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Артур Шопенгауэр
The World as Will and Idea (Vol. 2 of 3)

Chapter XIII.²⁵ On The Methods Of Mathematics

Euclid's method of demonstration has brought forth from its own womb its most striking parody and caricature in the famous controversy on the theory of parallels, and the attempts, which are repeated every year, to prove the eleventh axiom. This axiom asserts, and indeed supports its assertion by the indirect evidence of a third intersecting line, that two lines inclining towards each other (for that is just the meaning of “less than two right angles”) if produced far enough must meet – a truth which is supposed to be too complicated to pass as self-evident, and therefore requires a demonstration. Such a demonstration, however, cannot be produced, just because there is nothing that is not immediate. This scruple of conscience reminds me of Schiller's question of law: —

“For years I have used my nose for smelling. Have I, then, actually a right to it that can be proved?” Indeed it seems to me that the logical method is hereby reduced to absurdity. Yet it is just through the controversies about this, together with the vain attempts to prove what is directly certain as merely indirectly certain, that the self-sufficingness and clearness of intuitive evidence appears in contrast with the uselessness and difficulty of logical proof – a contrast which is no less instructive than amusing. The direct certainty is not allowed to be valid here, because it is no mere logical certainty following from the conceptions, thus resting only upon the relation of the predicate to the subject, according to the principle of contradiction. That axiom, however, is a synthetical proposition a priori, and as such has the guarantee of pure, not empirical, perception, which is just as immediate and certain as the principle of contradiction itself, from which all demonstrations first derive their certainty. Ultimately this holds good of every geometrical theorem, and it is quite arbitrary where we draw the line between what is directly certain and what has first to be demonstrated. It surprises me that the eighth axiom is not rather attacked. “Figures which coincide with each other are equal to each other.” For “coinciding with each other” is either a mere tautology or something purely empirical which does not belong to pure perception but to external sensuous experience. It presupposes that the figures may be moved; but only matter is movable in space. Therefore this appeal to coincidence leaves pure space – the one element of geometry – in order to pass over to what is material and empirical.

The reputed motto of the Platonic lecture-room, “Αγεωμετρητος μηδεις εισιτω,” of which mathematicians are so proud, was no doubt inspired by the fact that Plato regarded the geometrical figures as intermediate existences between the eternal Ideas and particular things, as Aristotle frequently mentions in his “Metaphysics” (especially i. c. 6, p. 887, 998, et Scholia, p. 827, ed. Berol.) Moreover, the opposition between those self-existent eternal forms, or Ideas, and the transitory individual things, was most easily made comprehensible in geometrical figures, and thereby laid the foundation of the doctrine of Ideas, which is the central point of the philosophy of Plato, and indeed his only serious and decided theoretical dogma. In expounding it, therefore, he started from geometry. In the same sense we are told that he regarded geometry as a preliminary exercise through which the mind of the pupil accustomed itself to deal with incorporeal objects, having hitherto in practical life had only to do with corporeal things (Schol. in Aristot., p. 12, 15). This, then, is the sense in which Plato recommended geometry to the philosopher; and therefore one is not justified in extending it further. I rather recommend, as an investigation of the influence of mathematics upon our mental powers, and their value for scientific culture in general, a very thorough and learned discussion, in the form of a review of a book by Whewell in the Edinburgh Review of January 1836. Its author, who afterwards published it with some other discussions, with his name, is Sir W. Hamilton, Professor of Logic and Metaphysics in Scotland. This work has also found a German translator, and has appeared by itself under the title, “Ueber den Werth und Unwerth der Mathematik” aus dem Englishen, 1836. The conclusion the author arrives at is that the value of mathematics is only indirect, and lies in the application to ends which are only attainable through them; but in themselves mathematics leave the mind where they find it, and are by no means conducive to its general culture and development, nay, even a decided hindrance. This conclusion is not only proved by thorough dianoiological investigation of the mathematical activity of the mind, but is also confirmed by a very learned accumulation of examples and authorities. The only direct use which is left to mathematics is that it can accustom restless and unsteady minds to fix their attention. Even Descartes, who was yet himself famous as a mathematician, held the same opinion with regard to mathematics. In the “Vie de Descartes par Baillet,” 1693, it is said, Liv. ii. c. 6, p. 54: “Sa propre expérience l'avait convaincu du peu d'utilité des mathématiques, surtout lorsqu'on ne les cultive que pour elles mêmes… Il ne voyait rien de moins solide, que de s'occuper de nombres tout simples et de figures imaginaires,” &c.

Chapter XIV. On The Association Of Ideas

The presence of ideas and thoughts in our consciousness is as strictly subordinated to the principle of sufficient reason in its different forms as the movement of bodies to the law of causality. It is just as little possible that a thought can appear in the mind without an occasion as that a body can be set in motion without a cause. Now this occasion is either external, thus an impression of the senses, or internal, thus itself also a thought which introduces another thought by means of association. This again depends either upon a relation of reason and consequent between the two; or upon similarity, even mere analogy; or lastly upon the circumstance that they were both first apprehended at the same time, which again may have its ground in the proximity in space of their objects. The last two cases are denoted by the word à propos. The predominance of one of these three bonds of association of thoughts over the others is characteristic of the intellectual worth of the man. The first named will predominate in thoughtful and profound minds, the second in witty, ingenious, and poetical minds, and the third in minds of limited capacity. Not less characteristic is the degree of facility with which one thought recalls others that stand in any kind of relation to it: this constitutes the activeness of the mind. But the impossibility of the appearance of a thought without its sufficient occasion, even when there is the strongest desire to call it up, is proved by all the cases in which we weary ourselves in vain to recollect something, and go through the whole store of our thoughts in order to find any one that may be associated with the one we seek; if we find the former, the latter is also found. Whoever wishes to call up something in his memory first seeks for a thread with which it is connected by the association of thoughts. Upon this depends mnemonics: it aims at providing us with easily found occasioners or causes for all the conceptions, thoughts, or words which are to be preserved. But the worst of it is that these occasioners themselves have first to be recalled, and this again requires an occasioner. How much the occasion accomplishes in memory may be shown in this way. If we have read in a book of anecdotes say fifty anecdotes, and then have laid it aside, immediately afterwards we will sometimes be unable to recollect a single one of them. But if the occasion comes, or if a thought occurs to us which has any analogy with one of those anecdotes, it immediately comes back to us; and so with the whole fifty as opportunity offers. The same thing holds good of all that we read. Our immediate remembrance of words, that is, our remembrance of them without the assistance of mnemonic contrivances, and with it our whole faculty of speech, ultimately depends upon the direct association of thoughts. For the learning of language consists in this, that once for all we so connect a conception with a word that this word will always occur to us along with this conception, and this conception will always occur to us along with this word. We have afterwards to repeat the same process in learning every new language; yet if we learn a language for passive and not for active use – that is, to read, but not to speak, as, for example, most of us learn Greek – then the connection is one-sided, for the conception occurs to us along with the word, but the word does not always occur to us along with the conception. The same procedure as in language becomes apparent in the particular case, in the learning of every new proper name. But sometimes we do not trust ourselves to connect directly the name of this person, or town, river, mountain, plant, animal, &c., with the thought of each so firmly that it will call each of them up of itself; and then we assist ourselves mnemonically, and connect the image of the person or thing with any perceptible quality the name of which occurs in that of the person or thing. Yet this is only a temporary prop to lean on; later we let it drop, for the association of thoughts becomes an immediate support.

The search of memory for a clue shows itself in a peculiar manner in the case of a dream which we have forgotten on awaking, for in this case we seek in vain for that which a few minutes before occupied our minds with the strength of the clearest present, but now has entirely disappeared. We grasp at any lingering impression by which may hang the clue that by virtue of association would call that dream back again into our consciousness. According to Kieser, “Tellurismus,” Bd. ii. § 271, memory even of what passed in magnetic-somnambular sleep may possibly sometimes be aroused by a sensible sign found when awake. It depends upon the same impossibility of the appearance of a thought without its occasion that if we propose to do anything at a definite time, this can only take place if we either think of nothing else till then, or if at the determined time we are reminded of it by something, which may either be an external impression arranged beforehand or a thought which is itself again brought about in the regular way. Both, then, belong to the class of motives. Every morning when we awake our consciousness is a tabula rasa, which, however, quickly fills itself again. First it is the surroundings of the previous evening which now reappear, and remind us of what we thought in these surroundings; to this the events of the previous day link themselves on; and so one thought rapidly recalls the others, till all that occupied us yesterday is there again. Upon the fact that this takes place properly depends the health of the mind, as opposed to madness, which, as is shown in the third book, consists in the existence of great blanks in the memory of past events. But how completely sleep breaks the thread of memory, so that each morning it has to be taken up again, we see in particular cases of the incompleteness of this operation. For example, sometimes we cannot recall in the morning a melody which the night before ran in our head till we were tired of it.

The cases in which a thought or a picture of the fancy suddenly came into our mind without any conscious occasion seem to afford an exception to what has been said. Yet this is for the most part an illusion, which rests on the fact that the occasion was so trifling and the thought itself so vivid and interesting, that the former is instantly driven out of consciousness. Yet sometimes the cause of such an instantaneous appearance of an idea may be an internal physical impression either of the parts of the brain on each other or of the organic nervous system upon the brain.

In general our internal process of thought is in reality not so simple as the theory of it; for here it is involved in many ways. To make the matter clear to our imagination, let us compare our consciousness to a sheet of water of some depth. Then the distinctly conscious thoughts are merely the surface; while, on the other hand, the indistinct thoughts, the feelings, the after sensation of perceptions and of experience generally, mingled with the special disposition of our own will, which is the kernel of our being, is the mass of the water. Now the mass of the whole consciousness is more or less, in proportion to the intellectual activity, in constant motion, and what rise to the surface, in consequence of this, are the clear pictures of the fancy or the distinct, conscious thoughts expressed in words and the resolves of the will. The whole process of our thought and purpose seldom lies on the surface, that is, consists in a combination of distinctly thought judgments; although we strive against this in order that we may be able to explain our thought to ourselves and others. But ordinarily it is in the obscure depths of the mind that the rumination of the materials received from without takes place, through which they are worked up into thoughts; and it goes on almost as unconsciously as the conversion of nourishment into the humours and substance of the body. Hence it is that we can often give no account of the origin of our deepest thoughts. They are the birth of our mysterious inner life. Judgments, thoughts, purposes, rise from out that deep unexpectedly and to our own surprise. A letter brings us unlooked-for and important news, in consequence of which our thoughts and motives are disordered; we get rid of the matter for the present, and think no more about it; but next day, or on the third or fourth day after, the whole situation sometimes stands distinctly before us, with what we have to do in the circumstances. Consciousness is the mere surface of our mind, of which, as of the earth, we do not know the inside, but only the crust.

But in the last instance, or in the secret of our inner being, what sets in activity the association of thought itself, the laws of which were set forth above, is the will, which urges its servant the intellect, according to the measure of its powers, to link thought to thought, to recall the similar, the contemporaneous, to recognise reasons and consequents. For it is to the interest of the will that, in general, one should think, so that one may be well equipped for all cases that may arise. Therefore the form of the principle of sufficient reason which governs the association of thoughts and keeps it active is ultimately the law of motivation. For that which rules the sensorium, and determines it to follow the analogy or other association of thoughts in this or that direction, is the will of the thinking subject. Now just as here the laws of the connection of ideas subsist only upon the basis of the will, so also in the real world the causal connection of bodies really subsists only upon the basis of the will, which manifests itself in the phenomena of this world. On this account the explanation from causes is never absolute and exhaustive, but leads back to forces of nature as their condition, and the inner being of the latter is just the will as thing in itself. In saying this, however, I have certainly anticipated the following book.

But because now the outward (sensible) occasions of the presence of our ideas, just as well as the inner occasions (those of association), and both independently of each other, constantly affect the consciousness, there arise from this the frequent interruptions of our course of thought, which introduce a certain cutting up and confusion of our thinking. This belongs to its imperfections which cannot be explained away, and which we shall now consider in a separate chapter.

Chapter XV. On The Essential Imperfections Of The Intellect

Our self-consciousness has not space but only time as its form, and therefore we do not think in three dimensions, as we perceive, but only in one, thus in a line, without breadth or depth. This is the source of the greatest of the essential imperfections of our intellect. We can know all things only in succession, and can become conscious of only one at a time, indeed even of this one only under the condition that for the time we forget everything else, thus are absolutely unconscious of everything else, so that for the time it ceases to exist as far as we are concerned. In respect of this quality our intellect may be compared to a telescope with a very narrow field of vision; just because our consciousness is not stationary but fleeting. The intellect apprehends only successively, and in order to grasp one thing must let another go, retaining nothing but traces of it, which are ever becoming weaker. The thought which is vividly present to me now must after a little while have escaped me altogether; and if a good night's sleep intervene, it may be that I shall never find it again, unless it is connected with my personal interests, that is, with my will, which always commands the field.

Upon this imperfection of the intellect depends the disconnected and often fragmentary nature of our course of thought, which I have already touched on at the close of last chapter; and from this again arises the unavoidable distraction of our thinking. Sometimes external impressions of sense throng in upon it, disturbing and interrupting it, forcing different kinds of things upon it every moment; sometimes one thought draws in another by the bond of association, and is now itself dislodged by it; sometimes, lastly, the intellect itself is not capable of fixing itself very long and continuously at a time upon one thought, but as the eye when it gazes long at one object is soon unable to see it any more distinctly, because the outlines run into each other and become confused, until finally all is obscure, so through long-continued reflection upon one subject our thinking also is gradually confused, becomes dull, and ends in complete stupor. Therefore after a certain time, which varies with the individual, we must for the present give up every meditation or deliberation which has had the fortune to remain undisturbed, but yet has not been brought to an end, even if it concerns a matter which is most important and pertinent to us; and we must dismiss from our consciousness the subject which interests us so much, however heavily our anxiety about it may weigh upon us, in order to occupy ourselves now with insignificant and indifferent things. During this time that important subject no longer exists for us; it is like the heat in cold water, latent. If now we resume it again at another time, we approach it like a new thing, with which we become acquainted anew, although more quickly, and the agreeable or disagreeable impression of it is also produced anew upon our will. We ourselves, however, do not come back quite unchanged. For with the physical composition of the humours and tension of the nerves, which constantly changes with the hours, days, and years, our mood and point of view also changes. Moreover, the different kinds of ideas which have been there in the meantime have left an echo behind them, the tone of which influences the ideas which follow. Therefore the same thing appears to us at different times, in the morning, in the evening, at mid-day, or on another day, often very different; opposite views of it now press upon each other and increase our doubt. Hence we speak of sleeping upon a matter, and for important determinations we demand a long time for consideration. Now, although this quality of our intellect, as springing from its weakness, has its evident disadvantages, yet, on the other hand, it affords the advantage that after the distraction and the physical change we return to our subject as comparatively new beings, fresh and strange, and thus are able to see it repeatedly in very different lights. From all this it is plain that human consciousness and thought is in its nature necessarily fragmentary, on account of which the theoretical and practical results which are achieved by piecing together such fragments are for the most part defective. In this our thinking consciousness is like a magic lantern, in the focus of which only one picture can appear at a time, and each, even if it represents the noblest objects, must yet soon pass away in order to make room for others of a different, and even most vulgar, description. In practical matters the most important plans and resolutions are formed in general; but others are subordinated to these as means to an end, and others again are subordinated to these, and so on down to the particular case that has to be carried out in concreto. They do not, however, come to be carried out in the order of their dignity, but while we are occupied with plans which are great and general, we have to contend with the most trifling details and the cares of the moment. In this way our consciousness becomes still more desultory. In general, theoretical occupations of the mind unfit us for practical affairs, and vice versâ.

In consequence of the inevitably distracted and fragmentary nature of all our thinking, which has been pointed out, and the mingling of ideas of different kinds thereby introduced, to which even the noblest human minds are subject, we really have only half a consciousness with which to grope about in the labyrinth of our life and the obscurity of our investigations; bright moments sometimes illuminate our path like lightning. But what is to be expected of heads of which even the wisest is every night the scene of the strangest and most senseless dreams, and which has to take up its meditations again on awakening from these? Clearly a consciousness which is subject to such great limitations is little suited for solving the riddle of the world; and such an endeavour would necessarily appear strange and pitiful to a being of a higher order whose intellect had not time as its form, and whose thinking had thus true completeness and unity. Indeed it is really wonderful that we are not completely confused by the very heterogeneous mixture of ideas and fragments of thought of every kind which are constantly crossing each other in our minds, but are yet always able to see our way again and make everything agree together. Clearly there must exist a simpler thread upon which everything ranges itself together: but what is this? Memory alone is not sufficient, for it has essential limitations of which I shall speak shortly, and besides this, it is exceedingly imperfect and untrustworthy. The logical ego or even the transcendental synthetic unity of apperception are expressions and explanations which will not easily serve to make the matter comprehensible; they will rather suggest to many:

“'Tis true your beard is curly, yet it will not draw you the bolt.”

Kant's proposition, “The I think must accompany all our ideas,” is insufficient; for the “I” is an unknown quantity, i. e., it is itself a secret. That which gives unity and connection to consciousness in that it runs through all its ideas, and is thus its substratum, its permanent supporter, cannot itself be conditioned by consciousness, therefore cannot be an idea. Rather it must be the prius of consciousness, and the root of the tree of which that is the fruit. This, I say, is the will. It alone is unchangeable and absolutely identical, and has brought forth consciousness for its own ends. Therefore it is also the will which gives it unity and holds together all its ideas and thoughts, accompanying them like a continuous harmony. Without it the intellect would no longer have the unity of consciousness, as a mirror in which now this and now that successively presents itself, or at the most only so much as a convex mirror whose rays unite in an imaginary point behind its surface. But the will alone is that which is permanent and unchangeable in consciousness. It is the will which holds together all thoughts and ideas as means to its ends, and tinges them with the colour of its own character, its mood, and its interests, commands the attention, and holds in its hand the train of motives whose influence ultimately sets memory and the association of ideas in activity; at bottom it is the will that is spoken of whenever “I” appears in a judgment. Thus it is the true and final point of unity of consciousness, and the bond of all its functions and acts; it does not itself, however, belong to the intellect, but is only its root, source, and controller.

From the form of time and the single dimension of the series of ideas, on account of which, in order to take up one, the intellect must let all the others fall, there follows not only its distraction, but also its forgetfulness. Most of what it lets fall it never takes up again; especially since the taking up again is bound to the principle of sufficient reason, and thus demands an occasion which the association of thoughts and motivation have first to supply; an occasion, however, which may be the more remote and smaller in proportion as our sensibility for it is heightened by our interest in the subject. But memory, as I have already shown in the essay on the principle of sufficient reason, is not a store-house, but merely a faculty acquired by practice of calling up ideas at pleasure, which must therefore constantly be kept in practice by use; for otherwise it will gradually be lost. Accordingly the knowledge even of the learned man exists only virtualiter as an acquired facility in calling up certain ideas; actualiter, on the other hand, it also is confined to one idea, and is only conscious of this one at a time. Hence arises a strange contrast between what he knows potentiâ and what he knows actu; that is, between his knowledge and what he thinks at any moment: the former is an immense and always somewhat chaotic mass, the latter is a single distinct thought. The relation resembles that between the innumerable stars of the heavens and the limited field of vision of the telescope; it appears in a striking manner when upon some occasion he wishes to call distinctly to his remembrance some particular circumstance in his knowledge, and time and trouble are required to produce it from that chaos. Rapidity in doing this is a special gift, but is very dependent upon day and hour; therefore memory sometimes refuses us its service, even in things which at another time it has readily at hand. This consideration calls us in our studies to strive more to attain to correct insight than to increase our learning, and to lay it to heart that the quality of knowledge is more important than its quantity. The latter imparts to books only thickness, the former thoroughness and also style; for it is an intensive quantity, while the other is merely extensive. It consists in the distinctness and completeness of the conceptions, together with the purity and accuracy of the knowledge of perception which forms their foundation; therefore the whole of knowledge in all its parts is penetrated by it, and in proportion as it is so is valuable or trifling. With a small quantity, but of good quality, one achieves more than with a very large quantity of bad quality.

The most perfect and satisfactory knowledge is that of perception, but it is limited absolutely to the particular, the individual. The combination of the many and the different in one idea is only possible through the conception, that is, through the omission of the differences; therefore this is a very imperfect manner of presenting things to the mind. Certainly the particular also can be directly comprehended as a universal, if it is raised to the (Platonic) Idea; but in this process, which I have analysed in the third book, the intellect already passes beyond the limits of individuality, and therefore of time; moreover it is only an exception.

These inner and essential imperfections of the intellect are further increased by a disturbance which, to a certain extent, is external to it, but yet is unceasing – the influence exerted by the will upon all its operations whenever it is in any way concerned in their result. Every passion, indeed every inclination and aversion, tinges the objects of knowledge with its colour. Of most common occurrence is the falsifying of knowledge which is brought about by wishes and hopes, for they picture to us the scarcely possible as probable and well nigh certain, and make us almost incapable of comprehending what is opposed to it: fear acts in a similar way; and every preconceived opinion, every partiality, and, as has been said, every interest, every emotion and inclination of the will, acts in an analogous manner.

To all these imperfections of the intellect we have finally to add this, that it grows old with the brain, that is, like all physiological functions, it loses its energy in later years, whereby all its imperfections are then much increased.

The defective nature of the intellect here set forth will not, however, surprise us if we look back at its origin and destiny as established by me in the second book. Nature has produced it for the service of an individual will. Therefore it is only designed to know things so far as they afford the motives of such a will, but not to fathom them or comprehend their true being. Human intellect is only a higher gradation of the intellect of the brutes; and as this is entirely confined to the present, our intellect also bears strong traces of this limitation, Therefore our memory and recollection is something very imperfect. How little of all that we have done, experienced, learnt, or read, can we recall! And even this little for the most part only laboriously and imperfectly. For the same reasons is it so very difficult for us to keep ourselves free from the impressions of the present. Unconsciousness is the original and natural condition of all things, and therefore also the basis from which, in particular species of beings, consciousness results as their highest efflorescence; wherefore even then unconsciousness always continues to predominate. Accordingly most existences are without consciousness; but yet they act according to the laws of their nature, i. e., of their will. Plants have at most a very weak analogue of consciousness; the lowest species of animals only the dawn of it. But even after it has ascended through the whole series of animals to man and his reason, the unconsciousness of plants, from which it started, still remains the foundation, and may be traced in the necessity for sleep, and also in all those essential and great imperfections, here set forth, of every intellect produced through physiological functions; and of another intellect we have no conception.