= A(ᵖ⁺𐞥⁺¹⁄ᵩ₊₁)xᵖy𐞥⁺¹ + A(ᵖ⁺𐞥⁺¹⁄ᵩ₊₁). (ᵖ⁄ᵩ₊₂)xᵖ⁻¹y𐞥⁺².
It is, thus, seen that the succession of terms follows the rule. Thus if
any integral power follows the rule, so also does the next higher power.
But the first power obviously follows the rule. Hence, all powers do so.
Such reasoning holds good of any collection of objects capable of being
ranged in a series which though it may be endless, can be numbered so
that each member of it receives a definite integral number. For instance,
all the whole numbers constitute such a numerable collection. Again, all
numbers resulting from operating according to any definite rule with
any finite number of whole numbers form such a collection. For they may
be arranged in a series thus. Let F be the symbol of operation. First
operate on 1, giving F(1) Then, operate on a second 1, giving F(1,1).
Next, introduce 2, giving 3rd, F(2); 4th, F(2,1); 5th, F(1,2); 6th,
F(2,2). Next use a third variable giving 7th, F(1,1,1); 8th, F(2,1,1);
9th, F(1,2,1); 10th, F(2,2,1); 11th, F(1,1,2); 12th, F(2,1,2); 13th,
F(1,2,2); 14th, F(2,2,2). Next introduce 3, and so on, alternately
introducing new variables and new figures; and in this way it is plain
that every arrangement of integral values of the variables will receive a
numbered place in the series.[72]
The class of endless but numerable collections (so called because they
can be so ranged that to each one corresponds a distinct whole number) is
very large. But there are collections which are certainly innumerable.
Such is the collection of all numbers to which endless series of
decimals are capable of approximating. It has been recognised since the
time of Euclid that certain numbers are surd or incommensurable, and
are not exactly expressible by any finite series of decimals, nor by a
circulating decimal. Such is the ratio of the circumference of a circle
to its diameter, which we know is nearly 3.1415926. The calculation of
this number has been carried to over 700 figures without the slightest
appearance of regularity in their sequence. The demonstrations that this
and many other numbers are incommensurable are perfect. That the entire
collection of incommensurable numbers is innumerable has been clearly
proved by Cantor. I omit the demonstration; but it is easy to see that to
discriminate one from some other would, in general, require the use of an
endless series of numbers. Now if they cannot be exactly expressed and
discriminated, clearly they cannot be ranged in a linear series.
It is evident that there are as many points on a line or in an interval
of time as there are of real numbers in all. These are, therefore,
innumerable collections. Many mathematicians have incautiously assumed
that the points on a surface or in a solid are more than those on a line.
But this has been refuted by Cantor. Indeed, it is obvious that for every
set of values of coördinates there is a single distinct number. Suppose,
for instance, the values of the coördinates all lie between 0 and + 1.
Then if we compose a number by putting in the first decimal place the
first figure of the first coördinate, in the second the first figure of
the second coördinate, and so on, and when the first figures are all
dealt out go on to the second figures in like manner, it is plain that
the values of the coördinates can be read off from the single resulting
number, so that a triad or tetrad of numbers, each having innumerable
values, has no more values than a single incommensurable number.
Were the number of dimensions infinite, this would fail; and the
collection of infinite sets of numbers having each innumerable
variations, might, therefore, be greater than the simple innumerable
collection, and might be called _endlessly infinite_. The single
individuals of such a collection could not, however, be designated, even
approximately, so that this is indeed a magnitude concerning which it
would be possible to reason only in the most general way, if at all.
Although there are but two grades of magnitudes of infinite collections,
yet when certain conditions are imposed upon the order in which
individuals are taken, distinctions of magnitude arise from that cause.
Thus, if a simply endless series be doubled by separating each unit into
two parts, the successive first parts and also the second parts being
taken in the same order as the units from which they are derived, this
double endless series will, so long as it is taken in that order, appear
as twice as large as the original series. In like manner the product of
two innumerable collections, that is, the collection of possible pairs
composed of one individual of each, if the order of continuity is to be
maintained, is, by virtue of that order, infinitely greater than either
of the component collections.
We now come to the difficult question, What is continuity? Kant confounds
it with infinite divisibility, saying that the essential character of
a continuous series is that between any two members of it a third can
always be found. This is an analysis beautifully clear and definite;
but unfortunately, it breaks down under the first test. For according
to this, the entire series of rational fractions arranged in the order
of their magnitude, would be an infinite series, although the rational
fractions are numerable, while the points of a line are innumerable.
Nay, worse yet, if from that series of fractions any two with all that
lie between them be excised, and any number of such finite gaps he made,
Kant’s definition is still true of the series, though it has lost all
appearance of continuity.
Cantor defines a continuous series as one which is _concatenated_ and
_perfect_. By a concatenated series, he means such a one that if any
two points are given in it, and any finite distance, however small, it
is possible to proceed from the first point to the second through a
succession of points of the series each at a distance from the preceding
one less than the given distance. This is true of the series of rational
fractions ranged in the order of their magnitude. By a perfect series, he
means one which contains every point such that there is no distance so
small that this point has not an infinity of points of the series within
that distance of it. This is true of the series of numbers between 0 and
1 capable of being expressed by decimals in which only the digits 0 and 1
occur.
It must be granted that Cantor’s definition includes every series that
is continuous; nor can it be objected that it includes any important
or indubitable case of a series not continuous. Nevertheless, it has
some serious defects. In the first place, it turns upon metrical
considerations; while the distinction between a continuous and a
discontinuous series is manifestly non-metrical. In the next place, a
perfect series is defined as one containing “every point” of a certain
description. But no positive idea is conveyed of what all the points
are: that is definition by negation, and cannot be admitted. If that
sort of thing were allowed, it would be very easy to say, at once, that
the continuous linear series of points is one which contains every point
of the line between its extremities. Finally, Cantor’s definition does
not convey a distinct notion of what the components of the conception of
continuity are. It ingeniously wraps up its properties in two separate
parcels, but does not display them to our intelligence.
Kant’s definition expresses one simple property of a continuum; but
it allows of gaps in the series. To mend the definition, it is only
necessary to notice how these gaps can occur. Let us suppose, then, a
linear series of points extending from a point, _A_, to a point, _B_,
having a gap from _B_ to a third point, _C_, and thence extending to
a final limit, _D_; and let us suppose this series conforms to Kant’s
definition. Then, of the two points, _B_ and _C_, one or both must be
excluded from the series; for otherwise, by the definition, there would
be points between them. That is, if the series contains _C_, though
it contains all the points up to _B_, it cannot contain _B_. What is
required, therefore, is to state in non-metrical terms that if a series
of points up to a limit is included in a continuum the limit is included.
It may be remarked that this is the property of a continuum to which
Aristotle’s attention seems to have been directed when he defines a
continuum as something whose parts have a common limit. The property may
be exactly stated as follows: If a linear series of points is continuous
between two points, _A_ and _D_, and if an endless series of points be
taken, the first of them between _A_ and _D_ and each of the others
between the last preceding one and _D_, then there is a point of the
continuous series between all that endless series of points and _D_, and
such that every other point of which this is true lies between this point
and _D_. For example, take any number between 0 and 1, as 0.1; then, any
number between 0.1 and 1, as 0.11; then any number between 0.11 and 1, as
0.111; and so on, without end. Then, because the series of real numbers
between 0 and 1 is continuous, there must be a _least_ real number,
greater than every number of that endless series. This property, which
may be called the Aristotelicity of the series, together with Kant’s
property, or its Kanticity, completes the definition of a continuous
series.
The property of Aristotelicity may be roughly stated thus: a continuum
contains the end point belonging to every endless series of points which
it contains. An obvious corollary is that every continuum contains its
limits. But in using this principle it is necessary to observe that a
series may be continuous except in this, that it omits one or both of the
limits.
Our ideas will find expression more conveniently if, instead of points
upon a line, we speak of real numbers. Every real number is, in one
sense, the limit of a series, for it can be indefinitely approximated
to. Whether every real number is a limit of a _regular_ series may
perhaps be open to doubt. But the series referred to in the definition
of Aristotelicity must be understood as including all series whether
regular or not. Consequently, it is implied that between any two points
an innumerable series of points can be taken.
Every number whose expression in decimals requires but a finite number of
places of decimals is commensurable. Therefore, incommensurable numbers
suppose an infinitieth place of decimals. The word infinitesimal is
simply the Latin form of infinitieth; that is, it is an ordinal formed
from _infinitum_, as centesimal from _centum_. Thus, continuity supposes
infinitesimal quantities. There is nothing contradictory about the idea
of such quantities. In adding and multiplying them the continuity must
not be broken up, and consequently they are precisely like any other
quantities, except that neither the syllogism of transposed quantity, nor
the Fermatian inference applies to them.
If A is a finite quantity and _i_ an infinitesimal, then in a certain
sense we may write A + _i_ = A. That is to say, this is so for all
purposes of measurement. But this principle must not be applied except
to get rid of _all_ the terms in the highest order of infinitesimals
present. As a mathematician, I prefer the method of infinitesimals to
that of limits, as far easier and less infested with snares. Indeed,
the latter, as stated in some books, involves propositions that are
false; but this is not the case with the forms of the method used by
Cauchy, Duhamel, and others. As they understand the doctrine of limits,
it involves the notion of continuity, and therefore contains in another
shape the very same ideas as the doctrine of infinitesimals.
Let us now consider an aspect of the Aristotelical principle which is
particularly important in philosophy. Suppose a surface to be part red
and part blue; so that every point on it is either red or blue, and, of
course, no part can be both red and blue. What, then, is the color of the
boundary line between the red and the blue? The answer is that red or
blue, to exist at all, must be spread over a surface; and the color of
the surface is the color of the surface in the immediate neighborhood of
the point. I purposely use a vague form of expression. Now, as the parts
of the surface in the immediate neighborhood of any ordinary point upon a
curved boundary are half of them red and half blue, it follows that the
boundary is half red and half blue. In like manner, we find it necessary
to hold that consciousness essentially occupies time; and what is
present to the mind at any ordinary instant, is what is present during a
moment in which that instant occurs. Thus, the present is half past and
half to come. Again, the color of the parts of a surface at any finite
distance from a point, has nothing to do with its color just at that
point; and, in the parallel, the feeling at any finite interval from the
present has nothing to do with the present feeling, except vicariously.
Take another case: the velocity of a particle at any instant of time is
its mean velocity during an infinitesimal instant in which that time
is contained. Just so my immediate feeling is my feeling through an
infinitesimal duration containing the present instant.