35. Continuity.
Up to this point our discussions have been based on the general concept of the thing, that is, of the individual experience differentiated from other experiences. Here the fact of being different, which, as a general experience, led to the corresponding elementary concept, appeared in the foreground in accordance with its generality. But in addition to it there is another general fact of experience, which has led to just as general a concept. It is the concept of continuity.
When, for example, we watch the diminution of light in our room as it grows dark in the evening, we can by no means say that we find it darker at the present moment than a moment before. We require a perceptibly long time to be able to say with certainty that it is now darker than before, and throughout the whole time we have never felt the increase of darkness from moment to moment, although theoretically we are absolutely convinced that this is the correct conception of the process.
This peculiar experience, our failure to perceive individual parts of a change, the reality of which we realize when the difference reaches a certain degree, is very general, and, like memory, is based upon a fundamental physiological fact. It has already been noted by Herbart, but its significance was first recognized by Fechner, and has since then become generally known in physiology and psychology under the name of threshold. Next to memory the threshold determines the fundamental lines of our psychic life.
The threshold therefore means that whatever state we are in a certain finite amount of difference or change must be stepped over before we can perceive the difference or change. This peculiarity appears in all our states or experiences. We have already given an example for the phenomena of light and darkness. The same is true of differences in color and of our judgments as to tone pitch and tone strength. Even the transition from feeling well to feeling ill is usually imperceptible, and it is only when the change occurs in a very brief time that we become conscious of it.
The physical causes of these psychic phenomena need be indicated only in brief. In all our experiences an existing chemico-physical state in our sense organs and in the central organ undergoes a change. Now experiments with physical apparatus have shown that such a process always requires a finite, though sometimes a very small, quantity of work, or, generally speaking, energy, before it can be brought about at all. Even the finest scale, sensitive to a millionth of a gram, remains stationary when only a tenth of a millionth is placed upon it, although we can see a body of such minute weight under the microscope. In the same way it requires a definite expenditure of energy in order to bring the sense organs, or the central organ, into action, and all stimuli less than this limit or threshold produce no experience of their presence.
By this the difficult concept of continuity is evoked in our experience. The transition from the light of day to the darkness of evening proceeds continuously, that is, at no point of the whole transition do we notice that the state just passed is different from the present one, while the difference over a wider extent of the experience is unmistakable. If we wish to bring vividly to our minds the contradiction to other habits of thought which this involves, we need only to represent to ourselves the following instance. I will compare the thing A at a certain time with the thing B, which is so constructed that though objectively different from A, the difference has not yet reached the threshold. From experience, therefore, I must take A to be equal to B. Then I compare B with a thing C, which is objectively different from B in the same way as A is from B, though here, too, the difference is still within the threshold, though very near it. I shall also have to take B as equal to C. But now if I compare A directly with C, the sum of the two differences oversteps the threshold value, and I find that A is different from C. This, then, is a contradiction of the fundamental principle that if A = B and B = C, A = C. This principle is valid for counted things, which, in consequence, are discontinuous, but not for continuous things susceptible by our senses. If in spite of this it is applied to continuous things or magnitudes in the narrower sense, we must bear in mind that it is just as much a case of an extrapolation to the non-existing ideal instance ([p. 46]) as in the case of the other general principles, which, though they are derived from experience, nevertheless, for practical purposes, transcend experience in their use.
The examples cited above prove also that these relations are by no means confined to the judgments we derive on the basis of immediate sensations. When by means of the scale we compare three weights, the differences of which lie within the limit of its sensitiveness but approach closely to it, we can arrive in a purely empirical and objective way also at the contradiction A = B, B = C, but A ≠ C. In weight and measurement, therefore, we hold fast to the principle that the relations cited have no claim to validity outside the limit of their possible errors. Accordingly, though the non-equation of A ≠ C can be observed, the difference of both values cannot be greater than at utmost the sum of the two threshold values.
These considerations also give us a means of appraising the oft-repeated statement that in contradistinction to the physical laws the mathematical laws are absolutely accurate. The mathematical laws do not refer to real things, but to imaginary ideal limit cases. Consequently they cannot be tested by experience at all, and the demands science makes on them lie in quite a different sphere. Their nature must be such that experience should approximate them infinitely, if certain definite well-known postulates are to be more and more fulfilled, and that the various abstractions and idealizations should be so chosen as not to contradict one another. Such contradictions have by no means always been avoided. But we must not regard them as inherent in the inner organization of our mind, as Kant did. These contradictions spring from careless handling of the concept technique, by which postulates elsewhere rejected are treated as valid. We have already come across an instance of such relations in the application of the concept of equality to unlimited groups ([p. 84]).
We must be guided by the same rules of precaution in answering the question whether the things felt as continuous—for example, space and time—are "truly" continuous, or whether in the last analysis they must not be conceived of as discontinuous. The various sense organs, and still more, the various physical apparatus with which we examine given states, are of very varying degrees of "sensibility," that is, the threshold for distinguishing the differences may be of very different magnitudes. Therefore, a thing which is discontinuous for a sensitive apparatus will behave as if it were continuous with a less sensitive apparatus. Accordingly, we shall find so many the more things continuous the less sharply developed our ability is to differentiate.
While this circumstance makes it possible that we should regard discontinuous things as continuous, time relations in certain circumstances produce the opposite effect. Even if in a process the change is continuous but very rapid, and the new state remains unchanged for a certain time, we easily conceive of this sequence as discontinuous. We cannot resist this view of the process when the change occurs in a shorter time than the threshold time of our mind for each step in the process. But since this threshold changes with our general condition, one and the same process can appear to us both continuous and discontinuous according to circumstances. Here, therefore, we have a cause through the operation of which, with advancing knowledge, more and more things will become recognized as continuous.
Now if we turn to experience, we find, as the sum total of our knowledge, that for the sake of expediency we approach everything with the presumption that it is continuous. This aggregate experience finds its expression in such sayings as "Nature makes no jumps," and similar proverbial generalizations. But we must emphasize the fact once more that in deciding matters in this way we deal solely with questions of expediency, not with questions of the nature of our mental capacity.