The natural numbers

Arithmetic begins with naming the sequence of numbers.  Otherwise, we cannot count.  We name the first number (which in English we call "one" ), the second ("two"), the third ("three"), and so on.  Ordinality -- first, second, third -- is fundamental.

These are the names of the natural numbers, both their cardinal and ordinal forms.  Having named those numbers, we can then make true statements about them, such as "Two plus two is four."

Every thing that exists, and has a name, and we can therefore call one, is a unit.  One apple, one orange, one axiom.  We count units, which is to say, we match them with the sequence of number-names:  One apple, two apples, three apples.  The natural number itself is the actual collection of

discrete units.  5 people, 10 pencils, 20 million stars.  The symbols '5,'  '10,'  '20' -- which are called numerals -- refer to those collections of units, those numbers. That is the nature of language and symbol. The symbol '5' or 'V' is not a number, any more than the word  'cat' will ever say "meeow"

Children in the elementary grades use "manipulatives" -- match sticks, blocks, etc. -- which are actual numbers, not their representatives. But that quickly outlives its usefulness, and it becomes conventional to speak of the numerals -- 1, 2, 3, and so on -- as the natural numbers; which they are not.

Mathematical existence

As far as mathematics is concerned, in what sense do these natural numbers exist -- and what do we even mean by the question?  Do we mean whether five apples or ten trillion stars exist?  Obviously, they do. Or do we mean that the idea of any natural number exists; that there is but one number 5, whose yet-to-be-named form, /////, we recognize in things?  The mathematician Leopold Kronecker once observed, "God made the natural numbers.  Man has made all the rest."

Apart from the idea of a natural number, or the physical existence of one, such as ten fingers, we have the clearest of models for what we call its mathematical existence.  It appears in Euclid's Elements, where a figure, such as a circle or a square, will "exist" only when we have drawn it.  For it is a principle of logic that we may not assume that what we have defined exists.  (We can define a unicorn, but does a unicorn exist?)  And so to simply define a "square" according to our idea of one is not enough. Rather, it is our ability to draw a square -- to produce it -- 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 which exist, that is, all which we have actually drawn.

Let us apply this to numbers.  We say that a number will exist, mathematically, when we name it.  Naming will be a form of producing it.  If we have not named it, whether in writing, speech, or thought, 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!)

Hence, expressions such as "all" natural numbers, or "every" natural number, will refer to every one that we name; which in practice is all we ever require.

Number as magnitude

Physical magnitudes -- lengths, say -- are not numbers, yet we assimilate them to the laws of numbers.  We measure them.  We cannot use the natural numbers, hence we need a kind of number that is itself a magnitude.  They will be the positive real numbers.  They will be the numbers of measurement.

Now what is a measurement?  Let PQ represent some physical

magnitude, say a length; and if it is the unit of measure, say 1 cm, and if RS is a magnitude of the same kind, then the real number n will be the measure of RS when, proportionally,

PQ : RS = 1 : n.

That is what a measurement is:  a name for a ratio to the unit of measure, which we name as 1.

Just as a natural number is a collection of units, which we are able to name, so a positive real number is a magnitude whose ratio to 1 we are able to name.  The real number is not the ratio itself, which is a relationship with respect to relative size (such as between the base of a triangle and its height). But the real number n  names that ratio.

The question, of course, is:  Will there always be a name?

The rational numbers

If AB represents the unit magnitude 1, then if AC is two thirds of

Irrational numbers and their order

Nameless numbers?

Arithmetic demands a name for a number.  We have a system for naming the natural numbers, and therefore we can name any rational number. 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 names are discrete.  On that point alone, it should be clear that there can be no arithmetical continuum -- a continuum of names is an absurdity.  Just as it is not possible to name every point on a line, so it not possible that corresponding to every point there is a number.  Any claim that there is such a number will have an inherent flaw, namely that there are "numbers" with no names.

In other words, no matter how a real number is defined -- whether as an infinite decimal (whatever that means), or as the limit of a rational sequence (whatever that means) -- the claim that the real numbers constitute a continuum will require the oxymoron, "nameless numbers." They will be "numbers" of which we have no knowledge -- which we could never order relative to any fraction -- and therefore do not exist.

It is not possible, in other words, to name -- to measure -- every length.  That is the tension between geometry and arithmetic.  That is the separation of geometry and arithmetic.

In fact, if AB is the unit length, and C is an arbitrary point on the

line, then, although AC will have a ratio to AB, the probability that there will be a number to name that ratio, that is, to name the length of AC, is virtually 0.

There is no arithmetical continuum.

Why the insistence on a continuum of numbers?  It was a logical attempt for the functions of calculus to be functions of continuous variables, and for the values of those variables to be "numbers," rather than the physical magnitudes length and time themselves.  The absence of such a continuum however is not a practical issue. We measure things with rational numbers, anyway.  Irrational numbers are an algebraic gateway to them.  As for the calculations of calculus, we come up with a number -- which is all we have ever done -- even though the theoretical explanation for what we are doing might be nonsense.  (At one time, mathematicians explained calculus in terms of "infinitesimals." And neither Newton nor Leibniz could give an intelligible definition of the derivative)

As for the enunciations of the theorems, they can stay as they are, with the understanding that by "all" real numbers, we mean all that we might name or write while doing a problem; that is, all that will actually exist.

The theory of real numbers, then, together with its associated set theory ("The set of real numbers," "The set of points on a line," etc.), are the most prominent current examples of fantasy mathematics.