would hold that the operation of the law is unquestionable— that it is merely concealed and not held in abeyance. And, finally, the principle of the lever we find actually operative in the iron crowbar, which, resting on one log, moves another. And let it be considered that apart from such concrete instances as these, the laws in question have no meaning for us. It was not from the consideration of infinite distances, or of the universe as a whole, or of the ultimate constituents of matter, that these laws were derived-so much is obvious. And their actual utility in the interpretation of our every-day life, as well as of our scientific experience, is enormous. To suppose that their true application is something utterly different from any application we ever actually make of them is trifling with common sense.

No, these laws, like other laws, are instruments by means of which we analyze phenomena. They are demonstrated, not from 'pure' instances, but from instances in which disturbing factors are as far as possible eliminated; and, both in the more simple and in the more complex instances, their significance is that of the description of a contributing factor in a total process. It is, indeed, to this fact that the exactness of the laws is due, for this is but complementary to the confessed insufficiency of the analysis. All inexactness is attributed to further, as yet undistinguished, conditions. But to say that the laws are approximately verified under approximately perfect conditions is to understate their experimental basis. They are verified with less and less average inexactness as the conditions approach perfection. M. Poincaré's theory takes no account of this all-important fact. And it must be added, that even though no single decisive test can ever overthrow these laws, yet, if with increasingly delicate observations the average error should ever fail to decrease, they would be regarded as disproved in their present form, and would have to be materially corrected.

On the whole, we find no sufficient reason for placing the principles of mechanics in an absolutely different category from those of economics. The essential difference appears to consist

in the nature of the abstraction which is made in the two sciences. The laws of economics are protected by an 'other things being equal,' where there is by no means a definite conception as to what these other things may possibly include. In mechanics there is no other things being equal.' The antecedent of each formula purports, at least, to set forth the precise conditions under which the consequent must follow. Aside from this we can only say that mechanical laws represent a far higher grade of universality and precision than economic laws have attained, or, very possibly, will ever attain.

The case of geometry and that of mechanics hang closely together. It is known that the principles of the two sciences are so related that considerable alterations can be made in either and sufficiently compensated by corresponding alterations in the other. A non-Euclidean geometry, coupled with its appropriate non-Newtonian mechanics, can describe our world as exactly as the Euclidean can do. In short, geometry is recognizedly a branch of applied mathematics—an experimental science which has long since reached the deductive stage. If mathematicians sometimes appear to take it otherwise, that is because they have redefined the term. It then no longer professes to treat of the space-relations of our experience, but is, as the phrase goes, a science of 'cross-classification.'

There remains only pure mathematics-that is to say, formal logic and the sciences of number and order deducible from formal logic as a possible obstacle to an evolutionary view of scientific validity. We are inclined to the belief that this also is no insuperable obstacle, that logic, like geometry and mechanics, represents a stage in the development of scientific universality, not the ideal consummation. The numerical formulas (such as Kant's notorious 7+ 5 = 12), upon whose a priori certainty so much stress was formerly laid, are in themselves, as has been definitely shown, analytical propositions and, indeed, absolute identities: the definitions of the two members of the equality can always be reduced to an identical form.1 The vital question is whether the under1Cf. L. Couturat, Les Principes des Mathematiques, p. 255, n. 3.

=

lying concept of number itself, and below it the concepts of implication and inclusion, are absolutely final. This we see no sufficient reason to believe. On the contrary, the utterly unexpected development which the concept of number has recently undergone through researches in the theory of infinite numbers is an index of the possibilities which may yet be in store. Nothing could ever have seemed more necessary than that if 2X X, X = 0; and yet we know today that there is a distinct class of other roots. The old number-theory, which was thought to be absolutely true, is seen to be true only within a certain limitation, namely, that the numbers considered be finite. It has been aufgehoben-refuted as absolute, and taken up and preserved as part of an ampler whole. For all that we know, the theory of today may be similarly aufgehoben tomorrow.

The classification of contemporary human races presents in temporal cross-section a picture of the evolution of humanity. The classification of the sciences presents in a like cross-section a picture of the evolution of human judgment. Of this evolution, we repeat, the pragmatist theory of truth has taken insufficient account. Nothing is more dangerously misleading than an indiscriminate induction from the various stages of a given development. That most, if not all, laws are approximate, that their validity is relative to the satisfaction of the particular wants of individual men, and hence that validity is determined by maximal individual satisfaction, is true enough to be exceedingly false. It is like Hume's theory-founded upon a similar sweeping inductionthat justice is whatever custom makes it. Whereas, for example, Locke had claimed that taxation without representation is unjust, Hume observes: "What authority any moral reasoning can have, which leads into opinions so wide of the general practice of mankind in every place but this single kingdom, it is easy to determine."1 Hume's induction was correct. He might even have added that in Great Britain the suffrage was strangely limited. And yet Locke was more than half right, because the norm which he described lay athwart the course of social evolu1Essay XXXIV.

tion. So, when the pragmatist interprets his doctrine as an individualism, we declare that we find the rationalist right as against him; for the latter merely describes as a realized, or definitely realizable, end an indefinitely distant ideal toward which the developing judgment tends.

Will it be said that the development which we have been tracing is not of truth, but of the capacity of the judgment for expressing truth, that truth itself is the eternal ideal toward which the whole development is tending? Well and good; we need not quarrel about terms. But, in the first place, let it be remembered, that the stages of this development are not past and gone. We cannot live by pure mathematics alone, enormously valuable as its conceptions are to us. Truths such as, "Johnny is a baby," and "William is still young," are still wonderfully important to us; and it is idle to say that they are not true. Whatever truth may mean for an absolute consciousness, for us it certainly includes all the grades that have been mentioned, and no doubt others which we have not distinguished. It is an utterly arbitrary use of terms to restrict it to the ultimate ideal. In the second place, we must beware of imagining that science as a whole is approaching the mathematical type-that the day is nearing, though still far distant, when all our encyclopedias shall be reduced to tables of formulæ. Take any particular field of concrete inquiry, and as investigation proceeds, a body of more and more general and precise propositions is accumulated within it. But even within the given field the looser, more vaguely limited propositions likewise accumulate. The evolution is a spreading-out and a filling-in, as well as a growth upward. The same is true of knowledge in general. Paradoxical as the statement may seem, each new stage in the advancement of science makes it more and more manifestly impossible that its highest type of judgment should ever be applied to express its entire content. There is a manifest increase in clearness and universality, but there is also a constant expansion of the confused and the contingent; and the importance of these in our world-view is assuredly not declining.

CHAPTER III

THE DEVELOPING CONCEPT AND ITS FUNCTIONS

I. THE CONCEPT OF THE OBJECT

It has been pointed out that pragmatists, explicitly or by implication, have recognized two aspects of meaning; on the one hand, the reference to conduct, the value of the idea, or what we have called its import; and, on the other hand, its content, consisting of its relations to certain other ideas and represented roughly by the terms genus and differentia. But, while they have done so much, they have not concerned themselves to bring out the very intimate relationship which the two aspects bear to each other. Had pragmatist writers faced this problem, they might have averted much of the criticism urged against them, and at the same time have opened the way to a very fruitful development of their theory.

That a very intimate relationship exists will readily appear upon consideration of a very simple case of the learning-process. We are fully aware that there is a certain danger in this procedure the same danger that is always incurred in the attempt to explain later and more complex features of an organism through reference to a simple and primitive type. On the one hand, there is the tendency to interpret the later type in terms far too simple to do it justice; and, on the other hand, there is the tendency, equally strong, to falsify the earlier type by reading into it characteristics which properly belong only to later stages of development. And yet these tendencies are not, we believe, so unavoidable, that we should forego the great advantage to be gained from the schematic clearness that is thus made possible.

Let us assume as the starting point of the process that an accustomed stimulus A is regularly met by the response B with satisfactory consequences. We assume the conscious experience