§4. The Divisions of Science

243. I still persist in leaving unnoticed a certain subbranch of theoretical science [the sciences of review]; and as for the practical sciences,^P1) I shall merely mention a few of them, just to give an idea of what I refer to under that name. I mean, then, all such well-recognized sciences now in actu, as pedagogics, gold-beating, etiquette, pigeon-fancying, vulgar arithmetic, horology, surveying, navigation, telegraphy, printing, bookbinding, paper-making, deciphering, ink-making, librarian's work, engraving, etc.¹⁾ In short, this is by far the more various of the two branches of science. I must confess to being utterly bewildered by its motley crowd, but fortunately the natural classification of this branch will not concern us in logic — at least, will not do so as far as I can perceive.

244. Now let us consider the relations of the classes of science to one another. We have already remarked that relations of generation must always be of the highest concern to natural classification, which is, in fact, no more nor less than an account of the existential, or natural, birth concerning relations of things; meaning by birth the relations of a thing to its originating final causes.

245. Beginning with Class I, mathematics meddles with every other science without exception. There is no science whatever to which is not attached an application of mathematics. This is not true of any other science, since pure mathematics has not, as a part of it, any application of any other science, inasmuch as every other science is limited to finding out what is positively true, either as an individual fact, as a class, or as a law; while pure mathematics has no interest in whether a proposition is existentially true or not. In particular, mathematics has such a close intimacy with one of the classes of philosophy, that is, with logic, that no small acumen is required to find the joint between them.

246. Next, passing to Class II, philosophy, whose business it is to find out all that can be found out from those universal experiences which confront every man in every waking hour of his life, must necessarily have its application in every other science. For be this science of philosophy that is founded on those universal phenomena as small as you please, as long as it amounts to anything at all, it is evident that every special science ought to take that little into account before it begins work with its microscope, or telescope, or whatever special means of ascertaining truth it may be provided with.

247. It might, indeed, very easily be supposed that even pure mathematics itself would have need of one department of philosophy; that is to say, of logic. Yet a little reflection would show, what the history of science confirms, that that is not true. Logic will, indeed, like every other science, have its mathematical parts. There will be a mathematical logic just as there is a mathematical physics and a mathematical economics. If there is any part of logic of which mathematics stands in need — logic being a science of fact and mathematics only a science of the consequences of hypotheses — it can only be that very part of logic which consists merely in an application of mathematics, so that the appeal will be, not of mathematics to a prior science of logic, but of mathematics to mathematics. Let us look at the rationale of this a little more closely. Mathematics is engaged solely in tracing out the consequences of hypotheses. As such, she never at all considers whether or not anything be existentially true, or not. But now suppose that mathematics strikes upon a snag; and that one mathematician says that it is evident that a consequence follows from a hypothesis, while another mathematician says it evidently does not. Here, then, the mathematicians find themselves suddenly abutting against brute fact; for certainly a dispute is not a rational consequence of anything. True, this fact, this dispute, is no part of mathematics. Yet it would seem to give occasion for an appeal to logic, which is generally a science of fact, being a science of truth; and whether or not there be any such thing as truth is a question of fact. However, because this dispute relates merely to the consequence of a hypothesis, the mere careful study of the hypothesis, which is pure mathematics, resolves it; and after all, it turns out that there was no occasion for the intervention of a science of reasoning.

248. It is often said that the truths of mathematics are infallible. So they are, if you mean practical infallibility, infallibility such as that of conscience. They appear even as theoretically infallible, if they are viewed through spectacles that cut off the rays of blunder. I never yet met with boy or man whose addition of a long column, of fifty to a hundred lines, was absolutely infallible, so that adding it a second time could in no degree increase one's confidence in the result, nor ought to do so. The addition of that column is, however, merely a repetition of 1 + 1 = 2; so that, however improbable it may be, there is a certain finite probability that everybody who has ever performed this addition of 1 and 1 has blundered, except on those very occasions on which we are accustomed to suppose (on grounds of probability merely) that they did blunder. Looked at in this light, every mathematical inference is merely a matter of probability. At any rate, in the sense in which anything in mathematics is certain, it is most certain that the whole mathematical world has often fallen into error, and that, in some cases, such errors have stood undetected for a couple of millennia. But no case is adducible in which the science of logic has availed to set mathematicians right or to save them from tripping. On the contrary, attention once having been called to a supposed inferential blunder in mathematics, short time has ever elapsed before the whole mathematical world has been in accord, either that the step was correct, or else that it was fallacious; and this without appeal to logic, but merely by the careful review of the mathematics as such. Thus, historically mathematics does not, as a priori it cannot, stand in need of any separate science of reasoning.

249. But mathematics is the only science which can be said to stand in no need of philosophy, excepting, of course, some branches of philosophy itself. It so happens that at this very moment the dependence of physics upon philosophy is illustrated by several questions now on the tapis. The question of non-Euclidean geometry may be said to be closed. It is apparent now that geometry, while in its main outlines, it must ever remain within the borders of philosophy, since it depends and must depend upon the scrutinizing of everyday experience, yet at certain special points it stretches over into the domain of physics. Thus, space, as far as we can see, has three dimensions; but are we quite sure that the corpuscles into which atoms are now minced have not room enough to wiggle a little in a fourth? Is physical space hyperbolic, that is, infinite and limited, or is it elliptic, that is, finite and unlimited? Only the exactest measurements upon the stars can decide. Yet even with them the question cannot be answered without recourse to philosophy. But a question at this moment under consideration by physicists is whether matter consists ultimately of minute solids, or whether it consists merely of vortices of an ultimate fluid. The third possibility, which there seems to be reason to suspect is the true one, that it may consist of vortices in a fluid which itself consists of far minuter solids, these, however, being themselves vortices of a fluid, itself consisting of ultimate solids, and so on in endless alternation, has hardly been broached. The question as it stands must evidently depend upon what we ought to conclude from everyday, unspecialized observations, and particularly upon a question of logic. Another still warmer controversy is whether or not it is proper to endeavor to find a mechanical explanation of electricity, or whether it is proper, on the contrary, to leave the differential equations of electrodynamics as the last word of science. This is manifestly only to be decided by a scientific philosophy very different from the amateurish, superficial stuff in which the contestants are now entangling themselves. A third pretty well defended opinion, by the way, is that instead of explaining electricity by molar dynamics, molar dynamics ought to be explained as a special consequence of the laws of electricity. Another appeal to philosophy was not long ago virtually made by the eminent electrician, the lamented Hertz, who wished to explain force, in general, as a consequence of unseen constraints. Philosophy alone can pronounce for or against such a theory. I will not undertake to anticipate questions which have not yet emerged; otherwise, I might suggest that chemists must ere long be making appeal to philosophy to decide whether compounds are held together by force or by some other agency. In biology, besides the old logico-metaphysical dispute about the reality of classifications, the momentous question of evolution has unmistakable dependence on philosophy. Then again, caryocinesis has emboldened some naturalists, having certain philosophical leanings, to rebel against the empire of experimental physiology. The origin of life is another topic where philosophy asserts itself; and with this I close my list, not at all because I have mentioned all the points at which just now the physical sciences are influenced by a philosophy, such as it is, but simply because I have mentioned enough of them for my present purpose.