The Natural Numbers and The Integers
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We identify manyness as one of the properties of things that occupy the world in which we live. In order to refine our description of that property, to answer the question How many?, we define the set of the natural numbers (the counting numbers) by saying that it comprises a well-ordered sequence of names, nothing more and nothing less. 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. Those rules must acknowledge the fact that we organize our set of names into subsets by tens (for example, the subset that we call one hundred comprises ten tens, the subset that we call one thousand comprises ten hundreds or ten tens of tens, and so on). We also have a pictorial representation of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, etc.) that, by use of place-value notation, reflects the ten-based naming rules. We call these the counting numbers (also the whole numbers and the positive integers) because we use them to mathematize our descriptions of the world and of the things in it by the process of counting, which gives collections of those objects their own numbers.
In order to constitute a well-ordered set, the natural numbers must (and do) possess four necessary properties:
1. Asymmetry; for elements X and Y and a relationship A, if X is A to Y, then Y cannot be A to X. If X is cousin to Y, then Y is cousin to X and asymmetry is not manifest. If X is child to Y, then Y is parent to X and asymmetry is manifest. In the set of the natural numbers the relationship A is before/less (or smaller) than and after/more (or greater) than and we have asymmetry.
2. Transitivity; for elements X, Y, and Z and a relationship B, if X is B to Y and Y is B to Z, then X is B to Z. If X is father to Y and Y is father to Z, then X is grandfather to Z and transitivity is not manifest. If X is subordinate to Y and Y is subordinate to Z, then X is subordinate to Z and transitivity is achieved. In the set of the natural numbers the relationship of before/less than and after/more than is transitive.
3. Connectedness; all pairs of elements in the set may be compared, making the rule"full". If we make up a freight train and specify that tank cars come before boxcars and boxcars come before flat cars, we do not have connectedness. True, we know that tank car A comes before boxcar M, but we do not know whether boxcar G comes before or after boxcar T. The set of the natural numbers possesses connectedness.
4. Initiality; every non-empty subset of the set has a first element. Most importantly, in this case certainly, the set itself has a first element, the number one. It then follows from that fact and the fact that the set of the natural numbers possesses the other three properties described above, that any subset that we identify within that set necessarily has a first element.
Thus the set of the natural numbers stands in the realm of Mathematics as a well-ordered set. Because we can so rearrange any countable set that we can match it with the set of the natural numbers or any subset thereof, we can make any countable set well-ordered. Indeed, that= s what we do when we count objects.
We have defined 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 manyness. 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. That fact leads to standard procedures for coordinating the elements of subsets by manipulating the numbers that we assign to them, beginning with the basic operation of combining numbers, the fundamental operations of arithmetic.
Let there exist a set of white discs and let us count them and give them the number N. Let there exist a set of black discs and let us count them and give them the number M. Combine those two sets and count the elements of the new set, the set of black and white discs, and thereby find the number P as the number of the elements in the set. Write the numbers in their proper sequence from left to right along a line and find N. Write one next to the successor of N (the next number in the sequence), write two next to that number's successor, and continue in that manner until you write M next to P. Thus we have addition;
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. Let there come to us a set of black and white discs, of which we have the number P. We remove the black discs, counting them as we do so, and produce an all-black set with number M. We want to determine the number of the elements comprising the all-white set without actually counting the white discs. On the number line write one next to P, write two next to the predecessor of P (the number that comes before P), and continue that process until you write M next to N+1, the successor of N. Now we know the number of the white discs. Thus we have subtraction;
Add M to both sides of that little equation and recall to mind Euclid= s second common notion ("If equals are added to equals, then the wholes are equal"). So we have as true to Mathematics
But we have above that
so we now have that -M+M gives us the number of elements in the empty set, zero;
We have N=P-M. Can we still have it if M is greater than P? On the number line draw one next to P and draw the numbers from right to left. You will draw P next to one and the successor of P next to zero (the predecessor of one). You continue drawing numbers along the line to the left of zero until you draw M next to whatever we represent with N. In this case N would represent something like a pre-emptive subtraction, something like a debt: it represents a number that we owe to our count. To avoid confusion, we cannot give these new numbers the same names that we give the natural numbers, but we also want to give them names sufficiently similar as to ease their use in calculation: therefore, we give them the same names as we give the natural numbers and the title"negative" (or "minus") because in the process of addition they negate part of the count. That makes the natural numbers "positive" (the numbers that we posit of sets by counting) and we call the combined set of the positive numbers and the negative numbers the set of the integers. With negative numbers we can describe holes, depths as distinct from heights, and other deficiencies. Thus the process of subtraction leads us to create a new kind of number and hints at how we might use it.
We will see this happen again. Whenever we produce a method of combining numbers, the reverse procedure will oblige us to invent a new kind of number. Thus Mathematics grows.
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