Mathematical Magnitude And ExperimentFebruary 20, 2022
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 theyscience and hypothesis 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 fa- miliar 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 ought to have of a quantity which should be measurable and almost tangible.
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.
It may happen that among the numbers of the first class there is one which is smaller than all the rest; if, for instance, we arrange in the first class all the numbers greater than 2, and 2 itself, and in the second class all the numbers smaller than 2, it is clear that 2 will be the smallest of all the numbers of thescience and hypothesis
first class. The number 2 may therefore be chosen as the symbol of this division.
It may happen, on the contrary, that in the second class there is one which is greater than all the rest. This is what takes place, for example, if the first class comprises all the numbers greater than 2, and if, in the second, are all the numbers less than 2, and 2 itself.
Here again the number 2 might be chosen as the symbol of this division. But it may equally well happen that we can find nei- ther in the first class a number smaller than all the rest, nor in the second class a number greater than all the rest.
Suppose, for instance, we place in the first class all the numbers whose squares are greater than 2, and in the second 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 there will be in the first class no number which is 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; and 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.—We are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment.
If that be so, the rough data of experiment, which are our sensations, could be measured. We might, indeed, 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 grammes and a weight B of 11 grammes produced identical sensations, that the weight B could no longer be distinguished from a weight C of 12 grammes, but that the weight A was readily distinguished from the weight C.
Thus the rough results of the experiments mayscience and hypothesis 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 mathe- matical 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.
The Creation of the Mathematical Continuum: First Stage.—
So far it would suffice, in order to account for facts, to intercalate between A and B a small number of terms which would remain discrete. What happens now if we have recourse to some instrument to make up for the weakness of our senses? If, for example, we use a microscope? Such terms as A and B, which before were indistinguishable from one another, appear now to be dis- tinct: but between A and B, which are distinct, is intercalated another new term D, which we can distinguish neither from A nor from B. Although we may use the most delicate methods, the rough results of our experi- ments will always present the characters of the physical continuum with the contradiction which is inherent in it. We only escape from it by incessantly intercalating new terms between the terms already distinguished, and this operation must be pursued indefinitely. We might con- ceive that it would be possible to stop if we could imagine an instrument powerful enough to decompose the physi- cal continuum into discrete elements, just as the telescope resolves the Milky Way into stars. But this we cannot imagine; it is always with our senses that we use our in- struments; it is with the eye that we observe the image magnified by the microscope, and this image must there- fore always retain the characters of visual sensation, and therefore those of the physical continuum. Nothing distinguishes a length directly observed from half that length doubled by the microscope. The whole is homogeneous to the part; and there is a fresh contradiction—or rather there would be one if the num- ber of the terms were supposed to be finite; it is clear that the part containing less terms than the whole cannot be similar to the whole. The contradiction ceases as soonscience and hypothesis
as the number of terms is regarded as infinite. There is nothing, for example, to prevent us from regarding the aggregate of integers as similar to the aggregate of even numbers, which is however only a part of it; in fact, to each integer corresponds another even number which is its double. But it is not only to escape this contradiction contained in the empiric data that the mind is led to create the concept of a continuum formed of an indefinite number of terms.
Here everything takes place just as in the series of the integers. We have the faculty of conceiving that a unit may be added to a collection of units.
Thanks to experiment, we have had the opportunity of exercising this faculty and are conscious of it; but from this fact we feel that our power is unlimited, and that we can count indef- initely, although we have never had to count more than a finite number of objects. In the same way, as soon as we have intercalated terms between two consecutive terms of a series, we feel that this operation may be continued without limit, and that, so to speak, there is no intrinsic reason for stopping. As an abbreviation, I may give the name of a mathematical continuum of the first order to every aggregate of terms formed after the same law as the scale of commensurable numbers. If, then, we intercalate new sets according to the laws of incommensurable numbers, we obtain what may be called a continuum of the second order.