Continuum

Continuum cannot be explained by geometry.

The geometer always seeks to represent shapes.

• But these are only instruments to him.
• He uses space in his geometry just as he uses chalk.

Too much importance must not be attached to accidents which are often just the whiteness of the chalk.

The pure analyst does not need to dread this pitfall.

• He has disengaged mathematics from all extraneous elements.
• He is in a position to answer our question:—“What is this continuum that mathematicians reason about.”

M. Tannery in his Introduction à la théorie des Fonctions d’une variable has answered this.

Between any 2 consecutive sets of integers, intercalate one or more intermediary sets.

• Then between these sets others again, and so on indefinitely.
• This gives us an unlimited number of terms
• These will be the numbers which we call fractional, rational, or commensurable.

But this is not yet all; between these terms, which, be it marked, are already infinite in number, other terms are intercalated, and these are called irrational or incommensurable.

The continuum thus conceived is no longer a collection of individuals arranged in a certain order. They are infinite in number, but external the one to the other.

This is not the ordinary conception in which it is supposed that between the elements of the continuum exists an intimate connection making of it one whole, in which the point has no existence previous to the line, but the line does exist previous to the point.

Multiplicity alone subsists, unity has disappeared—“the continuum is unity in multiplicity,” according to the celebrated formula.

The analysts have even less reason to define their continuum as they do, since it is always on this that they reason when they are particularly proud of their rigour. It is enough to warn the reader that the real mathematical continuum is quite different from that of the physicists and from that of the metaphysicians.

The mathematicians who are contented with this definition are the dupes of words, that the nature of each of these sets should be precisely indicated, that it should be explained how they are to be intercalated, and that it should be shown how it is possible to do it.

This, however, would be wrong; the only property of the sets which comes into the reasoning is that of preceding or succeeding these or those other sets; this alone should therefore intervene in the definition.

So we need not concern ourselves with the manner in which the sets are intercalated, and no one will doubt the possibility of the operation if he only remembers that “possible” in the language of geometers simply means exempt from contradiction. But our definition is not yet complete, and we come back to it after this rather long digression.

Definition of Incommensurables

The mathematicians of the Berlin school, and Kronecker in particular, have devoted themselves to constructing this continuous scale of irrational and fractional numbers using only the integer.

The mathematical continuum from this point of view would be a pure creation of the mind in which experiment would have no part.

The idea of rational number not seeming to present to them any difficulty, they have confined their attention mainly to defining incommensurable numbers.

But before reproducing their definition here, I must make an observation that will allay the astonishment which this will not fail to provoke in readers who are but little familiar with the habits of geometers.

Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change.

Matter does not engage their attention, they are interested by form alone.

If we did not remember it, we could hardly understand that Kronecker gives the name of incommensurable number to a simple symbol—that is to say, something very different from the idea we think we should have of a quantity which should be measurable and almost tangible.

Kronecker’s definition

Commensurable numbers may be divided into classes in an infinite number of ways, subject to the condition that any number whatever of the first class is greater than any number of the second.

The numbers of the first class might have one which is smaller than all the rest. For example:

• Class 1: all the numbers greater than 2, and 2 itself
• Class 2: all the numbers smaller than 2

2 will be the smallest of all the numbers of Class 1. 2 is therefore the symbol of this division.

On the contrary, Class 2 might have a number greater than all the rest. For example:

• Class 1: all the numbers greater than 2
• Class 2: all the numbers less than 2, and 2 itself.

Here again 2 might be chosen as the symbol of this division.

But it may equally well happen that we can find neither in Class 1 a number smaller than all the rest, nor in Class 2 a number greater than all the rest.

For example:

• Class 1 has all the numbers whose squares are greater than 2
• Class 2 has all the numbers whose squares are smaller than 2.

We know that in neither of them is a number whose square is equal to 2.

Evidently, Class 1 will have no number smaller than all the rest. For however near the square of a number may be to 2, we can always find a commensurable whose square is still nearer to 2.

From Kronecker’s point of view, the √ incommensurable number 2 is nothing but the symbol of this particular method of division of commensurable numbers. To each mode of repartition corresponds in this way a number, commensurable or not, which serves as a symbol.

But to be satisfied with this would be to forget the origin of these symbols; it remains to explain how we have been led to attribute to them a kind of concrete existence, and on the other hand, does not the difficulty begin with fractions?

Should we have the notion of these numbers if we did not previously know a matter which we conceive as infinitely divisible—i.e., as a continuum?

The Physical Continuum

Is the idea of the mathematical continuum simply drawn from experiment?

If that be so, the rough data of experiment, which are our sensations, could be measured.

We might be tempted to believe that this is so, for in recent times there has been an attempt to measure them, and a law has even been formulated, known as Fechner’s law, according to which sensation is proportional to the logarithm of the stimulus.

But if we examine the experiments by which the endeavour has been made to establish this law, we shall be led to a diametrically opposite conclusion.

It has, for instance, been observed that a weight A of 10 grams and a weight B of 11 grams produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grams, but that the weight A was readily distinguished from the weight C.

Thus the rough results of the experiments may be expressed by the following relations:

A = B, B = C, A < C,

which may be regarded as the formula of the physical continuum.

But here is an intolerable disagreement with the law of contradiction, and the necessity of banishing this disagreement has compelled us to invent the mathematical continuum. We are therefore forced to conclude that this notion has been created entirely by the mind, but it is experiment that has provided the opportunity.

We cannot believe that two quantities which are equal to a third are not equal to one another, and we are thus led to suppose that A is different from B, and B from C, and that if we have not been aware of this, it is due to the imperfections of our senses.