Chapter 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._]

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.]