4 THE MEASUREMENT OF THE CONTINUOUS
Until now, when speaking of number we have had in view whole number exclusively, [1] and logically this was so of necessity, since we were regarding numerical quantity strictly as discontinuous quantity: between two consecutive terms in the sequence of whole numbers there is always a perfectly definite interval, marked by the difference of a unit existing between these two numbers, which, when one keeps to the consideration of whole number, is in no way reducible. In reality, moreover, it is whole number alone that is true number, what one might call pure number; and the sequence of whole numbers, starting from the unit, continues increasing indefinitely without ever arriving at a final term, the supposition of which, as we have seen, would be contradictory; but it goes without saying that the sequence develops entirely in a single direction, and so the other, opposite direction-that of indefinite decrease cannot be represented by it, although from another point of view there is a certain correlation and a sort of symmetry between the considerations of indefinitely increasing and indefinitely decreasing quantities, as we shall demonstrate further on. However, people have not stopped at whole number, but have been led to consider various kinds of number; it is usually said that these are extensions
or generalizations of the idea of number, and this is true after a certain fashion; but at the same time these extensions are also distortions, and this modern mathematicians seem too easily to forget, since their 'conventionalism' leads them to misunderstand the origin and raison d'être of these numbers. In fact, numbers other than whole numbers always appear above all as the representation of the results of operations that would be impossible were one to keep to the point of view of pure arithmetic, which in all rigor is the arithmetic of whole numbers alone: thus a fractional number, for example, is no more than the representation of the result of a division that cannot in fact be made, that is, one that must be declared arithmetically impossible, and this, moreover, is implicitly recognized when it is said, according to ordinary mathematical terminology, that one of the two numbers in question is not divisible by the other. Here we should point out that the definition commonly given to fractional numbers is absurd; fractions can in no way be 'parts of a unit', as is said, for the true arithmetical unit is necessarily indivisible and without parts; and from this results the essential discontinuity of number, which is formed from the unit; but let us see whence this absurdity arises.
Indeed, one does not arbitrarily consider the results of the aforementioned operations thus, instead of regarding them purely and simply as impossible; generally speaking, it is in consequence of the application made of number-discontinuous quantity-to the measurement of magnitudes belonging to the order of continuous quantity, as, for example, spatial magnitudes. Between these two modes of quantity is a difference of nature such that a correspondence between the two cannot be perfectly established; to remedy this to a certain point, at least insofar as it is possible, one seeks to reduce, as it were, the intervals of this discontinuity constituted by the sequence of whole numbers, by introducing other numbers between its terms, and fractional numbers first of all, which would be meaningless apart from this consideration. It is then easy to understand that the absurdity we just pointed out concerning the definition of fractions arises quite simply from a confusion of the arithmetical unit with what are called 'units of measurement', units that are such only by convention, and that in reality are magnitudes
of another sort than number, notably geometric magnitudes. The unit of length, for example, is only a certain length chosen for reasons foreign to arithmetic, and the number 1 is made to correspond to it in order to be able to measure all other lengths by reference thereto; but all length, even when so represented by the unit, is by its very nature as continuous magnitude no less always and indefinitely divisible. Comparing it to other lengths that are not exact multiples of it, one might thus have to consider parts of this unit of measurement, which would in no way be parts of the arithmetical unit on that account; and it is only thus that the consideration of fractional numbers is really introduced, as a representation of the ratios of magnitudes that are not exactly divisible by one another. The measurement of a magnitude is indeed no more than the numerical expression of its ratio to another magnitude of the same kind taken as the unit of measurement, or, basically, as the term of comparison; and this is why the ordinary method of measuring geometric magnitudes is essentially founded on division.
It must be said, moreover, that in spite of this method something of the discontinuous nature of number is always bound to remain, preventing one from thus obtaining a perfect equivalent to the continuous; reduce the intervals as much as one likes-which finally is to say, reduce them indefinitely, rendering them smaller than any quantity that can be given in advance-but they will never be done away with entirely. To make this clearer, let us take the simplest example of a geometric continuum, a straight line: we shall consider half a straight line, extending indefinitely in a certain direction, [2] and let us agree to make each of its points correspond to a number expressing the distance of the point from the origin, represented by zero, as its distance from itself is obviously nothing; starting from this origin, the whole numbers will then correspond to the successive extremities of all segments equal to each other and to the unit of length; the points contained between these will be representable
only by fractional numbers, since their distances from the origin are not exact multiples of the unit of length. It goes without saying that, taking fractional numbers with greater and greater denominators, hence smaller and smaller differences, the intervals between the points to which these numbers correspond will be reduced in the same proportion; in this way the intervals can be decreased indefinitely, theoretically at any rate, since the possible denominators of the fractional numbers are themselves whole numbers, the sequence of which increases indefinitely. [3] We say theoretically because in fact the multitude of fractional numbers is indefinite, and one could never use them all, but let us suppose that ideally all the possible fractional numbers could be made to correspond to the points on the half of the line in consideration. Despite the indefinite decrease of the intervals, a multitude of points to which no number will correspond will still remain on this line. At first this might seem strange and even paradoxical, but it is nevertheless easily demonstrated, for such a point can be obtained by means of a very simple geometric construction. Let us construct a square having for its side the line segment with extremities at the points 0 and 1 , and let us draw the diagonal of the square starting from the origin, then a circle having for its center the origin and for its radius this diagonal; the point at which this circle cuts the straight line cannot be represented by any whole or fractional number, since its distance from the origin is equal to the diagonal of the square, which is incommensurable with its side, that is, with the unit of length. Thus, the multitude of fractional numbers, despite an indefinite decrease of their differences, still does not suffice to fill, so to speak, the intervals between the points contained in the line, [4] which amounts to saying that this multitude is not a real and adequate equivalent to linear continuity; in order to express the measurement of certain lengths, one is thus forced to introduce still other kinds of numbers, what are called incommensurable numbers, that is, those having no
common measure with the unit. Such are the irrational numbers, which represent the results of arithmetically impossible extractions of roots, as, for example, the square root of a number that is not a perfect square; thus in the preceding example, the ratio of the diagonal of the square to its side, and consequently the point having a distance from the origin equal to this diagonal, can be represented only by the irrational number √2, which is indeed incommensurable, for there exists no whole or fractional number the square of which is equal to 2 ; and besides these irrational numbers there are still other incommensurable numbers, the geometrical origin of which is obvious, as, for example, the number π, which represents the ratio of the circumference of a circle to its diameter.
Without entering further into the question of the 'composition of the continuous', it will thus be seen that number, however far the notion is extended, is never perfectly applicable to it; finally this application always amounts to replacing the continuous with a discontinuity, the intervals of which can be very small, and can even become smaller and smaller still by an indefinite series of successive divisions, but without ever being done away with, for in reality there is no 'final term' to which the divisions might lead, since a continuous quantity, however small it might be, will always remain indefinitely divisible. It is to these divisions of the continuous that the consideration of fractional numbers properly corresponds; but, and this is particularly important to note, a fraction, however minute it might be, is always a determined quantity, and however small one supposes the difference between two fractions there is always an equally determined interval. Now the property of indefinite divisibility that characterizes continuous magnitudes obviously demands that one always be able to take elements as small as one wishes, and that the intervals existing between these elements can likewise be rendered less than any given quantity; but-and it is here that we see the insufficiency of fractional numbers, and even, we can say, of number altogether-in order that there really be continuity, these elements and these intervals must not be conceived of as something determined. Consequently, the most perfect representation of continuous quantity will be obtained by the consideration not of fixed and determined magnitudes such as those just
discussed, but on the contrary of variables, for then their variability can itself be regarded as accomplished in a continuous fashion; and these quantities must be capable of indefinite decrease by virtue of their variability, without ever canceling themselves out or reaching a 'minimum', which would be no less contradictory than 'final terms' of the continuous: here, precisely, as we shall see, is the true notion of infinitesimal quantities.