Auguste Comte - Collected Quotes

"Geometry is a true natural science: - only more simple, and therefore more perfect than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logical science, independent of observation. Everybody studied by geometers presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone." (Auguste Comte, “Course of Positive Philosophy”, 1830)  

"In mathematics we find the primitive source of rationality; and to mathematics must the biologists resort for means to carry on their researches."  (Auguste Comte, “Course of Positive Philosophy”, 1830) 

“[…] in order to observe, our mind has need of some theory or other. If in contemplating phenomena we did not immediately connect them with principles, not only would it be impossible for us to combine these isolated observations, and therefore to derive profit from them, but we should even be entirely incapable of remembering facts, which would for the most remain unnoted by us.” (Auguste Comte, “Course of Positive Philosophy”, 1830)

"It must ever be remembered that the true positive spirit first came forth from the pure sources of mathematical science; and it is only the mind that has imbibed it there, and which has been face to face with the lucid truths of geometry and mechanics, that can bring into full action its natural positivity, and apply it in bringing the most complex studies into the reality of demonstration. No other discipline can fitly prepare the intellectual organ." (Auguste Comte, “Course of Positive Philosophy”, 1830) 

“The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.” (Auguste Comte, “Course of Positive Philosophy”, 1830) 

"The domain of physics is no proper field for mathematical pastimes. The best security would be in giving a geometrical training to physicists, who need not then have recourse to mathematicians, whose tendency is to despise experimental science. By this method will that union between the abstract and the concrete be effected which will perfect the uses of mathematical, while extending the positive value of physical science. Meantime, the uses of analysis in physics is clear enough. Without it we should have no precision, and no co-ordination; and what account could we give of our study of heat, weight, light, etc.? We should have merely series of unconnected facts, in which we could foresee nothing but by constant recourse to experiment; whereas, they now have a character of rationality which fits them for purposes of prevision." (Auguste Comte, “The Positive Philosophy”, 1830)

 “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 phenomena, in becoming special, become complex.” (Auguste Comte, “The Positive Philosophy”, 1830)

"There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations." (Auguste Comte, “The Positive Philosophy”, 1830)

“To understand a science it is necessary to know its history.” (Auguste Comte, “The Positive Philosophy”, 1830)

"We may therefore define Astronomy as the science by which we discover the laws of the geometrical and mechanical phenomena presented by the heavenly bodies." (Auguste Comte, “The Positive Philosophy”, 1830)

“[Algebra] has for its object the resolution of equations; taking this expression in its full logical meaning, which signifies the transformation of implicit functions into equivalent explicit ones. In the same way arithmetic may be defined as destined to the determination of the values of functions. […] We will briefly say that Algebra is the Calculus of functions, and Arithmetic is the Calculus of Values.” (Auguste Comte, “Philosophy of Mathematics”, 1851)

“Every science consists in the coordination of facts; if the different observations were entirely isolated, there would be no science.” (Auguste Comte, “Philosophy of Mathematics”, 1851)

"Mathematical Analysis is […] the true rational basis of the whole system of our positive knowledge." (Auguste Comte, "System of Positive Polity", 1851)

 "No science can be really understood apart from its special history, which again cannot be separated from the general history of Humanity." (Auguste Comte, "System of Positive Polity", 1851)

"The formation of the differential equations proper to the phenomena is, independent of their integration, a very important acquisition, on account of the approximations which mathematical analysis allows between questions, otherwise heterogeneous [...]" (Auguste Comte, "System of Positive Polity", 1851)

“[…] it is only through Mathematics that we can thoroughly understand what true science is. Here alone can we find in the highest degree simplicity and severity of scientific law, and such abstraction as the human mind can attain.” (Auguste Comte)