Chapter 12

The question as to whether history is the foundation or the crown of thought.

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.

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 pastnatural 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 simpleintuitions and perceptions, to the historical consideration of reality. Thenaturalistwho makes a discovery, in so far as he is a discoverer of truth, is ahistoricaldiscoverer; 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 fromother empirical sciences) to the descriptive and casuistic portions of feudal law. When the book is no longer read, theindexalso falls into disuse.

The prejudice as to the non-historicity of nature.

The strangest of statements, thatnature 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'unelongue 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 wedo 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.

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—hisPhilosophy 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 falsifythe 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.

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 apreparationfor 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 ofthe pure concepts; a negation that need not be confuted, because the whole of our exposition of Logic is its explicit confutation.

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.

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 hisphilosophizing 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 theempirical stageof 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 wellas 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 levelof 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, asextraneous 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.

[1]Nov. Org.I. §§ 81, 116; and II. in fine.

[2]SeeThe Philosophy of the Practical,pt. i. sect. i.

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

The idea of a mathematical science of nature.

The conception of amathematical science of natureis 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.

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 toits truth) is a science "in which it can never be knownwhatwe are talking about, nor whether what we are talking about betrue" 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 aslanguage,sometimes aslogic.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 humanspirit 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 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 betweenourselves 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.

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 area prioriora 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 ownspirit, in isolation from external nature. This means, whether they like it or no, that they too distinguish them profoundly froma posteriorior experimental knowledge. Thea prioricharacter of mathematical principles is made manifest by every attack upon it.

Contradictory nature of these a priori principles. Their unthinkability,

But when they are recognized as being nota posterioriand empirical, buta priori,difficulties are not thereby at an end. The apriority of those principles possesses other most singular characteristics, which render them unlike thea prioriknowledge 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 isimpossible 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, theirrational, 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 followin 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 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 sucha priorivalidity as they have. They area 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.

Identification of mathematics with abstract pseudoconcepts.

Now, a function which organizes theoretic contradictions without thinking them, and so without falling into contradictions, is not a theoretic, but a practical function, and is perfectly well known to us as that particular productive form of the practical spirit which createspseudoconcepts. But since those contradictions area prioriand nota 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 pseudoconcepts, which areabstractconcepts, which we have already defined as altogether void of truth and also void of representation, as analytica prioriand not synthetica 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 convention, 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 createsfictitious beings, which are neither representations nor concepts, but rather concepts treated as representations. 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 falsea priorisynthesis, which is the enumeration of single objects.

The ultimate end of mathematics: to enumerate and consequently to aid the determination of the single. Its place.

Applying thus to mathematics what has been said of the judgment of enumeration, it is now clear that it facilitates the manipulation of knowledge as to individual reality. Calculation indeed presupposes: (i) perceptions (individual judgments); (2) classifications (judgments of classification); and only by means of these latter does it attain to the first. But it must attain to thefirst, 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 magnato 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 anappendix magnaor anindex locupletissimusto 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,appendixandindex,with text and preface.

Particular questions concerning mathematics.

It does not form part of the task that we have undertaken further to investigate the constitution 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 casein 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 infinitesimals, 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 employed in mathematics. If three dimensions of space are arbitrary but convenient, four, five andndimensions will be arbitrary, and the onlyquestion that can be discussed will be whether they are convenient. Of this the philosopher knows nothing, as indeed he is surea prioriis the case.

Rigour of mathematics and rigour of philosophy. Loves and hates of the two forms.

Practical convenience suggests the postulates to mathematics; but the purity of the elements that it manipulates gives to them the rigour of demonstrations, the force of truth. It is a 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 demonstrations and the arbitrariness of the foundations explain how philosophers have been in turn attracted and repelled by mathematics. Mathematics operating with pure concepts is a truesimia philosophiae(as it was said of the devil that he wassimia 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 recognized 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, notwithstanding its rigorous procedure; and of constructing empty formulæ 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.[2]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."[3]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 of reducing the empirical sciences to mathematics, and empirical limits of the mathematical science of nature.

The nature of mathematics being explained, we can now resume the thread of the narrative, left hanging loose, and discover how inadmissible is the claim for a mathematical science of nature, which should be the true end and the inner soul 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 unrealizableideal, 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 formulæ 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 universalapplicabilityof 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 beensaid,descriptivesciences (and this time it has been well said, as it prevents the failure to recognize the intuitive elements, of which they are composed).

Decreasing utility of mathematics in the most lofty spheres of the real.

We have already illustrated the slight perceptibility of differences (or the slight interest that we take in individual differences), as we gradually descend into what is called nature or inferior reality. On this is founded the illusion that nature is invariable and without history. And it also explains why mathematics has seemed more applicable to theglobus naturalisthan to theglobus intellectualis,and in theglobus naturalis,to mineralogy more than to zoology, to physics more than to biology. Still, mathematics is equally applicable to theglobus intellectualis,as, for instance, in Economics and Statistics. And, on the other hand, it is inapplicable to both spheres, when they are considered in their effective truth and unity as thehistory of natureor thehistory of reality,in which nothing is repeated and therefore nothing is equal and identical. Beneath that difference of applicability there is nothing but a consideration of utility. If the grains of sand on which we tread can be considered (although they are not) equal to one another, it happens less frequently that we regardthose with whom we associate and act in the same light. Hence thedecreasing utilityof naturalistic constructions (and of mathematical calculation), as we gradually approach human life and the historical situation in which we find ourselves. Decreasing but never non-existent, for otherwise, neither empirical sciences (grammars, books on moral conduct, psychological types, etc.) nor calculations (statistics, economic calculations, etc.,) would continue in use. A constructor of machines needs little intuition, but much physics and mechanics. A leader of men needs very little mathematics, little empirical science, but much intuitive and perceptive faculty for the vices and value of the human individuals with whom he has to do. But both little and much are empirical determinations; the Spirit, which is the whole spirit in every particular man and at every particular instant of life, is never composed of measurable elements.

[1]Introduction to Philosophy,Italian tr., Vidossich, p. 272.

[2]There is a curious collection of judgments adverse to mathematics in Hamilton,Fragments philosophiques,tr. Plisse, Paris, 1840, pp. 283-370.

[3]Autobiography inWorks,Ferrari, 2nd edition, iv. p. 336.

The theory of the forms of knowledge and the doctrine of the categories.

The explanations given as to the various forms of knowledge are also explanations concerning the categories of the theoretic and theoretic-practical spirit: the intuition, the concept, historicity, type, number; and also quality and quantity and qualitative quantity, space, time, movement, and so on. They form part of that doctrine of the categories, in which the account of philosophy in the strict sense is completed. To ask what mathematics or history is, means to search for the corresponding categories; to ask what is the relation between history and mathematics, and in general how the various forms of knowledge are related to one another, means to develop genetically all these forms, which is precisely what we have attempted.

The problem of the classification of the sciences and its practical nature.

But the difficult enquiry as to the forms of knowledge as categories has not been much in favour in recent times. Another problem has, onthe other hand, acquired vogue. It has seemed more easy, but that is not so, because though artfully disguised, it is at bottom identical with the preceding problem. Instead of putting the question in the manner indicated above, which implies seeking out the constitution of the theoretic spirit, a modest request has been made for a classification of the various forms of knowledge, aclassification of the sciences.

Scant confidence in philosophic thought, and excessive confidence in naturalistic methods, have so operated that, unable to renounce the necessity of dominating the chaos of the various competing sciences and not wishing to have recourse to philosophic systematization, an attempt has been made to classify the sciences like minerals, vegetables, and animals. Even now there exist writers occupying professorships who claim to be specialists in classifying sciences. Volumes on this theme appear with an unprofitable frequency and abundance.

False philosophic character that it assumes.

Certainly, if such writers and professors were to proceed in an altogether empirical manner, corresponding with their declarations, nothing could be said against their labours, beyond advising them not to discuss them philosophically in order that they may not waste time inmisunderstandings, and to recognize their slight utility. But, as a fact, none of them contains himself within empirical limits, but each gives some philosophic and rational basis to the classification which he proposes. Thus there appear bipartitions of the sciences intoconcreteandabstract,intohistoricalandtheoromatic(or nomotechnical), into sciences of thesuccessiveand sciences of thecoexistent,or intorealandformal;ortripartitions,into sciences offact,oflawand ofvalue; intophenomenalist, geneticandsystematicsciences; and into similar partitions and groups, of which some are old acquaintances and correspond to functions of the spirit that we have already distinguished, while others, on the contrary, must be held to be false, because they confuse under the same name functions that are different and divide functions that are unique. But all of them, true or false, leave the empirical and direct themselves to the problem of Logic and of theoretic Philosophy. This is not the place to criticize them, because substantially it has already been done in the course of the exposition of our theories; and what is left would reduce itself to a criticism of minute errors, which finds a more suitable place in reviews dealing with books of the day thanin philosophic treatises. So true is it that those classificatory systems pass with the day that witnessed their birth.

Coincidence of that problem with the search for the categories, when understood in a strictly philosophic sense.

We are concerned only to demonstrate more clearly that the demand inherent in such attempts is identical with that which leads to the establishing of a doctrine of the categories or a philosophic system. It is indeed possible to discover now and then in the demands for a classification of the sciences, two demands, the one limited, the other wider. The first takes the form of a demand for a classification of the forms of knowledge, as in the Baconian system, and in the others which repeat the type. Here the sciences are divided according to the three faculties, memory (natural and civil history), imagination (narrative, dramatic and parabolical poetry), and reason (theology, philosophy of nature and philosophy of man). The other tends to a classification not according to gnoseological forms alone, but according to objects, according to all the real principles of being, as in the system of Comte and in those derived from it. Now a classification of the first kind coincides with researches relating to the forms of the theoretic spirit, and the problems that it exposes cannot be solved save by penetrating into the problemsof these forms. Otherwise it is not possible to say if, for example, the Baconian classification be exact or no, and if not, where it should be corrected. But in passing to the other form of classification, according to objects or to the real principles of being, we pass from the sea to the ocean, because that coincides with the entire philosophic system. The classification of Comte, for example, is his positivism itself, and it is not possible to accept or refute or evaluate the one, without accepting or refuting or submitting to examination the other. There are people who ingenuously believe that they can understand things by representing them on a sheet of paper, in the form of a genealogical tree or of a table rich in graphic signs of inclusion and exclusion. But when we seriously engage upon the work, we perceive that in order to draw up the tree and construct the table, it is above all things needful to have understood them. The pen falls from the hand and the head is obliged to bend itself in meditation, when it does not prefer to abandon the dangerous game and amuse itself in other ways.

Back to Index Next