Part 22

History, which has philosophy for its foundation, becomes in its turn foundation in the natural sciences. This explains why, with the controversy as to whether history be a science or an art, there has always been inextricably connected the other question as to whether history be the foundation of science or science the foundation of history. The question finds a solution in the solution of the ambiguity of the term "science," which is used indifferently, sometimes in the sense of philosophy, sometimes in that of the natural sciences. If science is understood as philosophy, history is not its foundation, indeed philosophy is the foundation of history. Both mingle and are identified in the sense already explained. If science is understood as naturalistic science, then history is its necessary foundation or precedent. Certainly, naturalistic classifications are also reflected in historical narrative; but, as we have seen, they do not perform a constitutive function in it; they are of merely subsidiary assistance.

[Sidenote: _Naturalists and historical research._]

But since history is the foundation of the natural sciences, and the special treatment of perceptive material or historical data by these sciences does not possess theoretic value, but is valuable merely as a convenient classification, it is clear that the whole content of truth of the natural sciences (the measure of truth and reality that at bottom they contribute) is history. Therefore it is not without reason that the natural sciences or some of them have been called in the past _natural history._ History is the hot and fluid mass, which the naturalist cools and solidifies by pouring it into formal classes and types. Previous to this manipulation, the naturalist must have thought as a historian. The matter thus cooled and solidified for preservation and for transport has no theoretic value, save in so far as it can again be rendered hot and fluid. Similarly, on the other hand, it is necessary to revise continually the classifications adopted, returning to the observation of facts, to simple intuitions and perceptions, to the historical consideration of reality. The _naturalist_ who makes a discovery, in so far as he is a discoverer of truth, is a _historical_ discoverer; and revolutions in the natural sciences represent progress in historical knowledge. Lamarckianism and Darwinism may serve as an example of this. Naturalists (and we use the word in its ordinary meaning, applying it to those who explore this "fair family of plants and animals," and what is called in general the physical world) feel themselves somewhat humiliated when described as classifiers careless of truth. But if such classification is exactly what the natural sciences accomplish from the gnoseological point of view, yet naturalists as individuals and as corporations of students exercise a far more substantial and fruitful function. The historical foundation of the life of the natural sciences is also found in the fact that a change of historical conditions sometimes renders, if not wholly useless, at least less useful, certain classifications made with the object of controlling conditions of life remote from us, or perceptions concerning life that have now been abandoned. This has occurred with regard to the classifications of alchemy and of astrology, and also (passing on to examples from other empirical sciences) to the descriptive and casuistic portions of feudal law. When the book is no longer read, the _index_ also falls into disuse.

[Sidenote: _The prejudice as to the non-historicity of nature._]

The strangest of statements, that _nature has no history,_ comes from forgetting the historical foundation of the natural sciences, from ignorance that it constitutes their sole truth, and from attributing theoretic importance to classifications which have merely practical importance. In this case, nature signifies that reality, from man downwards, which is empirically called inferior reality. But how, if it is reality, is it without history? How, if it is reality, is it not becoming? And further, the thesis is confuted by all the most attentive studies of so-called inferior reality. To limit ourselves to the animal kingdom, a century before Darwin the acute intellect of the Abbé Galiani shook itself free of this prejudice as to the immobility of animals. He remarks in certain places about cats: "_A-t-on des naturalistes bien exacts qui nous disent que les chats, il y a trois mille ans, prenaient les souris, préservaient leurs petits, connaissaient la vertu médicinale de quelques herbes, ou, pour mieux dire, de l'herbe, comme ils font à présent? ... Mes recherches sur les mœurs des chattes m'ont donné des soupçons très forts qu'elles sont perfectibles; mais au bout d'une longue traînée de siècles, je crois que tous que les cliats savent est l'ouvrage de quarante à cinquante mille ans. Nous n'avons que quelques siècles d'histoire naturelle: ainsi le changement qu'ils auront subi dans ce temps, est imperceptible."_[3] This slight perceptibility of the relative changes of what is called nature or inferior reality has contributed to that prejudice (not to mention the confusion between the fixity that belongs to naturalistic classifications and reality, which is always in motion). Nature appears to be motionless, just because of the slight interest that we take in the shadings of its phenomena and in their continuous variation. But not only is nature not motionless, but it is not even true that it proceeds (as the poet says) "with steps so slow that it seems to stand still." The movement of nature or inferior reality is fast or slow, neither in less nor greater degree than human reality, according to the various arbitrary constructions of empirical concepts which are adopted, and according to the variable and arbitrary standards of measurement which are applied to them. We watch with vigilant eye every social movement that can cause a variation in the price of grain or the value of Stock Exchange securities; but we do not surprise with equally vigilant eye the revolutions that are prepared in the bosom of the earth or among the green-clad herbs of the field.

[Sidenote: _The philosophic foundation of the natural sciences, and the efficacy of the philosophy that they contain._]

But if history is the foundation of the natural sciences, it follows from this that those sciences are always based upon a philosophy. This is indubitable, for the naturalist, however much he be a naturalist, is above all things a man, and a man without a philosophy (or what comes to the same thing, without a religion) has not yet been found. This does not mean that the natural sciences are philosophy. Their special task is classification, and here they are just as independent and autonomous as philosophy is incompetent. But philosophy is competent in philosophy, and so we see that those naturalists who possess philosophic culture avoid the prejudices, errors, and absurdities that spring from bad philosophies, and to which other naturalists are prone. For instance, if the chemist Professor Ostwald had possessed a better philosophy, he would not have abandoned his good chemistry for that doubtful mixture of things--his _Philosophy of Nature._ And had Ernest Haeckel made an elementary study of philosophy, he would never have given up his researches upon micro-organisms, in order to solve the riddles of the universe and to falsify the natural sciences. Let us limit ourselves to these instances, for our life of to-day supplies innumerable examples of philosophizing men of science, who are as pernicious to science as they are to philosophy and to culture. The antithesis between science and philosophy, of which so many speak, is a dream. The antithesis is between philosophy and philosophy, between true philosophy and that which is very imperfect and yet very arrogant, and manifestly active in the brains of many scientists, though it has nothing to do with the discoveries made in laboratories and observatories.

[Sidenote: _Action of the natural sciences upon philosophy, and errors in conceiving such relation._]

The action of philosophy upon the natural sciences is not constitutive of them, but preparatory. The action of the natural sciences upon philosophy is not even preparatory, but merely incidental and subsidiary, having for its end simplicity of exposition and of memorizing, just as in history. A very common error, derived from a too hasty analysis of the forms of spiritual life, is that of looking upon the empirical and natural sciences as a _preparation_ for philosophy. But in the achievement of the natural sciences, philosophy has been cold-shouldered, and to recover it we must seek pure intuition, which is the necessary and only precedent of logical thought.

Still worse is it, when the natural sciences are considered, not only as preparation, but just as a first sketch, or a chiselling of the marble block, from which philosophy will carve the statue. For this view denies without being aware of it, either the autonomy of the natural sciences, or that of philosophy, according as either the philosophic method or the naturalistic method is held to be the method of truth.

Indeed, in the first case, if the natural sciences be of a philosophic nature and represent a first approximation to philosophy, they must disappear when philosophy is evolved, as the provisional disappears before the definite, as the proof before the printed book. This would mean that natural sciences as such do not exist and that what really exists is philosophy. In the second case, if philosophy have the same nature as the natural sciences, the further development of the first sketch will always be the work of the naturalistic method, however refined and however increased in power we may please to imagine it. Thus, what would really exist would never be philosophy, but always the natural sciences. This erroneous conception therefore reduces itself to a denial, either of the natural sciences or of philosophy; either of the pseudoconcepts or of the pure concepts; a negation that need not be confuted, because the whole of our exposition of Logic is its explicit confutation.

[Sidenote: _Motive of these errors: naturalistic philosophy._]

The genesis of such a psychological illusion resides in the fact that the natural sciences seem to be tormented with the thirst for full and real truth, and philosophy, on the other hand, to be intent solely upon correcting the perversions and inexactitudes of the empirical and natural sciences. But it is a question of likeness or appearance only, because the thirst for truth belongs not to the natural sciences, but to philosophy, which lives in all men, and also in the naturalist. And the philosophic perversions and inexactitudes which have to be corrected do not form part of the natural sciences (which as such affirm neither the true nor the false), but to that philosophy which the naturalist forms and into which he introduces the prejudices derived from his special business.

[Sidenote: _Philosophy as destroyer of naturalistic philosophy, but not of the natural sciences. Autonomy of these._]

The proof of the theory here maintained is that even when philosophy engages in strife with naturalistic prejudices, it dissolves those prejudices, but does not and could not dissolve the sciences which had suggested them. Indeed, a philosopher becoming again a naturalist, cultivates those sciences successfully, just as his philosophizing does not forbid his going into the garden and there scenting and pruning the plants. The naturalistic sciences of language and of art, of morality, of rights and of economics (to take instances from the intellectual world, which seem to have closer contact with philosophy), are not only what is called the _empirical stage_ of the corresponding philosophic disciplines, but persist and will persist side by side with them, because they render services which cannot be replaced. Thus there is no philosophy of language and of art which can expel from their proper spheres, even if it does expel them from its own, empirical Linguistic, Grammar, Phonetics, Morphology, Syntax, and Metric, with their empirical categories, which are useful to memory. Nor can they eliminate the classifications of artistic and literary kinds, and those of the arts according to what are called means of expression, by means of which it is possible to arrange books on shelves, statues and pictures in museums, and our knowledge of artistic-literary history in our memories. Psychology, an empirical and natural science, certainly does not make us understand the activity of the spirit; but it permits us to summarize and to remember very many effective manifestations of the spirit, by classifying as well as may be the species or classes of facts of representation (sensations, intuitions, perceptions, imaginings, illusions, concepts, judgments, arguments, poems, histories, systems, etc.), facts of sentiment, and volitional facts (pleasure, pain, attraction, repulsion, mixed feelings, desires, inclinations, nostalgias, will, morality, duties, virtue, family, judicial, economic, political, religious life, etc.), or by classifying these same facts according to groups of individuals (the Psychology of animals, of children, of savages, of criminals, and of man, both in his normal and abnormal conditions). This wholly extrinsic mode of consideration, which is now prevalent in Psychology, is the source of the remark that it has risen (or has sunk?) _to the level_ of a natural science, and that its method is mechanical, determinist, positive, antiteleological. Sociology, understood not as a philosophic science (--there is no such thing--), but as an empirical science, classifies as well as may be the forms of family and the forms of production, the forms of religion, of science and of art, political and social forms, and constructs series of classifications to summarize the principal forms which human history has assumed in the course of its development. The philosopher expels these classifications from philosophy, as extraneous elements causing pathological processes; but that same philosopher, in so far as he is a complete man, and in so far as he provides for the economy of his internal life and for more easy communication with his fellows, must fashion and avail himself of the empirical. Having ideally destroyed the adjective and the adverb, the epic and the tragic kinds, the virtues of courage and of prudence, the monogamous and the polygamous family, the dog and the wolf, he must yet speak when necessary of adjectives and adverbs, of epics and tragedies, of courage and of prudence, of families formed in this or that way, of the species "dog," as though it were clearly distinguished from the species "wolf."

Thus is confirmed the autonomy and the peculiar nature of the empirical or natural sciences, indestructible by philosophy as philosophy is indestructible by them.

[Footnote 1: _Nov. Org._ I. §§ 81, 116; and II. in fine.]

[Footnote 2: See _The Philosophy of the Practical,_ pt. i. sect. i.]

[Footnote 3: Letter to d'Epinay, October 12, 1776.]

VI

MATHEMATICS AND THE MATHEMATICAL SCIENCE OF NATURE

[Sidenote: _The idea of a mathematical science of nature._]

The conception of a _mathematical science of nature_ is at variance with the thesis that recognizes the ineliminable historical foundation of the natural sciences and the consequences which follow from it. It is claimed that this mathematical science, in expressing the ideal and end of the natural sciences, would express also their true nature, which is not empirical but abstract, not synthetic but analytic, not inductive but deductive. The mathematical conception of the natural sciences would imply perfect mechanism, the reduction of all phenomena to quantity without quality, the representation of each phenomenon by means of a mathematical formula, which should be its adequate definition.

[Sidenote: _Various definitions of mathematics._]

But the nature of mathematics cannot be considered a mystery in our time. Mathematics (as has lately been said with a subtlety equal to its truth) is a science "in which it can never be known _what_ we are talking about, nor whether what we are talking about be _true_" These affirmations are made one after the other by all mathematicians who are conscious of their own methods. In what sense can a process that merits such a description be called a science? A science that states no sort of truth does not belong to the theoretic spirit, since it is not even poetry; and a science which is not related to anything is not even an empirical science, which is always related to a definite group of representations. For this reason, others incline to consider mathematics sometimes as _language,_ sometimes as _logic._ But mathematics is neither language in general nor any special language; it is not language in the universal sense, co-extensive with expression and with art; nor is it a historically given language, which would be a contingent fact; nor a class of languages (phonetic, pictorial, or musical language, etc.), which would be an approximate and empirical definition, inapplicable in a function like mathematics, which expresses its own original nature. It is not logic, because there is only one logic, and thought thinks always as thought. If it is maintained, on the other hand, that the human spirit has also a special logic, which is that of mathematicizing, a return is made to the problem to be solved, namely, what is mathematicizing? that is to say, this logic, which is not the logic of thought, because it does not give truth, and is not the logic of the empirical sciences, because it does not depend upon representations.

[Sidenote: _Mathematical process._]

Any sort of arithmetical operation can serve as an example of mathematical process. Let us take the multiplication: 4×4 = 16. The sign = (equals) indicates identity: 4×4 is identical with 16, as it is identical with an infinite number of such formulæ, since there can be infinite definitions of every number. What do we learn from such an equivalence concerning the reality, phenomenal or absolute, to which the human mind aspires? Nothing at all. But we learn how to substitute 16 for 8×2, for 9+7, for 21-5, for 32÷2, for 4², for √256, and so on. One or the other substitution is of service, according to circumstances. When, for instance, some one promises to pay us 4 lire daily, and we wish to know the total amount of lire, that is to say, the object that we shall have at our disposal after four days, we shall carry out the operation 4×4=16. Again, when we have 32 lire to divide into equal parts between ourselves and another, we shall have recourse to the formula: 32÷2 = 16. Mathematics as Mathematics does not know, but establishes formulæ of equality; it does not subserve knowing, but counting and calculating what is already known.

[Sidenote: _Apriority of mathematical principles._]

For counting and calculating Mathematics requires formulæ, and to establish these it requires certain fundamental principles. These are called in turn definitions, axioms, and postulates. Thus arithmetic requires the number series, which beginning from unity, is obtained by always adding one unit to the preceding number. Geometry requires the conception of three dimensional spaces, with the postulates connected with it. Mechanics requires certain fundamental laws, such as the law of inertia, by which a body in motion, which is not submitted to the action of other forces, covers in equal times equal spaces. There has been much dispute as to whether these principles are _a priori_ or _a posteriori,_ pure or experimental; but the dispute must henceforth be considered settled in favour of the former alternative. Even empiricists distinguish mathematical principles from natural or empirical principles, as at least (to use their expression) _elementary experiences,_ as experiences which man completes in his own spirit, in isolation from external nature. This means, whether they like it or no, that they too distinguish them profoundly from _a posteriori_ or experimental knowledge. The _a priori_ character of mathematical principles is made manifest by every attack upon it.

[Sidenote: _Contradictory nature of these a priori principles. Their unthinkability,_]

But when they are recognized as being not _a posteriori_ and empirical, but _a priori,_ difficulties are not thereby at an end. The apriority of those principles possesses other most singular characteristics, which render them unlike the _a priori_ knowledge of philosophy, the consciousness of universals and of values, for instance, of logical or of moral value. For if it is impossible to think that the concepts of the true and of the good are not true, on the other hand it is _impossible to think that the principles of mathematics are trice._ Indeed, when closely considered, they prove to be all of them altogether false. The number series is obtained by starting from unity and adding always one unit; but in reality, there is no fact which can act as the beginning of a series, nor is any fact detachable from another fact, in such a way as to generate a discrete series. If mathematics abandons the discrete for the continuous, it comes out of itself, because it abandons quantity for quality, the irrational, which is its kingdom, for the rational. If it remains in the discrete, it posits something unreal and unthinkable. Space is characterized as constituted of three or more dimensions; but reality gives, not this space, thus constituted, made up of dimensions, but spatiality, that is to say, thinkability, intuitibility in general, living and organic extension, not mechanical and aggregated. Its character is not to have three dimensions, one, two, three, but to be spatiality, in which all the other dimensions are in the one, and so there are not distinguishable and enumerable dimensions. And if the three or more dimensions as attributes of space prove to be unthinkable, and also the point without extension, the line without superficies, and the superficies without solidity--so too in consequence are all the concepts derived from them, such as those of geometrical figures, none of which has, or can have, reality. No triangle has, or can have, the sum of its angles equal to two right angles, because no triangle has existence. Hence those geometrical concepts are not completely expressed in any real fact, since they are in none, thereby differing from the philosophic concepts, which are all in every instant and are not completely expressed in any instant. Similar results follow in the case of the principles of Mechanics. No body can be withdrawn from the action of external forces, because every body is connected with all the others in the universe; hence the law of inertia is unthinkable.

[Sidenote: _and not intuitible._]

As they are unthinkable, so are the principles of mathematics unimaginable; they have therefore been ill defined as imaginary entities, for they would in that case lose such _a priori_ validity as they have. They are _a priori,_ but without the character of truth--they are organized contradictions. Had mathematics (said Herbart) to die because of the contradictions of which it is composed, it would have died long ago.[1] But it does not die of them, because it does not set itself to think them, as a venomous animal does not die of its own poison, because it does not inoculate itself. Were it to pretend to think them and to give them as true, those contradictions would all become falsities.

[Sidenote: _Identification of mathematics with abstract pseudoconcepts._]