Part 57
Dr. Carus adduces the instance of a geometrical proposition, namely, “that two congruent regular tetrahedrons, when put together, will form a hexahedron.” (¶ 25.) This, he says, seems to be “a very wonderful thing”; for why should not a larger tetrahedron be formed, just as two heaps of flour make a large heap of flour? Yet, he continues, the probability that the two tetrahedrons do always make a hexahedron is 1, “which means certainty” (¶ 27). But as it happens, the proposition, in the form stated is quite erroneous. What is true is this. If two tetrahedra are so placed that one face of each is coincident with one face of the other, while all the other faces are inclined to one another, and if of the 8 faces, the 2 that are coincident are not counted, there remain to be counted 8-2=6 faces. But there is nothing more wonderful about this than that 8-2=6, which is an easy corollary from definitions. Very few propositions in mathematics that appear “marvellous” will hold water; and those few excite our astonishment only because the real complexity of the conditions are masked in an intuitional presentation of them.
Dr. Carus holds (¶ 15) that formal knowledge is absolutely universal, exact, and necessary. In some cases, as where he says that, given the number of dimensions of space, the entire geometry could be deduced (¶ 35), the boasted infallibility will prove on examination to be downright error. In all other cases, the propositions only relate to ideal constructions, and their applicability to the real world is at the best doubtful, and, as I think, false; while in their ideal purity, they are not synthetical.
Thus, my good friend and antagonist holds that the combination of oxygen and hydrogen to produce water is not “different in principle” from that of the tetrahedra to produce a hexahedron (¶ 26). There is all the difference between the ideal and the real; which to my Scotistic mind is very important. But this is not the only passage in which he speaks as if form were the principle of individuation.
§ 9. Dr. Carus’s position is even weaker than that of Kant, who makes space, for example, a necessary form of thought (in a broad sense of that term). But Dr. Carus appears to consider space as an absolute reality. For he says (¶ 119) that “every single point of space has its special and individual qualities.” Here again form is made the principle of individuation; whence the queer phrase, “individual qualities.”
§ 10. Dr. Carus argues that whatever is unequivocally determinate is necessary. (¶ 124.) Were the determination spoken of real dynamic determination, this would be a mere truism. But the expression used, _eindeutig bestimmt_, merely expresses a mathematical determination, and therefore no real necessity ensues. The equation
(Transcriber’s Note: Italics have been removed from the formulæ for readability.)
x² - 23x + 132¼ = 0
determines _x_ to be either 11·477 or 11·523. In this sense, _x_ has necessarily one value or the other. The equation
x² - 12x + 6 = 0
determines _x_ to be either 11·477 or 0·523. Together, the two equations uniquely determine _x_ to be 11·477. This shows how much that argument amounts to.
§ 11. By “sameness,” Dr. Carus means equivalence for a given purpose. (¶¶ 102, 106.) By the “idea of sameness,” he means (¶¶ 77, 96) the principle that things having a common character are for some purpose equivalent. This, he says, “has a solid basis in the facts of experience.” By a “world of sameness” (¶ 113), he seems to mean one in which any two given concrete things are in some respect equivalent. He argues (¶ 122) that a “world of sameness is a world in which necessity rules.” I do not see this. It seems to me so bald a _non sequitur_, that I cannot but suppose the thought escapes my apprehension. If there were anything in the argument, it would seem to be a marvellously expeditious way of settling the whole dispute; and therefore it would have been worth the trouble of stating, so as to bring it within the purview of minds like mine.
§ 12. My candid opponent sometimes endorses emphatically the Leibnizian principle. “Necessitarianism must be founded on something other than observation. Observation is _a posteriori_; it has reference to single facts, to particulars; yet the doctrine of necessity ... is of universal application. The doctrine of necessity ... is of an _a priori_ nature.” (¶ 11.) “Millions of single experiences ... cannot establish a solid belief in necessity.” (¶ 14.) “No amount of experience is sufficient to constitute causation by a mere synthesis of sequences.” (¶ 22.) “Millions of millions of cases” constitute “no proof” that a proposition “is always so.” (¶ 29.)
Nevertheless, he holds that the law of “the conservation of matter and energy” so conclusively proves necessary causation, that the obstinacy of Hume, himself, could not have withstood the argument. (¶ 23.) One wonders, then, what is supposed to prove this “law of the conservation of matter and energy,” if no amount of experience can prove it.
But the _a priori_ itself can “be based on the firm ground of experience.” (¶ 14.) In that case, it is not prior to experience, after all! “The idea of necessity is based upon the conception of sameness, and ... the existence of sameness is a fact of experience.” (¶ 87.) If absolute necessity can be irrefragably demonstrated from the fact that two things are alike, it is a pity Dr. Carus should not state this demonstration in a form, that I, and men like me, can understand. That would be more to the purpose than merely saying it can be proved. Absolute chance is rejected as “involving a violation of laws well established by _positive evidence_.” (¶ 149.)
All these _denials_ that absolute necessity can be established and absolute chance refuted by experiential evidence, mixed with as clear _assertions_ of the same things, when taken together, have the appearance of an attempt, as the politicians say, to “straddle” the question.
§13. But the ingenious Doctor seeks to bolster up necessity by introducing the confused notion of “causation.”
I do not know where the idea originated that a cause is an instantaneous state of things, perfectly determinative of every subsequent state. It seems to be at the bottom of Kant’s discourse on the subject; yet it accords neither with the original conception of a cause, nor with the principles of mechanics. The original idea of an efficient cause is that of an agent, more or less like a man. It is prior to the effect, in the sense of having come into being before the latter; but it is not transformed into the effect. In this sense, it may happen that an event is a cause of a subsequent event; seldom, however, is it the principal cause. Far less are events the only causes. The modern mechanical conception, on the other hand, is that the relative positions of particles determine their accelerations at the instants when they occupy those positions. In other words, if the positions of all the particles are given at _two_ instants (together with the law of force), then the positions at all other instants may be deduced.[109] This doctrine conflicts with Kant’s second analogy of experience, as interpreted by him, in no less than four essential particulars. In the first place, far from involving any principle that could properly be termed generation, or _Erzeugung_, which is Kant’s word for the sequence of effect from cause, the modern mechanical doctrine is a doctrine of persistence, and, as I have repeatedly explained, positively prohibits any real growth. In the second place, one state of things (i. e. one configuration of the system) is not sufficient to determine a second, it is two that determine a third. To whomsoever may think that this is an inconsiderable divergence of opinions, let me say, study the logic of relatives, and you will think so no longer. In the third place, the two determining configurations, according to mechanics, may be taken at almost any two instants, and the determined configuration be taken at any third instant we like. _There is no mechanical truth in saying that the past determines the future, rather than the future the past._ We habitually follow tradition in continuing to use that form of expression, but every mathematician knows that it is nothing but a form of expression. We continue, for convenience, to talk of mechanical phenomena as if they were regulated, in the same manner in which our intentions regulate our actions (which is essentially a determination of the future by the past), although we are quite aware that it is not really so. Remark how Kant reasons:
“If it is a necessary law of our sensibility, and consequently a _formal condition_ of all perceptions, that the preceding time determines the following, (since I can only come to the following through the preceding,) then is it also an indispensable _law of the empirical representation_ of the time-series that the appearances of the preceding time determine every occurrence in the following.”
What this leads to is a causality like that of mental phenomena, where it _is_ the past which determines the future, and _not_ (in the same sense) the reverse; but the doctrine of the conservation of energy consists precisely in the denial that anything like this occurs in the domain of physics. Had Kant studied the psychological phenomena more attentively and generalised them more broadly, he would have seen that in the mind causation is not absolute, but follows such a curve as is traced in my essay towards “The Law of Mind” (_The Monist_, Vol. II, 550). Does our judicious editor deem it ungracious to find fault with Kant for not doing so much more than he did, considering what that hero-like achievement was? We must seem to carp, as long as thinkers can hold that achievement for sufficient. In the fourth place, Kant’s “Analogy” ignores that continuity which is the life-blood of mathematical thought. He deals with those awkward chunks of phenomenon, called “events.” He represents one such “event” as determined by certain others, definitely, while the rest have nothing to do with it. It is impossible to cement such thought as this into hermetic continuity with the refined conceptions of modern dynamics. The statement that every instantaneous state of things determines precisely all subsequent states, and not at all any previous states, could, I rather think, be shown to involve a contradiction.
The notion which Dr. Carus holds of a cause seems to be that it is a state, embracing all the positions and velocities of all the masses at one instant, the effect being a similar state for any subsequent instant. (¶¶ 21, 24.) This breaks at once with common parlance, with dynamics, and with philosophical logic. In common parlance, we do not say that the position and upward velocity of a missile is the cause of its being at a subsequent instant lower down and moving with a greater downward velocity.[110] In dynamics, it is the fixed force, gravitation, or whatever else, together with those relative positions of the bodies that determine the intensity and direction of the forces, that is regarded as the cause. But these causes are not previous to, but simultaneous with, their effects, which are the instantaneous accelerations. Finally, logic opposes our calling one of two states which equally determine one another (as any two states of a system do, if the velocities are taken to be included in these states) _the_ determinator, or cause, simply because of the circumstance that it precedes the other in time,—a circumstance that is upon the principles of dynamics plainly insignificant and irrelevant.
Everybody will make slips in the use of words that have been on his lips from before the time when he learned to think; but the practice which I endeavor to follow in regard to the word _cause_, is to use it in the Aristotelian sense of an _efficient cause_, in all its crudeness. In short, I refuse to use it at all as a philosophical word. When my conception is of a dynamical character, I endeavor to employ the accepted terminology of dynamics;[111] and when my idea is a more general and logical one, I prefer to speak of the _explanation_.
§ 14. Dr. Carus thinks the element of necessity in causation can be demonstrated by considering the process as a transformation. “It is a sequence of two states which belong together as an initial and final aspect of one and the same event.” (¶ 21. Compare ¶¶ 20, 24.) He neglects to explain how he brings under this formula the inward causation of the will and character, as set forth by him in ¶¶ 163-167.
It is unnecessary for me to reply, at length, to an argument so manifestly inconclusive. On the one hand, it conflicts with the principle that absolute necessity cannot be proved from experience; and on the other hand, it leaves room for an imperfect necessity.
Professor Tait has done an ill office to thought in countenancing the idea that the conservation of energy is of the same nature as the “conservation,” or rather perduration, of matter. Dr. Carus says (¶ 121) that
“The law of the conservation of matter and energy rests upon the experience (corroborated by experiments) that causation is transformation. It states that the total amount of matter and the total amount of energy remain constant. There is no creation out of nothing and no conversion of something into nothing.”
The historical part of this statement contains only a small grain of truth; but that I will not stop to criticise. The point I wish to make is that the law of the conservation of energy is here represented under a false aspect. The true substance of the law is that the accelerations, or rates of change, of the motions of the particles at any instant depend solely on their relative positions at those instants. The equation which expresses the law under this form is a differential equation of the second order; that is, it involves the rates of change of the rates of change of positions, together with the positions themselves. Now, because of the purely analytic proposition of the differential calculus that
Dₜ²s = ½Dₛ(Dₜs)²,
the first integral of the differential equation of the second order, that is, the differential equation of the first order which expresses the same state of things, equates half the sum of the masses, each multiplied by the square of its velocity, to a function of the relative positions of the particles _plus_ an arbitrary constant.[112] In order to fix our ideas, let us take a very simple example, that of a single particle accelerated towards an infinite plane, at a rate proportional to the _n_ᵗʰ power of its distance from the plane. In this case, if _s_ be the distance, the second differential equation will be
Dₜ²s = -asⁿ,
and the first integral of it will be
(Dₜs)² = -(²ᵃ⁄ₙ₊₁)sⁿ⁺¹ + C.
By the first law of motion, and the Pythagorean proposition, the part of the velocity-square depending on the horizontal component is also constant.
The arbitrary constant, _C_, plainly has its genesis in the fact that forces do not determine velocities, but only accelerations. Its value will be fixed as soon as the velocity at any instant is known. This quantity would exist, just the same, and be independent of the time, and would therefore be “conserved” whether the forces were “conservative,” that is, simply positional, or not. Now, this constant is the energy; or rather, the energy is composed of this constant increased by another which is absolutely indeterminable, being merely supposed large enough to make the sum positive.
Thus, the law of energy does not prescribe that the total amount of energy shall remain constant; for this would be so in any case by virtue of the second law of motion; but what it prescribes is that the total energy diminished by the living force shall give a remainder which depends upon the relative positions of the particles and not upon the time or the velocities. It is also to be noticed that the energy has no particular magnitude, or quantity. Furthermore, in transformations of kinetical energy into positional energy, and the reverse, the different portions of energy do not retain their identity, any more than, in book-keeping, the identity of the amounts of different items is preserved. In short, the conservation of energy, (I do not mean the _law_ of conservation,) is a mere result of algebra. Very different is it with the “conservation” of matter. For, in the first place, the total mass is a perfectly definite quantity; and, in the second place, in all its transformations, not only is the _total_ amount constant, but all the different parts preserve their identity. To speak, therefore, of “the conservation of matter and energy,” is to assimilate facts of essentially contrary natures; and to say that the law of the conservation of energy makes the total amount of energy constant is to attribute to this law a phenomenon really due to another law, and to overlook what this law really does determine, namely, _that the total energy less the kinetic energy gives a remainder which is exclusively positional_.
§ 15. Dr. Carus does not make it clear what he means by _chance_. He does, indeed, say (¶¶ 145, 146):
“What is chance?
“Chance is any event not especially intended, either not calculated, or, with a given and limited stock of knowledge, incalculable.”
This defines what he means by a chance event, in the concrete; what he understands by probability, we are left to conjecture. But from what he says in ¶ 147, I infer that he regards it as dependent upon the state of our ignorance, and therefore nothing real.
I am, therefore, much puzzled when I find him expressing a conviction (¶¶ 88, 156) that chance plays an important part in the real world. He explains very distinctly that “when we call a throw of the dice pure chance, we mean that the incidents which condition the turning up of these or those special faces of the dice have not been, or cannot be, calculated.” (¶ 147.) This is the commonest, because the shallowest, philosophy of chance. Even Venn might teach him better than that. However, according to that view, when he writes of “the important part that chance plays in the world,—not absolute chance ... but that same chance of which the throw of the die is a typical instance” (¶ 88), he can be understood to mean no more than that many things happen which we are not in condition to calculate or predict. This is not playing a part _in the world_, one would say—at least, not in the natural world; it is only playing a part in our ignorance.
Dr. Carus frequently uses phrases which make us suspect he penetrates deeper. Thus, he says, “we do not believe in absolute chance, but we believe in chance” (¶ 144); and again, “Every man is the architect of his own fortune—but not entirely. There are sometimes coincidences determining the fates of men.” (¶ 161.) But when we remark the consecution of ¶¶ 137-162, we feel pretty sure he really sees no further. To do so would have been to perceive that indefinitely varied specificalness _is_ chance.
For a long time, I myself strove to make chance that diversity in the universe which laws leave room for, instead of a violation of law, or lawlessness. That was truly believing in chance that was not absolute chance. It was recognising that chance does play a part in the real world, apart from what we may know or be ignorant of. But it was a transitional belief which I have passed through, while Dr. Carus seems not to have reached it.
As for absolute chance, Dr. Carus makes the momentous admission that it is “not unimaginable” (¶ 150). If so, its negation, or absolute necessity, cannot be a formal principle.
§ 16. But it is time for me to leave the consideration of Dr. Carus’s system and to take up his strictures upon mine. His philosophy is one eminently enlightened by modern ideas, which it synthetises to an unusual extent. It is distinguished for its freedom from the vice of one-sidedness, and displays every facet of the gem of philosophical inquiry, except the one on which it rests, the question of absolute law. Its prominent faults, which I feel sure must have struck every competent reader, are that it shows little trace of meditation upon the thoughts of the great idealists, and that there is a certain want of congruity between different elements of it. How strangely it sounds, for instance, to find an apriorian, and one who is dinging “formal laws” so perpetually into our ears, one who holds that “in order to weave the woof of the _a posteriori_ elements into coherent cloth, we want the warp of the _a priori_” (¶ 15), to find this man declaring for a positivism “which accepts no doctrine, theory, or law, unless it be a formulation of facts,” and proclaiming that “the whole business of science is to systematise the samenesses of experience, and to present them in convenient formulas” (¶ 120). Now there is just one way of bringing such warring elements into harmony, and curing the greatest defect of the system,—and it is a way which would also bring the whole into far better concordance with natural science. It is to lop off the heads of all absolute propositions Whose subject is not the Absolute, and reduce them to the level of probable and approximate statements. Were that defalcation performed, Dr. Carus’s philosophy would, in its general features, offer no violent opposition to my opinions. Moreover, the Doctor has at heart the conciliation of religion and science. I confess such serious concern makes me smile; for I think the atonement he desires is a thing which will come to pass of itself when time is ripe, and that our efforts to hasten it have just that slight effect that our efforts to hasten the ripening of apples on a tree may have. Besides, natural ripening is the best. Let science and religion each have stout faith in itself, and refuse to compromise with alien and secondary purposes, but push the development of its own thought on its own line; and then, when reconcilement comes,—as come it surely will,—it will have a positive value, and be an unmixed good. But since our accomplished editor thinks himself called upon to assist in this birth of time, let me ask him whether of all the conditions of such peace, the first is not that religious thought should abandon that extravagant absoluteness of assertion which is proper to the state of intellectual infancy, but which it has so long been too timid to let go? This pragmatical and unneeded absoluteness it is which is most deeply contrary to the method, the results, and the whole spirit of science; and no error can be greater than to fancy that science, or scientific men, rest upon it or readily tolerate it.