logical trea ed by vols. ii, and iii. treatises announced, one on Ma in all its completeness this capital elaboration of the doctrinal The Sociosystem of Positivism; I must limit myself to giving a clear tise furnishidea of it. The two preceding volumes have done the work adequately, so far as regards the last of the preliminary sciences, for they are a systematic exposition of Sociology, on the basis laid in the Philosophie. Of the three works which are to follow Three new the present, two are meant to be analogous constructions for the two extreme sciences of the Positive series, and will systematise, thematics, the first, Mathematics; the second, Morals. The renovation of the intermediate sciences, by a synthetical treatment, has been served. satisfactorily explained in the first volume of this work, more particularly as regards Biology. Enough then if I now give a sketch of the aperçus or the summaries, corresponding to this systematic comprehension of the whole range of the intellect, adopting the encyclopædic order. one on Morals, the third re Cosmology requires the Synthesis, in especial Subjective as the contrary ten dency is so In Cosmology-more than elsewhere-it is important to inaugurate the subjective synthesis, as it is in Cosmology that objective analysis has most consistency, nay most dignity. When we enter on the study of vital phenomena, the indivisibility which is the normal characteristic of all real investiga- natural. tions, so forces itself upon the attention that, in spite of the existing anarchy, the most ordinary thinkers are found always open to systematic suggestions. For in Biology we are too near the terminus of speculation, man, to ignore or despise the true aim of Positive theories, each problem soon tending to evidence the irrationality of all conceptions from which this aim is eliminated. Whereas the domain of inorganic matter may be kept so perfectly distinct as a study, that sound speculations could be entered upon and accepted within its limits, whilst in all the other departments of human thought, the fictions of Theology maintained undisputed empire. Man is, it is true, by his constitution, subject to all the laws, without exception, cf the material world, but the search after these laws has no immediate reference to man; it is always confined to some part or other of his environment. Over and above our need of the knowledge of this milieu, if success is to be attained in cosmological researches, it is requisite that we pursue them in regard to the simpler cases, even when the results have reference exclusively to the more complex. Further, the study of matter is favourable to dispersion, as dealing with an existence without unity, in a milieu which as a whole is beyond our grasp. Futility of an objective synthesis sufficiently recognised This is the explanation of the great persistence of th analytical regime in the study of the material world, even wit eminent thinkers who would never have endured it in othe branches of enquiry. And the same result would recur in the view to pre- study, if, from inattention on the part of the priesthood and the to allow the human point of vail. The Mathe- public, it were ever again to be exclusively pursued. Events in which man is directly a simple spectator, will always offer scope for the aberrations of theoricians, alarmed at the continuity of exertion demanded by the subjection of all phenomena to the subjective method. The futility of an objective synthesis is however by this time so freely admitted, that true thinkers may accept, in Cosmology, the human point of view as paramount, as alone adapted to connect everything. The divergence natural to this department of science has gone so far, since the old discipline succumbed to the anarchy of scientific academies, that its need of coordination, nay even of elimination, becomes undeniable. Appealing nobly to moral and social considerations, the priesthood of Positivism will find it no difficult task to secure general respect and love for the only system capable of protecting the feeble powers of our intelligence from being wasted on puerile investigations. All that is required is that the rationality, as well as the dignity, of our abstract enquiries, be always vindicated, as a result of the definitive fusion of science in religion. The attainment of this result is the great aim of the several works above mentioned, and the accomplishment of which is in the main reserved for my successors. Taking the most critical for myself, I hope soon to show to what an extent mathematical science, grown almost out of our grasp as a whole, gains in coherence and dignity, under the synthetical discipline instituted by the Positive religion. For the present, I must limit myself to some hints bearing on this typical result, whilst I explain the plan and the general spirit of my next treatise. A single volume will suffice for this work; originally I thought it would require two, an abstract and a concrete volume, when I announced it in 1842 at the end of my Philosophy, and even when I repeated the promise in 1851 in the general preface of the present work. So decided a condensation will scarcely surprise those who can appreciate the synthetical determination indicated at the opening of the construction I am now ending-the determination to make the moderr in Novem parts of the calculus a component part of general geometry. By carrying out this project, the new volume, which I hope to publish before the Published end of 1856, will definitively systematise the philosophy of Ma- ber, 1856. thematics. Between a religious introduction and a synthetical Component conclusion, seven chapters will put into their proper shape the work. calculus, arithmetical and algebraic; preliminary geometry; algebraic geometry; differential geometry; integral geometry; and general mechanics. The title of this forthcoming work, System of Positive Logic,' or Treatise of Mathematical Philosophy,' is in itself sufficient to express its chief object, which I proceed summarily to explain. object of to exhibit this Treatise the general laws of reasoning. That object is a deduction from the natural combination of two aphorisms, both indisputable: the study of methods is inseparable from that of the doctrines; all important observations must be shown to hold in the simplest cases. Combine these two principles and the practical inference is that the general laws of the reasoning process are best traced in the abstract study of the form of existence common to all objects equally—the form in which existence is reduced to its simplest attributes, number, extension, motion. Although this systematic delimitation of the Vol. 1, province of Mathematics is exclusively due to Positivism, yet the confusion indicated by its plural name has never concealed the fact, that the initial science alone is sufficiently simple in character to be suited to the exposition of these laws. It would seem, however, that by thus simplifying in the highest degree the speculations of Positive Mathematics, we preclude ourselves from finding in them well-marked types of all the processes of logic, several of which seem exclusively reserved for the higher studies. Alarm may be felt, lest the field of Mathematics be sufficient only as regards deduction and coordination, the two processes spontaneously developed in Mathematics, with a perfection thought to be unattainable elsewhere. Induction and generalisation-these, it would appear, can be satisfactorily appreciated only in the departments in which their several forms successively had their origin. But, in establishing the normal state, we must not rest in the blind repetition of the course followed during the preparatory period. Those on whom the Great Being devolves the task of transmitting to all its servants the general results of the intellectual developement of the race, must more and more emancipate themselves from the obligations which were binding during the p. 463. The higher cesses logical pro though no derived from ties may be Mathema illustrated by them. All the logi cal processes found in Mathematics. The Logic of Images, both well developed in Mathematics. original movement. There is coming forward an increasing difference between the dogmatic exposition and the historical creation, as human thought becomes more synthetical; as has long been traceable in mathematical teaching, notwithstanding its extreme imperfection. Whilst bound ever to respect the natural correspondence between the education of the individual and the evolution of the race, the intrinsic uniformity of the Positive method enables us to introduce its leading artifices in simpler sciences than those which originally gave them birth. In a special treatise, Analytical Geometry,' I have already shown that it is possible in Mathematics to teach that branch of inductive logic which seems most peculiar to biology, viz., the comparative method and the theory of taxonomy. Its two capital forms, the formation of natural groups, and even the institution of hierarchical series, are perfectly available for the normal classification of surfaces from the point of view of their generation. Although the simplicity of this particular case naturally disqualified it at first for spontaneously exhibiting these general processes, it constitutes a strong ground for choosing it as the proper place for their systematic investigation. The capabilities of Mathematics are, it is to be noticed, recognised as regards the less eminent forms of inductive logic, observation, and even experiment; which find large scope in Mathematics, in spite of the tendency of geometricians to look on their science as purely deductive. Lastly, the most exalted processes, historical filiation and the subjective method, may fitly, by virtue of their evidently universal applicability, be introduced into the science of Mathematics, and the use of them there is decisive of the matter in hand. Admit these capacities, however, and yet they seem inthe Logic of adequate to show the logical completeness of Mathematics, if we confront them with the systematisation of Positive logic foreshadowed in the first volume of the present work. For, if not confining ourselves to the special processes of induction or deduction, we press beyond to the general means we use for contemplation and meditation, the most instinctive, the most ancient, and the most powerful of those means seems to have no place in Mathematics. Mathematical speculation is, more than any other, adapted to display the logical power of signs, and signs are the chief resource to which the prejudices of pedants would reduce us in reasoning. Simultaneously with signs, in the very first beginnings of Mathematics, the use of images is spontaneously introduced. After a long separate existence-signs prevailing in the calculus, images in geometry--the final and irrevocable combination of these two resources was effected by Descartes in his capital reform of mathematical science. It is the only branch in which the combination has as yet been effected. But it is an inadequate expression of the real system of Positive logic, and must remain so till the influence of feeling give completeness to the fusion. The revolution effected in Mathematics by the most important of my precursors, carries with it, then, for me, an obligation to base the regeneration of science on the power of affection as an intellectual instrument. The combination thus formed of signs, images, and feelings, must, if it is to be definitive, be worked out in regard to the simplest sphere of science, and the one farthest removed from man. In no other way can the pure reason be raised to the level of practical reason, for the latter has always been able without effort to avail itself of the above combination in its concrete researches. When this has been done, we shall have removed the great danger of abstraction, and be able to use freely its inherent powers for our generalisations and coordinations, without imperilling the natural alliance of synthesis and sympathy. Such a regeneration is destined to be at once the consequence and the condition of the definitive fusion of science in religion. The Logic of quisite. Comte's task to accom plish this. complished these Sub vol. i. Prior to the publication of my forthcoming work, it is im- The task acpossible to judge a reform so opposed to the actual tendencies in the Synof scientific men, nay, even of the general public. But minds jective, suitably disposed may even now forecast its practicability, guided by the convergence of the observations on this head which have found their place in the several volumes of the present work. In especial, it should be evidenced by a judicious combination How accom between the ultimate fusion of Fetichism in Positivism and the moral reaction of mathematical studies. Whatever dryness it is sought to retain in Mathematics the necessary commencement of rational Positivity--no efforts can prevent a healthy mind from drawing from them deep and salutary emotions, as it submits to the influence on its affections of a demonstrated order. The efficacy of the irresistible convictions thus formed, their efficacy in raising and purifying our plished. (1) Moral mathemati reactions of cal studies. |