An Introduction to Mathematics · Alfred North Whitehead
One great peculiarity of mathematics is the
set of allied ideas which have been invented in connection with the integral numbers from which we started. These ideas may be called extensions or generalizations of number. In the first place there is the idea of fractions. The earliest treatise on arithmetic which we possess was written by an Egyptian priest, named Ahmes, between 1700 B.C. and 1100 B.C.,
and it is probably a copy of a much older work. It deals largely with the properties of fractions. It appears, therefore, that this concept was developed very early in the history of mathematics. Indeed the subject is a very obvious one. To divide a field into three equal parts, and to take two of the parts, must be a type of operation which had often occurred. Accordingly, we need not be surprised that the men of remote civilizations were familiar with the idea of two-thirds, and
with allied notions. Thus as the first generalization of number we place the concept of fractions. The Greeks thought of this subject rather in the form of ratio, so that a Greek would naturally say that a line of two feet in length bears to a line of three feet in length the ratio of 2 to 3. Under the influence of our algebraic notation we would more often say that one line was two-thirds of the other in length, and would think of two-thirds as a numerical multiplier.
In connection with the theory of ratio, or
fractions, the Greeks made a great discovery, which has been the occasion of a large amount of philosophical as well as mathematical thought. They found out the existence of "incommensurable" ratios. They proved, in fact, during the course of their geometrical investigations that, starting with a line of any length, other lines must exist whose lengths do not bear to the original length the ratio of any pair of integers--or, in other words, that lengths exist which are not any exact fraction of the original length.
For example, the diagonal of a square cannot be expressed as any fraction of the side of the same square; in our modern notation the length of the diagonal is sqrt(2) times the length of the side. But there is no fraction which exactly represents sqrt(2). We can approximate
to sqrt(2) as closely as we like, but we never exactly reach its value. For example, 49/25 is just less than 2 , and 9/4 is greater than 2 , so that sqrt(2) lies between 7/5 and 3/2. But the best systematic way of approximating to sqrt(2) in obtaining a series of decimal fractions, each bigger than the last, is by the ordinary method of extracting the square root; thus the series is 1 , 14/10 , 141/100 , 1414/1000 , and so on.
Ratios of this sort are called by the Greeks incommensurable. They have excited from the time of the Greeks onwards a great deal of philosophic discussion, and the difficulties connected with them have only recently been cleared up.
We will put the incommensurable ratios
with the fractions, and consider the whole set of integral numbers, fractional numbers, and incommensurable numbers as forming one class of numbers which we will call "real numbers." We always think of the real numbers as arranged in order of magnitude, starting from zero and going upwards, and becoming indefinitely larger and larger as we proceed. The real numbers are conveniently
represented by points on a line. Let OX be
any line bounded at O and stretching away indefinitely in the direction OX. Take any convenient point, A , on it, so that OA represents the unit length; and divide off lengths AB , BC , CD , and so on, each equal to OA. Then the point O represents the number 0 , A the number 1 , B the number 2 , and so on. In fact the number represented by any point is the measure of its distance from O , in terms of the unit length OA. The points between O and A represent the proper fractions and the incommensurable numbers less than 1 ; the middle point of OA represents 1/2 , that of AB represents 3/2 , that of BC represents 5/2 , and so on. In this way every point on OX represents some one real number, and every real number is represented by some one point on OX.
The series (or row) of points along OX ,
starting from O and moving regularly in the direction from O to X , represents the real numbers as arranged in an ascending order
of size, starting from zero and continually increasing as we go on.
All this seems simple enough, but even at
this stage there are some interesting ideas to be got at by dwelling on these obvious facts. Consider the series of points which represent the integral numbers only, namely, the points, O , A , B , C , D , etc. Here there is a first point O , a definite next point, A , and each point, such as A or B , has one definite immediate predecessor and one definite immediate successor, with the exception of O , which has no predecessor; also the series goes on indefinitely without end. This sort of order is called the type of order of the integers; its essence is the possession of next-door neighbours on either side with the exception of No. 1 in the row. Again consider the integers and fractions together, omitting the points which correspond to the incommensurable ratios. The sort of serial order which we now obtain is quite different. There is a first term O ; but no term has any immediate predecessor or immediate successor. This is easily seen to be the case, for between any two fractions we can always find another fraction intermediate in value. One very simple way of doing this is to add the fractions together and to halve the result. For example,
between 2/3 and 3/4 , the fraction 1/2(2/3 + 3/4) , that is 17/24 , lies; and between 2/3 and 17/24 the
fraction 1/2(2/3 + 17/24) , that is 33/48 , lies; and so on indefinitely. Because of this property the series is said to be "compact." There is no end point to the series, which increases indefinitely without limit as we go along the line OX. It would seem at first sight as though the type of series got in this way from the fractions, always including the integers, would be the same as that got from all the real numbers, integers, fractions, and incommensurables taken together, that is, from all the points on the line OX. All that we have hitherto said about the series of fractions applies equally well to the series of all real numbers. But there are important differences which we now proceed to develop. The absence of the incommensurables from the series of fractions leaves an absence of endpoints to certain classes. Thus, consider the incommensurable sqrt(2). In the series of real numbers this stands between all the numbers whose squares are less than 2 , and all the numbers whose squares are greater than 2. But keeping to the series of fractions alone and not thinking of the incommensurables, so that we cannot bring in sqrt(2) , there is no fraction which has the property of dividing off the series into two parts in this way, i.e. so that all the members on one side have their squares less than 2 , and on the other side greater than 2. Hence in the series of fractions
there is a quasi-gap where sqrt(2) ought to come. This presence of quasi-gaps in the series of fractions may seem a small matter; but any mathematician, who happens to read this, knows that the possible absence of limits
or maxima to a class of numbers, which yet does not spread over the whole series of numbers, is no small evil. It is to avoid this difficulty that recourse is had to the incommensurables, so as to obtain a complete series with no gaps.
There is another even more fundamental difference between the two series. We can rearrange the fractions in a series like that of the integers, that is, with a first term, and such that each term has an immediate successor and (except the first term) an immediate predecessor. We can show how this can be done. Let every term in the series of fractions and integers be written in the fractional form by writing 1/1 for 1 , 2/1 for 2 , and so on for all the integers, excluding 0. Also for the moment we will reckon fractions which are equal in value but not reduced to their lowest terms as distinct; so that, for example, until further notice 2/3 , 4/6 , 6/9 , 8/12 , etc., are all reckoned as distinct. Now group the fractions into classes by adding together the numerator and denominator of each term. For the sake of brevity call this sum of the numerator and denominator of a fraction its index. Thus 7 is
the index of 4/3 , and also of 3/4 , and of 2/5. Let the fractions in each class be all fractions which have some specified index, which may therefore also be called the class index. Now arrange these classes in the order of magnitude of their indices. The first class has the index 2 , and its only member is 1/1 ; the second class has the index 3 , and its members are 1/2 and 2/1 ; the third class has the index 4 , and its members are 1/3 , 2/2 , 3/1 ; the fourth class has the index 5 , and its members are 1/4 , 2/3 , 3/2 , 4/1 ; and so on. It is easy to see that the number of members (still including fractions not in their lowest terms) belonging to any class is one less than its index. Also the members of any one class can be arranged in order by taking the first member to be the fraction with numerator 1 , the second member to have the numerator 2 , and so on, up to (n - 1) where n is the index. Thus for the class of index n , the members appear in the order
1/n - 1, 2/n - 2, 3/n - 3..., n - 1/1. The members of the first four classes have in fact been mentioned in this order. Thus the whole set of fractions have now been arranged in an order like that of the integers. It runs thus 1/1, 1/2, 2/1, 1/3, [2/2], 3/1, 1/4, 2/3, 3/2, 4/1..., ; n - 1/1, 1/n - 1, 2/n - 2, 3/n - 3..., n - 1/1, 1/n, and so on.
Now we can get rid of all repetitions of fractions of the same value by simply striking them out whenever they appear after their first occurrence. In the few initial terms written down above, 2/2 which is enclosed above in square brackets is the only fraction not in its lowest terms. It has occurred before as 1/1. Thus this must be struck out. But the series is still left with the same properties, namely, (a) there is a first term, (b) each term has next-door neighbours, (c) the series goes on without end.
It can be proved that it is not possible to
arrange the whole series of real numbers in this way. This curious fact was discovered by Georg Cantor, a German mathematician still living; it is of the utmost importance in the philosophy of mathematical ideas. We are here in fact touching on the fringe of the great problems of the meaning of continuity and of infinity.
Another extension of number comes from
the introduction of the idea of what has been variously named an operation or a step, names which are respectively appropriate from slightly different points of view. We will start with a particular case. Consider
the statement 2 + 3 = 5. We add 3 to 2 and obtain 5. Think of the operation of adding 3 : let this be denoted by +3. Again 4 - 3 = 1. Think of the operation of subtracting 3 : let this be denoted by -3. Thus instead of considering the real numbers in themselves, we consider the operations of adding or subtracting them: instead of sqrt(2) , we consider +sqrt(2) and -sqrt(2) , namely the operations of adding sqrt(2) and of subtracting sqrt(2). Then we can add these operations, of course in a different sense of addition to that in which we add numbers. The sum of two operations is the single operation which has the same effect as the two operations applied successively. In what order are the two operations to be applied? The answer is that it is indifferent, since for example 2 + 3 + 1 = 2 + 1 + 3; so that the addition of the steps +3 and +1 is commutative.
Mathematicians have a habit, which is puzzling to those engaged in tracing out meanings, but is very convenient in practice, of using the same symbol in different though allied senses. The one essential requisite for a symbol in their eyes is that, whatever its possible varieties of meaning, the formal laws for its use shall always be the same. In
accordance with this habit the addition of operations is denoted by + as well as the addition of numbers. Accordingly we can write (+3) + (+1) = +4; where the middle + on the left-hand side denotes the addition of the operations +3 and +1. But, furthermore, we need not be so very pedantic in our symbolism, except in the rare instances when we are directly tracing meanings; thus we always drop the first + of a line and the brackets, and never write two + signs running. So the above equation becomes 3 + 1 = 4, which we interpret as simple numerical addition, or as the more elaborate addition of operations which is fully expressed in the previous way of writing the equation, or lastly as expressing the result of applying the operation +1 to the number 3 and obtaining the number 4. Any interpretation which is possible is always correct. But the only interpretation which is always possible, under certain conditions, is that of operations. The other interpretations often give nonsensical results.
This leads us at once to a question, which must have been rising insistently in the
reader's mind: What is the use of all this elaboration? At this point our friend, the practical man, will surely step in and insist on sweeping away all these silly cobwebs of the brain. The answer is that what the mathematician is seeking is Generality. This is an
idea worthy to be placed beside the notions of the Variable and of Form so far as concerns
its importance in governing mathematical procedure. Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct. These three notions, of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science.
Let us see how generality is gained by the introduction of this idea of operations. Take the equation x + 1 = 3 ; the solution is x = 2. Here we can interpret our symbols as mere numbers, and the recourse to "operations" is entirely unnecessary. But, if x is a mere number, the equation x + 3 = 1 is nonsense. For x should be the number of things which remain when you have taken 3 things away from 1 thing; and no such procedure is possible. At this point our idea of algebraic form steps in, itself only generalization under another aspect. We consider, therefore, the
general equation of the same form as x + 1 = 3. This equation is x + a = b , and its solution is x = b - a. Here our difficulties become acute; for this form can only be used for the numerical interpretation so long as b is greater than a , and we cannot say without qualification that a and b may be any constants. In other words we have introduced a limitation on the variability of the "constants" a and b , which we must drag like a chain throughout all our reasoning. Really prolonged mathematical investigations would be impossible under such conditions. Every equation would at last be buried under a pile of limitations. But if we now interpret our symbols as "operations," all limitation vanishes like magic. The equation x + 1 = 3 gives x = +2 , the equation x + 3 = 1 gives x = -2 , the equation x + a = b gives x = b - a which is an operation of addition or subtraction as the case may be. We need never decide whether b - a represents the operation of addition or of subtraction, for the rules of procedure with the symbols are the same in either case.
It does not fall within the plan of this work to write a detailed chapter of elementary algebra. Our object is merely to make plain the fundamental ideas which guide the formation of the science. Accordingly we do not further explain the detailed rules by which the "positive and negative numbers" are
multiplied and otherwise combined. We have explained above that positive and negative numbers are operations. They have also been called "steps." Thus +3 is the step by which we go from 2 to 5 , and -3 is the step backwards by which we go from 5 to 2. Consider the line OX divided in the way explained in the earlier part of the chapter, so that its points represent numbers. Then +2
is the step from O to B , or from A to C , or (if the divisions are taken backwards along OX' ) from C' to A' , or from D' to B' , and so on. Similarly -2 is the step from O to B' , or from B' to D' , or from B to O , or from C to A.
We may consider the point which is reached by a step from O , as representative of that step. Thus A represents +1 , B represents +2 , A' represents -1 , B' represents -2 , and so on. It will be noted that, whereas previously with the mere "unsigned" real numbers the points on one side of O only, namely along OX , were representative of numbers, now with steps every point on the whole line stretching on both sides of O is representative of a step. This is a pictorial representation of the superior generality introduced by the positive and negative numbers, namely the
operations or steps. These "signed" numbers are also particular cases of what have been called vectors (from the Latin veho, I
draw or carry). For we may think of a particle as carried from O to A , or from A to B.
In suggesting a few pages ago that the practical man would object to the subtlety involved by the introduction of the positive and negative numbers, we were libelling that excellent individual. For in truth we are on the scene of one of his greatest triumphs. If the truth must be confessed, it was the practical man himself who first employed the actual symbols + and -. Their origin is not very certain, but it seems most probable that they arose from the marks chalked on chests of goods in German warehouses, to denote excess or defect from some standard weight. The earliest notice of them occurs in a book published at Leipzig, in A.D. 1489. They seem first to have been employed in mathematics by a German mathematician, Stifel, in a book
published at Nuremburg in 1544 A.D. But then it is only recently that the Germans have come to be looked on as emphatically a practical nation. There is an old epigram which assigns the empire of the sea to the English, of the land to the French, and of the clouds to the Germans. Surely it was from the clouds that the Germans fetched + and - ;
the ideas which these symbols have generated are much too important for the welfare of humanity to have come from the sea or from the land.
The possibilities of application of the positive and negative numbers are very obvious. If lengths in one direction are represented by positive numbers, those in the opposite direction are represented by negative numbers. If a velocity in one direction is positive, that in the opposite direction is negative. If a rotation round a dial in the opposite direction to the hands of a clock (anti-clockwise) is positive, that in the clockwise direction is negative. If a balance at the bank is positive, an overdraft is negative. If vitreous electrification is positive, resinous electrification is negative. Indeed, in this latter case, the terms positive electrification and negative electrification, considered as mere names, have practically driven out the other terms. An endless series of examples could be given. The idea of positive and negative numbers has been practically the most successful of mathematical subtleties.
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End of part 6 of 19
Continue readingChapter VII: Imaginary Numbers
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