Appendix

THE MATHEMATICAL EXISTENCE

OF

NUMBERS

Is there an arithmetical continuum?

THE WORLD IS FULL of things to count. Photons, electrons, atoms, molecules, stars, people. The universe creates itself as distinct things. This is the principle of the unit, which is the form in which we perceive, and of which Euclid spoke.

The UNIT is that by virtue of which every thing that exists is called one.

-- Book VII, Definition 1.

We distinguish those collections without the slightest effort; for the mind that counts is a creation of the same universe which creates that which is counted.  The term "natural number" expresses the idea we have of those units, those collections of same rather than different things. That ability to distinguish -- "These two are the same. These two are different" --is so fundamental, that it is not possible to define the term "natural number" with a simpler concept.  If we tried, the concept would certainly require more effort.  What is more, we would be going in circles, because the concept would surely already involve the concept of number

We perceive numbers, collections of units.  Until we give the idea of those numbers a name, however, there is no mathematics.  And upon naming the natural numbers, we can then proceed to the problem of measuring.  The following, while not a definition, plainly describes all the numbers of mathematics.

We have the clearest criterion for what we call mathematical or logical existence in Euclid's Elements, where a figure, such as a circle or a square, will exist only when we have drawn it.  Obviously, we have the idea of a square, and according to that idea, we frame a definition.  But it is our ability to draw a square -- to agree that what we have drawn will refer to the ideal square -- which shows that it is more than just an idea.  The square that we can actually produce is the square that mathematically exists.  (See the commentary on the definitions of Euclid's Elements.)  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.

As for numbers, we deal with them completely through their names. We say, then, that a number will exist mathematically at the moment we name it, whether in writing, speech, or thought.  Naming will be a form of producing it, of bringing a verbal symbol for that idea into this world.  As for the natural numbers in particular, no matter which one we name, we can always name a larger one.  And it is thanks to the system of positional numeration based on the powers of 10, that we can.  (That system is like an instrument of construction.)  A number, for mathematics, has a potential existence.  But it will not have an actual existence until we name it.  Which in practice is all that is necessary.

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

Expressions such as "all" natural numbers or "all" real numbers, then, will mean all that we might name.  The opposite assertion, that an actual infinity of natural numbers exists all at once now, calls for belief more typical of the demands of religion.  Mathematics can get along perfectly well without such unrealizable ideas. The objects of mathematics can be those that we actually experience, that is, experience as more than just an idea.  If mathematics is in any sense a science, then what more scientific meaning could we give to saying that something exists?  "Fourteen" That actually exists.  Now.

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.  For, if we are to measure a length L, say, then we expect to name a real number x such that, proportionally,

L is to the unit of measure  as  x is to 1.

The reals fall into two categories:  rational and irrational.  A rational number essentially 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 because it has a name.  It must satisfy a property of any number, which is that we must be able to place it with respect to order.  Our knowledge of 8 is that it is more than 7 and less than 9.  An irrational number must pass a similar test. We must be able to decide whether it is less than or greater than any rational number we might name.  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 place with respect to order in this way, guarantees its mathematical existence as a number.

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

Our idea of length is that it is continuous (Lesson 1).  And since we think of measuring length as the distance from the origin O along the x-axis,

the thought was that the values of x must reflect the continuum of lengths by being a continuum of numbers.  One way to express that is to say that, corresponding to each endpoint P of OP, there is a real number x, the coordinate of P, which is the measured length of OP.  In other words, we must be able to measure every length.

But will that be possible?  Will it be possible to name the ratio that every length will have to a unit of measure?

No.  It is impossible to name every point in a continuum – a continuum of names is an absurdity. Names are discrete.  And nameless numbers do not exist, not even potentially.  There is no arithmetical continuum.

(This simple argument is called the semantic rejection.)

That is the tension between geometry and arithmetic, a tension realized by Pythagoras with his discovery of what we call the irrational, and brought to a head with the introduction of coordinate geometry, which has been the guiding methodology since the 17th century. Geometry is of the continuous, while arithmetic belongs to what is discrete, in that it requires names.  It will not be possible to assign a number to every length as its measure.  There is no arithmetical continuum.

Infinite decimals?

"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."

"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

Alice in Wonderland

The symbol for an infinite decimal --

.24059165378. . .

-- was invented as a way to claim that the real numbers constitute a continuum.  But a decimal is a way of representing a number.  And numbers have names.  The name of this decimal --

.2405

-- is "Two thousand four hundred five ten-thousandths."

But an infinite decimal has no name.  It is not that we will never finish naming it. We cannot even begin Infinite decimals, therefore, are not numbers.  An infinite decimal will never be the result of a measurement; we cannot name the sum of infinite decimals; we cannot name their difference; we cannot name their product; and we cannot name their quotient.  Infinite decimals are not numbers.

If a student in an arithmetic class were to say, "Although I cannot name that sum, teacher, that sum exists; and to know that is sufficient," then the student might deserve an A in metaphysics but in arithmetic she would certainly fail.

The symbol for an infinite decimal, although it is called a real number, is intended to refer to a point on what is called the real line.  But a point is a concept completely different from a number.  And to achieve their identity by postulation -- "To every point on the line there corresponds a point on the line" -- is both a tautology and an acknowledgement of defeat.

We can try to make sense of an infinite decimal, however, as being an abbreviation for a limit.  By

.24059165378. . .

we mean the limit of the sequence,

.2,  .24  .240  .2405, . . .

Each term of the sequence is a decimal. Each one has a name.  And the limit of that sequence, if it exists, will itself have a name, because a limit is a number.

Now to generate such a sequence, such a decimal expansion, requires a method, an algorithm.  But to create an algorithm, we must first name the number -- the limit -- whose decimal expansion we want to compute
π, , arctan 7.  Such an algorithm exists for the decimal expansion of π:

π 3.141592653589793

To suppose, however, that there could be algorithms for computing a continuum of real numbers, would require a continuum of algorithms. Again that is absurd.  Algorithms are discrete.  And in the absence of an algorithm, it will be impossible to place a supposedly infinite decimal, such as .24059165378. . . , with respect to order relative to any rational number.

In the absence of an algorithm,  .24059165378. . .  is nothing but a sequence of made-up digits followed by three dots.  It is not the symbol of 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.  For a computer at least, such "numbers" do not exist.

Why the obsession with a continuum of numbers?  It appeared to be demanded by coordinate geometry:  For every point on the x-axis there must be a number which is its coordinate.  But in the actual practice of calculus, it never comes up When we do a calculation, we name a number.  That is all anyone has ever done or ever will do, even though the theoretical explanation in terms of an arithmetical continuum might be nonsense.  At one time, mathematicians explained -- and did -- calculus in terms of "infinitesimals."  And neither Newton nor Leibniz could come up with an unobjectionable way to calculate the derivative -- neither of them had the idea of a limit.  Yet each of them showed that the derivative of x² is 2x.

Here we join the philosophical question of the relationship between what are called logical foundations and truth. Is the Pythagorean theorem, for example, true because, or only until, we prove it? And as any student of logic knows: A false hypothesis can lead to a true conclusion.

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.  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 © 2010 Lawrence Spector

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