An Introduction to Mathematics · Alfred North Whitehead
Geometry, like the rest of mathematics, is
abstract. In it the properties of the shapes and relative positions of things are studied. But we do not need to consider who is observing the things, or whether he becomes acquainted with them by sight or touch or hearing. In short, we ignore all particular sensations. Furthermore, particular things such as the Houses of Parliament, or the terrestrial globe are ignored. Every proposition refers to any things with such and such geometrical properties. Of course it helps our imagination to look at particular examples of spheres and cones and triangles and squares. But the propositions do not merely apply to the actual figures printed in the book, but to any such figures.
Thus geometry, like algebra, is dominated by the ideas of "any" and "some" things. Also, in the same way it studies the interrelations of sets of things. For example, consider any two triangles ABC and DEF.
What relations must exist between some of
the parts of these triangles, in order that the triangles may be in all respects equal? This is one of the first investigations undertaken in all elementary geometries. It is a study
of a certain set of possible correlations between the two triangles. The answer is that the triangles are in all respects equal, if:-- Either, (a) Two sides of the one and the included angle are respectively equal to two sides of the other and the included angle:
Or, (b) Two angles of the one and the side joining them are respectively equal to two angles of the other and the side joining them:
Or, (c) Three sides of the one are respectively equal to three sides of the other.
This answer at once suggests a further enquiry. What is the nature of the correlation between the triangles, when the three angles of the one are respectively equal to the three angles of the other? This further investigation leads us on to the whole theory of similarity
(cf. Chapter XIII.), which is another type of correlation.
Again, to take another example, consider the internal structure of the triangle ABC. Its sides and angles are inter-related--the greater angle is opposite to the greater side, and the base angles of an isosceles triangle are equal. If we proceed to trigonometry this correlation receives a more exact determination in the familiar shape sin A/a = sin B/b = sin C/c, a^2 = b^2 + c^2 - 2bc cos A , with two similar formulæ.
Also there is the still simpler correlation between the angles of the triangle, namely, that their sum is equal to two right angles; and between the three sides, namely, that the sum of the lengths of any two is greater than the length of the third.
Thus the true method to study geometry is to think of interesting simple figures, such as the triangle, the parallelogram, and the circle, and to investigate the correlations between their various parts. The geometer has in his mind not a detached proposition, but a figure with its various parts mutually inter-dependent. Just as in algebra, he generalizes the triangle into the polygon, and the side into
the conic section. Or, pursuing a converse route, he classifies triangles according as they are equilateral, isosceles, or scalene, and polygons according to their number of sides, and conic sections according as they are hyperbolas, ellipses, or parabolas.
The preceding examples illustrate how the fundamental ideas of geometry are exactly the same as those of algebra; except that algebra deals with numbers and geometry with lines, angles, areas, and other geometrical entities. This fundamental identity is one of the reasons why so many geometrical truths can be put into an algebraic dress. Thus if A , B , and C are the numbers of degrees respectively in the angles of the triangle ABC , the correlation between the angles is represented by the equation A + B + C = 180°; and if a , b , c are the number of feet respectively in the three sides, the correlation between the sides is represented by a < b + c , b < c + a , c < a + b. Also the trigonometrical formulæ quoted above are other examples of the same
fact. Thus the notion of the variable and the correlation of variables is just as essential in geometry as it is in algebra.
But the parallelism between geometry and algebra can be pushed still further, owing to the fact that lengths, areas, volumes, and
angles are all measurable; so that, for example, the size of any length can be determined by the number (not necessarily integral) of times which it contains some arbitrarily known unit, and similarly for areas, volumes, and angles. The trigonometrical formulæ, given above, are examples of this fact. But it receives its crowning application in analytical geometry. This great subject is often misnamed as Analytical Conic Sections, thereby
fixing attention on merely one of its subdivisions. It is as though the great science of Anthropology were named the Study of Noses, owing to the fact that noses are a prominent part of the human body.
Though the mathematical procedures in geometry and algebra are in essence identical and intertwined in their development, there is necessarily a fundamental distinction between the properties of space and the properties of number--in fact all the essential difference between space and number. The "spaciness" of space and the "numerosity" of number are essentially different things, and must be directly apprehended. None of the applications of algebra to geometry or of geometry to algebra go any step on the road to obliterate this vital distinction.
One very marked difference between space and number is that the former seems to be so much less abstract and fundamental than the
latter. The number of the archangels can be counted just because they are things. When we once know that their names are Raphael, Gabriel, and Michael, and that these distinct names represent distinct beings, we know without further question that there are three of them. All the subtleties in the world about the nature of angelic existences cannot alter this fact, granting the premisses.
But we are still quite in the dark as to their relation to space. Do they exist in space at all? Perhaps it is equally nonsense to say that they are here, or there, or anywhere, or everywhere. Their existence may simply have no relation to localities in space. Accordingly, while numbers must apply to all things, space need not do so.
The perception of the locality of things would appear to accompany, or be involved in many, or all, of our sensations. It is independent of any particular sensation in the sense that it accompanies many sensations. But it is a special peculiarity of the things which we apprehend by our sensations. The direct apprehension of what we mean by the positions of things in respect to each other is a thing sui generis, just as are the apprehensions of sounds, colours, tastes, and smells. At first sight therefore it would appear that mathematics, in so far as it includes geometry in its scope, is not abstract in the sense in
which abstractness is ascribed to it in Chapter I.
This, however, is a mistake; the truth being
that the "spaciness" of space does not enter into our geometrical reasoning at all. It enters into the geometrical intuitions of mathematicians in ways personal and peculiar to each individual. But what enter into the reasoning are merely certain properties of things in space, or of things forming space, which properties are completely abstract in the sense in which abstract was defined in Chapter I.; these properties do not involve any peculiar space-apprehension or space-intuition or space-sensation. They are on exactly the same basis as the mathematical properties of number. Thus the space-intuition which is so essential an aid to the study of geometry is logically irrelevant: it does not enter into the premisses when they are properly stated, nor into any step of the reasoning. It has the practical importance of an example, which is essential for the stimulation of our thoughts. Examples are equally necessary to stimulate our thoughts on number. When we think of "two" and "three" we see strokes in a row, or balls in a heap, or some other physical aggregation of particular things. The peculiarity of geometry is the fixity and overwhelming importance of the one particular example which occurs to our
minds. The abstract logical form of the propositions when fully stated is, "If any collections of things have such and such abstract properties, they also have such and such other abstract properties." But what appears before the mind's eye is a collection of points, lines, surfaces, and volumes in the space: this example inevitably appears, and is the sole example which lends to the proposition its interest. However, for all its overwhelming importance, it is but an example.
Geometry, viewed as a mathematical science, is a division of the more general science of order. It may be called the science of dimensional order; the qualification "dimensional" has been introduced because the limitations, which reduce it to only a part of the general science of order, are such as to produce the regular relations of straight lines to planes, and of planes to the whole of space.
It is easy to understand the practical importance of space in the formation of the scientific conception of an external physical world. On the one hand our space-perceptions are intertwined in our various sensations and connect them together. We normally judge that we touch an object in the same place as we see it; and even in abnormal cases we touch it in the same space as we see it, and this is the real fundamental fact which ties together our various sensations. Accordingly,
the space perceptions are in a sense the common part of our sensations. Again it happens that the abstract properties of space form a large part of whatever is of spatial interest. It is not too much to say that to every property of space there corresponds an abstract mathematical statement. To take the most unfavourable instance, a curve may have a special beauty of shape: but to this shape there will correspond some abstract mathematical properties which go with this shape and no others.
Thus to sum up: (1) the properties of space which are investigated in geometry, like those of number, are properties belonging to things as things, and without special reference to any particular mode of apprehension: (2) Space-perception accompanies our sensations, perhaps all of them, certainly many; but it does not seem to be a necessary quality of things that they should all exist in one space or in any space.
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End of part 16 of 19
Continue readingChapter XVII: Quantity
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