Comte began with mathematics, the tool of the mind. Accurate thinking is always done in terms of mathematics. With mathematics as its chief tool, the mind of man can go anywhere in its thinking. Mathematics is the most powerful instrument which the mind may use in the investigation of natural laws.[XIII-7]
Mathematics is not a constituent member of the group of sciences. It is the basis of them all. It holds the first place in the hierarchy of the sciences, and is the best point of departure in all education, whether general or special.[XIII-8] It is the oldest and most perfect of all the sciences.[XIII-9]
Mathematics is the science which measures precisely the relations between objects and ideas. It is the science.[XIII-10] The Greeks had no other. Its definition is the definition of all science. Its function is that of ascertaining relationships, a process which is basic to scientific thinking in all fields. Education that is based on any other method is faulty, inexact, and unreliable. It is only through mathematics that we can understand science.
The highest form of mathematics is calculus. There is no scientific inquiry in which calculus is not used. Even the physician in prescribing for the cure of a disease, must provide for the mixing together of different quantities of different medicines, so that, when taken at determined intervals of time, they will possess the right qualities for bringing the human body back to its normal state. Calculus is the branch of science which has the highest intellectual dignity. In it the proportion of reasoning to observation is greater than elsewhere.
With mathematics as the tool, the classification of knowledge may proceed. All natural phenomena fall into two grand divisions: inorganic and organic. The inorganic are more general and should be considered first. Inorganic phenomena are of two classes: astronomical and terrestrial. Astronomical phenomena are the most general of all. The stars and planets appear under the least varied aspects.[XIII-12] Astronomy is the science by which the movements of the heavenly bodies, including the earth, are measured. How can we thoroughly understand any terrestrial phenomena without considering the nature of the earth and its relation to the other units of the solar system?[XIII-13]
Terrestrial physics includes two fields: physics proper and chemistry. Material bodies may be regarded in either their physical or chemical aspects. Physics is more general than chemistry; it deals with masses rather than elements. Chemical phenomena depend upon the laws of physics, without being influenced by them in turn. Chemical action is conditioned by the laws of weight, heat, electricity. The study of inorganic phenomena thus falls under three scientific heads: astronomy, physics, and chemistry.
Organic phenomena include two types: individual and group. The first refers to the function and structure of all individual forms in the plant and animal worlds. It is general physiology, or, in modern terms, biology. It involves the study of all life and the general laws pertaining to the individual units of life.
Biology rests on chemistry, because it is in chemistry that all reliable knowledge about nutrition or secretion is found. Biology is indebted to physics for knowledge concerning the weight of, temperature of, and related facts about living organisms. Biological laws are partially determined by astronomical factors. If the earth were to rotate faster than it does, the course of physiological phenomena would be accelerated, and the length of life would be shortened.[XIII-14] If the orbit of the earth were to become as eccentric as that of a comet, changes of a fatal nature would occur to all life on the earth. If there were no inclination of the earth’s axis, the seasons would be unknown, and the geographical distribution of living species would be vastly different from the present situation. All accurate work in biological studies is mathematical in character. Thus biology, the science of organic phenomena, is dependent on all the preceding divisions on the scale of knowledge.
The study of gregarious or associative life is a special field. Comte called this science social physics, and for it invented the specific term, sociology. It rests in turn upon biological, chemical, physical, astronomical knowledge and uses mathematics as its tool. Comte virtually defines six sciences: mathematics, astronomy, physics, chemistry, biology, and sociology. He treats of transcendental biology, which is the basis of modern psychology. Comte urged that no science could be effectually studied without competent knowledge concerning the sciences on which it depends. It is necessary not only to have a general knowledge of all the sciences but that they should be studied in order—this is Comte’s dictum to the student of sociology. Comte insisted that one general science could not develop beyond a given point until the preceding science has passed a given stage.
Each of the six general sciences has passed through the three stages of thought. Mathematics, which has advanced furthest into the positive stage, is still connected with superstition, such as that which hovers round the number 13. The other general sciences are less further along. Sociology, the latest science to develop, Comte hoped by his works to push over into the positive stage.