was never considered in the kind of way that is capable of yielding a
precise answer. Philosophers were content with some vague dictum such
as, "Number is unity in plurality." A typical definition of the kind
that contented philosophers is the following from Sigwart's _Logic_
(Sec. 66, section 3): "Every number is not merely a _plurality_, but a
plurality thought _as held together and closed, and to that extent as a
unity_." Now there is in such definitions a very elementary blunder, of
the same kind that would be committed if we said "yellow is a flower"
because some flowers are yellow. Take, for example, the number 3. A
single collection of three things might conceivably be described as "a
plurality thought as held together and closed, and to that extent as a
unity"; but a collection of three things is not the number 3. The number
3 is something which all collections of three things have in common, but
is not itself a collection of three things. The definition, therefore,
apart from any other defects, has failed to reach the necessary degree
of abstraction: the number 3 is something more abstract than any
collection of three things.
Such vague philosophic definitions, however, remained inoperative
because of their very vagueness. What most men who thought about numbers
really had in mind was that numbers are the result of _counting_. "On
the consciousness of the law of counting," says Sigwart at the beginning
of his discussion of number, "rests the possibility of spontaneously
prolonging the series of numbers _ad infinitum_." It is this view of
number as generated by counting which has been the chief psychological
obstacle to the understanding of infinite numbers. Counting, because it
is familiar, is erroneously supposed to be simple, whereas it is in fact
a highly complex process, which has no meaning unless the numbers
reached in counting have some significance independent of the process by
which they are reached. And infinite numbers cannot be reached at all in
this way. The mistake is of the same kind as if cows were defined as
what can be bought from a cattle-merchant. To a person who knew several
cattle-merchants, but had never seen a cow, this might seem an admirable
definition. But if in his travels he came across a herd of wild cows, he
would have to declare that they were not cows at all, because no
cattle-merchant could sell them. So infinite numbers were declared not
to be numbers at all, because they could not be reached by counting.
It will be worth while to consider for a moment what counting actually
is. We count a set of objects when we let our attention pass from one to
another, until we have attended once to each, saying the names of the
numbers in order with each successive act of attention. The last number
named in this process is the number of the objects, and therefore
counting is a method of finding out what the number of the objects is.
But this operation is really a very complicated one, and those who
imagine that it is the logical source of number show themselves
remarkably incapable of analysis. In the first place, when we say "one,
two, three ..." as we count, we cannot be said to be discovering the
number of the objects counted unless we attach some meaning to the words
one, two, three, ... A child may learn to know these words in order, and
to repeat them correctly like the letters of the alphabet, without
attaching any meaning to them. Such a child may count correctly from the
point of view of a grown-up listener, without having any idea of numbers
at all. The operation of counting, in fact, can only be intelligently
performed by a person who already has some idea what the numbers are;
and from this it follows that counting does not give the logical basis
of number.
Again, how do we know that the last number reached in the process of
counting is the number of the objects counted? This is just one of those
facts that are too familiar for their significance to be realised; but
those who wish to be logicians must acquire the habit of dwelling upon
such facts. There are two propositions involved in this fact: first,
that the number of numbers from 1 up to any given number is that given
number--for instance, the number of numbers from 1 to 100 is a hundred;
secondly, that if a set of numbers can be used as names of a set of
objects, each number occurring only once, then the number of numbers
used as names is the same as the number of objects. The first of these
propositions is capable of an easy arithmetical proof so long as finite
numbers are concerned; but with infinite numbers, after the first, it
ceases to be true. The second proposition remains true, and is in fact,
as we shall see, an immediate consequence of the definition of number.
But owing to the falsehood of the first proposition where infinite
numbers are concerned, counting, even if it were practically possible,
would not be a valid method of discovering the number of terms in an
infinite collection, and would in fact give different results according
to the manner in which it was carried out.
There are two respects in which the infinite numbers that are known
differ from finite numbers: first, infinite numbers have, while finite
numbers have not, a property which I shall call _reflexiveness_;
secondly, finite numbers have, while infinite numbers have not, a
property which I shall call _inductiveness_. Let us consider these two
properties successively.
(1) _Reflexiveness._--A number is said to be _reflexive_ when it is not
increased by adding 1 to it. It follows at once that any finite number
can be added to a reflexive number without increasing it. This property
of infinite numbers was always thought, until recently, to be
self-contradictory; but through the work of Georg Cantor it has come to
be recognised that, though at first astonishing, it is no more
self-contradictory than the fact that people at the antipodes do not
tumble off. In virtue of this property, given any infinite collection of
objects, any finite number of objects can be added or taken away without
increasing or diminishing the number of the collection. Even an infinite
number of objects may, under certain conditions, be added or taken away
without altering the number. This may be made clearer by the help of
some examples.
Imagine all the natural numbers 0, 1, 2, 3, ... to be written down in a
row, and immediately beneath them write down the numbers 1, 2, 3,
4, ..., so that 1 is under 0, 2 is under 1, and so on. Then every number
in the top row has a number directly under it in the bottom row, and no
number occurs twice in either row. It follows that the number of numbers
in the two rows must be the same. But all the numbers that occur in the
bottom row also occur in the top row, and one more, namely 0; thus the
number of terms in the top row is obtained by adding one to the number
of the bottom row. So long, therefore, as it was supposed that a number
must be increased by adding 1 to it, this state of things constituted a
contradiction, and led to the denial that there are infinite numbers.
0, 1, 2, 3, ... _n_ ...
1, 2, 3, 4, ... _n_ + 1 ...
The following example is even more surprising. Write the natural numbers
1, 2, 3, 4, ... in the top row, and the even numbers 2, 4, 6, 8, ... in
the bottom row, so that under each number in the top row stands its
double in the bottom row. Then, as before, the number of numbers in the
two rows is the same, yet the second row results from taking away all
the odd numbers--an infinite collection--from the top row. This example
is given by Leibniz to prove that there can be no infinite numbers. He
believed in infinite collections, but, since he thought that a number
must always be increased when it is added to and diminished when it is
subtracted from, he maintained that infinite collections do not have
numbers. "The number of all numbers," he says, "implies a contradiction,
which I show thus: To any number there is a corresponding number equal
to its double. Therefore the number of all numbers is not greater than
the number of even numbers, _i.e._ the whole is not greater than its
part."[49] In dealing with this argument, we ought to substitute "the
number of all finite numbers" for "the number of all numbers"; we then
obtain exactly the illustration given by our two rows, one containing
all the finite numbers, the other only the even finite numbers. It will
be seen that Leibniz regards it as self-contradictory to maintain that
the whole is not greater than its part. But the word "greater" is one
which is capable of many meanings; for our purpose, we must substitute
the less ambiguous phrase "containing a greater number of terms." In
this sense, it is not self-contradictory for whole and part to be equal;
it is the realisation of this fact which has made the modern theory of
infinity possible.
[49] _Phil. Werke_, Gerhardt's edition, vol. i. p. 338.
There is an interesting discussion of the reflexiveness of infinite
wholes in the first of Galileo's Dialogues on Motion. I quote from a
translation published in 1730.[50] The personages in the dialogue are
Salviati, Sagredo, and Simplicius, and they reason as follows:
"_Simp._ Here already arises a Doubt which I think is not to be
resolv'd; and that is this: Since 'tis plain that one Line is given
greater than another, and since both contain infinite Points, we must
surely necessarily infer, that we have found in the same Species
something greater than Infinite, since the Infinity of Points of the
greater Line exceeds the Infinity of Points of the lesser. But now, to
assign an Infinite greater than an Infinite, is what I can't possibly
conceive.
"_Salv._ These are some of those Difficulties which arise from
Discourses which our finite Understanding makes about Infinites, by
ascribing to them Attributes which we give to Things finite and
terminate, which I think most improper, because those Attributes of
Majority, Minority, and Equality, agree not with Infinities, of which we
can't say that one is greater than, less than, or equal to another. For
Proof whereof I have something come into my Head, which (that I may be
the better understood) I will propose by way of Interrogatories to
_Simplicius_, who started this Difficulty. To begin then: I suppose you
know which are square Numbers, and which not?
"_Simp._ I know very well that a square Number is that which arises from
the Multiplication of any Number into itself; thus 4 and 9 are square
Numbers, that arising from 2, and this from 3, multiplied by themselves.
"_Salv._ Very well; And you also know, that as the Products are call'd
Squares, the Factors are call'd Roots: And that the other Numbers, which
proceed not from Numbers multiplied into themselves, are not Squares.
Whence taking in all Numbers, both Squares and Not Squares, if I should
say, that the Not Squares are more than the Squares, should I not be in
the right?
"_Simp._ Most certainly.
"_Salv._ If I go on with you then, and ask you, How many squar'd Numbers
there are? you may truly answer, That there are as many as are their
proper Roots, since every Square has its own Root, and every Root its
own Square, and since no Square has more than one Root, nor any Root
more than one Square.
"_Simp._ Very true.
"_Salv._ But now, if I should ask how many Roots there are, you can't
deny but there are as many as there are Numbers, since there's no Number
but what's the Root to some Square. And this being granted, we may
likewise affirm, that there are as many square Numbers, as there are
Numbers; for there are as many Squares as there are Roots, and as many
Roots as Numbers. And yet in the Beginning of this, we said, there were
many more Numbers than Squares, the greater Part of Numbers being not
Squares: And tho' the Number of Squares decreases in a greater
proportion, as we go on to bigger Numbers, for count to an Hundred
you'll find 10 Squares, viz. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, which
is the same as to say the 10th Part are Squares; in Ten thousand only
the 100th Part are Squares; in a Million only the 1000th: And yet in an
infinite Number, if we can but comprehend it, we may say the Squares are
as many as all the Numbers taken together.
"_Sagr._ What must be determin'd then in this Case?
"_Salv._ I see no other way, but by saying that all Numbers are
infinite; Squares are Infinite, their Roots Infinite, and that the
Number of Squares is not less than the Number of Numbers, nor this less
than that: and then by concluding that the Attributes or Terms of
Equality, Majority, and Minority, have no Place in Infinites, but are
confin'd to terminate Quantities."
[50] _Mathematical Discourses concerning two new sciences relating to
mechanics and local motion, in four dialogues._ By Galileo Galilei,
Chief Philosopher and Mathematician to the Grand Duke of Tuscany. Done
into English from the Italian, by Tho. Weston, late Master, and now
published by John Weston, present Master, of the Academy at Greenwich.
See pp. 46 ff.
The way in which the problem is expounded in the above discussion is
worthy of Galileo, but the solution suggested is not the right one. It
is actually the case that the number of square (finite) numbers is the
same as the number of (finite) numbers. The fact that, so long as we
confine ourselves to numbers less than some given finite number, the
proportion of squares tends towards zero as the given finite number
increases, does not contradict the fact that the number of all finite
squares is the same as the number of all finite numbers. This is only an
instance of the fact, now familiar to mathematicians, that the _limit_
of a function as the variable _approaches_ a given point may not be the
same as its _value_ when the variable actually _reaches_ the given
point. But although the infinite numbers which Galileo discusses are
equal, Cantor has shown that what Simplicius could not conceive is true,
namely, that there are an infinite number of different infinite numbers,
and that the conception of _greater_ and _less_ can be perfectly well
applied to them. The whole of Simplicius's difficulty comes, as is
evident, from his belief that, if _greater_ and _less_ can be applied, a
part of an infinite collection must have fewer terms than the whole; and
when this is denied, all contradictions disappear. As regards greater
and less lengths of lines, which is the problem from which the above
discussion starts, that involves a meaning of _greater_ and _less_ which
is not arithmetical. The number of points is the same in a long line and
in a short one, being in fact the same as the number of points in all
space. The _greater_ and _less_ of metrical geometry involves the new
metrical conception of _congruence_, which cannot be developed out of
arithmetical considerations alone. But this question has not the
fundamental importance which belongs to the arithmetical theory of
infinity.
(2) _Non-inductiveness._--The second property by which infinite numbers
are distinguished from finite numbers is the property of
non-inductiveness. This will be best explained by defining the positive
property of inductiveness which characterises the finite numbers, and
which is named after the method of proof known as "mathematical
induction."
Let us first consider what is meant by calling a property "hereditary"
in a given series. Take such a property as being named Jones. If a man
is named Jones, so is his son; we will therefore call the property of
being called Jones hereditary with respect to the relation of father and
son. If a man is called Jones, all his descendants in the direct male
line are called Jones; this follows from the fact that the property is
hereditary. Now, instead of the relation of father and son, consider the
relation of a finite number to its immediate successor, that is, the
relation which holds between 0 and 1, between 1 and 2, between 2 and 3,
and so on. If a property of numbers is hereditary with respect to this
relation, then if it belongs to (say) 100, it must belong also to all
finite numbers greater than 100; for, being hereditary, it belongs to
101 because it belongs to 100, and it belongs to 102 because it belongs
to 101, and so on--where the "and so on" will take us, sooner or later,
to any finite number greater than 100. Thus, for example, the property
of being greater than 99 is hereditary in the series of finite numbers;
and generally, a property is hereditary in this series when, given any
number that possesses the property, the next number must always also
possess it.
It will be seen that a hereditary property, though it must belong to all
the finite numbers greater than a given number possessing the property,
need not belong to all the numbers less than this number. For example,
the hereditary property of being greater than 99 belongs to 100 and all
greater numbers, but not to any smaller number. Similarly, the
hereditary property of being called Jones belongs to all the descendants
(in the direct male line) of those who have this property, but not to
all their ancestors, because we reach at last a first Jones, before whom
the ancestors have no surname. It is obvious, however, that any
hereditary property possessed by Adam must belong to all men; and
similarly any hereditary property possessed by 0 must belong to all
finite numbers. This is the principle of what is called "mathematical
induction." It frequently happens, when we wish to prove that all finite
numbers have some property, that we have first to prove that 0 has the
property, and then that the property is hereditary, _i.e._ that, if it
belongs to a given number, then it belongs to the next number. Owing to
the fact that such proofs are called "inductive," I shall call the
properties to which they are applicable "inductive" properties. Thus an
inductive property of numbers is one which is hereditary and belongs to
0.
Taking any one of the natural numbers, say 29, it is easy to see that it
must have all inductive properties. For since such properties belong to
0 and are hereditary, they belong to 1; therefore, since they are
hereditary, they belong to 2, and so on; by twenty-nine repetitions of
such arguments we show that they belong to 29. We may _define_ the
"inductive" numbers as _all those that possess all inductive
properties_; they will be the same as what are called the "natural"
numbers, _i.e._ the ordinary finite whole numbers. To all such numbers,
proofs by mathematical induction can be validly applied. They are those
numbers, we may loosely say, which can be reached from 0 by successive
additions of 1; in other words, they are all the numbers that can be
reached by counting.
But beyond all these numbers, there are the infinite numbers, and
infinite numbers do not have all inductive properties. Such numbers,
therefore, may be called non-inductive. All those properties of numbers
which are proved by an imaginary step-by-step process from one number to
the next are liable to fail when we come to infinite numbers. The first
of the infinite numbers has no immediate predecessor, because there is
no greatest finite number; thus no succession of steps from one number
to the next will ever reach from a finite number to an infinite one, and
the step-by-step method of proof fails. This is another reason for the
supposed self-contradictions of infinite numbers. Many of the most
familiar properties of numbers, which custom had led people to regard as
logically necessary, are in fact only demonstrable by the step-by-step
method, and fail to be true of infinite numbers. But so soon as we
realise the necessity of proving such properties by mathematical
induction, and the strictly limited scope of this method of proof, the
supposed contradictions are seen to contradict, not logic, but only our
prejudices and mental habits.
The property of being increased by the addition of 1--_i.e._ the
property of non-reflexiveness--may serve to illustrate the limitations
of mathematical induction. It is easy to prove that 0 is increased by
the addition of 1, and that, if a given number is increased by the
addition of 1, so is the next number, _i.e._ the number obtained by the
addition of 1. It follows that each of the natural numbers is increased
by the addition of 1. This follows generally from the general argument,
and follows for each particular case by a sufficient number of
applications of the argument. We first prove that 0 is not equal to 1;
then, since the property of being increased by 1 is hereditary, it
follows that 1 is not equal to 2; hence it follows that 2 is not equal
to 3; if we wish to prove that 30,000 is not equal to 30,001, we can do
so by repeating this reasoning 30,000 times. But we cannot prove in this
way that _all_ numbers are increased by the addition of 1; we can only
prove that this holds of the numbers attainable by successive additions
of 1 starting from 0. The reflexive numbers, which lie beyond all those
attainable in this way, are as a matter of fact not increased by the
addition of 1.
The two properties of reflexiveness and non-inductiveness, which we have
considered as characteristics of infinite numbers, have not so far been
proved to be always found together. It is known that all reflexive
numbers are non-inductive, but it is not known that all non-inductive
numbers are reflexive. Fallacious proofs of this proposition have been
published by many writers, including myself, but up to the present no
valid proof has been discovered. The infinite numbers actually known,
however, are all reflexive as well as non-inductive; thus, in
mathematical practice, if not in theory, the two properties are always
associated. For our purposes, therefore, it will be convenient to ignore
the bare possibility that there may be non-inductive non-reflexive
numbers, since all known numbers are either inductive or reflexive.
When infinite numbers are first introduced to people, they are apt to
refuse the name of numbers to them, because their behaviour is so
different from that of finite numbers that it seems a wilful misuse of
terms to call them numbers at all. In order to meet this feeling, we
must now turn to the logical basis of arithmetic, and consider the
logical definition of numbers.
The logical definition of numbers, though it seems an essential support
to the theory of infinite numbers, was in fact discovered independently
and by a different man. The theory of infinite numbers--that is to say,
the arithmetical as opposed to the logical part of the theory--was
discovered by Georg Cantor, and published by him in 1882-3.[51] The
definition of number was discovered about the same time by a man whose
great genius has not received the recognition it deserves--I mean
Gottlob Frege of Jena. His first work, _Begriffsschrift_, published in
1879, contained the very important theory of hereditary properties in a
series to which I alluded in connection with inductiveness. His
definition of number is contained in his second work, published in 1884,
and entitled _Die Grundlagen der Arithmetik, eine logisch-mathematische
Untersuchung ueber den Begriff der Zahl_.[52] It is with this book that
the logical theory of arithmetic begins, and it will repay us to
consider Frege's analysis in some detail.
[51] In his _Grundlagen einer allgemeinen Mannichfaltigkeitslehre_ and
in articles in _Acta Mathematica_, vol. ii.
[52] The definition of number contained in this book, and elaborated
in the _Grundgesetze der Arithmetik_ (vol. i., 1893; vol. ii., 1903),
was rediscovered by me in ignorance of Frege's work. I wish to state
as emphatically as possible--what seems still often ignored--that his
discovery antedated mine by eighteen years.