Numerous NumbersBy Bobby Neal Winters
What is man, that thou art mindful of him?
And the son of man, that thou visitest him?
The Psalmist asked the timeless question “What is man?” thousands of years ago. The answers have come back in many forms. Darwin said man is an animal; Freud said man is a sick animal. Others would say that man is an animal sick enough to care about math.
At least some of us.
Some of us care about mathematics. Some of us care about numbers. The modern, mathematically-minded psalmist might ask: What is number that man art mindful of it?
For most people, that is a truly strange question. Numbers are those things that are written on your bills. You write them in your check register; you add them up at the end of the month; none of them has more than two decimals.
Other people had encountered numbers in a somewhat more sophisticated way. They’ve been in science classes and have encountered Avogadro’s Number, Pi, and the speed of light. Still these numbers are, in most minds, yoked, nay, identified, with their decimal expansions. Our teachers do tell us--and the sicker ones of us do care--that Pi can’t be completely captured by its finite decimal expansion, but for most the distinction is not made between the decimal expansion for the number and the number itself.
In a certain way of looking at the world, the failure of making that distinction is not a bad thing. If you putty over the difference between the two, you can build the pyramids, create the hydrogen bomb, and work on cold fusion. If make the distinction, you might not be worthy any activity besides mathematics.
Mathematicians are careful about making such distinctions and precise about language because they need to prove their assertions. Mathematicians prove their assertions not only so that people will believe them but so their students will understand.
One means of laying the ground work for proof is setting up a system of axioms. Those of you who’ve been through a course in geometry have experienced a system of axioms. Axioms are statements about the objects in your system that allow you to do proofs. What can be done for geometry can be done for the real numbers as well.
This is called a synthetic description of the real numbers. My aim is to stay as un-technical as possible so I won’t go too deeply into detail, but the axioms for the real numbers state the properties of the four arithmetic operations and how they deal with each other and with the order properties of the real numbers. These axioms can be packed into the phrase that the real numbers are a complete, ordered field.
Dealing with mathematical objects synthetically, i.e. by listing properties in the form of axioms is clean. It can be tricky because sometimes one must be rather clever. It is much like trying to tie your shoes when you are too fat to see your feet: you have to be patient and have a good imagination.
There is also the danger that the object you are describing with your axioms might not actually exist.
There is a joke about a woman who went into a store to find a husband. She came to two doors. The one on the left said choose this door for men who are kind and the one on the right said choose this door for men who are kind and make a good salary. She chose the one on the right.
She then came into a small hall that again had two doors. The one one the left said choose this door for men who are kind and make a good salary and the one on the right said choose this door for men who are kind, make a good salary, and are handsome. She again choose the one on the right and again she was in a room with two doors.
This time the one on the left was one like she had just chosen but the one on the right said choose this door for men who are kind, make a good salary, are handsome, and are fantastic lovers. Very excitedly, she chose the door on the right and found herself back out on the street.
Whatever point the one who made this joke had, mine is that sometimes you can put so many conditions upon an object, making them rarer and rarer, until they disappear entirely.
Mathematicians like to have at least one non-trivial example of whatever class of objects they are talking about. These examples have to be described in terms of other well-understood mathematical objects and the language of set theory. This is referred to as making a model.
One means of creating a model of a the real numbers is to begin with the rational numbers. As I said earlier, the real numbers are a complete ordered field. The rational numbers are simply an ordered field; this is to say they lack the property of completeness.
Completeness, in the way of mathematical words, has a very precise, very technical definition. One can discern from the meaning of the ordinary English word completeness that a complete ordered field, such as the real numbers, has something that an ordered field that is not complete, such as the rational numbers, lacks. What is this?
A quick and--to the cognescenti--smart-alecky answer to this is the irrational numbers such as the square root of two and Pi. This is smart-alecky because it ignores a the very real need that the rational numbers have for those irrational numbers. The incompleteness of the rational numbers--again in the English sense of the word--signifies a lack, a deficiency. This lack can be described in two different ways.
The least technical of these two ways involves the existence of least upper bounds. The set of positive rational numbers whose square is no more than two does not have a least upper bound that is a rational number. This fact--in different language--was discovered by the Pythagoreans in ancient Greece some time in the Sixth Century B.C.
The more technical of these two ways involves certain sequences of numbers. You may have heard of infinite sequences of numbers such as ½, ¼, ⅛, and so forth. This sequence of numbers converges to zero. There is a certain type of sequences that are referred to as being Cauchy. All sequences that converge are Cauchy, but not all Cauchy sequences of rational numbers converge to rational numbers. Again, one can easily find Cauchy sequences of rational numbers than converge to Pi and to the square root of two.
What mathematicians do in these two cases is to construct models based on the rational numbers. In the first case, special sets of rational are created and the arithmetic functions are extended to those sets. The objects in this model are no longer rational numbers but sets of rational numbers. In the second case, the objects in the model are sets of sequences of rational numbers.
These two different models of the real numbers are clearly different from each other in terms of what they are, but both of them satisfy the axioms. Each as a complete, ordered field. We call that field the real numbers, and there is a very precise mathematical sense in which that definite article is justified.
But I am becoming a mystic. There are more numbers than we can know. Our need for numbers springs from the world around us in numerous ways and I wonder if by drilling down to one idea of the real numbers if we are missing other things.
But the timeis late, and I want to go home.