The Positive Philosophy of Auguste Comte

"There is no inquiry which is not finally reducible to a question of Numbers..."

"Nothing can appear less like a mathematical inquiry than the study of living bodies in a state of disease; yet, in studying the cure... we are endeavouring to ascertain the quantities of the different agents which are to modify the organism, in order to bring it to its natural state..."

"Kant has divided human ideas into the two categories of quantity and quality, which, if true, would destroy the universality of Mathematics; but Descartes' fundamental conception of the relation of the concrete to the abstract in Mathematics abolishes this division, and proves that all ideas of quality are reducible to ideas of quantity. He had in view geometrical phenomena... but his successors have included... first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal application, and admit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion, if only we were always capable (which we are very far from being) of first discovering, and then resolving it."

"The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably restricted, in proportion as phenomeua, in becoming special, become complex."

"[T]he reduction [to mathematics] cannot be made by us except in the case of the simplest and most general phenomena. ...[A]t the utmost, it is only the phenomena of the first three classes... of Inorganic Physics,—that we can even hope to subject to the process."

"By the rapidity of their changes, and their incessant numerical variations, vital phenomena are, practically, placed in opposition to mathematical processes."

"Social phenomena, being more complicated still, are even more out of the question, as subjects for mathematical analysis."

"It is not that a mathematical basis does not exist in these cases... but that our faculties are too limited for the working of problems so intricate."

"To the popular mind it may appear strange... that we know so much as we do about the planets. But... that class of phenomena is the most simple of all within our cognizance. The most complex problem... they present is the influence of a third body acting in the same way on two which are tending towards each other in virtue of gravitation; and this is a more simple question than any terrestrial problem... We have, however, attained only approximate solutions..."

"[T]he high perfection to which solar astronomy has been brought... is owing to our having profited by... facilities... accidental, which... our planetary system presents. The planets which compose it are few; their masses are very unequal, and much less than that of the sun; they are far distant from each other; their forms are nearly spherical; their orbits are nearly circular, and only slightly inclined in relation to each other; and so on."

"Their perturbations are, in consequence, inconsiderable... and all we have to do is usually to take into the account, together with the influence of the sun on each planet, the influence of one other planet, capable, by its size and its nearness, of occasioning perceptible derangements."

"If any of the conditions mentioned above had been different, though the law of gravitation had existed as it is, we might not... have discovered it."

"The most difficult sciences must remain, for an indefinite time, in that preliminary state which prepares for the others the time when they too may become capable of mathematical treatment. Our business is to study phenomena... abstaining from introducing considerations of quantities, and mathematical laws... beyond our power to apply."

"We owe to Mathematics both the origin of Positive Philosophy and its Method. When this method was introduced into the other sciences, it was natural that it should be urged too far. But each science modified the method by the operation of its own peculiar phenomena."

"We must next pass in review the three great sciences of which [Mathematical Science] is composed,—the Calculus, Geometry, and Rational Mechanics."

"The historical development of the Abstract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics."

"The Concrete portions of the science depend on the Abstract, which are wholly independent..."

"The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon... and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others."

"It is only by forming a true idea of an that we can lay down the real line of separation between the concrete and the abstract part of mathematics."

"[I]t almost impossible to explain the difficulty we find in establishing the relation of the concrete to the abstract which meets us in every great mathematical question..."

"[F]unctions must... be divided into Abstract and Concrete; the first of which alone can enter into true equations."

"Every is a relation of equality between two abstract functions of the magnitudes... including the primary magnitudes and all auxiliary magnitudes which may... facilitate the discovery of the equations..."

"This distinction may be established by both the à priori and à posteriori methods; by characterizing each kind of function, and by enumerating all the abstract functions yet known..."

"A priori; Abstract functions express a mode of dependence between magnitudes which may be conceived between numbers alone, without the need of pointing out any phenomena... while Concrete functions are those whose expression requires a specified... case of physics, geometry, mechanics, etc."

"Most functions were concrete in their origin,—even those which are at present the most purely abstract; and the ancients discovered... through geometrical definitions elementary algebraic properties of functions, to which a numerical value was not attached till long afterwards, rendering abstract to us what was concrete to the old geometers."

"[[w:Trigonometric functions|[C]ircular functions]], both direct and inverse... are still sometimes concrete, sometimes abstract, according to the point of view from which they are regarded."

"A posteriori; the distinguishing character, abstract or concrete, of a function having been established, the question of any determinate function being abstract, and therefore able to enter into true analytical equations, becomes a simple question of fact, as we are acquainted with the elements which compose all the abstract functions at present known. We say we know them all, though analytical functions are infinite in number, because we are here speaking, it must be remembered,— of the elements— of the simple, not of the compound."

"We have ten elementary formulas; and, few as they are, they may give rise to an infinite number of analytical combinations. There is no reason for supposing that there can never be more. We have more than Descartes had, and even Newton and Leibnitz; and our successors will doubtless introduce additions, though there is so much difficulty attending their augmentation, that we cannot hope that it will proceed very far."

"It is the insufficiency of this very small number of analytical elements which constitutes our difficulty in passing from the concrete to the abstract."

"In order to establish the equations of phenomena, we must conceive of their mathematical laws by the aid of functions composed of these few elements."

"[T]hese elements of our analysis have been supplied to us by the mathematical consideration of the simplest phenomena of a geometrical origin, which can afford us à priori no rational guarantee of their fitness to represent the mathematical laws of all other classes of phenomena."

"[W]e have considered the Calculus as a whole. We must now consider its divisions... the Algebraic Calculus, and the Arithmetical Calculus, or Arithmetic, taking care to give them the most extended logical sense, and not the restricted one... usually received."

"[E]very question of Mathematical Analysis presents two successive parts, perfectly distinct... The first stage is the transformation of the proposed equations, so as to exhibit the mode of formation of unknown quantities by the known. This constitutes the algebraic question. Then ensues the task of finding the values of the formulas thus obtained. ...this is the arithmetical question."

"Thus the algebraic and the arithmetical calculus differ in their object. They differ also in their view of quantities,—Algebra considering quantities in regard to their relations, and Arithmetic in regard to their values."

"In practice, it is not always possible... to separate the processes entirely in obtaining a solution; but the radical difference of the two operations should never be lost sight of."

"Algebra... is the Calculus of Functions, and Arithmetic the Calculus of Values."

"We have seen that the division of the Calculus is into two branches. It remains... to compare the two... to learn their respective extent, importance, and difficulty."

"Functions being divided into simple and compound... when we become able to determine the value of simple functions, there will be no difficulty with the compound."

"In the algebraic relation, a compound function plays a very different part from... the elementary functions which constitute it; and this is the source of our chief analytical difficulties. But it is quite otherwise with the Arithmetical Calculus."

"[T]here can be no new arithmetical operations otherwise than by the creation of new analytical elements, which must... for ever be extremely small."

"The domain of arithmetic then is, by its nature, narrowly restricted, while that of algebra is rigorously indefinite."

"Still, the domain of arithmetic is... extensive... for there are many questions treated as incidental in the midst of a body of analytical researches, which... are... arithmetical. Of this kind are the construction of a table of logarithms, and the calculation of trigonometrical tables, and some distinct and higher procedures; in short, every operation which has for its object the determination of the values of functions."

"[W]e must also include... the Theory of Numbers, the object of which is to discover the properties inherent in different numbers, in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic."

"Though the domain of arithmetic is thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which mathematical science essentially consists."

"Determinations of values are, in fact, nothing else than real transformations of the functions to be valued. These transformations have a special end; but... are essentially of the same nature as all taught by analysis."

"[T]he calculus of values might be regarded as a particular application of the calculus of functions, arithmetic thereby disappearing, as a distinct section, from the domain of abstract mathematics."

"[W]e will now see how the establishment of the s of phenomena has been achieved."

"The first means of remedying the difficulty of the small number of analytical elements seems to be to create new ones. But... this resource is illusory."

"[T]he introduction of another elementary abstract function into analysis supposes the simultaneous creation of a new arithmetical operation; which is certainly extremely difficult. ...We have... no idea how to proceed to create new elementary abstract functions. Yet, we must not... conclude that we have reached the limit... Special improvements in mathematical analysis have yielded us some partial substitutes, which have increased our resources: but... the augmentation... cannot proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establishment of equations."

"As it is impossible to find the equations directly, we must seek... corresponding ones between other auxiliary quantities, connected with the first according to a certain determinate law, and from the relation... ascend to that of the primitive magnitudes. This is the... transcendental analysis... our finest instrument for the mathematical exploration of natural phenomena."