# Numerous Numbers

By Bobby Neal WintersWhat is man, that thou art mindful of him?

And the son of man, that thou visitest him?

--The Psalms

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.

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