relation. 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, philo sophy has been cold-shouldered, and to recover it we must seek pure intuition, which is the necessary and only precedent of logical thought.
NATURAL SCIENCES 357 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 philo sophic 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 philo sophy, 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 LOGIC PART Motive of these errors: naturalistic philosophy.
Philosophy as destroyer of naturalistic philosophy, but not of the natural sciences. Autonomy of these.
the pure concepts; a negation that need not be confuted, because the whole of our exposition of Logic is its explicit confutation.
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 hot form part of the natural sciences (which as such affirm neither the true nor the false), but to that philo-, sophy which the naturalist forms and into which he introduces the prejudices derived from his special business.
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, culti vates those sciences successfully, just as his ii NATURAL SCIENCES 359 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 philo sophy), 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 360 LOGIC PART as may be the species or classes of facts of re presentation (sensations, intuitions, perceptions, imaginings, illusions, concepts, judgments, argu ments, poems, histories, systems, etc.), facts of sentiment, and volitional facts (pleasure, pain, attraction, repulsion, mixed feelings, desires, in clinations, 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 ii NATURAL SCIENCES 361 extraneous elements causing pathological pro cesses; but that same philosopher, in so far as he is a complete man, and in so far as he pro vides 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.
VI MATHEMATICS AND THE MATHEMATICAL SCIENCE OF NATURE The idea of a THE conception of a mathematical science of science of nature is at variance with the thesis that nature.
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. various But the nature of mathematics cannot be definitions of mathematics, considered a mystery in our time. Mathematics (as has lately been said with a subtlety equal to PART ii MATHEMATICS 363 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 repre sentations. 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 mathe matics, 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 364 LOGIC PART 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. Mathematical Any sort of arithmetical operation can serve process.
process.
as an example of mathematical process, Let us take the multiplication: 4x4=16. The sign = (equals) indicates identity: 4 x 4 is identical with 1 6, as it is identical with an infinite number of such formulae, 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 8x2, for 9 + 7, for 21-5, for 32-^-2, for 42, for x/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 4x4=16. Again, when we have 32 lire to divide into equal parts between our- MATHEMATICS 365 selves and another, we shall have recourse to the formula: 32-^-2 = 1 6. Mathematics as Mathe matics does not know, but establishes formulae of equality; it does not subserve knowing, but counting and calculating what is already known.
For counting and calculating Mathematics Apriority of . mathematical requires formulae, and to establish these it principles. 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 sub mitted 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 366 LOGIC PART 366 LOGIC PART 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. contradictory But when they are recognized as being not nature of these a priori a posteriori and empirical, but a priori, difficulties principles.
are not tnereby at an end- The apriority of those principles possesses other most singular characteristics, which render them unlike the a priori knowledge of philosophy, the conscious ness of universals and of values, for instance, of logical or of moral value. For if it is im possible 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 true. 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 MATHEMATICS 367 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 aggre gated. 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 enumer able 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 PART PART 368 LOGIC 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. and not As they are unthinkable, so are the principles intuitible.
of mathematics unimaginable; they have there fore 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 con tradictions. 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 contradic tions would all become falsities. identification Now, a function which organizes theoretic -with abstract contradictions without thinking them, and so pseudoconcepts.
without falling into contradictions, is not a theoretic, but a practical function, and is per fectly well known to us as that particular pro ductive form of the practical spirit which creates 1 Introduction to Philosophy, Italian tr., Vidossich, p. 272.
„ MATHEMATICS 369 pseudoconcepts. But since those contradictions are a priori and not a posteriori, pure and not representative, mathematics cannot consist of those pseudoconcepts which are representative or empirical concepts. It remains, therefore, that it consists of the other form of pseudo- concepts, which are abstract concepts, which we have already defined as altogether void of truth and also void of representation, as analytic a priori and not synthetic a priori. And we have demonstrated how, in the falsification or practical reduction of the pure concept, concreteness without universality, that is to say, mere generality, belongs to empirical concepts, and universality without concreteness, that is to say, abstraction, 'to abstract concepts.
Such indeed are the fictions of mathematics; — they have universality without concreteness, and therefore feigned universality. Inversely to the natural sciences, which give the value of the concept to representations of the singular, although they succeed in doing so only by con vention, mathematics gives the value of the single to concepts, also succeeding in this only by convention. Thus it divides spatiality into dimensions, individuality into numbers, movement into motion and rest, and so on. It also creates LOGIC PART The ultimate end of mathematics: to enumerate and consequently to aid the determination of the single. Its place.
fictitious beings, which are neither representations nor concepts, but rather concepts treated as re presentations. It is a devastation, a mutilation, a scourge, penetrating into the theoretical world, in which it has no part, being altogether innocuous, because it affirms nothing of reality and acts as a simple practical artifice. The general purpose of that artifice is known; it is to aid memory. And the particular mnemonic purpose of this is at once evident; it is to aid the recall to memory of series of representations, previously collected in empirical concepts and thus rendered homogeneous. That is to say, they serve to supply the abstract concepts, which make possible the judgment of enumeration; to construct instruments for counting and calculating and for composing that sort of false a priori synthesis, which is the enumeration of single objects.
Applying thus to mathematics what has been said of the judgment of enumeration, it is now clear that it facilitates the manipulation of know ledge as to individual reality. Calculation indeed presupposes: (i) perceptions (individual judg ments); (2) classifications (judgments of classifica tion); and only by means of these latter does it attain to the first. But it must attain to the MATHEMATICS 371 first, because were there no single things to recall to the mind, calculation would be vain. Quantification would be sterile fencing, if it did not eventually arrive at qualification.
Mathematics is sometimes conceived as the special instrument of the natural sciences, appendix magna to the natural sciences, as Bacon called it; but from what has been said, we must not forget that both taken together, because co operating, constitute an appendix magna or an index locupletissimus to history, which is full knowledge of the real. It is further altogether erroneous to present mathematics as a prologue to all knowledge of the real, to philosophy and to the sciences, for this confuses head with tail, appendix and index, with text and preface.
It does not form part of the task that we have Particular undertaken further to investigate the constitution concerning mathematics.
of mathematics and to determine whether there be one or several mathematical sciences; if one be fundamental and the others derived from it; if the Calculus include in itself Geometry and Mechanics, or if all three can be co-ordinated and unified in general mathematics; if Geometry and Mechanics be pure mathematics, or if they do not introduce representative and contingent elements (as seems to be without doubt the case 372 LOGIC 372 LOGIC in mathematical Physics); and so on. Suffice it that we have established the nature of mathematical science and furnished the criterion according to which it can be discerned if a given formation be mathematics or natural science, if it be pure or applied mathematics (concept or judgment of enumeration, scheme of calculation, or calculation in the act). And for this reason we shall not enter into the solution of particular questions, like those concerning the number of possible fundamental operations of arithmetic, or concerning the nature of the calculus of in finitesimals, and whether, in this, there be any place for non-mathematical concepts, that is, the philosophic, not the quantitative infinite, or, again, concerning the number of the dimensions of space. As to the use of mathematics, it concerns the mathematician who knows his business to see what arbitrary distinctions it suits him to introduce, and what arbitrary unifications to produce, in order to attain certain ends. For the philosopher, these unifications and those distinctions, if transported into philosophy, are all alike false, and all can be legitimate, if em ployed in mathematics. If three dimensions of space are arbitrary but convenient, four, five and n dimensions will be arbitrary, and the only MATHEMATICS 373 MATHEMATICS 373 question that can be discussed will be whether they are convenient. Of this the philosopher knows nothing, as indeed he is sure a priori is the case.
Practical convenience suggests the postulates Rigour of mathematics to mathematics; but the purity of the elements and rigour of philosophy.
that it manipulates gives to them the rigour of j^ves and hates demonstrations, the force of truth. It is curious force, that has a weakness for point of support, — the non-truth of the postulate, and reduces itself to a perpetual tautology, by which it is recorded that what has been granted has been granted. But the rigour of the demonstra tions and the arbitrariness of the foundations explain how philosophers have been in turn attracted and repelled by mathematics. Mathe matics operating with pure concepts is a true simia philosophiae (as it was said of the devil that he was simia Dei], and philosophers have sometimes seen in it the absoluteness of thought and have saluted it as sister or as the first-born of philosophy. Other philosophers have recog nized the devil in that divine form, and have addressed to it the far from pleasant words that saints and ascetics used to employ on similar occasions. Hence mathematics has been accused of not being able to justify its own principles.
374 LOGIC PART 374 LOGIC PART notwithstanding its rigorous procedure; and of constructing empty formulae and of leaving the mind vacant. It has been accused of promoting superstition, since the whole of concrete reality lies outside its conventions, an unattainable mystery; and of being too difficult for lofty spirits, just because it is too easy.1 Gianbattista Vico confessed that having applied himself to the study of Geometry, he did not go beyond the fifth proposition of Euclid, since " that study, proper to minute intellects, is not suitable to minds already made universal by metaphysic."2 But these accusations are not accusations, and simply confirm the peculiar nature of those spiritual formations, eternal as the nature of the spirit is eternal. impossibility The nature of mathematics being explained, of reducing the empirical we can now resume the thread ot the narrative, sciences to mathematics, left hanging loose, and discover how inadmissible and empirical limits is tne claim for a mathematical science of nature, of the mathematical wnich should be the true end and the inner soul science of the empirical and natural sciences. It is said that this mathematical science presides, as an ideal, over all the particular natural sciences, but it should be added, as an unrealized and unrealiz- 1 There is a curious collection of judgments adverse to mathematics in Hamilton, Fragments philosophiques, tr. Plisse, Paris, 1840, pp. 283-370.
2 Autobiography in Works, Ferrari, 2nd edition, iv. p. 336.
MATHEMATICS 375 able ideal, and therefore rather an illusion and a mirage than an ideal. It is urged that this ideal has been partially realized, and that therefore nothing prevents its being altogether realized. But, indeed, whoever looks closely will see that it has not been even partially realized, because mathematical formulae of natural facts are always affected by the empirical and approximate character of the naturalistic concepts which they use, and by the intuitive element upon which these are based. When it is sought to establish in all its rigour the ideal of the mathematical science of nature, it becomes necessary to assume as a point of departure elements that are distinct, but perfectly identical and therefore unthinkable; quantity without quality, which are nothing but those mathematical fictions of which we have spoken. The idea of a mathematical science is thus resolved into the idea simply of mathematics, and the much-vaunted universality of that science is the universal applicability of mathematics, wherever there are things and facts to number, to calculate and to measure. The natural sciences will never lose their inevitable intuitive and historical foundation, whatever progress may be made in the calculus and in the application of the calculus. They will remain, as has been 376 LOGIC PART said, destriptive sciences (and this time it has been well said, as it prevents the failure to recognize the intuitive elements, of which they are composed). Decreasing We have already illustrated the slight perceptiutility of mathematics bility of differences (or the slight interest that in the most