and distinguish the ordered aspects of other things (typically things that are not numbers). Against enormous precedent, we could try to find alternatives to greater than and less than such as yeaer than and nayer than, but there might always be unintended connotations. The connotations of size, magnitude or distance originate in our applications and not in the nature of numbers.

The term amount is an undefined term. Amounts are often characterized by numbers, but also by terms like more, greater than, many, less, less than or few. An amount of water, weight or heat can be measured or specified, as can the amount of many sorts of things. In the case of individual things, like people or cups, the term number has been widely used as a synonym for the term amount. We can preserve this tradition only by now assigning the second meaning (the synonym) to the term number. I do it reluctantly. I can only hope that the number of context dependent meanings does not rise.

The concept of the one-to-one correspondence has facilitated comparison of sets. We may recite the names of positive integers, in their defined order, as we place a sting of these numbers in a one-to-one correspondence with objects. Integers can be used for counting the size of sets:

Definition - A set is said to be counted if and only if a name of a number is assigned to its size.

We use the term amount in a somewhat more general way, often without reference to sets, and often directly tied to numbers:

Definition - An amount is said to be quantified if and only if a name of a number is assigned it.

Since a number is a non-set element in an ordered set containing only non-set elements, the term number in the definition just stated will apply to new kinds of sets and numbers yet to be defined. Indeed, there is a need to quantify amounts that cannot be quantified with integers alone.

Another Point of Language - Anything may be known by its name or by any one of its names. This means that, in almost all contexts, saying "a name of a number is assigned" and saying "a number is assigned" means the same thing to us. If we are talking about names for numbers, there might be a possibility of confusion, but in most cases there is little chance of it.

As Stated Earlier - N is zero, or N = zero if and only if N is the center element of an integer set.
As Stated Earlier - N is one, or N = one if and only if N is the starting element of the set of positive integers of an integer set.
Definition - N is two if and only if the starting element of a set P of positive integers of an integer set is associated with N, or N is next in order from the starting element of P.

Although it exists in the common language, we have not made a formal connection between "1 thing" and "a single thing" or "0 things" and "nothing" or "2 things" and "a pair of things." So here it is:

Definition - Zero things = nothing.
Definition - One thing = a single thing.
Definition - Two things = a pair of things.

Likewise (because of the meaning of thing), zero elements equals no elements, one element equals a single element and two elements equals a pair of elements. In order to count the elements in a set, we only need to assign the name of an integer to the size of that set. If the name is to be N, then either a one-to-one correspondence with the string from 1 to N in a set of positive integers needs to be shown, or the set is empty and N is required to be zero. So, what about negative amounts?

Accountants can describe both dollar profits, and dollar losses with integers. They name negative amounts with negative integers. There are points of possible confusion. Any string of negative integers (and any string is a set) has a size. This size is always represented by a whole number. There are no sets defined as containing a negative number of elements. There are no anti-elements to balance actual elements. Each association between any given element and the element that is next in order from it is part of a succession of such associations that does not proceed consistently toward zero or away from zero. The succession proceeds away from zero within the positive integers and toward zero within the negative integers. Graphically, if an element p is associated with an element q, element q is just to the right of p whether p is negative, zero or positive. In the graphical sense, the positive direction is toward the right, and the negative direction is toward the left. The accountant assigns negative numbers to debts because then the sum will reveal the balance. The physicist assigns negative numbers to opposing forces so that the sum of the forces will reveal the balance. The description of kinds or directions as being positive or negative is up to those applying the math; but given these choices, the arithmetic of integers is determined by the mathematical definitions involved. Everybody applying math anticipates that 7 minus 12 is equal to a negative number, and that -2 plus 8 is a positive number; but whether a negative result means a debt, or a move downward, or something else is not something that the meanings defined in the arithmetic itself can determine.

While the size of sets cannot be defined as negative, the negative integers are non-set elements that have been defined. Integers, unlike set sizes, have both absolute value and sign. Elements in a set can be counted; but some amounts are not described by counting the elements in a set. We may, however, quantify an account balance or the sum of various forces by assigning a number, whether positive, zero or negative. However, this set-size restriction should not distract us from the realization that sets have provided a path to the creation of integers. In turn, integers provide us with a means of counting and quantifying all kinds of things in the world.

Now we need a way to deal with parts of things. A tomato may weight only part of a pound or cost only part of a dollar. The business of counting and quantifying often requires distinctions other than "a single thing" or "a pair of things" or "many things." Quantifying goes beyond whole numbers to negative numbers, and soon to numbers that can describe parts of things. The remainder of the present chapter is devoted to defining such numbers.

Counting Sets

Definition - T is a partition of M if and only if the following statements are true:
M is an ordered set.
T is an ordered set of strings.
Every element in T is a string in M.
Every element in M belongs to one and only one string in T.
If S1 and S2 are strings in T, then S1 and S2 have a one-to-one correspondence.
If S1 and S2 are strings in T, then the intersection of S1 and S2 is empty.
The concatenation of all of the strings in T is equal to M.
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Definition - C is the concatenation of all of the elements in V if and only if the following statements are true:
C is an ordered set and V is an ordered set.
Each element in V is a string.
If A is associated with B in V, then the ending element of A in C is associated with the starting element of B in C.
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Definition - C is a counting set and Q is the integer set interleaved in C and M is the mantissa set of C if and only if the following statements are true:
Q is an integer set.
M is an ordered set.
M is not an integer set.
The intersection of Q and M is empty.
T is a partition of M.
V is an ordered set of strings such that if string A is in V, then A is the concatenation of x in Q and S in T.
If A is the concatenation of x in Q and S in T, and B is the concatenation of y in Q and R is T, and A is associated with B in V, then x is associated with y in Q and S is associated with R in T.
C is the concatenation of all of the elements in V.
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Definition - The elements in Q are integers in C if and only if C is counting set and Q is the integer set interleaved in C.
Definition - Q is the set of integers in C if and only if the elements in Q are integers in counting set C.
Definition - S is the string from n to m among the integers in C if and only if S is the string from n to m in Q, and Q is the integer set interleaved in C.

Graphical Representation of Example - The dots in the representation of counting set C below are elements in the mantissa set of C:

..........-3..........-2..........-1..........0..........1..........2..........3..........

The integers along this line are the integers of Q, the integer set interleaved in C.

Unless stated otherwise, the integers referred to below are named in base 10.

Three terms have been defined in the above definition. The same set of statements could have been repeated to form the definition of each term separately. This as was done earlier in the case of a shorter definition which gave us the integers and the major subsets of integers in Chapter II.

Integer set Q is a subset of counting set C. Q is interleaved in C in such a way as to include one integer from Q at regular intervals along the succession of elements in C. These integers from Q appear in C in the same order that they appear in Q from the starting element of Q to the ending element of Q. The elements between integers from Q in C are non-set elements that come from ordered set M. The elements between integers from Q in C appear in the same order that they do in M.

Definition - Element y is between x and z in S and z is greater than y if and only if the following statements are true:
S is an ordered set of non-set elements.
S contains elements x, y, and z.
Element y is in the string from x to z
Element y is not equal to x, and element y is not equal to z.
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Definition - S is a string between elements x and y in Q if and only if the following statements are true:
Q is an ordered set.
Elements x and y are in Q.
S is a string in Q.
Each element in S is between x and y in Q.
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Definition - Element n is an integer in counting set C if and only if the following statements are true:
C is a counting set.
Q is an integer set interleaved in C.
Element n is in Q.
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Definition - Element x is a counting set integer if and only if x is an integer in a counting set.

An integer in a counting set is not associated with the integer next in order within an integer set. Each integer in a counting set is associated with a non-integer element in the counting set and some non-integer element is associated with that integer. If x is associated with y in integer set Q, then a succession of non-integer elements intervene between x and y in counting set C. These elements have a definite order because they are strings obtained from ordered set M. The purpose of this structure is to provide numbers (non-set elements in an ordered set of such elements) that might be used to represent amounts different from those represented by integers.

Definition - Element c is a counting number if and only if c is an element in a counting set.
Definition - R is a set of positive counting numbers if and only if the following statements are true:
C is a counting set.
Q is the integer set interleaved in C.
Element x is an integer in counting set C.
Element x = 0 in C.
Element x in C is associated with element s in C.
Element e is the ending element of C.
R is the string from s to e in C.
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Notice that s is the starting element of R.

Definition - L is a set of negative counting numbers if and only if the following statements are true:
C is a counting set.
Q is the integer set interleaved in C.
Element x is an integer in counting set C.
Element x = 0 in C.
Element e in C is associated with element x in C.
Element s is the starting element of C.
L is the string from s to e in C.
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Notice that e is the ending element of L.

Definition - Numbers n and m are symmetric counting numbers if and only if the following statements are true:
C is a counting set containing m and n.
S is a string in C such that n is the starting element of string S, and 0 is the ending element of S.
W is a string in C such that 0 is the starting element of string W, and m is the ending element of W.
S has a one-to-one correspondence with W.
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Definition - Counting number z is the negative of counting number y if and only if z and y are symmetric counting numbers.
Definition - The absolute value of counting number x = v if and only if the following statements are true:
C is a counting set containing x.
If x is in the set of positive counting numbers of C, or of x = 0 in C, then v = x.
If x is in the set of negative counting number of C, then x and v are symmetric.
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The absolute value of x is non-negative whether x is positive, zero or negative.

Definition - Counting numbers x and y are of the same sign if and only if the following statements are true:
C is a counting set containing x and y.
Neither x nor y is positive; or, neither x nor y is negative.
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Definition - The sum of counting number x and counting number y equals counting number z if and only if the following statements are true:
C is a counting set containing x and y.
Integer j is the starting element of C, and k is the ending element of C.
If y is not negative, the string in C from x to z has a one-to-one correspondence with the string in C from 0 to y.
If y is not positive, the string in C from z to x has a one-to-one correspondence with the string in C from y to 0.
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Definition - The difference between counting numbers x and y is d or y subtracted from x is d or x minus y is d if and only if d is the sum of counting number x and the negative of counting number y.
Definition - S is a unit string if and only if the following statements are true:
C is a counting set and Q is the integer set interleaved in C.
S is a string in C.
Elements n and m are integers in counting set C, and n is associated with m in Q.
Element n is the starting element of S.
Element z is the ending element of S.
Element z is associated with m in C.
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Definition - Counting set C has a unit string size equal to s if and only if the following statements are true:
Integer set Q is interleaved in C and contains s.
The size of each unit string in C is s.
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Definition - S is the unit string of integer x if and only if x is the starting element of unit string S.
Definition - S is a unit string mantissa if and only if the following statements are true:
C is a counting set.
T is a unit string in C.
Element x is the counting set integer in T.
If y is not in T, then y is not in S.
If m is in T and m is not equal to x, then m is in S.
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Selecting a Counting Set

In this explanation, the term part is used in the sense that a penny is a part of a dollar, or an ounce is part of a pound, or an inch is part of a foot.

An integer in a counting set is followed by a collection of non-integer elements. That integer together with the other elements (the string mantissa), comprises the a unit string. This leaves open the question of how many non-integer elements are in the unit string.

The accountant, wishing to quantify amounts in dollars and pennies (dollars and cents), may appreciate a counting set that has 100 elements per unit string. After all, there are 100 pennies (100 cents) in a dollar. However, there are 16 ounces in a pound and there are 12 inches in a foot. A surveyor who is paid in dollars and cents but who must measure distances in feet and inches might prefer a counting set that has a total of 120 elements per unit string. The grosser might argue that standard unit string should be 16 times this number. One could imaging a unit string size that is the product of every integer up to1000, or a very much greater number, just so that nobody is left out. Some do think of parts that way, conceiving of a number line that has every possible relationship as an abstract given. Engineers and scientist tend to select a unit string size that is a high exponent of a small number like 16, or 10 or 2. This makes the quantifying of some things approximate, but approximations can be arbitrarily accurate.

Here is an example counting set:

Definition - C is a base 10 counting set if and only if the following statements are true:
C is a counting set and Q is the integer set interleaved in C.
S is a unit sting in C.
N is a positive integer in Q.
The size of S is 10N
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To generalize this concept, we have the following:

Definition - C is a base b counting set if and only if the following statements are true:
C is a counting set and Q is the integer set interleaved in C.
S is a unit sting in C.
N is a positive integer in Q.
Positive integer b in Q is greater than 1.
The size of S is bN
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Names for Counting Numbers

Consider 31 cents. This would be somewhere between no dollars and 1 dollar. If b = 10 and N = 2, then the size of the unit string is 100. The 31st element in the unit string having a starting element equal to zero would be a good number to represent 31 cents. There are always bN - 1 non-integers in a unit string. In this case, there would be 102 - 1 = 99 non-integers in the unit string, ranging from the first non integer to the 99th non integer. This suggests a naming system:

Definition - The greatest integer in counting set C that is not greater than x is j if and only if the following statements are true:
C is a counting set and Q is the integer set interleaved in C.
Element x is in C.
Element j is an integer in counting set C.
If j is associated with n in Q, then either x is between j and n in C or x = j.
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Definition - The name of counting number x is Y.Z if and only if the following statements are true:
C is a counting set and Q is the integer set interleaved in C.
C is a base b counting set and the elements in Q are given base b names.
The size of the unit string in C is bN where N is a positive integer in Q.
Element x is in C.
Elements j and k are in Q.
Element j is an integer in counting set C.
The greatest integer in counting set C that is not greater than x is j.
The size of the string from 1 to k in Q equals the size of the string from j to x in C.
Y is the base b name of j.
If x equals j, then Z is a name comprised of N numerals, all of them 0.
If x is not equal to j, then Z is the base b name of k.
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Consider three dollars and 31 cents. The base 10 name for 31 aptly names the number of parts because any numeral except the numeral on the extreme right represents 10 times the amount as would the same numeral immediately to its right. This holds all the way to the left as j is reached. As j = 3, it represent 10 times the amount that the 3 in 31 represents. The dot only tells us where 1 is - that is, where whole things end and parts begin. In this case, X.Y is 3.31 (the amount of dollars, or the number of dollars).

Notation - If the name of counting number x is Y.Z, and if Y and Z are base 10 names, the notation of x that explicitly reveals the base is Y.Z[base 10]. The dollar amount of the previous paragraph would be 3.31[base 10]. The period between X and Y is traditionally called a decimal point, regardless of the base.