An Introduction to Mathematics · Alfred North Whitehead
Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathematicians.
The ideas of any and of some are introduced into algebra by the use of letters, instead of the definite numbers of arithmetic. Thus, instead of saying that 2 + 3 = 3 + 2 , in algebra we generalize and say that, if x and y stand for any two numbers, then x + y = y + x. Again, in the place of saying that 3 > 2 , we generalize and say that if x be any number there exists some number (or numbers) y such that y > x. We may remark in passing that this latter assumption--for when put in its strict ultimate form it is an assumption--is
of vital importance, both to philosophy and to mathematics; for by it the notion of infinity is introduced. Perhaps it required the introduction of the arabic numerals, by which the use of letters as standing for definite numbers has been completely discarded in mathematics, in order to suggest to mathematicians the technical convenience of the use of letters for the ideas of any number and some number. The Romans would have stated the number of the year in which this is written in the form MDCCCCX., whereas we write it 1910, thus leaving the letters for the other usage. But this is merely a speculation. After the rise of algebra the differential calculus was invented by Newton and
Leibniz, and then a pause in the progress
of the philosophy of mathematical thought occurred so far as these notions are concerned; and it was not till within the last few years that it has been realized how fundamental any and some are to the very nature of mathematics, with the result of opening out still further subjects for mathematical exploration.
Let us now make some simple algebraic statements, with the object of understanding exactly how these fundamental ideas occur.
(1) For any number x , x + 2 = 2 + x ;
(2) For some number x , x + 2 = 3 ;
(3) For some number x , x + 2 > 3.
The first point to notice is the possibilities contained in the meaning of some, as here used. Since x + 2 = 2 + x for any number x , it is true for some number x. Thus, as here used, any implies some and some does not exclude any. Again, in the second example, there is, in fact, only one number x , such that x + 2 = 3 , namely only the number 1. Thus the some may be one number only. But in the third example, any number x which is greater than 1 gives x + 2 > 3. Hence there are an infinite number of numbers which answer to the some number in this case. Thus some may be anything between any and one only, including both these limiting cases.
It is natural to supersede the statements (2) and (3) by the questions:
(2') For what number x is x + 2 = 3 ;
(3') For what numbers x is x + 2 > 3.
Considering (2'), x + 2 = 3 is an equation, and
it is easy to see that its solution is x = 3 - 2 = 1. When we have asked the question implied in the statement of the equation x + 2 = 3 , x is called the unknown. The object of the solution of the equation is the determination of the unknown. Equations are of great importance in mathematics, and it seems as
though (2') exemplified a much more thoroughgoing and fundamental idea than the original statement (2). This, however, is a complete mistake. The idea of the undetermined
"variable" as occurring in the use of "some" or "any" is the really important one in mathematics; that of the "unknown" in an equation, which is to be solved as quickly as possible, is only of subordinate use, though of course it is very important. One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations. The same remark applies to the solution of the inequality (3') as compared to the original statement (3).
But the majority of interesting formulæ,
especially when the idea of some is present, involve more than one variable. For example, the consideration of the pairs of numbers x and y (fractional or integral) which satisfy x + y = 1 involves the idea of two correlated variables, x and y. When two variables are present the same two main types of statement occur. For example, (1) for any pair of numbers, x and y , x + y = y + x , and (2) for some pairs of numbers, x and y , x + y = 1.
The second type of statement invites consideration of the aggregate of pairs of numbers which are bound together by some fixed relation--in the case given, by the relation x + y = 1. One use of formulæ of the first type, true for any pair of numbers, is that by them formulæ of the second type can be
thrown into an indefinite number of equivalent forms. For example, the relation x + y = 1 is equivalent to the relations y + x = 1, (x - y) + 2y = 1, 6x + 6y = 6, and so on. Thus a skilful mathematician uses that equivalent form of the relation under consideration which is most convenient for his immediate purpose.
It is not in general true that, when a pair of terms satisfy some fixed relation, if one of the terms is given the other is also definitely determined. For example, when x and y satisfy y^2 = x , if x = 4 , y can be ±2 , thus, for any positive value of x there are alternative values for y. Also in the relation x + y > 1 , when either x or y is given, an indefinite number of values remain open for the other.
Again there is another important point to be noticed. If we restrict ourselves to positive numbers, integral or fractional, in considering the relation x + y = 1 , then, if either x or y be greater than 1 , there is no positive number which the other can assume so as to satisfy the relation. Thus the "field" of the relation for x is restricted to numbers less than 1 , and similarly for the "field" open to y. Again, consider integral numbers only, positive or negative, and take the relation
y^2 = x , satisfied by pairs of such numbers. Then whatever integral value is given to y , x can assume one corresponding integral value. So the "field" for y is unrestricted among these positive or negative integers. But the "field" for x is restricted in two ways. In the first place x must be positive, and in the second place, since y is to be integral, x must be a perfect square. Accordingly, the "field" of x is restricted to the set of integers 1^2 , 2^2 , 3^2 , 4^2 , and so on, i.e., to 1 , 4 , 9 , 16 , and so on.
The study of the general properties of a relation between pairs of numbers is much facilitated by the use of a diagram constructed as follows:
Draw two lines OX and OY at right angles; let any number x be represented by x units
(in any scale) of length along OX , any number y by y units (in any scale) of length along OY. Thus if OM , along OX , be x units in length, and ON , along OY , be y units in length, by completing the parallelogram OMPN we find a point P which corresponds to the pair of numbers x and y. To each point there corresponds one pair of numbers, and to each pair of numbers there corresponds one point. The pair of numbers are called the coordinates of the point. Then the points whose coordinates satisfy some fixed relation can be indicated in a convenient way, by drawing a line, if they all lie on a line, or by shading an area if they are all points in the area. If the relation can be represented by an equation such as x + y = 1 , or y^2 = x , then the points lie on a line, which is straight in the former case and curved in the latter. For example, considering only positive numbers, the points whose coordinates satisfy x + y = 1 lie on the straight line AB in Fig. 1, where 0A = 1 and OB = 1. Thus this segment of the straight line AB gives a pictorial representation of the properties of the relation under the restriction to positive numbers.
Another example of a relation between two variables is afforded by considering the variations in the pressure and volume of a given mass of some gaseous substance--such as air
or coal-gas or steam--at a constant temperature. Let v be the number of cubic feet in its volume and p its pressure in lb. weight per square inch. Then the law, known as Boyle's law, expressing the relation between p and v as both vary, is that the product pv is constant, always supposing that the temperature does not alter. Let us suppose, for example, that the quantity of the gas and its other circumstances are such that we can put pv = 1 (the exact number on the right-hand side of the equation makes no essential difference).
Then in Fig. 2 we take two lines, OV and OP , at right angles and draw OM along OV to represent v units of volume, and ON along OP
to represent p units of pressure. Then the point Q , which is found by completing the parallelogram OMQN , represents the state of the gas when its volume is v cubic feet and its pressure is p lb. weight per square inch. If the circumstances of the portion of gas considered are such that pv = 1 , then all these points Q which correspond to any possible state of this portion of gas must lie on the curved line ABC , which includes all points for which p and v are positive, and pv = 1. Thus this curved line gives a pictorial representation of the relation holding between the volume and the pressure. When the pressure is very big the corresponding point Q must be near C , or even beyond C on the undrawn part of the curve; then the volume will be very small. When the volume is big Q will be near to A , or beyond A ; and then the pressure will be small. Notice that an engineer or a physicist may want to know the particular pressure corresponding to some definitely assigned volume. Then we have the case of determining the unknown p when
v is a known number. But this is only in particular cases. In considering generally the properties of the gas and how it will behave, he has to have in his mind the general form of the whole curve ABC and its general properties. In other words the really fundamental idea is that of the pair of variables
satisfying the relation pv = 1. This example illustrates how the idea of variables is fundamental,
both in the applications as well as in the theory of mathematics.
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End of part 2 of 19
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