Appendix

THE MATHEMATICAL EXISTENCE

OF

REAL NUMBERS

Is there an arithmetical continuum?

THE IDEA OF NUMBER is so fundamental, so much a part of our mind and how we perceive, that it is an irreducible understanding.  It is not possible to define number.  If we tried, we would be going in circles, because the words we used would already contain the idea of number

Even the rules of calculation -- the rules of addition, subtraction, multiplication, and division -- contain within them the idea of number. Therefore we cannot define a number as that which obeys those rules.
(In the rule 'ab = ba,' how many distinct symbols do you see? And are they in any order?)

The following, then, while not a definition, is nonetheless true:

We have the clearest criterion for what we call mathematical existence in Euclid's Elements, where a figure, such as a circle or a square, will exist only when we have drawn it.  A "square" for example will not exist simply because we have the idea of one and, according to that idea, are able to frame a definition.  Rather, it is our ability to draw a square -- to agree that what we have drawn is a square -- which shows that it is more than just an idea.  As with everything in life that begins as an idea, we must bring it into this world. If we cannot, then it is nothing but an idea, which is to say, a fantasy.

Moreover, statements with the word "all" or "every" -- such as "All right angles are equal" -- refer to all that exist, that is, all that we have actually drawn.

We say, then, that a number will exist mathematically at the moment that we name it.  Naming will be a form of producing it.  If we have not named it, whether in writing, speech, or thought, then it does not yet exist.

("Do you mean to say that the number 100 does not exist mathematically until I name it?"  That is correct, and you have just named it)

Expressions such as "all" natural numbers, or "all" real numbers, will refer then to every one that we name; which in practice is all that is necessary.

The idea of a presently existing, actually infinite "set" of numbers is nothing but an idea. What is more, not only does a theory of counting and measuring not need such a philosophical concept, but that concept does not lead to where it was intended, as we will see below,.

See Euclid's enunciation of the theorem that there is no last prime.

The perimeter and area of a square  or of a circle  are not numbers. Yet we assimilate them to numbers. We measure them, which is to say, we name the ratio each has to a unit of measure.  For that purpose the real numbers were created. The reals fall into two categories:  rational and irrational.  A rational number effectively is a nameable number. (Topic 2 of Precalculus.)  Apart from unique irrational numbers such as π and e, names for the irrationals come from the categories of functions:  roots, sines, arcsines, logarithms, and so on.

But an irrational number will exist not only on being named.  It must satisfy a property of any number, which is that we must know how to place it with respect to order.  Our knowledge of 8 is that it is more than 7 and less than 9.  As for an irrational number, we must be able to place it with respect to order relative to any rational number.  Is it less than or greater than 2.71828103594612074?  The verbal or symbolic name of an irrational -- π, e, -- does not answer that. Only a rational approximation will. And such an approximation will depend on the existence of a method, an algorithm, to actually produce one.

For example,

1.414213562373095 < < 1.414213562373096.

That we can "name" to as many decimal places as we please guarantees its mathematical existence as a number.

We say, then, that the sentence "This irrational number exists" means

Now, a graph may be continuous because it is a line.  For the same reason, the x-axis is continuous,  When we now say that a function f(x) is "continuous" at the value x = c, then like any defined term, "continuous" will mean what we say it means, namely that when c and f(c) are numbers that exist, then as x approaches c as a limit -- approaches in the form of a discrete sequence of rational numbers (Definition 2.1) -- then f(x) approaches f(c).

*

Let us not forget the second requirement for the existence of an irrational number, which is that there be a step-by-step algorithm to approximate it, typically by a decimal expansion.  π is an example:

π 3.141592653589793

Here, we begin with a precisely defined number, and we compute its decimal expansion.  But to imitate that in reverse, to begin with what we say is an actually infinite sequence of random digits,

0.2985620957. . . ,

will not qualify it as a number -- unless of course there is a method for determining each next digit.  If there is not, then we have no way of placing that "number" with respect to order.  And such an indeterminate sequence of digits will definitely be nameless.  It will not be a number.

In fact, the English mathematician and father of artificial intelligence Alan Turing proved the following:

To compute the decimal expansion of a real number, it is possible to create an algorithm for only a countable number of them.

In other words, only a countable number of irrationals are computable.  Anyone who depends on a computer program must face that fact.  As far as a computer is concerned, it has no knowledge of  "most" irrational numbers, even though they have names and symbols; e.g. It can never order them relative to any rational number.  For a computer at least, such "numbers" do not exist.

In short, inasmuch as measurements -- numbers that we can know and name -- are the essence of the physical sciences, the theory of real numbers is not a theory of measurement.  In that theory, "numbers" are abstract elements not recognizable as characterized above.  Together with its associated set theory ("The set of real numbers," "The set of points on a line"), the theory of real numbers is the most prominent current example of fantasy mathematics.

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Copyright © 2009 Lawrence Spector

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