This case, however, can be referred to the general one first described, by treating the characteristics belonging together as being one characteristic, so that the group is defined solely by the independent characteristics. Then, according to the definition, we can, without losing our connection with experience, carry out that formal manifoldness of all possible related groups which yields what is called a classification of the corresponding things.
If for the determination of a group a definite number of independent characteristics is taken, say, a, b, c, d, and e, then we have at first the narrowest or poorest group abcde. By the elimination of one characteristic we obtain the five groups, bcde, acde, abde, abce, and abcd. If we omit one other characteristic we get ten different groups abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. Likewise, there are ten groups with two characteristics each, and finally five groups with one characteristic each. All these groups are related. There is a science, the Theory of Combinations, which gives the rules by which, in given elements or characteristics, the kind and number of the possible groups can be found. The theory of combinations enables us to obtain a complete table and survey of all possible complex concepts which can be formed from given simple ones (whether they be really elementary concepts, or only relatively so). When in any field of science the fundamental concepts have been combined in this manner, a complete survey can be had of all the possible parts of this science by means of the theory of combinations.
In order to present this process vividly to our minds, let us take as an example the science of the chemical combination of substances which form an important part of chemistry. There are about eighty elements in chemistry, and this science has to treat of
a) each of the eighty elements by itself
b) all substances containing two elements and no more
c) all substances containing three elements
d, e, f, etc.) the substances containing four, five, and six, etc., elements,
until finally we reach a group (not existing in experience) embracing substances formed of all the elements. That there is no such substance in the present scope of human knowledge has, of course, no significance for the structure of the scheme. What is significant is the fact that the scheme really embraces and arranges all possible substances in such a way that we cannot conceive of any case in which a newly discovered substance cannot after examination immediately be classed with one of the existing groups.
To cite an example from another science. Physics, it will be recalled, may be considered to be the science of the different kinds of energy. This science, accordingly, is divided first into the study of the properties of each energy, and then into the study of the relations of two energies, of three energies, of four energies, etc. Here, too, we may say that in the end there can be no physical phenomenon which cannot be placed in one of the groups so obtained.
Of course, neither in chemistry nor in physics does this mean that each new case will fall within the scheme obtained by the exhaustive combination of elementary concepts (whether chemical elements or kinds of energy) known at the time. It is quite possible that a new thing under investigation contains a new elementary concept, so that on account of it the scheme must be enlarged through the embodiment of this new element. But simultaneously a corresponding number of new groups appear in the scheme, and the investigator's attention is directed to the fact that he still has a reasonable prospect, in favorable circumstances, of discovering these new things also. Thus combinatory schematization serves not only to bring the existing content of science into such order that each single thing has its assigned place, but the groups which have thereby been found to be vacant, to which as yet nothing of experience corresponds, also point to the places in which science can be completed by new discoveries.
From the above presentation it is apparent how from the two concepts "thing" and "association" alone a great manifoldness of various and regular forms can be developed. They are purely empirical relations, for the fact that several things can be combined in the graded series described above according to a fixed rule does not follow merely from the two concepts, but must be experienced. But, on the other hand, both concepts are so general that the experiences obtained in some cases can be applied to all possible experiences and may serve the purpose of classifying and making a general survey of them.
The above statements, however, have by no means exhausted the possibilities. For it has been tacitly assumed that in the combination of several things the sequence according to which this combination takes place should not condition a difference of the result. This is true of a number of things, but not of all. In order, therefore, to exhaust the possibilities the theory of combinations must be extended also to cases in which the sequence is to be taken account of, so that the form ab is regarded as different from ba.
We will not undertake to work out the results of this assumption. It is obvious that the manifoldness of the various cases is much greater than if we neglect the sequence. On this point we have one more observation to make, that further causes for diversity exist. It is true that a chemical combination is not influenced by the sequence in which its elements enter the combination, but there do occur with the same elements differences in their quantitative relations, and thereby a new complexity is introduced into the system, so that two or more similar elements can form different combinations according to the difference in the quantitative relations. Still, even with this, the actual manifoldness is not exhausted, for from the same elements and with the same quantitative relations there can arise different substances called isomeric, which, for all their similarity, possess different energy contents. But the first scheme is not demolished, nor does it become impracticable because of this increase of manifoldness. What simply happens is that several different things instead of one appear in the same group of the original scheme, the systematic classification of which necessitates a further schematization by the use of other characteristics.