Hence when, following Bergson’s advice, we “have recourse to an extended image” and picture, say, twelve dots such as are obtained by throwing double sixes at dice, we have still not obtained a picture of the number 12. The number 12, in fact, is something more abstract than any picture. Before we can be said to have any understanding of the number 12, we must know what different collections of twelve units have in common, and this is something which cannot be pictured because it is abstract. Bergson only succeeds in making his theory of number plausible by confusing a particular collection with the number of its terms, and this again with number in general.
The confusion is the same as if we confused a particular young man with youth, and youth with the general concept “period of human life,” and were then to argue that because a young man has two legs, youth must have two legs, and the general concept “period of human life” must have two legs. The confusion is important because, as soon as it is perceived, the theory that number or particular numbers can be pictured in space is seen to be untenable. This not only disproves Bergson’s theory as to number, but also his more general theory that all abstract ideas and all logic are derived from space.
But apart from the question of numbers, shall we admit Bergson’s contention that every plurality of separate units involves space? Some of the cases that appear to contradict this view are considered by him, for example successive sounds. When we hear the steps of a passer-by in the street, he says, we visualise his successive positions; when we hear the strokes of a bell, we either picture it swinging backwards and forwards, or we range the successive sounds in an ideal space. But these are mere autobiographical observations of a visualizer, and illustrate the remark we made before, that Bergson’s views depend upon
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the predominance of the sense of sight in him. There is no logical necessity to range the strokes of a clock in an imaginary space: most people, I imagine, count them without any spatial auxiliary. Yet no reason is alleged by Bergson for the view that space is necessary. He assumes this as obvious, and proceeds at once to apply it to the case of times. Where there seem to be different times outside each other, he says, the times are pictured as spread out in space; in real time, such as is given by memory, different times interpenetrate each other, and cannot be counted because they are not separate.
The view that all separateness implies space is now supposed established, and is used deductively to prove that space is involved wherever there is obviously separateness, however little other reason there may be for suspecting such a thing. Thus abstract ideas, for example, obviously exclude each other: whiteness is different from blackness, health is different from sickness, folly is different from wisdom. Hence all abstract ideas involve space; and therefore logic, which uses abstract ideas, is an offshoot of geometry, and the whole of the intellect depends upon a supposed habit of picturing things side by side in space. This conclusion, upon which Bergson’s whole condemnation of the intellect rests, is based, so far as can be discovered, entirely upon a personal idiosyncrasy mistaken for a necessity of thought, I mean the idiosyncrasy of visualizing successions as spread out on a line. The instance of numbers shows that, if Bergson were in the right, we could never have attained to the abstract ideas which are supposed to be thus impregnated with space; and conversely, the fact that we can understand abstract ideas (as opposed to particular things which exemplify them) seems sufficient to prove that he is wrong in regarding the intellect as impregnated with space.
One of the bad effects of an anti-intellectual philosophy, such asthat of Bergson, is that it thrives upon the errors and confusions of the intellect. Hence it is led to prefer bad thinking to good, to declare every momentary difficulty insoluble, and to regard every foolish mistake as revealing the bankruptcy of intellect and the triumph of intuition. There are in Bergson’s works many allusions to mathematics and science, and to a careless reader these allusions may seem to strengthen his philosophy greatly. As regards science, especially biology and physiology, I am not competent to criticize his interpretations. But as regards mathematics, he has deliberately preferred tra-
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ditional errors in interpretation to the more modern views which have prevailed among mathematicians for the last eighty years. In this matter, he has followed the example of most philosophers. In the eighteenth and early nineteenth centuries, the infinitesimal calculus, though well developed as a method, was supported, as regards its foundations, by many fallacies and much confused thinking. Hegel and his followers seized upon these fallacies and confusions, to support them in their attempt to prove all mathematics self-contradictory. Thence the Hegelian account of these matters passed into the current thought of philosophers, where it has remained long after the mathematicians have removed all the difficulties upon which the philosophers rely. And so long as the main object of philosophers is to show that nothing can be learned by patience and detailed thinking, but that we ought rather to worship the prejudices of the ignorant under the title of “reason” if we are Hegelians, or of “intuition” if we are Bergsonians, so long philosophers will take care to remain ignorant of what mathematicians have done to remove the errors by which Hegel profited.
Apart from the question of number, which we have already considered, the chief point at which Bergson touches mathematics is his rejection of what he calls the “cinematographic” representation of the world. Mathematics conceives change, even continuous change, as constituted by a series of states; Bergson, on the contrary, contends that no series of states can represent what is continuous, and that in change a thing is never in any state at all. The view that change is constituted by a series of changing states he calls cinematographic; this view, he says, is natural to the intellect, but is radically vicious. True change can only be explained by true duration; it involves an interpenetration of past and present, not a mathematical succession of static states. This is what is called a “dynamic” instead of a “static” view of the world. The question is important, and in spite of its difficulty we cannot pass it by.
Bergson’s position is illustrated–and what is to be said in criticism may also be aptly illustrated–by Zeno’s argument of the arrow. Zeno argues that, since the arrow at each moment simply is where it is, therefore the arrow in its flight is always at rest. At first sight, this argument may not appear a very powerful one. Of course, it will be said, the arrow is where it is at one moment, but at another moment it is somewhere else, and this is just what constitutes motion. Certain
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difficulties, it is true, arise out of the continuity of motion, if we insist upon assuming that motion is also discontinuous. These difficulties, thus obtained, have long been part of the stock-in-trade of philosophers. But if, with the mathematicians, we avoid the assumption that motion is also discontinuous, we shall not fall into the philosopher’s difficulties. A cinematograph in which there are an infinite number of pictures, and in which there is never a next picture because an infinite number come between any two, will perfectly represent a continuous motion. Wherein, then, lies the force of Zeno’s argument?
Zeno belonged to the Eleatic school, whose object was to prove that there could be no such thing as change. The natural view to take of the world is that there are things which change; for example, there is an arrow which is now here, now there. By bisection of this view, philosophers have developed two paradoxes. The Eleatics said that there were things but no changes; Heraclitus and Bergson said there were changes but no things. The Eleatics said there was an arrow, but no flight; Heraclitus and Bergson said there was a flight but no arrow. Each party conducted its argument by refutation of the other party. How ridiculous to say there is no arrow! say the “static” party. How ridiculous to say there is no flight! say the “dynamic” party. The unfortunate man who stands in the middle and maintains that there is both the arrow and its flight is assumed by the disputants to deny both; he is therefore pierced, like Saint Sebastian, by the arrow from one side and by its flight from the other. But we have still not discovered wherein lies the force of Zeno’s argument.
Zeno assumes, tacitly, the essence of the Bergsonian theory of change. That is to say, he assumes that when a thing is in a process of continuous change, even if it is only change of
