Part 23

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 creates 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 pseudoconcepts, 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 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 creates fictitious 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 false _a priori_ synthesis, which is the enumeration of single objects.

[Sidenote: _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 the 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.

[Sidenote: _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 case 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 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 and _n_ dimensions will be arbitrary, and the only 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.

[Sidenote: _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 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 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.

[Sidenote: _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 unrealizable 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 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 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 said, _descriptive_ sciences (and this time it has been well said, as it prevents the failure to recognize the intuitive elements, of which they are composed).

[Sidenote: _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 the _globus naturalis_ than to the _globus intellectualis,_ and in the _globus naturalis,_ to mineralogy more than to zoology, to physics more than to biology. Still, mathematics is equally applicable to the _globus 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 the _history of nature_ or the _history 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 regard those with whom we associate and act in the same light. Hence the _decreasing utility_ of 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.

[Footnote 1: _Introduction to Philosophy,_ Italian tr., Vidossich, p. 272.]

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

[Footnote 3: Autobiography in _Works,_ Ferrari, 2nd edition, iv. p. 336.]

VII

THE CLASSIFICATION OF THE SCIENCES

[Sidenote: _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.

[Sidenote: _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, on the 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, a _classification 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.

[Sidenote: _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 in misunderstandings, 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 into _concrete_ and _abstract,_ into _historical_ and _theoromatic_(or nomotechnical), into sciences of the _successive_ and sciences of the _coexistent,_ or into _real_ and _formal;_ or _tripartitions,_ into sciences of _fact,_ of _law_ and of _value_; into _phenomenalist, genetic_ and _systematic_ sciences; 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 than in philosophic treatises. So true is it that those classificatory systems pass with the day that witnessed their birth.

[Sidenote: _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 problems of 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.

[Sidenote: _Forms of knowledge and literary-didactic forms._]

And this is just the occasion to make clear the distinction that we have on several occasions employed, between forms of knowledge and literary or didactic forms of knowledge, between the orders of knowledge and books. The arrangement of books is not always determined solely by the demand for the strict treatment of a determinate problem; very frequently, its motive is supplied by the practical need of having certain different pieces of knowledge collected together, in order not to be obliged to go and search for them in several places, that is to say, in their true places. Thus, side by side with scientific treatises properly so-called, are to be found scholastic compilations and manuals. Such are Geographies, Pedagogies, juridical or philological Encyclopædias, Natural Histories, and so on. Authors, even outside strictly scholastic limits, used formerly to consider it convenient sometimes to isolate, sometimes to unite certain orders of knowledge, and to baptize the mutilation or mixture with a particular name. It is evident that when dealing with these hybrid compilations and formations the philosopher and the historian of the sciences, who seek not books, but ideas, must carry out a series of analyses and syntheses, of disassociations and associations, without allowing themselves to be seduced by the authority of the writers or by the solidity of these mixtures, which have become traditional.

[Sidenote: _Prejudices arising from these last._]

But it is not an easy matter. Those mixtures are no longer ingenuous, nor are the practical motives that have determined them apparent. Around them has grown up a dense forest of philosophemes, of capricious distinctions, of false definitions, of imaginary sciences, of prejudices of every sort. Any one who has succeeded in discerning the genuine connections and attempts to separate the interlaced boughs, to isolate the trees and to show the different roots, any one who sets an axe to those wild tree-trunks, is horrified by cries and complaints, not less resonant than those that drove Tancred from the enchanted wood. And there is the traditionalist who admonishes us severely not to divide _natural_ groupings and not to introduce among them our own _caprice._ Thus he calls the capricious natural and the natural capricious. "What?" (has recently written the shocked Professor Wundt) "for the excellent reason that the search for the individual is historical search, must Geology be considered history and research relating to the glacial epoch be abandoned to the amiable interest of the historian?" And others lament that the ancient _richness_ of the sciences is destroyed by these simplifications, and call the confusion richness.

[Sidenote: _Methodical prologues to Scholastic Manuals and their powerlessness._]

It is true that in order to obviate the evil of confusion and the defective consciousness of the various kinds of research which have been mingled together, many authors are in the habit of prefixing to their books theoretic introductions, about the _method,_ as they call it, of their science. The special logic of the individual disciplines is to be sought (they say) in the books that treat of these. Manuals in the German language are especially notable for this arrangement, preceded, as they are, by the heaviest introductions, which occupy a great part of the volume or of the volumes of the book. They present a contrast to French and English books, which usually enter at once _in medias res._ This arrangement seems preferable: the German type has against it the sensible observation of Manzoni, that one book at a time is enough, when it is not more than enough. He who opens a historical book in order there to learn the particulars of an event, or a book on economics in order to learn how an economic institution works, should not be obliged to read the theory of historica events and disquisitions on the place of Economics in the system of the sciences. _"Il s'agit d'un chapon et non point d'Aristote,"_ as the judge in the _Plaideurs_ said to the advocate who went back in his speech to the _Politics_ of Aristotle. But, besides the literary contamination, there is also here the other inconvenience, that science and the theory of the sciences being different operations and demanding different aptitudes and preparations, the specialist who is competent in the first is usually not at all competent in the second; though he may be believed to be so, owing to a confusion of names. Why, indeed, should an expert on banking and Stock Exchange business be versed in the gnoseology of economic science? The affirmation of competence in the one on the strength of competence in the other constitutes a true and proper sophism _a dicto simpliciter ad dictum secundum quid._

[Sidenote: _The capricious multiplication of the sciences._]