# proof of the irrationality of root 2

Irrational numbers are those numbers which can not be expressed as a ratio between two integers. The first irrational number to be confronted by the Greeks was the square root of 2. Irrational numbers are still regarded as real numbers but they are not commensurate to any quantity. So they are in-commensurate numbers. Greek found that if the length of square figure is one then its diagonal is the square root of 2. They were puzzled by this weird number and its properties.

The diagonal of this square can not be multiple of any integer. On the other hand you can see that on a number line square root 2 can be represented as the distance which is the same as the diagonal of the square. All this was very troublesome for mathematics and mathematician in the ancient time. However we will proceed to the proof of the irrationality of the square root of two (2) :

Suppose root 2 is equal to the ratio of m to n. So

root 2 = m/n , if we square it we get 2n^2 = m^2 . That means that m^2 is an even number. Now let m = 2p so m^2 = 4p^2 . Thus we have 4p^2 = 2n^2 where p is the half of m. Hence 2p^2 = n^2 , and therefore n/p will also be equal to square root of 2. But we can repeat the argument : if n = 2q, p/q will also be the square root of 2 , and so on , through an unending series of numbers each of which is half of its predecessors. But this is clearly impossible, if we divide a number continuously by two 2 we must reach an odd number after a finite number of steps. Or we may put the argument in this way by assuming that the m/n we start with is in its lowest terms; in that case m and n can not both be even as we saw they were supposed to be.

So we must conclude that square root of 2 can not be represented as a ratio of two numbers. If we do we come to a contradiction as we have shown.

square root of two(2) can be expressed using circular and hyperbolic angle too. Here is an example .

Where the symbol ^ is the exponential.

Real numbers and irrational number:

As stated, no fraction will express exactly the length of the diagonal of a square whose side is one inch long. This problem was purely geometrical but the same problem arose as regards the solution of equations, though here it took a wider form, since it involved complex numbers.

It is clear that fractions can be found which approach nearer and nearer to having their square equal to 2. We can form an ascending series of fractions all of which have their squares less than 2, but differing from 2 in their later members by less than any assigned amount. This is to say, suppose I assign some small amount in advance, say one-billionth, it will be found that all the terms of our series after certain one, say, the tenth, have squares that differ from 2 by less than this amount. And if I had assigned a still smaller amount, it might have been necessary to go further along the series, but we should have reached sooner or later a term in the series, say, twentieth, after which all terms would have had squares differing from 2 by less than this still smaller amount. If we set to work to extract the square root of 2 by the usual arithmetical rule, we shall obtain an unending decimal which, taken to so-an-so many places, exactly fulfils the above conditions. We can equally form an descending series of fractions whose squares are all greater than 2, but greater by continually smaller amounts as we come to the later terms of the series, and differing , sooner or later, by less than any assigned amount. In this way we seem to be drawing a cordon around the square root of 2, and , it seems difficult to prove that, it can permanently escape us. But , it is not by this method that we will actually reach the square root of 2.

DEDEKIND’S CUT AND IRRATIONAL NUMBER:

If we divide all ratios into two classes, according as their squares are less than 2 or not, we find that , among those whose squares are not less than 2 , all have their squares greater than 2. There is no maximum to the ratios whose square is less than 2, and no minimum to those whose square is greater than 2. There is no lower limit short of zero to the difference between the numbers whose square is little greater than 2. We can, in short, divide all ratios into two classes such that all the terms in one class are less than all in the other, there is no maximum to the one class, and there is no minimum to the other. Between these two classes , where root 2 ought to be there, there is actually nothing. We have not actually caught root 2. The cordon, though we have drawn as tight as possible , has been drawn in the wrong place and has not caught 2 .

The above method of dividing all the terms of a series into two classes, of which the one wholly precedes the other, was brought into prominence by Dedekind, and is therefore called a “Dedekind cut” . With respect to what happens at the point of section, there are four possibilities.

If we divide all ratios into two classes such that all of the terms of one class is greater than all the terms of the other then there may arise four case: 1) there may be a minimum to upper the upper section and a maximum to the lower section 2) there may be a maximum to the one and no minimum to the other 3) there may be no maximum to the one , but a minimum to the other 4) there may be neither a maximum to the one nor a minimum to the other. Of these four cases, the first can be illustrated by any series in which there are consecutive terms: in the series of integers, the lower class may contain a maximum number n and the upper class can contain a minimum n+1 . The second of these cases will be illustrated in the series of ratios if we take as our lower section all the ratios up to and including 1, and in our upper section all the ratios greater than 1. The third case is illustrated if we take for our lower section all ratios from 1 upward (including 1 itself) . The fourth case, as we have seen , is illustrated if we put in our lower section all the ratios whose square is less than 2 , and in our upper section all the ratios whose square is greater than 2.

We may neglect the first of our four cases, since it only arises in series where there are consecutive terms. In the second of our four cases, we say that the maximum of the lower section is the lower limit of the upper section, or of any set of terms chosen out of the upper section in such a way that no term of the upper section is the upper limit of the lower section, or of any set of terms chosen out of the lower section is after all of them. In the fourth case, we say that there is a “gap”: neither the upper section or lower section has a limit or last term. In this case, we may say that we have an irrational section, for the class of ratios which do not have limit corresponds to irrationals.

What delayed the true theory of irrational was a erroneous belief that there must be “limits” of series of ratios. The notion of “limit” is of the utmost importance , and before proceeding further it will be well to define it.

A term x is said to be an “upper limit” of a class s with respect to a relation P if (1) s has no maximum in P , (2) every member of s which belongs to the field of P precedes x , (3) every member of the field of P which precedes x precedes some member of s . ( by “precedes ” we mean “has the relation P to”).

This presupposes the following definition of “maximum” : –

A term x is said to be maximum of a class s with respect to a relation P if x is a member of s and of the field P and does not have the relation P to any other member of s.

These definitions do not demand that the terms to which they are applied should be quantitative. For example, given a series of moments of time arranged by earlier and later, their maximum (if any) will be the last of the moments.; but if they are arranged by later and earlier, their maximum (if any) will be the first of the moments.

The “minimum” of a class with respect to P is its maximum with respect to the converse of P; and the “lower limit” with respect to P is the “upper limit” with respect to the converse of P.

The notions of limit and maximum do not essentially demand that the relation in respect to which they are defined should be serial, but they have few important applications except to cases when the relation is serial and quasi-serial. A notion which is often important is the notion “upper limit or maximum” to which we may give the name “upper boundary”. Thus the “upper boundary ” of a set of terms chosen out of a series is their last member if they have one, but , if not it is the first term after all of them, if there is such a term. If there is neither a maximum nor a limit, there is no upper boundary. The “lower boundary” is the lower limit or minimum. Readers at this moment, should be aware that limit and maximum are not the same concept or the same thing.

Reverting to the four kinds of Dedekind section, we see that in the case of the first three kinds each section has a boundary( upper or lower as the case may be), while in the fourth kind neither has a boundary. It is also clear that, whenever the lower section has an upper boundary , the upper section has a lower boundary. In the second and third cases, the two boundaries are identical; in the first , they are consecutive terms of the series.

A series is called “Dedekindian” when every section has a boundary, upper or lower as the case may be.

We have seen that the series of ratios in order of magnitude is not Dedekindian.

From the habit of being influenced by spatial imagination, people have supposed that series must have limits in cases where it seems odd if they do not. Thus, perceiving that there was no rational limit to the ratios whose square is less than 2 , they allowed themselves to “postulate” an irrational limit, which was to fill the Dedekindian gap. Dedekind, in the above mentioned work, set up the axiom that the gap must always be filled, i.e that every section must have a boundary . It is for this reason that series where his axiom is verified are called “Dedekindian.” But there are an infinite number of series for which it is not verified.

It is clear that an irrational Dedekind cut in some way “represents” an irrational. But this is not true to say that an irrational must be the limit of a set of ratios. Just as ratios whose denominator is 1 is not identical with integers, so those rational numbers which can be greater or less than irrationals, or can have irrationals as their limits, must not be identified with ratios. We have to define a new kind of numbers called “real numbers” of which some will be rational and some irrationals. Those that are rationals correspond to ratios, in the same kind of way in which the ration n/1 corresponds to the integer n; but they are not the same as ratios. In order to decide what they are to be , let us observer that an irrational is represented by an irrational cut, and a cut is represented by its lower section. Let us confine ourselves to cuts in which the lower section has no maximum; in this case we call the lower section a “segment”. Then those segments that correspond to ratios are those that consist of all ratios less than a certain ratio that they correspond to, which is their boundary; while those that represent irrationals are those that have no boundary. Segments , both those that have boundaries and those that do not, are such that , of any two pertaining to one series one is the part of the other. In that case all those segments can be arranged by the relation of whole and part. A series in which there are Dedekind gaps, i.e in which there are segments that have no boundary, will give rise to more segments than its terms, since each term will define a segment having that term for boundary , and the segments without boundaries will be extra.

We are now in a position to define a real number and an irrational number.

A real number is a segment of the series of ratios in order of magnitude.

An ‘irrational number” is a segment of the series of ratios which has no boundary.

A “rational real number ” is a segment of the series of ratios which has a boundary.

Thus a rational real number consists of all ratios less than a certain ratio, and it is the rational real number corresponding to that ratio. The real number 1, for instance , is the class of proper fractions.

In this way we come up with a new theory of real numbers.

In the cases in which we naturally supposed that an irrational must be the limit of a set of ratios, the truth is that it is the limit of a set of corresponding set of rational real numbers in the series of segments order by whole and part. For example root 2 is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. More simply still, root 2 is the segment consisting of all the ratios whose square is less than 2.

It is easy to prove that the series of segments of any series is Dedekindian. For, given any set of segments, their boundary will be their logical sum, i.e the class of all those terms that belong to at least one segment of the set.

The above definition of real numbers is an example of “construction” as against “postulation” of which we had another example in the definition of cardinal numbers. The great advantage of this method is that it requires no new assumptions, but enables us to proceed deductively from original apparatus of logic.

There is no difficulty in defining addition and multiplication for real numbers as above defined. Given two real numbers u and v , each being a class of ratios, take any member of u and any member of v and add them together according to the rule for the addition of ratios. Form the class of all such sums obtainable by varying the selected members of u and v. This gives a new class of ratios, and it is easy to prove that this new class of ratios is a segment of the series of ratios. We define as the sum of u and v. We may state the definition more shortly as follows:

The arithmetical sum of two real numbers is the class of the arithmetical sums of a member of the one and a member of the other chosen in all possible ways.

We can define the arithmetical product of two real numbers in exactly the same way, by multiplying a member of the one by a member of the other in all possible ways. The class of ratios thus generated is defined as the product of the two real numbers. (in all such definitions, the series of ratios is to be defined as excluding 0 and the infinity).

There is no difficulty in extending our definitions to positive and negative real numbers and their addition and multiplication.

It remains to give the definition of complex numbers.

Complex numbers:

Complex numbers, though capable of a geometrical interpretation , are not demanded by geometry in the same imperative way in which irrationals are demanded . A “complex” number means a number involving the square root of a negative number, whether integral, fractional, or real. Since the square of a negative number is positive, a number whose square is to be negative has to be a new sort of number. using the letter i for the square root of -1 , any number involving the square root of a negative number can be expressed in the form x+iy , where x and y are real. The part yi is called the “imaginary” part of this number, x being the “real ” part. ( The reason for the phrase “real numbers” is that they are contrasted with such as are “imaginary”). Complex numbers have been for a long time habitually used by mathematicians, in spite of the absence of any precise definition. It has be simply assumed that they would obey the usual arithmetical rules, and on this assumption their employment has been found profitable. They are required less for geometry than for algebra and analysis. We desire , for example, to be able to say that every quadratic equation has two roots, and every cubic equation has three, and so on. But if we are confined to real numbers, such an equation x^2 + 1 = 0 has no roots, and such as x^3 -1 = 0 has only one. Every generalization of number has first presented itself as needed in order that subtraction might be always possible, since otherwise a-b would be meaningless if a were less than b; fractions were needed | in order that division might be always possible; and complex numbers are needed in order that extraction of roots and solution of equations may be always possible. But extensions of number are not created by mere need for them: they are created by the definition, and it is to the definition of complex numbers that we must now turn our attention.

A complex number may regarded and defined as simply an ordered couple of real numbers. Here, as elsewhere, many definitions are possible. All that is necessary is that the definitions adopted shall lead to certain properties. In the case of complex numbers, if they are defined as ordered couples of real numbers, we secure at once some of the properties required, namely, that two real numbers are required to determine a complex number, and that among these we can distinguish a first and a second, and two complex numbers are identical only when the first real number involved in the one is identical with the first in the second and the second to the second. What is needed further can be secured by defining the rules of addition and multiplication. We are to have

(x+iy)+(x``+iy`) = (x+x`) + (y+y``)i

(x+iy)(x`` + i y`) = (xx` - yy`) + i(xy` + yx``)

The ethiops say that their gods are pug-nosed and black

while the Thracians say that theirs have blue eyes and red hair.

Yet if cattle or horse or lions had hands and could draw

And could sculpture like men, then the horses would draw their gods like horses, and cattle like cattle, and each would then shape Bodies of Gods in the likeness, each kind, of its own,.

The gods did not reveal, from the beginning ,

All things to us, but in the course of time,

Through seeking we may learn, and know things better…

these things , we conjecture , are somehow like the truth,

but as for certain truth, no man has known it,

nor will he know it; neither of the gods,

nor yet all the

Mathematics is a study which, when we start from  its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater and greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced. It is the fact of pursuing this opposite direction that characterises mathematical philosophy as opposed to ordinary mathematics. But it should be understood that the distinction is one, not in the subject matter, but in the state of mind of the investigator. Early Greek geometers, passing from the empirical rules of Egyptian land-surveying to the general propositions by which those rules were found to be justifiable, and thence to Euclid’s axioms and postulates, were engaged in mathematical philosophy, according to the above definition; but when once the axioms and postulates had been reached, their deductive employment, as we find it in Euclid, belonged to mathematics in the | ordinary sense. The distinction between mathe-  matics and mathematical philosophy is one which depends upon the interest inspiring the research, and upon the stage which the research has reached; not upon the propositions with which the research is concerned. We may state the same distinction in another way. The most obvious and easy things in mathematics are not those that come logically at the beginning; they are things that, from the point of view of logical deduction, come somewhere in the middle. Just as the easiest bodies to see are those that are neither very near nor very far, neither very small nor very great, so the easiest conceptions to grasp are those that are neither very complex nor very simple (using “simple” in a logical sense). And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics. We shall find that by analysing our ordinary mathematical notions we acquire fresh insight, new powers, and the means of reaching whole new mathematical subjects by adopting fresh lines of advance after our backward journey. It is the purpose of this book to explain mathematical philosophy simply and untechnically, without enlarging upon those portions which are so doubtful or difficult that an elementary treatment is scarcely possible. A full treatment will be found in Principia Mathematica;  the treatment in the present volume is intended merely as an introduction. To the average educated person of the present day, the obvious starting-point of mathematics

Definition of Number The question “What is a number?” is one which  has been often asked, but has only been correctly answered in our own time. The answer was given by Frege in , in his Grundlagen der Arithmetik.  Although this book is quite short, not difficult, and of the very highest importance, it attracted almost no attention, and the definition of number which it contains remained practically unknown until it was rediscovered by the present author in . In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry. Many philosophers, when attempting to define number, are really setting to work to define plurality, which is quite a different thing. Number is The same answer is given more fully and with more development in his Grundgesetze der Arithmetik, vol. i., . what is characteristic of numbers, as man is what is characteristic of men. A plurality is not an instance of number, but of some particular number. A trio of men, for example, is an instance of the number , and the number  is an instance of number; but the trio is not an instance of number. This point may seem elementary and scarcely worth mentioning; yet it has proved too subtle for the philosophers, with few exceptions

Reverting now to the generation of series by the
relation R between consecutive terms, we see that,
if this method is to be possible, the relation “proper
R-ancestor” must be an aliorelative, transitive, and
connected. Under what circumstances will this occur?
It will always be transitive: no matter what
sort of relation R may be, “R-ancestor” and “proper
R-ancestor” are always both transitive. But it is
only under certain circumstances that it will be
an aliorelative or connected. Consider, for example,
the relation to one’s left-hand neighbour at a
round dinner-table at which there are twelve people.
If we call this relation R, the proper R-posterity
of a person consists of all who can be reached by
going round the table from right to left. This includes
everybody at the table, including the person
himself, since j twelve steps bring us back to our 37
starting-point. Thus in such a case, though the relation
“proper R-ancestor” is connected, and though
R itself is an aliorelative, we do not get a series because
“proper R-ancestor” is not an aliorelative. It
is for this reason that we cannot say that one person
comes before another with respect to the relation
“right of” or to its ancestral derivative.
The above was an instance in which the ancestral
relation was connected but not contained in
diversity. An instance where it is contained in diversity
but not connected is derived from the ordinary
sense of the word “ancestor.” If x is a proper ancestor
of y, x and y cannot be the same person; but it
is not true that of any two persons one must be an
ancestor of the other.
The question of the circumstances under which
series can be generated by ancestral relations derived
from relations of consecutiveness is often important.
Some of the most important cases are the
following: Let R be a many-one relation, and let us
confine our attention to the posterity of some term x.
When so confined, the relation “proper R-ancestor”
must be connected; therefore all that remains to ensure
its being serial is that it shall be contained in
diversity. This is a generalisation of the instance of
the dinner-table. Another generalisation consists in
taking R to be a one-one relation, and including the
ancestry of x as well as the posterity. Here again,
the one condition required to secure the generation
of a series is that the relation “proper R-ancestor”
shall be contained in diversity.
The generation of order by means of relations
of consecutiveness, though important in its own
sphere, is less general than the method which uses a
transitive relation to define the order. It often happens
in a series that there are an infinite number of
intermediate terms between any two that may be
selected, however near together these may be. Take,
for instance, fractions in order of magnitude. Between
any two fractions there are others—for example,
the arithmetic mean of the two. Consequently
there is no such thing as a pair of consecutive fractions.
If we depended j upon consecutiveness for 38
defining order, we should not be able to define the
order of magnitude among fractions. But in fact the
relations of greater and less among fractions do not
demand generation from relations of consecutiveness,
and the relations of greater and less among
fractions have the three characteristics which we
need for defining serial relations. In all such cases
the order must be defined by means of a transitive
relation, since only such a relation is able to leap
over an infinite number of intermediate terms. The
method of consecutiveness, like that of counting for
discovering the number of a collection, is appropriate
to the finite; it may even be extended to certain
infinite series, namely, those in which, though the
total number of terms is infinite, the number of
terms between any two is always finite; but it must
not be regarded as general. Not only so, but care
must be taken to eradicate from the imagination all
habits of thought resulting from supposing it general.
If this is not done, series in which there are
no consecutive terms will remain difficult and puzzling.
And such series are of vital importance for
the understanding of continuity, space, time, and
motion.
There are many ways in which series may be
generated, but all depend upon the finding or construction
of an asymmetrical transitive connected
relation. Some of these ways have considerable importance.
We may take as illustrative the generation
of series by means of a three-term relation which
we may call “between.” This method is very useful
in geometry, and may serve as an introduction to
relations having more than two terms; it is best introduced
in connection with elementary geometry.
Given any three points on a straight line in ordinary
space, there must be one of them which is
between the other two. This will not be the case
with the points on a circle or any other closed curve,
because, given any three points on a circle, we can
travel from any one to any other without passing
through the third. In fact, the notion “between” is
characteristic of open series—or series in the strict
sense—as opposed to what may be called j “cyclic” 39
series, where, as with people at the dinner-table,
a sufficient journey brings us back to our startingpoint.
This notion of “between” may be chosen as
the fundamental notion of ordinary geometry; but
for the present we will only consider its application
to a single straight line and to the ordering of the
points on a straight line.2 Taking any two points a,
b, the line (ab) consists of three parts (besides a and
b themselves):
(1) Points between a and b.
(2) Points x such that a is between x and b.
(3) Points y such that b is between y and a.
Thus the line (ab) can be defined in terms of the
relation “between.”
In order that this relation “between” may arrange
the points of the line in an order from left
to right, we need certain assumptions, namely, the
following:—
2Cf. Rivista di Matematica, iv. pp. 55ff.; Principles of Mathematics,
p. 394 (§375).
(1) If anything is between a and b, a and b are
not identical.
(2) Anything between a and b is also between b
and a.
(3) Anything between a and b is not identical
with a (nor, consequently, with b, in virtue of (2)).
(4) If x is between a and b, anything between a
and x is also between a and b.
(5) If x is between a and b, and b is between x
and y, then b is between a and y.
(6) If x and y are between a and b, then either x
and y are identical, or x is between a and y, or x is
between y and b.
(7) If b is between a and x and also between a and
y, then either x and y are identical, or x is between b
and y, or y is between b and x.
These seven properties are obviously verified
in the case of points on a straight line in ordinary
space. Any three-term relation which verifies them
gives rise to series, as may be seen from the following
definitions. For the sake of definiteness, let us
assume j that a is to the left of b. Then the points 40
of the line (ab) are (1) those between which and b, a
lies—these we will call to the left of a; (2) a itself; (3)
those between a and b; (4) b itself; (5) those between
which and a lies b—these we will call to the right of
b. We may now define generally that of two points
x, y, on the line (ab), we shall say that x is “to the
left of” y in any of the following cases:—
(1) When x and y are both to the left of a, and y is
between x and a;
(2) When x is to the left of a, and y is a or b or
between a and b or to the right of b;
(3) When x is a, and y is between a and b or is b or
is to the right of b;
(4) When x and y are both between a and b, and y
is between x and b;
(5) When x is between a and b, and y is b or to the
right of b;
(6) When x is b and y is to the right of b;
(7) When x and y are both to the right of b and x
is between b and y.
It will be found that, from the seven properties
which we have assigned to the relation “between,”
it can be deduced that the relation “to the left of,”
as above defined, is a serial relation as we defined
that term. It is important to notice that nothing in
the definitions or the argument depends upon our
meaning by “between” the actual relation of that
name which occurs in empirical space: any threeterm
relation having the above seven purely formal
properties will serve the purpose of the argument
equally well.
Cyclic order, such as that of the points on a circle,
cannot be generated by means of three-term
relations of “between.” We need a relation of four
terms, which may be called “separation of couples.”
The point may be illustrated by considering a journey
round the world. One may go from England to
New Zealand by way of Suez or by way of San Francisco;
we cannot j say definitely that either of these 41
two places is “between” England and New Zealand.
But if a man chooses that route to go round the
world, whichever way round he goes, his times in
England and New Zealand are separated from each
other by his times in Suez and San Francisco, and
conversely. Generalising, if we take any four points
on a circle, we can separate them into two couples,
say a and b and x and y, such that, in order to get
from a to b one must pass through either x or y, and
in order to get from x to y one must pass through
either a or b. Under these circumstances we say that
the couple (a, b) are “separated” by the couple (x, y).
Out of this relation a cyclic order can be generated,
in a way resembling that in which we generated an
open order from “between,” but somewhat more
complicated.3
The purpose of the latter half of this chapter
has been to suggest the subject which one may call
“generation of serial relations.” When such relations
have been defined, the generation of them from
other relations possessing only some of the properties
required for series becomes very important, especially
in the philosophy of geometry and physics.
But we cannot, within the limits of the present volume,
do more than make the reader aware that such
a subject exists.
3Cf. Principles of Mathematics, p. 205 (§194), and references
there given.
Chapter V
Kinds of Relations
A great part of the philosophy of mathematics is 42
concerned with relations, and many different kinds
of relations have different kinds of uses. It often
happens that a property which belongs to all relations
is only important as regards relations of certain
sorts; in these cases the reader will not see the
bearing of the proposition asserting such a property
unless he has in mind the sorts of relations for
which it is useful. For reasons of this description, as
well as from the intrinsic interest of the subject, it
is well to have in our minds a rough list of the more
mathematically serviceable varieties of relations.
We dealt in the preceding chapter with a supremely
important class, namely, serial relations.
Each of the three properties which we combined in
defining series—namely, asymmetry, transitiveness,
and connexity—has its own importance. We will
begin by saying something on each of these three.
Asymmetry, i.e. the property of being incompatible
with the converse, is a characteristic of the very
greatest interest and importance. In order to develop
its functions, we will consider various examples.
The relation husband is asymmetrical, and so
is the relation wife; i.e. if a is husband of b, b cannot
be husband of a, and similarly in the case of wife.
On the other hand, the relation “spouse” is symmetrical:
if a is spouse of b, then b is spouse of a.
Suppose now we are given the relation spouse, and
we wish to derive the relation husband. Husband is
the same as male spouse or spouse of a female; thus
the relation husband can j be derived from spouse ei- 43
ther by limiting the domain to males or by limiting
the converse domain to females. We see from this
instance that, when a symmetrical relation is given,
it is sometimes possible, without the help of any
further relation, to separate it into two asymmetrical
relations. But the cases where this is possible are
rare and exceptional: they are cases where there are
two mutually exclusive classes, say and , such
that whenever the relation holds between two terms,
one of the terms is a member of and the other is
a member of —as, in the case of spouse, one term
of the relation belongs to the class of males and one
to the class of females. In such a case, the relation
with its domain confined to will be asymmetrical,
and so will the relation with its domain confined
to . But such cases are not of the sort that occur
when we are dealing with series of more than two
terms; for in a series, all terms, except the first and
last (if these exist), belong both to the domain and
to the converse domain of the generating relation,
so that a relation like husband, where the domain
and converse domain do not overlap, is excluded.
The question how to construct relations having
some useful property by means of operations upon
relations which only have rudiments of the property
is one of considerable importance. Transitiveness
and connexity are easily constructed in many cases
where the originally given relation does not possess
them: for example, if R is any relation whatever, the
ancestral relation derived from R by generalised induction
is transitive; and if R is a many-one relation,
the ancestral relation will be connected if confined
to the posterity of a given term. But asymmetry is a
much more difficult property to secure by construction.
The method by which we derived husband from
spouse is, as we have seen, not available in the most
important cases, such as greater, before, to the right
of, where domain and converse domain overlap. In
all these cases, we can of course obtain a symmetrical
relation by adding together the given relation
and its converse, but we cannot pass back from this
symmetrical relation to the original asymmetrical
relation except by the help of some asymmetrical j relation. Take, for example, the relation greater: the 44
relation greater or less—i.e. unequal—is symmetrical,
but there is nothing in this relation to show that
it is the sum of two asymmetrical relations. Take
such a relation as “differing in shape.” This is not
the sum of an asymmetrical relation and its converse,
since shapes do not form a single series; but
there is nothing to show that it differs from “differing
in magnitude” if we did not already know that
magnitudes have relations of greater and less. This
illustrates the fundamental character of asymmetry
as a property of relations.
From the point of view of the classification of relations,
being asymmetrical is a much more important
characteristic than implying diversity. Asymmetrical
relations imply diversity, but the converse
is not the case. “Unequal,” for example, implies
diversity, but is symmetrical. Broadly speaking, we
may say that, if we wished as far as possible to dispense
with relational propositions and replace them
by such as ascribed predicates to subjects, we could
succeed in this so long as we confined ourselves to
symmetrical relations: those that do not imply diversity,
if they are transitive, may be regarded as asserting
a common predicate, while those that do imply
diversity may be regarded as asserting incompatible
predicates. For example, consider the relation
of similarity between classes, by means of which we
defined numbers. This relation is symmetrical and
transitive and does not imply diversity. It would be
possible, though less simple than the procedure we
adopted, to regard the number of a collection as a
predicate of the collection: then two similar classes
will be two that have the same numerical predicate,
while two that are not similar will be two that have
different numerical predicates. Such a method of replacing
relations by predicates is formally possible
(though often very inconvenient) so long as the relations
concerned are symmetrical; but it is formally
impossible when the relations are asymmetrical, because
both sameness and difference of predicates
are symmetrical. Asymmetrical relations are, we
may j say, the most characteristically relational of 45
relations, and the most important to the philosopher
who wishes to study the ultimate logical nature of
relations.
Another class of relations that is of the greatest
use is the class of one-many relations, i.e. relations
which at most one term can have to a given term.
Such are father, mother, husband (except in Tibet),
square of, sine of, and so on. But parent, square root,
and so on, are not one-many. It is possible, formally,
to replace all relations by one-many relations by
means of a device. Take (say) the relation less among
the inductive numbers. Given any number n greater
than 1, there will not be only one number having the
relation less to n, but we can form the whole class of
numbers that are less than n. This is one class, and
its relation to n is not shared by any other class. We
may call the class of numbers that are less than n
the “proper ancestry” of n, in the sense in which we
spoke of ancestry and posterity in connection with
mathematical induction. Then “proper ancestry”
is a one-many relation (one-many will always be
used so as to include one-one), since each number
determines a single class of numbers as constituting
its proper ancestry. Thus the relation less than can
be replaced by being a member of the proper ancestry
of. In this way a one-many relation in which the
one is a class, together with membership of this
class, can always formally replace a relation which
is not one-many. Peano, who for some reason always
instinctively conceives of a relation as one-many,
deals in this way with those that are naturally not
so. Reduction to one-many relations by this method,
however, though possible as a matter of form, does
not represent a technical simplification, and there
is every reason to think that it does not represent a
philosophical analysis, if only because classes must
be regarded as “logical fictions.” We shall therefore
continue to regard one-many relations as a special
kind of relations.
One-many relations are involved in all phrases
of the form “the so-and-so of such-and-such.” “The
King of England,” j “the wife of Socrates,” “the fa- 46
ther of John Stuart Mill,” and so on, all describe
some person by means of a one-many relation to a
given term. A person cannot have more than one
father, therefore “the father of John Stuart Mill” described
some one person, even if we did not know
whom. There is much to say on the subject of descriptions,
but for the present it is relations that
we are concerned with, and descriptions are only
relevant as exemplifying the uses of one-many relations.
It should be observed that all mathematical
functions result from one-many relations: the logarithm
of x, the cosine of x, etc., are, like the father
of x, terms described by means of a one-many relation
(logarithm, cosine, etc.) to a given term (x).
The notion of function need not be confined to numbers,
or to the uses to which mathematicians have
accustomed us; it can be extended to all cases of
one-many relations, and “the father of x” is just as
legitimately a function of which x is the argument
as is “the logarithm of x.” Functions in this sense
are descriptive functions. As we shall see later, there
are functions of a still more general and more fundamental
sort, namely, propositional functions; but
for the present we shall confine our attention to descriptive
functions, i.e. “the term having the relation
R to x,” or, for short, “the R of x,” where R is any
one-many relation.
It will be observed that if “the R of x” is to describe
a definite term, x must be a term to which
something has the relation R, and there must not
be more than one term having the relation R to x,
since “the,” correctly used, must imply uniqueness.
Thus we may speak of “the father of x” if x is any
human being except Adam and Eve; but we cannot
speak of “the father of x” if x is a table or a chair or
anything else that does not have a father. We shall
say that the R of x “exists” when there is just one
term, and no more, having the relation R to x. Thus
if R is a one-many relation, the R of x exists whenever
x belongs to the converse domain of R, and not
otherwise. Regarding “the R of x” as a function in
the mathematical j sense, we say that x is the “argu- 47
ment” of the function, and if y is the term which has
the relation R to x, i.e. if y is the R of x, then y is the
“value” of the function for the argument x. If R is a
one-many relation, the range of possible arguments
to the function is the converse domain of R, and the
range of values is the domain. Thus the range of
possible arguments to the function “the father of
x” is all who have fathers, i.e. the converse domain
of the relation father, while the range of possible
values for the function is all fathers, i.e. the domain
of the relation.
Many of the most important notions in the logic
of relations are descriptive functions, for example:
converse, domain, converse domain, field. Other examples
will occur as we proceed.
Among one-many relations, one-one relations are
occasion to speak of one-one relations in connection
with the definition of number, but it is necessary to
be familiar with them, and not merely to know their
formal definition. Their formal definition may be
derived from that of one-many relations: they may
be defined as one-many relations which are also
the converses of one-many relations, i.e. as relations
which are both one-many and many-one. One-many
relations may be defined as relations such that, if x
has the relation in question to y, there is no other
term x0 which also has the relation to y. Or, again,
they may be defined as follows: Given two terms x
and x0, the terms to which x has the given relation
and those to which x0 has it have no member in common.
Or, again, they may be defined as relations
such that the relative product of one of them and its
converse implies identity, where the “relative product”
of two relations R and S is that relation which
holds between x and z when there is an intermediate
term y, such that x has the relation R to y and y has
the relation S to z. Thus, for example, if R is the
relation of father to son, the relative product of R
and its converse will be the relation which holds
between x and a man z when there is a person y,
such that x is the father of y and y is the son of z. It
is obvious that x and z must be j the same person. 48
If, on the other hand, we take the relation of parent
and child, which is not one-many, we can no longer
argue that, if x is a parent of y and y is a child of
z, x and z must be the same person, because one
may be the father of y and the other the mother.
This illustrates that it is characteristic of one-many
relations when the relative product of a relation and
its converse implies identity. In the case of one-one
relations this happens, and also the relative product
of the converse and the relation implies identity.
Given a relation R, it is convenient, if x has the relation
R to y, to think of y as being reached from x by
an “R-step” or an “R-vector.” In the same case x will
be reached from y by a “backward R-step.” Thus we
may state the characteristic of one-many relations
with which we have been dealing by saying that an
R-step followed by a backward R-step must bring
us back to our starting-point. With other relations,
this is by no means the case; for example, if R is the
relation of child to parent, the relative product of
R and its converse is the relation “self or brother
or sister,” and if R is the relation of grandchild to
grandparent, the relative product of R and its converse
is “self or brother or sister or first cousin.” It
will be observed that the relative product of two
relations is not in general commutative, i.e. the relative
product of R and S is not in general the same
relation as the relative product of S and R. E.g. the
relative product of parent and brother is uncle, but
the relative product of brother and parent is parent.
One-one relations give a correlation of two classes,
term for term, so that each term in either class
has its correlate in the other. Such correlations are
simplest to grasp when the two classes have no
members in common, like the class of husbands
and the class of wives; for in that case we know
at once whether a term is to be considered as one
from which the correlating relation R goes, or as one
to which it goes. It is convenient to use the word
referent for the term from which the relation goes,
and the term relatum for the term to which it goes.
Thus if x and y are husband and wife, then, with
respect to the relation j “husband,” x is referent and 49
y relatum, but with respect to the relation “wife,” y
is referent and x relatum. We say that a relation and
its converse have opposite “senses”; thus the “sense”
of a relation that goes from x to y is the opposite of
that of the corresponding relation from y to x. The
fact that a relation has a “sense” is fundamental,
and is part of the reason why order can be generated
by suitable relations. It will be observed that the
class of all possible referents to a given relation is
its domain, and the class of all possible relata is its
converse domain.
But it very often happens that the domain and
converse domain of a one-one relation overlap. Take,
for example, the first ten integers (excluding 0), and
we now have the integers
2; 3; 4; 5; 6; 7; 8; 9; 10; 11:
These are the same as those we had before, except
that 1 has been cut off at the beginning and 11 has
been joined on at the end. There are still ten integers:
they are correlated with the previous ten by
the relation of n to n+1, which is a one-one relation.
ten integers, we could have doubled each of them,
thus obtaining the integers
2; 4; 6; 8; 10; 12; 14; 16; 18; 20:
Here we still have five of our previous set of integers,
namely, 2, 4, 6, 8, 10. The correlating relation in
this case is the relation of a number to its double,
which is again a one-one relation. Or we might have
replaced each number by its square, thus obtaining
the set
1; 4; 9; 16; 25; 36; 49; 64; 81; 100:
On this occasion only three of our original set are
left, namely, 1, 4, 9. Such processes of correlation
may be varied endlessly.
The most interesting case of the above kind is
the case where our one-one relation has a converse
domain which is part, but j not the whole, of the 50
domain. If, instead of confining the domain to the
first ten integers, we had considered the whole of the
inductive numbers, the above instances would have
illustrated this case. We may place the numbers
concerned in two rows, putting the correlate directly
under the number whose correlate it is. Thus when
the correlator is the relation of n to n + 1, we have
the two rows:
1; 2; 3; 4; 5; : : : n : : :
2; 3; 4; 5; 6; : : : n + 1 : : :
When the correlator is the relation of a number to
its double, we have the two rows:
1; 2; 3; 4; 5; : : : n : : :
2; 4; 6; 8; 10; : : : 2n : : :
When the correlator is the relation of a number to
its square, the rows are:
1; 2; 3; 4; 5; : : : n : : :
1; 4; 9; 16; 25; : : : n2 : : :
In all these cases, all inductive numbers occur in the
top row, and only some in the bottom row.
Cases of this sort, where the converse domain
is a “proper part” of the domain (i.e. a part not the
whole), will occupy us again when we come to deal
with infinity. For the present, we wish only to note
that they exist and demand consideration.
Another class of correlations which are often important
is the class called “permutations,” where
the domain and converse domain are identical. Consider,
for example, the six possible arrangements of
three letters:
a, b, c
a, c, b
b, c, a
b, a, c
c, a, b
c, b, a j
Each of these can be obtained from any one of the 51
others by means of a correlation. Take, for example,
the first and last, (a, b, c) and (c, b, a). Here a is
correlated with c, b with itself, and c with a. It is
obvious that the combination of two permutations
is again a permutation, i.e. the permutations of a
given class form what is called a “group.”
These various kinds of correlations have importance
in various connections, some for one purpose,
some for another. The general notion of one-one
correlations has boundless importance in the philosophy
of mathematics, as we have partly seen already,
but shall see much more fully as we proceed.
One of its uses will occupy us in our next chapter.
Chapter VI
Similarity of Relations
We saw in Chapter II. that two classes have the 52
same number of terms when they are “similar,” i.e.
when there is a one-one relation whose domain is
the one class and whose converse domain is the
other. In such a case we say that there is a “one-one
correlation” between the two classes.
In the present chapter we have to define a relation
between relations, which will play the same
part for them that similarity of classes plays for
classes. We will call this relation “similarity of relations,”
or “likeness” when it seems desirable to use
a different word from that which we use for classes.
How is likeness to be defined?
We shall employ still the notion of correlation:
we shall assume that the domain of the one relation
can be correlated with the domain of the other,
and the converse domain with the converse domain;
but that is not enough for the sort of resemblance
which we desire to have between our two relations.
What we desire is that, whenever either relation
holds between two terms, the other relation shall
hold between the correlates of these two terms. The
easiest example of the sort of thing we desire is a
map. When one place is north of another, the place
on the map corresponding to the one is above the
place on the map corresponding to the other; when
one place is west of another, the place on the map
corresponding to the one is to the left of the place on
the map corresponding to the other; and so on. The
structure of the map corresponds with that of j the 53
country of which it is a map. The space-relations
in the map have “likeness” to the space-relations in
the country mapped. It is this kind of connection
between relations that we wish to define.
We may, in the first place, profitably introduce
a certain restriction. We will confine ourselves, in
defining likeness, to such relations as have “fields,”
i.e. to such as permit of the formation of a single
class out of the domain and the converse domain.
This is not always the case. Take, for example, the
relation “domain,” i.e. the relation which the domain
of a relation has to the relation. This relation
has all classes for its domain, since every class is
the domain of some relation; and it has all relations
for its converse domain, since every relation has a
domain. But classes and relations cannot be added
together to