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`

`)i

`) + (y+y`+iy`

`) = (x+x

(x+iy)(x`

`)

` + yx` + i y``

`) = (xx``

` - yy`

`) + i(x`y`

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

would be the series of whole numbers, , , , , … etc. | Probably only a person with some mathematical knowledge would think of beginning with instead of with , but we will presume this degree of knowledge; we will take as our starting-point the series: , , , , … n, n + , … and it is this series that we shall mean when we speak of the “series of natural numbers.” It is only at a high stage of civilisation that we could take this series as our starting-point. It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number : the degree of abstraction involved is far from easy. And the discovery that is a number must have been difficult. As for , it is a very recent addition; the Greeks and Romans had no such digit. If we had been embarking upon mathematical philosophy in earlier days, we should have had to start with something less abstract than the series of natural numbers, which we should reach as a stage on our backward journey. When the logical foundations of mathematics have grown more familiar, we shall be able to start further back, at what is now a late stage in our analysis. But for the moment the natural numbers seem to represent what is easiest and most familiar in mathematics. But though familiar, they are not understood. Very few people are prepared with a definition of what is meant by “number,” or “,” or “.” It is not very difficult to see that, starting from , any other of the natural numbers can be reached by repeated additions of , but we shall have to define what we mean by “adding ,” and what we mean by “repeated.” These questions are by no means easy. It was believed until recently that some, at least, of these first notions of arithmetic must be accepted as too simple and primitive to be defined. Since all terms that are defined are defined by means of other terms, it is clear that human knowledge must always be content to accept some terms as intelligible without definition, in order | to have a starting-point for its definitions. It is not clear that there must be terms which are incapable of definition: it is possible that, however far back we go in defining, we always might go further still. On the other hand, it is also possible that, when analysis has been pushed far enough, we can reach terms that really are simple, and therefore logically incapable of the sort of definition that consists in analysing. This is a question which it is not necessary for us to decide; for our purposes it is sufficient to observe that, since human powers are finite, the definitions known to us must always begin somewhere, with terms undefined for the moment, though perhaps not permanently. All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. That all traditional pure mathematics can be derived from the natural numbers is a fairly recent discovery, though it had long been suspected. Pythagoras, who believed that not only mathematics, but everything else could be deduced from numbers, was the discoverer of the most serious obstacle in the way of what is called the “arithmetising” of mathematics. It was Pythagoras who discovered the existence of incommensurables, and, in particular, the incommensurability of the side of a square and the diagonal. If the length of the side is inch, the number of inches in the diagonal is the square root of , which appeared not to be a number at all. The problem thus raised was solved only in our own day, and was only solved completely by the help of the reduction of arithmetic to logic, which will be explained in following chapters. For the present, we shall take for granted the arithmetisation of mathematics, though this was a feat of the very greatest importance. | Having reduced all traditional pure mathemat- ics to the theory of the natural numbers, the next step in logical analysis was to reduce this theory itself to the smallest set of premisses and undefined terms from which it could be derived. This work was accomplished by Peano. He showed that the entire theory of the natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic. These three ideas and five propositions thus became, as it were, hostages for the whole of traditional pure mathematics. If they could be defined and proved in terms of others, so could all pure mathematics. Their logical “weight,” if one may use such an expression, is equal to that of the whole series of sciences that have been deduced from the theory of the natural numbers; the truth of this whole series is assured if the truth of the five primitive propositions is guaranteed, provided, of course, that there is nothing erroneous in the purely logical apparatus which is also involved. The work of analysing mathematics is extraordinarily facilitated by this work of Peano’s. The three primitive ideas in Peano’s arithmetic are: , number, successor. By “successor” he means the next number in the natural order. That is to say, the successor of is , the successor of is , and so on. By “number” he means, in this connection, the class of the natural numbers. He is not assuming that we know all the members of this class, but only that we know what we mean when we say that this or that is a number, just as we know what we mean when we say “Jones is a man,” though we do not know all men individually. The five primitive propositions which Peano assumes are: () is a number. () The successor of any number is a number. () No two numbers have the same successor. | () is not the successor of any number. We shall use “number” in this sense in the present chapter. Afterwards the word will be used in a more general sense. () Any property which belongs to , and also to the successor of every number which has the property, belongs to all numbers. The last of these is the principle of mathematical induction. We shall have much to say concerning mathematical induction in the sequel; for the present, we are concerned with it only as it occurs in Peano’s analysis of arithmetic. Let us consider briefly the kind of way in which the theory of the natural numbers results from these three ideas and five propositions. To begin with, we define as “the successor of ,” as “the successor of ,” and so on. We can obviously go on as long as we like with these definitions, since, in virtue of (), every number that we reach will have a successor, and, in virtue of (), this cannot be any of the numbers already defined, because, if it were, two different numbers would have the same successor; and in virtue of () none of the numbers we reach in the series of successors can be . Thus the series of successors gives us an endless series of continually new numbers. In virtue of () all numbers come in this series, which begins with and travels on through successive successors: for (a) belongs to this series, and (b) if a number n belongs to it, so does its successor, whence, by mathematical induc- tion, every number belongs to the series. Suppose we wish to define the sum of two numbers. Taking any number m, we define m + as m, and m + (n + ) as the successor of m + n. In virtue of () this gives a definition of the sum of m and n, whatever number n may be. Similarly we can define the product of any two numbers. The reader can easily convince himself that any ordinary elementary proposition of arithmetic can be proved by means of our five premisses, and if he has any difficulty he can find the proof in Peano. It is time now to turn to the considerations which make it necessary to advance beyond the standpoint of Peano, who | represents the last perfection of the “arithmetisation” of mathematics, to that of Frege, who first succeeded in “logicising” mathematics, i.e. in reducing to logic the arithmetical notions which his predecessors had shown to be sufficient for mathematics. We shall not, in this chapter, actually give Frege’s definition of number and of particular numbers, but we shall give some of the reasons why Peano’s treatment is less final than it appears to be. In the first place, Peano’s three primitive ideas— namely, “,” “number,” and “successor”—are capable of an infinite number of different interpre- tations, all of which will satisfy the five primitive propositions. We will give some examples. () Let “” be taken to mean , and let “number” be taken to mean the numbers from onward in the series of natural numbers. Then all our primitive propositions are satisfied, even the fourth, for, though is the successor of , is not a “number” in the sense which we are now giving to the word “number.” It is obvious that any number may be substituted for in this example. () Let “” have its usual meaning, but let “number” mean what we usually call “even numbers,” and let the “successor” of a number be what results from adding two to it. Then “” will stand for the number two, “” will stand for the number four, and so on; the series of “numbers” now will be , two, four, six, eight … All Peano’s five premisses are satisfied still. () Let “” mean the number one, let “number” mean the set , , , , , … and let “successor” mean “half.” Then all Peano’s five axioms will be true of this set. It is clear that such examples might be multiplied indefinitely. In fact, given any series x, x, x, x, … xn, … | which is endless, contains no repetitions, has a be- ginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano’s axioms. This is easily seen, though the formal proof is somewhat long. Let “” mean x, let “number” mean the whole set of terms, and let the “successor” of xn mean xn+. Then () “ is a number,” i.e. x is a member of the set. () “The successor of any number is a number,” i.e. taking any term xn in the set, xn+ is also in the set. () “No two numbers have the same successor,” i.e. if xm and xn are two different members of the set, xm+ and xn+ are different; this results from the fact that (by hypothesis) there are no repetitions in the set. () “ is not the successor of any number,” i.e. no term in the set comes before x. () This becomes: Any property which belongs to x, and belongs to xn+ provided it belongs to xn, belongs to all the x’s. This follows from the corresponding property for numbers. A series of the form x, x, x, … xn, … in which there is a first term, a successor to each term (so that there is no last term), no repetitions, and every term can be reached from the start in a finite number of steps, is called a progression. Progressions are of great importance in the principles of mathematics. As we have just seen, every progression verifies Peano’s five axioms. It can be proved, conversely, that every series which verifies Peano’s five axioms is a progression. Hence these five axioms may be used to define the class of progressions: “progressions” are “those series which verify these five axioms.” Any progression may be taken as the basis of pure mathematics: we may give the name “” to its first term, the name “number” to the whole set of its terms, and the name “successor” to the next in the progression. The progression need not be composed of numbers: it may be | composed of points in space, or moments of time, or any other terms of which there is an infinite supply. Each different progression will give rise to a different in- terpretation of all the propositions of traditional pure mathematics; all these possible interpretations will be equally true. In Peano’s system there is nothing to enable us to distinguish between these different interpretations of his primitive ideas. It is assumed that we know what is meant by “,” and that we shall not suppose that this symbol means or Cleopatra’s Needle or any of the other things that it might mean. This point, that “” and “number” and “successor” cannot be defined by means of Peano’s five axioms, but must be independently understood, is important. We want our numbers not merely to verify mathematical formulæ, but to apply in the right way to common objects. We want to have ten fingers and two eyes and one nose. A system in which “” meant , and “” meant , and so on, might be all right for pure mathematics, but would not suit daily life. We want “” and “number” and “successor” to have meanings which will give us the right allowance of fingers and eyes and noses. We have already some knowledge (though not sufficiently articulate or analytic) of what we mean by “” and “” and so on, and our use of numbers in arithmetic must conform to this knowledge. We cannot secure that this shall be the case by Peano’s method; all that we can do, if we adopt his method, is to say “we know what we mean by ‘’ and ‘number’ and ‘successor,’ though we cannot explain what we mean in terms of other simpler concepts.” It is quite legitimate to say this when we must, and at some point we all must; but it is the object of mathematical philosophy to put off saying it as long as possible. By the logical theory of arithmetic we are able to put it off for a very long time. It might be suggested that, instead of setting up “” and “number” and “successor” as terms of which we know the meaning although we cannot define them, we might let them | stand for any three terms that verify Peano’s five axioms. They will then no longer be terms which have a meaning that is definite though undefined: they will be “variables,” terms concerning which we make certain hypotheses, namely, those stated in the five axioms, but which are otherwise undetermined. If we adopt this plan, our theorems will not be proved concerning an ascertained set of terms called “the natural numbers,” but concerning all sets of terms having certain properties. Such a procedure is not fallacious; indeed for certain purposes it represents a valuable generalisation. But from two points of view it fails to give an adequate basis for arithmetic. In the first place, it does not enable us to know whether there are any sets of terms verifying Peano’s axioms; it does not even give the faintest suggestion of any way of discovering whether there are such sets. In the second place, as already observed, we want our numbers to be such as can be used for counting common objects, and this requires that our numbers should have a definite meaning, not merely that they should have certain formal properties. This definite meaning is defined by the logical theory of arithmetic

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

a specially important class. We have already had

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

add 1 to each; thus instead of the first ten integers

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.

Or, again, instead of adding 1 to each of our original

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