(2) Fusion of Fetichism and Positivism and

inclinations, must come to be indisputable even without personal experience, for anyone who feels how great need we have to subject ourselves to external laws. Of the three species of natural laws, physical, intellectual, and moral, the second are so suited to the sphere of Mathematics that it is looked on as not able to admit others. This is an unwarrantable exaggeration, which disappears when we trace to Mathematics, on the one hand, the origin of physical laws, especially in mechanics, or even in geometry; and when, on the other hand, we see that the laws of the intellect are only unveiled in the speculations of Sociology. Still, after correcting this scientific prejudice, we cannot but be struck with the inherent aptitude of Mathematics to verify and give a true conception of intellectual order. If so, we feel that they have a corresponding aptitude to manifest, and even enlarge the sphere of moral laws, so natural is the connection of these last with intellectual laws. This, then, is the conception we should form of the true aim of mathematical education, as furnishing a complete basis for the systematisation of Positivism, a basis for the doctrine no less than for the method.

Human reason in its maturity will adopt Fetichity as the complement of Positivity, and by so doing will open the field its results on of mathematical speculation to the familiar influence of the

Mathema

tics.

emotions, inadmissible at an earlier stage of its culture, as it was necessary to avoid the risk, the imminent risk, of pernicious illusions. The simple fact that Positivism radically precludes all objective error as to causes, allows us without scrupie to enlarge the sphere of subjective vitality, which we instinctively attribute to all beings of whatever kind. Far from checking this propensity, Positivism sanctions and gives it a systematic direction, as a powerful aid not merely in language and art, but also in thought, especially in abstract thought, where it lends the image the support of feeling. Emancipated from the prejudices of science, the Positivist will be more fetichist than the Fetichist, for he will extend to phenomena the tendency which the Fetichist confined to bodies. Enough if the emotions we imagine have in all cases a real resting-place, it is indifferent whether it be abstract or concrete; the essential is that they be not attributed to fictitious beings. On this single condition, Positive reason is guaranteed against a relapse into Theology, and so is free to act on a tendency as favourable to

media.

the intellect as it is to the heart. And the regime here in- Subjective dicated is peculiarly adapted to Mathematics, for it is in this department that the institution of subjective milieus has its origin, by availing ourselves of which we shall be able to advance abstraction, by endowing with life curves, and even equations.

state of

heralded by

the practice of the higher minds who vated them neously with the superior

have culti

simulta

sciences.

We have a foreshadowing of this ultimate condition of the This final earliest and best cultivated of human speculations in the Mathematics growing tendency of the most eminent mathematicians to combine the cultivation of Mathematics with their meditations on higher subjects. In defiance of modern anarchy, Descartes and Leibnitz made it their aim at one and the same time to advance Mathematics and to regenerate Philosophy. Their worthy successor, Lagrange, would have prolonged this noble spectacle, the scale of which was always being enlarged from Thales to Pascal, had he not confined his high systematic genius within the limits of Mathematics, in the midst of a demolition in which it was not for him to take part. And although such co-existence cannot take the place of a combination, it heralds it and prepares it, by showing us the highest minds, and such minds are always favourable to scientific unity, engaged in cultivating simultaneously the two extremes of the domain of speculation. It cannot be that this tendency should disappear at the very time appointed for its systematisation; so I have ground for the hope that my synthesis of Mathematics will be rejected only by the geometricians, or rather the algebraists, from their incapacity to rise above the existing academical regime.

Unheeding their futile opposition, I will remove its only plausible ground by the rejection in toto of their troublesome claim to the intellectual presidency. Since their triumph over the physicians, they invoke, utterly without justification, in support of this noxious domination, the principle, in itself indisputable, of generality. Previous to the advent of Positivism there was no refuting the academical sophisms from the inability to distinguish the two forms, objective and subjective, which the rule may wear, and which make it issue in opposite modes of discipline. But since the second volume of this work, this capital distinction has been too fully and clearly stated for me to dread any involuntary mistakes on that point. The properties common to many beings do not, by virtue of their

The princi

ple of sub

jective gene

rality will

annul their

opposition.

The simplicity of Mathematics adapts

them for the

systematisa

tion of Posi

tive Logic.

In their true place

they will

gain in dig

source of improve

ment in the

ing.

being common, present difficulties in point of abstraction; on the contrary, the study of them is made easier by this community, as it is an evidence of their simplicity. It is only when abstraction has to deal with very complicated notions, as in the higher sciences, that by the nature of the case it increases the difficulty and the merit of our inductions, and still more of our deductions, though the objects which form their sphere be fewer in number. So, it is generality in the subjective sense that justly claims the intellectual presidency, for it is competent to raise in Morals, in Sociology, and even in Biology, systematic constructions, in utility, in difficulty, and even in perfection, surpassing those of Mathematics.

But the incorporation of science with religion, by ensuring the prevalence of encyclopædic culture, puts an end once for all to discussions which depended for their importance on the regime of specialism. It is exclusively by virtue of their greater simplicity that the domain of Mathematics offers the best field for the developement of Positive logic in its final systematic form. When cultivated in this spirit, there can be no revival of the unreasonable claim to precedence on the part of a science which, by its very nature, is confined to the most elementary subjects of human contemplation.

When restricted to its true object, a logical rather than a scientific one, this fundamental branch of science acquires a

nity, as the dignity which it could not have whilst vainly claiming supremacy. Its capacity for systematising true logic will be shown art of think more fully by our drawing from it a general improvement of the art of thinking. This conclusive result of the treatise under consideration will be explained in detail in its synthetical conclusion; for the present I only anticipate so far as to give an idea of the nature of the progress contemplated.

Syn. Subj. p. 755.

A species of universal Algebra.

It consists in the creation of a species of universal Algebra, calculated to facilitate thought, whatever be the subject on which thought is exercised, in as great a degree as ordinary algebra facilitates our meditations upon quantity. Without here explaining this new algorithm, I simply announce that it will condense alphabetic writing, as its predecessor condensed hieroglyphical writing. So that the writing of Sociocracy will thus receive an improvement, the equivalent to that which the Theocracy introduced in its writing. By such a creation alone will systematic Positivity be able to offer, as it comes to

183

guage com

embrace all departments of thought, resources no less perfect than when confined to the simplest speculations. Then, and Human lanthen only, will human language be constituted in its normal pleted. plenitude, for then only will the signs which are the best medium for communication have become the best medium for mental labour.

In this way, the logic of Mathematics, made synthetical by the introduction of feeling, will in its turn react upon the From the scientific general advance of abstract reason. point of view, the definitive systematisation of the first step in the abstract encyclopædia, carries with it results of equal importance, as it gives us an elementary general conception of the whole order of things, moral, intellectual, and physical. This triple system of laws will consequently find recognition, not merely as regards the study of motion, but also in that of extension, and even of number, on the ground of the necessary relation between the object and the subject, a relation more appreciable in the more simple abstraction. The earliest phase of our initiation in science will thus elicit from the doctrine a moral influence of a kind to complete that derived from the worship, more particularly from the personal worship. In fact the worship developes the fundamental instinct of veneration, by its accustoming us to be fond of order: of order imposed by will; of order of our own institution; of order enforced by external necessity. Now the synthesis of Mathematics should exercise an equivalent influence, with the terms of the progression inverted. Although this reversed course is less pure and less noble, it forms irresistible convictions, which tend to consolidate the discipline arising from the worship, as they bring with them a profound sense of the value of this threefold submission, which thenceforth is as precious in the eye of reason as of feeling.

The

The proximate publication of the work, the character of which I have been explaining, left it still incumbent on me to point out here the nature and the object of the first of the seven volumes, constituting the Abstract Encyclopædia,' which is to condense the definitive system of the Positive doctrine. foundation once duly laid by the execution of this volume, it will be impossible any longer to dispute the feasibility of reducing the normal exposition of true science to seven volumes, each volume devoted to one of the seven sciences of the ency

Scientific Mathematics renovated.

influence of

when thus

sufficient for

One volume

matical

the Mathe

Synthesis.

This realised

by few till the task is accom

plished.

Condensation of Cosmology.

Second
Volume.

clopædic hierarchy, each also composed of seven chapters. On comparing the volume on Mathematics with the mathematical phase of Positive education, as laid down in the plan given in the 'General View,' there may seem reason to fear that so short a book will be insufficient for such a science. For each chapter of the mathematical synthesis offers, on an average, a condensation three times as great as that effected in the other volumes. But, besides that the science, owing to its greater simplicity, admits of more concentration in its written exposition, though its oral teaching must be much fuller; two other reasons combine to explain this exceptional condensation. Easier, older, and more independent, the speculations of Mathematics have naturally been more exposed to idle digressions, so as to require expurgation on a vaster scale. As the end they have in view is to develope method rather than science, their culture demands more time and even effort than any other, but this is no reason why their systematic exposition in writing should occupy more space.

Be this as it may, it is only synthetical thinkers duly trained, and such at the present day are extremely rare, who will consider practicable so great a condensation, previous to my effecting it. But this first step once taken, it will no longer be possible to reject the concentration of science requisite if feeling is to preponderate, activity to have free play. Hence naturally, I attached peculiar importance to this explanation, as in no other way could I make it clear, to what an extent the admirable wish of Diderot comes to be, after the lapse of a century, attainable in a satisfactory degree, nay even in a degree beyond the hopes originally entertained.

Such remarks as I have to offer on the rest of Cosmology may be more brief. For Biology, though in the first volume I treated it in some detail, so as to prepare the way for its definitive systematisation, it still requires as much explanation as the whole of the inorganic sciences together. But in this place, it is Morals to which the fullest developement must be given, as soon as I have pointed out the form which Sociology will definitively take by the condensation of the present treatise. Of the seven fundamental sciences, Astronomy is actually nearest its final state, so as to require merely coordination and some elimination, for which the way has been prepared by my first volume. My treatise on Astronomy, published in 1844, the