The mentioned aspiration and the tendence of all thought, of all science, to assume the character of mathematics admit of many illus trations, which at the same time serve to show clearly the ultimate distinction between that thought which is mathematical and that which is not. One or two such illustrations must suffice.
There is no more common or more im portant scientific notion than that of function. The term is applied to either of two variable things (including constants conceived as special variables, whose variation is zero in amount) so related that to any value or state of either there corresponds one (or more) value (values) or state (states) of the other. Any two corresponding values or states are said to constitute a pair of values or states, and any two functionally related variables may be called a function-pair. Examples of such func tion-pairs abound on every hand, as the radius and the area of a circle; the corresponding values of x and y in an equation, 2x-3y-F50; elasticity of medium and velocity of sound or other undulation; the amount of sodium chlo ride formed and the time occupied by the reac tion that generates it; the prosperity of a given community and, ceteris paribus, the intelligence of its patriotism. Indeed it may be that there is no thing which is not in some sense a func tion of every other. Be this as it may, one thing is very certain, namely, a very great part, if not the whole, of our thinking is pri marily concerned with functional relationships, deals, that is, immediately or mediately with pairs or systems of corresponding values or states or changes. Behold, for example, how the parallelistic psychology searches for corre lations between psychical and physical phe nomena. Witness, too, the sociologist seeking to determine a law of correspondence between the homogeneity of a population and its peace fulness; the anthropologist attempting to find a formula correlating mental power and brain weight; the physicist's determinations of de pendence between pressure and' volume of a i gas; and so on and so on. It is, then, here, in the immense and wondrously diversified do main of correspondence, the answering of value to value, of change to change, of condition to condition, of state to state, that the knowing activity, the intellect, finds its field.
What is it precisely that we seek to do by means of a correlation? The answer is: when one or more facts are given or known, to pass with absolute certainty to the correlative fact or facts. For example, if y = sin x, then, if a value be assigned to x or y, the corresponding value or values of v or x are determined by the equation, or definition, of the functional corre spondence between the assemblage of values which x may take and the assemblage that v may assume. To effect the desired transition from the given or known to the dependent ungiven or unknown obviously requires one or more formula or equations which shall serve to define precisely the manner of corre spondence, the law of dependence. Where do such defining formula or equations come from? Strictly speaking, they are never found, they are always assumed, assumed immediately or else mechately, that is, in the latter case, de rived by assumed logical processes from such as belong to the former case. This statement is valid in every field of logical thought. In every field it is true that from nothing assumed nothing can be derived. Now, nothing is easier than to write down a perfectly definite formula that does not tell how cheerfulness depends on climate or retentiveness on interest or the volume of a cube on its dimensions. The men tioned inapplicability or "inutility') does not, however, at all tend to invalidate the formula regarded as defining a certain law of correla tion. Indeed a given formula may be per fectly intelligible in itself ; it, alone or joined with others in similar case, may state a perfectly intelligible law of correspondence, which, nevertheless, may have no validity whatever in the physical or in the sensuous universe. What is it, then, that guides in the choice of for mula? For such determination or choice seems hardly referable to chance alone. The answer is that, broadly speaking, choice is determined by curiosity, and curiosity is itself not determined by choice, but is rather matter of native gift or predilection.