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At base the things that we call numbers comprise an ordered sequence of names. We begin with a small set of names (in English; one, two, three, four, five, six, seven, eight, nine, ten) and establish rules for making up new names in the sequence. We also have a pictorial representation of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.) that, by the use of place representation reflects the naming rules, which rules we have based on powers of ten. We call these the counting numbers (also the natural numbers, the whole numbers, and the positive integers) because we use them to mathematize our descriptions of the world and the things in it by counting objects to give collections of those objects numbers.
We define numbers by their property of ordinality, their property of having a definite order. But we use them for their property of cardinality, their property of many-ness. We make the transition from ordinality to cardinality through the process of counting, either explicit or implicit. If we have a set of objects, we match the objects, one for one, with the number names, beginning with"one", in their proper sequence and take the last number name that we match with an object in the set to be the number of the objects in the set.
If we look at two of these number names, represented by the letters A and B, and if we know that A comes before B in the sequence of names, then we say that A is less than or smaller than B or that B is greater than or bigger than A. Now we know as an axiom that we have a smallest natural number, because we know that we have no natural number smaller than"one". We then ask whether we can find a largest natural number, one for which we cannot find any larger numbers. Assume that we have found just such a largest number and that we have a pile of objects whose number matches that presumed largest number. We need only put one more object on the pile to justify making up a new number name, one that is larger than our presumed largest natural number. And we can repeat that process. Because we cannot reach a finish to that process, we say that the natural numbers extend to infinity, to the presumed number Aleph-Null. We say that the set of the natural numbers has a cardinality of Aleph-Null or, more, commonly, that it has infinite size.
Having thus defined the set of numbers, we now want to begin to develop standard procedures for coordinating certain subsets with other subsets; that is, we want to begin to define the concept of a function. In order to do that we must first define the basic operations of combining numbers, the fundamental operations of arithmetic.
Let's begin with the fact that we can count subsets of the numbers themselves; in particular, we can count how many names we have in any sequence of successive number names. Thus, for example, I count four elements in the subset "seven, eight, nine, ten" and I do so by using a one-to-one matching of the names; that is, I match "one" with "seven", "two" with "eight", "three" with "nine", and "four" with "ten". Of course, for any such sequence beginning with the element "one" the number of elements in the subset coincides with the last name in the sequence. This fact gives us the axiomatic basis for determining the number of things in a collection by counting them.
So start with a sequence that begins with"one" and ends with the name N. Then consider the sequence that begins with the name that comes immediately after N (defined as the successor of N) and ends with the name M and assume that counting gives us the name P as the number of elements in that latter sequence. Now we know that if we combine a set that has N elements with a set that has P elements, we obtain a set that has M=N+P elements. We thus define the operation of addition.
By defining numbers as an ordered sequence of names and by defining counting as the one-to-one matching of the elements of a set with those names beginning with"one", we make our definition of addition axiomatically true to mathematics. That means that it remains true whenever we use it in a mathematical proposition, even though we did not infer it in the context of a mathematical proposition outside itself. We base our judgment of that truth on the knowledge that nothing can change the sequence of names that we have used to define the numbers; once we have established that sequence, then all of mathematics consists of our using the relationships among the positions of the names in the sequence to spin out yet other relationships.
Having thus defined addition as the means of determining the number of elements comprising a set made by combining the elements of at least two smaller sets, we want to reverse the process. Given a set, we match each of its elements with one and only one of the names in the number sequence from"one" to M and we say that M names the number of the elements in the set. We remove a subset of elements from that set and count them, getting N as the number of the elements in the subset. We then count P as the number of the elements left in the original set, so we have
M-N = P.
In essence we have two sets; one containing P elements and one that contained M elements and then had N elements removed.
Add N to both sides of that equation and get
M-N+N = P+N = M.
Here I implicitly invoke Euclid's Common Notion #2 (If equals are added to equals, then the wholes are equal). If set A equals set B (that is, they match, element for element, and have no elements in common) and if set C equals set D, then we write A with C and B with D and know that
A+C = B+D,
in accordance with Euclid. The fact that M-N+N = M gives us a new number, the number of the empty set, zero.
We have P-M = N. Can we have M>P? The statement -N+N = 0 implies that we can say yes to that question if we take the symbol -N to represent a number in its own right rather than a number combined with the operation of subtraction. We can then have (+N)+(-N) = 0.
If we conceive the natural numbers as markers that we draw next to a line extending to the right, then the new numbers comprise a set of markers that we draw next to the line extending to the left. We can give those new numbers any names that we like, but we have our best choice in using the same names as we use for the natural numbers modified with the word"negative": thus, we can say "positive so-much plus negative so-much equals zero". Those new numbers, the negative numbers, allow us to express debts or deficiencies. Now we can describe holes, depths as distinct from heights, and so on. And, as with the positive integers, we can extend the set of negative numbers indefinitely.
Euclid's Common Notions
In this essay I have implicitly used what Euclid called common notions. Although Euclid included them in his"Elements", his treatise on plane geometry, we can just as well interpret them as the axioms of arithmetic. In order, they tell us that:
1. Things which equal the same thing also equal one another;
2. If equals are added to equals, then the wholes are equal;
3. If equals are subtracted from equals, then the remainders are equal;
4. Things which coincide with one another equal one another; and
5. The whole is greater than the part.
Now consider what those axioms mean for arithmetic.
We say that equality constitutes a binary relation on a set of magnitudes and that it also constitutes an equivalence relation in having the properties of
1. Reflexivity: for each number X, we have X=X;
2. Symmetry: if X=Y, then Y=X; and
3. Transitivity: if X=Y and Y=Z, then X=Z.
Item #3 on that short list coincides with Euclid's first common notion.
Next assume the existence in mathematics of a binary operation called addition and assume that it satisfies the axioms
1. Substitution of equals: for any numbers X, Y, and Z, if X=Y, then X+Z=Y+Z;
2. Associativity: for each number X, Y, and Z, (X+Y)+Z=X+(Y+Z); and
3. Commutativity: for numbers X and Y, X+Y=Y+X.
Item #1 on that list coincides with Euclid's second common notion.
Assume the existence in mathematics of a binary operation called subtraction and assume that it satisfies the axioms
1. Substitution of equals: for any numbers X, Y, and Z, if X=Y, then we must have X-Z=Y-Z; and
2. Non-commutativity: for numbers X and Y, X-Y≠ Y-X.
Item #1 on that list coincides with Euclid's third common notion.
We also have the law of trichotomy: for numbers X and Y we must have one and only one of the following true to mathematics:
B. X=Y, or
Note that while we cannot negate the first four common notions without creating absurdities, we can negate the fifth and get a non-Euclidean arithmetic, the arithmetic, it turns out, of infinity.
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