Volume 3 / Part 4 — THE MONIST. (continued 2)

Note that the necessity of the means has reference to an end still to be attained, and in so far itself hypothetical or contingent, while the contingent circumstances are no longer needed precisely because they have resulted in a definite outcome (which, accordingly, is now a fact, and, in that sense, necessary) and we begin to see how completely necessity and chance are bound up with each other.

Their correlation may thus be stated: If we are to reach an end we must take certain means; while so far as we want an undefined end, an end in general, conditions which accompany it are mere accidents. Whichever way the relationship be stated, the underlying truth is that we are dealing with only partial phases of fact, which, having been unduly separated from each other through their erection into distinct wholes, have now to be brought back into their real unity.

In the first place, then, if I am to reach an end, certain means must be used. Here the end is obviously postulated; save as it is begged (presupposed), the necessity of the means has no sense. If, when starving, I am to live I must steal a dinner, but, having stolen, the logical but unsympathetic judge may question the relevancy (that is, the necessity) of my end, and thus cut the ground out from under the necessity of my means. My end requires its justification, the establishing of its validity, before the necessity of the means is anything more than hypothetical. The proximate end must be referred to a more ultimate and inclusive end to get any solid ground. Here we have our choice: we may deny the existence of any organic whole in life and keep chasing in a never-ending series, the progressus ad infinitum, after an end valid in itself. In this case we never get beyond a hypothetical necessity—something is necessary if we are to have something else, the necessity being relative to the implied doubt. Or, being convinced that life is a whole and not a series merely, we may say there is one comprehensive end which gives its own validity to the lesser ends in so far as they constitute it. While, on the other alternative, we reach only a hypothetical necessity, on this we reach none at all. The comprehensive end is no end at all in the sense of something by itself to be reached by means external to it. Any such end would be simply one in the infinite series and would be itself hypothetical. Whenever minor ends cease to be in turn means to further ends it is because they have become parts, constituent elements, of the higher end and thus ceased to be steps towards an end and beyond and outside of themselves. Given a final (i. e., inclusive) end, eating and drinking, study and gossip, play and business, cease to be means towards an end and become its concrete definition, its analytic content. The minor activities state the supreme activity in its specific factors.

Our dilemma is the choice between an end which itself has no existence save upon presupposition of another end, (is contingent,) and an end which as an end in itself simply is.

The externality of means to end is merely a symptom of lack of specification or concreteness in the end itself. If I am going to invent some improvement in a type-writer, the necessity of going through certain preliminary steps is exactly proportionate to the indefiniteness of my conception of what the improvement is to be; when the end is realised, the operations which enter into the realisation cease to be means necessary to an end and become the specific content of that end. The improvement is a fact, having such and such elements defining it. If I simply want, in general, to get my mail I must take this path (there being but one road); but if my end is not thus general, if it is individualised with concrete filling, the walk to the office may become a part of the end, a part of the actual fact. In so far, of course, it loses all aspect of necessitation. It simply is. And in general, so far as my end is vague, or abstract, so far as it is not specified as to its details, so far the filling up of its empty schema to give it particularity (and thus make it fact) appears as a means necessary to reach an end outside itself. The growth in concreteness of the end itself is transformed into ways of effecting an end already presupposed. Or, to state it in yet one other way, determination in the sense of definition in consciousness is hypostatised into determination in the sense of a physical making.

The point may come out more clearly if we consider it with the emphasis on chance instead of upon necessity. The usual statement that chance is relative to ignorance seems to me to convey the truth though not in the sense generally intended—viz., that if we knew more about the occurrence we should see it necessitated by its conditions. Chance is relative to ignorance in the sense rather that it refers to an indefiniteness in our conception of what we are doing. In our consciousness of our end (our acts) we are always making impossible abstractions; we break off certain phases of the act which are of chief interest to us, without any regard to whether the concrete conditions of action—that is, the deed in its whole definition—permits any such division. Then, when in our actual doing the circumstances to which we have not attended thrust themselves into consciousness—when, that is to say, the act appears in more of its own specific nature—we dispose of those events, foreign to our conscious purpose, as accidental; we did not want them or intend them—what more proof of their accidental character is needed? The falling of a stone upon a man’s head as he walks under a window is “chance,” for it has nothing to do with what the man proposed to do, it is no part of his conception of that walk. To an enemy who takes that means of killing him, it is anything but an accident, being involved in his conscious purpose. It is “chance” when we throw a two and a six; for the concreteness of the act falls outside of the content of our intention. We intended a throw, some throw, and in so far the result is not accidental, but this special result, being irrelevant to our conception of what we were to do, in so far is contingent. The vagueness or lack of determinateness in our end, the irrelevancy of actual end to conscious intent, chance, are all names for the same thing. And if I am asked whether a gambler who has a hundred dollars upon the outcome does not intend to throw double sixes, I reply that he has no such intention—unless the dice are loaded. He may hope to make that throw, but he cannot intend it save as he can define that act—tell how to do it, tell, that is, just what the act is. Or, once more, if I intend to get my mail and there are four paths open to me it is chance which I take, just in proportion to the abstractness of my end. If I have not defined it beyond the mere “general result” of getting mail, anything else is extraneous and in so far contingent. If the end is individualised to the extent, say, of getting the mail in the shortest possible time, or with the maximum of pleasant surroundings, or with the maximum of healthy exercise, the indifferency of the “means,” and with it their contingency, disappears. This or that path is no longer a mere means which may be taken to get a result foreign to its own value; the path is an intrinsic part of the end.

In so far as a man presents to himself an end in general, he sets up an abstraction so far lacking in detail as (taken per se) to exclude the possibility of realisation. In order to exist as concrete or individual (and of course, nothing can exist except as individual or concrete) it must be defined or particularised. But so far as consciousness is concerned the original vague end is the reality; it is all that the man cares about and hence constitutes his act. The further particularisation of the end, therefore, instead of appearing as what it really is, viz., the discovery of the actual reality, presents itself as something outside that end. This externality to the end previously realised in consciousness is, taken as mere externality, contingency, or accident; taken as none the less so bound up with the desired end that it must be gone through before reaching that end, it is necessary. Chance, in other words, stands for the irrelevancy as the matter at first presents itself to consciousness; necessity is the required, but partial, negation of this irrelevancy. Let it be complete, instead of partial, and we have the one real activity defined throughout. With reference to this reality, conditions are neither accidental nor necessary, but simply constituting elements—they neither may be nor must be, but just are. What is irrelevant is now not simply indifferent; it is excluded, eliminated. What is relevant is no longer something required in order to get a result beyond itself; it is incorporated into the result, it is integral.

It now remains to connect the two parts of our discussion, the logical and the practical consideration of necessity, and show that, as suggested, logical necessity rests upon teleological—that, indeed, it is the teleological read backwards. The logical process of discovering and stating the reality of some event simply reverses the process which the mind goes through in setting up and realising an end. Instead of the killing of an enemy as something to be accomplished, we have the fact of a murder to be accounted for. Just as on the practical side, the end, as it first arises in consciousness, is an end in general and thus contrasts with the concrete end which is individualised; so the fact, as at first realised in consciousness, is a bare fact, and thus contrasts with the actual event with its complete particularisation. The actual fact, the murder as it really took place, is one thing; the fact as it stands in consciousness, the phases of the actual event which are picked out and put together, is another thing. The fact of knowledge, it is safe to say, is no fact at all; that is, if there had been in reality no more particularisation, no more of detail, than there is consciousness, the murder would never have happened. But just as, practically, we take the end in general to be the real thing, (since it is the only thing of any direct interest,) so in knowledge we take the bare fact as abstracted from the actual whole, as the fact. Just as the end of the savage is merely to kill his enemy, so the “fact” is merely the dead body with the weapon sticking in it. The fact, as it stands in consciousness, is indeterminate and partial, but, since it is in consciousness by itself, it is taken as a whole and as the certain thing. But as the abstractness of the “end in general” is confessed in the fact that means are required in order to make it real—to give it existence—so the unreal character of the “fact” is revealed in the statement that the causes which produced it are unknown and have to be discovered. The bare fact thus becomes a result to be accounted for: in this conception the two sides are combined; the “fact” is at once given a certain reality of its own while at the same time the lack of concreteness is recognised in the reference to external causes.

The gradual introduction of further factors, under the guise of causes accounting for the effect, defines the original vague “fact,” until, at last, when it is accounted for, we have before us the one and only concrete reality. This done, we no longer have an effect to be accounted for, and causes which produce it, but one fact whose statement or description is such and such. But intermediate between the isolation and the integration is the stage when necessity appears. We have advanced, we will suppose, from the bare fact of the murder to the discovery of a large amount of “circumstantial” evidence regarding that fact. We hear of a man who had a quarrel with the deceased; he cannot account for himself at the time when the murder must have been committed; he is found to have had a weapon like that with which the murder must have been committed. Finally we conclude he must have been the murderer. What do these “musts” (the “must” of the time, weapon, and murderer) mean? Are they not obviously the gradual filling-in of the previously empty judgment, through bringing things at first unconnected into relation with each other? The existence of the man M. N. is wholly isolated from the “fact” of the murder till it is learned that he had a grudge against the murdered man; this third fact, also distinct per se, brought into connection with the others (the “fact” of the murder and of the existence of M. N.) compels them to move together; the result is at first the possibility, later, as the points of connection get more and more marked and numerous, the “necessity,” that M. N. is the murderer. Further, it is clear that this “must” marks not a greater certainty or actuality than a mere “is” would indicate, but rather a doubt, a surmise or guess gradually gaining in certainty. When the fact is really made out to our satisfaction, we drop the “must” and fall back on the simple is. Only so long as there is room for doubt, and thus for argument do we state that the time and weapon must have been such and such. So when we finally conclude that the murderer must have been M. N., it means that we have woven a large number of facts, previously discrete, into such a state of inter-relationship that we do not see how to avoid denying their discreteness and incorporating them all into one concrete whole, or individual fact. That we still say “must” shows, however, that we have not quite succeeded in overcoming the partial and indefinite character of the original “fact.” Had we succeeded in getting the whole fact before us the judgment would take this form: The murder is a fact of such and such definite nature, having as its content such and such precise elements. In this comprehensive whole all distinction of effect to be accounted for and causes which produce clean disappears. The idea of necessity, in a word, comes in only while we are still engaged in correcting our original error, but have not surrendered it root and branch; this error being that the fragment of reality which we grasp is concrete enough to warrant the appellation “fact.”

A great deal of attention has been directed to the category of cause and effect. One striking feature of the ordinary consideration is, that it takes for granted the matter most needing investigation and aims the inquiry at the dependent member of the firm. The effect seems to be so clearly there, while the cause is so obviously something to be searched for that the category of effect is assumed, and it is supposed that only the idea of causation is in need of examination. And yet this abstraction of certain phases of fact, the erection of the parts thus abstracted into distinct entities, which, though distinct, are still dependent in their mode of existence, is precisely the point needing examination. It is but another instance of the supreme importance of our practical interests. The effect is the end, the practical outcome, which interests us; the search for causes is but the search for the means which would produce the result. We call it “means and end” when we set up a result to be reached in the future and set ourselves upon finding the causes which put the desired end in our hands; we call it “cause and effect” when the “result” is given, and the search for means is a regressive one. In either case the separation of one side from the other, of cause from effect, of means from end, has the same origin: a partial and vague idea of the whole fact, together with the habit of taking this part (because of its superior practical importance) for a whole, for a fact.

I hope now to have made good my original thesis: that the idea of necessity marks a certain stage in the development of judgment; that it refers to a residuum, in our judgments and thus in our objects, of indeterminateness or vagueness, which it replaces without wholly negating; that it is thus relative to “chance” or contingency; that its value consists wholly in the impulse given judgment towards the is, or the concrete reality defined throughout. The analysis has been long; the reader may have found it not only tedious, but seemingly superfluous, since, as he may be saying to himself, no one nowadays regards necessity as anything but a name for fixed uniformities in nature, and of this view of the case nothing has been said. I hope, however, that when we come to a consideration of necessity as equivalent to uniformity, it will be found that the course of this discussion has not been irrelevant, but the sure basis for going further.

John Dewey.

FOOTNOTES:

THE ISSUES OF “SYNECHISM.”

In a late number of The Monist, (Vol. II, No. 4,) there appears a singularly acute and profound article, from the pen of one of the ablest of American logicians and mathematicians, Mr. Charles S. Peirce. Its subject is “The Law of Mind”—the idea of continuity. The writer tells us, (p. 534,) “the tendency to regard continuity, in the sense in which I shall define it, as an idea of prime importance in philosophy, may conveniently be termed Synechism.” With this synechistic philosophy, as applied to mind, the paper is occupied, to the exclusion, for the nonce, of Mr. Peirce’s companion doctrine of Tychism,⁠[67] which was dealt with, by him, in the January, 1891, and April, 1892, issues of The Monist. These conceptions are, both of them, to be viewed as essential to philosophy as a whole, but the latter is; for the present, allowed to drop out of sight, in order to allow of the due elaboration of the former.⁠[68]

THE FORMULA OF SYNECHISM.

The formula of Synechism, with which the article begins, is as follows:

“Logical analysis applied to mental phenomena shows that there is but one law of mind, namely, that ideas tend to spread continuously, and to affect certain others which stand to them in a peculiar relation of affectibility. In this spreading they lose intensity, and especially the power of affecting others, but gain generality, and become welded with other ideas.” (Vol. II, No. 4, p. 534.)

The individuality and continuity of ideas are, then, shown respectively to involve no contradiction; an idea once past—in the sense of an event in an individual consciousness—is not wholly past, it is only going—“infinitesimally past, less past than any assignable past date.” Thus the conclusion is reached that “the present is connected with the past by a series of real, infinitesimal steps.” Again, “We are forced to say that we are immediately conscious through an infinitesimal interval of time. This is all that is requisite.” (Ibid., pp. 535-536.)

All that it is necessary to say at the outset is, that this view is supported by an elaborate inquiry into the nature of infinity and continuity in general, into which, for the purpose of the present paper, it is not needful to enter. And this for two reasons: (1) The synechistic philosophy, by itself, does not profess to be monistic. Its expounder does not, even if his Tychism were not in reserve, profess to carry it beyond the realm of mind, with all that is implied in such a reservation. Now, it is the bearing of Mr. Peirce’s Synechism upon a monistic solution of the universe with which the present article is concerned. And (2) Mr. Peirce’s method of treatment, though precise and logical in the direction of its own path, is too purely technical to be summarised for the general reader’s benefit. But withal, Synechism is far too fertile, not so much in respect of what it makes clear, as suggestively, and, if the expression may be allowed, obliquely, to be passed over without comment. Its excogitator is eminently frank; he does not conceal the difficulties which, ever and anon, occur in his statement. Sometimes his theory seems a trifle too wide for the facts encountered, sometimes rather too scanty to contain them. Such phrases as the following: “No, I think we can only hold”—p. 552; “we are driven to perceive”—p. 555; “this obliges me to say”—p. 557; “the principle with which I set out requires me to maintain”—p. 558; “the only answer that I can, at present, make is”—p. 559, etc., etc., do every credit to the writer’s candor, but they would scarcely occur in an exposition, which, in the mind of its author, made the rough places altogether plain. Synechism, even with Tychism in the background, probably does not, in Mr. Peirce’s own mind, completely solve the world-riddle, at least, as yet. Still these very pauses themselves, on the part of a thinker of such ability, are eminently suggestive. To use his own words: “the present paper is intended to show what Synechism is, and what it leads to.” Let us emphasise this latter clause, as likely to be more fruitful than the former.

MR. PEIRCE’S POSITIVISM.

Mr. Peirce, in spite of his theory of chance, is, in his Synechism, almost severely a positivist;⁠[69] but his positivism, like most of that current nowadays, does not go deep enough. He is positivist, after he has got externality—fertile in excitations—comfortably disposed around his subject; and vibrations, undulations, attractions, etc., ready to play upon the thousand-stringed harp, but not before. For, “we must not tax introspection,” he tells us, p. 548, “to make a phenomenon manifest, which essentially involves externality,” when the real problem at issue is: Is there externality, in the vulgar sense, at all, or is it only that rationalised externality which circumspection, within the limits of egoity, reveals? Now, upon this a good deal hinges. At all events the difference in question, or, rather, that there is a difference, has been mooted, to say the least. And, this being the case, it is a little tedious, when the really vital point of the spatial extension of feelings is being debated, to have this illustration brought in, (p. 548,): “Moreover, our own feelings are focused in attention to such a degree, that we are not aware that ideas are not brought to an absolute unity. Just as nobody, not instructed by special experiment, has any idea how very, very little of the field of vision is distinct.” Why, that is reasoning in a circle, if some systems are true; and it is a begging of the question, if they are the reverse.

If the system of so-called objective reality were, at sight, wholly veracious, if everything existed just as it seems, this positivism of Mr. Peirce’s might be workable. Then no one would seek to go beneath the process of the apparent, the actually visible, for a rationale. But modern science teaches, in its very primer, that many things are, and act, quite otherwise than as they seem to be, and do. Appearances rationalised are alone to be accepted. The sun does not “rise” and “set,” as it seems to do. The earth is not, as it appears to be, an immovable plane, and so on. And, this once allowed, where is the principle to end? If the superficial judgment may be thus corrected, or reversed, it is liable to revision or reversal ad infinitum, unless reason be shown to the contrary. It may thus be disputed whether our author is quite in order in writing, as he does, and using the statement to support his theory—“Precisely how primary sensations, as colors and tones, are excited, we cannot tell, in the present state of psychology.... As far as sight and hearing are in question, we know that they are only excited by vibrations of inconceivable complexity; and the chemical senses are probably not more simple.” (P. 557.)

To argue, we cannot tell precisely how they are excited, but we know that they are excited, is somewhat feminine; seeing that the said “excitement” is not patent on the surface of ordinary perception. And, this being the case, the excitement, or its mode rather, not being given immediately, but only mentally annexed, Mr. Peirce is not consistently positivist. It is equally open to an opponent to “annex” something else of his own to the “given” thing, or altogether to deny the necessity of anything whatever being thus annexed. In any case that (if anything) which is sought to be annexed must stand the test of positivism; we must know if such a thing is, and what it is precisely. And this is just what Mr. Peirce cannot do for us. He cannot tell us exactly what the “excitant” of feelings is; he can only guess what it is “something like,” viz.: the feelings themselves. Hence the following:

“The principle with which I set out [that of continuity] requires me to maintain that these feelings are communicated to the nerves by continuity, so that there must be something like them in the excitants themselves. If this seems extravagant, it is to be remembered that it is the sole possible way of reaching any explanation of sensation, which otherwise must be pronounced a general fact absolutely inexplicable and ultimate. Now absolute inexplicability is a hypothesis which sound logic refuses, under any circumstances, to justify.” (P. 558.—The italics are not in the original.)

There must be something like the feelings in the excitants of the feelings. Now, this point is worthy of the closest attention. Note that “the excitant” alone is mentioned. Vibrations excite sight and hearing. Yet, from what follows, it is plain that Synechism is not inconsistent with belief in a fixed objective. “Even the least psychical of peripheral sensations, that of pressure, has, in its excitation, conditions which, though apparently simple, are seen to be complicated enough when we consider the molecules and their attractions,” pp. 557-558. Can there, then, be any doubt that we have here three distinct things: (1) a subjective, (2) an “excitant,” and (3) an objective; the middle term being a vehicle of communication between the first and third? It does not affect this presentation of Mr. Peirce’s position that, at an earlier stage of his argument, he speaks of matter—synonymous, presumably, with the objective—as being “not completely dead, but merely mind, hide-bound with habits,” as “partially deadened” or “effete,” mind; or that the editor of The Monist says that, with Mr. Peirce, “mind is the beginning of all.” (The Monist, Vol. III, No. 1, p. 95.) The question, at present, is not regarding origins, but regarding co-existences. So that there is a distinct hiatus here, arising from the confusion of the stimulant, or excitant, of sensation with the objective itself.⁠[70] Now, the stimulant of sensation is never the object perceived. Hence, once an objective is admitted, a trinity of entities is unavoidable, since still less can the “stimulant” be the subject. This special difficulty, in the present writer’s opinion, is inseparable from dualism in every form. How it besets Mr. Peirce’s theory is evident from his hazarded suggestion: “There must be something like the feelings in the excitants.” He thus uses only two of his cosmical terms, and gives the third the go-by! All dualism halts, but surely there is here a palpable stumble.

In a recent article in The Open Court⁠[71] I have pointed out the vanity of introducing a vehicle of communication between object and subject, especially emphasising the fact that, once this intermediate term is brought in, the veritable objective disappears. “Once you bring in vibrations,” I remarked, “you practically provide a second object, which is really a part of the subject, and, in order to do this, you have taken from the original objective all that composed it.”⁠[72] (The Open Court, p. 3361.)

Is it any wonder, then, that Mr. Peirce should suppose the excitants to be “something like” the excited feelings? Since he, practically, surrenders the objective, what could more closely resemble the subjective than the subjective itself? If he had adopted the position of Hume, and made impressions and ideas all-in-all, his principle of continuity might hold. But this he does not do, since (1) he implicitly admits the objective element, and (2) even if he did not do this, there must be something other than the idea or feeling in his system, since, otherwise, there could be no ground for the charge of seeming “extravagance,” which, he admits, may be leveled against, at least one of, his conclusions.

FEELINGS SPATIALLY EXTENDED.

This leads us to Mr. Peirce’s conclusions regarding subjective spatial extension—the spatial extension of feelings—as the result of observation of irritated protoplasm. Our attention is directed to an excited mass of protoplasm,—an amœba, or a slime-mould,—which “does not differ in any radical way from the contents of a nerve-cell, though its functions may be less specialised.” (P. 547.) The irritation is induced when, say, the amœba is “quiescent and rigid,” and we note its behaviour under it. That feeling passes from one part of this amorphous continuum of protoplasm to another, we are led to believe. And this conclusion follows: “Whatever there is in the whole phenomenon to make us think there is feeling in such a mass of protoplasm,—feeling, but plainly no personality,—goes logically to show that that feeling has a subjective, or substantial, spatial extension, as the excited state has.” This is a chain of reasoning. Let us examine its links. We have:

(1) The behaviour of the amœba under immediate, mechanical irritation—the spread, or spatial extension, of the state of irritation.

(2) We are asked to identify this spread-out irritation, this field of excitation, with “feeling” on the part of the amœba, because there is “no doubt that it feels when it is excited.”

(3) From the spatial extension of the irritation, thus identified with feeling, we are asked to conclude that the feeling, in the amœba, has a subjective, spatial extension as the excited state has, and, finally, passing from the feeling of the amœba to our own feelings, by inference, we are asked to admit:

(4) Not that we have necessarily a feeling of bigness, but that “the feeling [inferentially arrived at from the spread-out irritation on the part of the amœba] as a subject of inhesion is big.” (P. 548.)

After this, we are disposed to agree with Mr. Peirce when he says: “This is, no doubt, a difficult idea to seize”; not, as he goes on to say, “for the reason that it is a subjective, not an objective, extension,” but on the ground that the reasoning involves, plainly, not only the subjective and objective, but what Clifford calls the “ejective,” as well, and this assumption, inter alia, that the last-named lies on the same plane as the former. Never, surely, was the conclusion that feelings have spatial extension more easily reached. It is only when we find that in (1) we are dealing with the objective pure and simple, observed phenomena; that in (2) the connection between irritation or excitation, and feeling is assumed, in the object, because feeling, subjectively, is found to accompany irritation; that (3) as the irritation, in the amœba, is spread out, so is the feeling to be viewed; and (4) that, as the feeling of the amœba, so is our feeling to be considered, viz.: that the feeling, “as a subject of inhesion, is big,” we are led to say after all this, that, by such a process, anything, or everything, could be demonstrated,—the field of spatial extension, for example, having no more claim to be assumed than the point at which the irritation admittedly begins. Why should the middle stage of the irritation be selected in preference to the initial and final ones? The irritation originates in a point, spreads, and then dies out. Thus our feeling, (we purposely use Mr. Peirce’s nomenclature,) or idea, of an elephant, is unquestionably, as a subject of inhesion, “big.” But only for a time, and not at first. Really, our idea, or feeling—in Synechism—of an elephant, must logically commence as a minute speck, and return to this vanishing-point again. There is no other way out of it. For must not the analogy of the irritated amœba be followed throughout, and if not, why not?

DUALISM AND THE WAY OUT.

The crux of philosophy, from the time of Hume to the present day, has been, what may be summarised as, the consciousness of succession as succession. The hours pass over the mental dial, but, though one succeeds the other, something is needed besides the succession of the terms of the series to give consciousness of the series as a series, to give the synthesis of the day made up of hours. Hume virtually gave up the problem in eviscerating the subjective. Prof. T. H. Green only missed the point at issue when he placed his eternal consciousness, which was to “have and to hold” the terms of the cosmical series, as it were in solution, for the human organism, out of time altogether. Mr. Peirce puts the matter boldly when he says: “An idea once past is gone forever, [in the sense of an event in an individual consciousness,] and any supposed recurrence of it is another idea.” (P. 534.) In order, then, that an idea past may be present really, and not vicariously, the notion that consciousness necessarily occupies an interval of (finite) time must be given up; since, to put it briefly, a second past is as much past as a year. According to Mr. Peirce then, and his contention is supported by an elaborate inquiry into the nature of infinity and continuity generally, “we are immediately conscious through an infinitesimal interval of time.” For the complete rationale, reference must necessarily be made to the article itself.

Even the above outline, however, is sufficient to show that, here as elsewhere, Mr. Peirce’s dualism is his snare. Nothing but this could lead to a disintegration so complete as the following:

“In this infinitesimal interval, not only is consciousness continuous in a subjective sense, that is, considered as a subject, or substance, having the attribute of duration; but also, because it is immediate consciousness, its object is ipso facto continuous.” (P. 536.)

This is to admit, practically, that there is something in consciousness other than the consciousness itself. And this is evident, because at one and the same time, (whether an interval of finite time, or an infinitesimal interval,—whether an “instant” or a “moment,”—does not matter,) these two entities are different. For:

“This mediate perception is objectively, or as to the object represented, spread over the four instants; but subjectively, or as itself the subject of duration, it is completely embraced in the second moment.” (Ibid.)

But this “mediate” and “immediate” cannot simultaneously exist, unless there is something else to which they do so exist. It is only paltering with us in a double sense to speak of “instant” and “moment” in this connection. The one may pass into the other, but there is “a time when” (it matters not whether the interval be finite or infinitesimal) they do not coexist. Hence, they are not the same, but different.

According to Mr. Peirce’s notation, for all ordinary purposes we may write, if a is a finite quantity, and i an infinitesimal, a + i = a. “That is to say, this is so for all purposes of measurement.” Be it so; the infinitesimal may be neglected for purposes of calculation. But such a formula can only be experimental. The theory which embodies it cannot avail for a world-scheme; to admit it would be to grant that a thing is, and is not, at one and the same time. Surely the most superficial reader will see that, to put it popularly, a world-scheme admits of no alternative subject to accept, or to reject, a neglectable quantity.

And this is not the only instance of dualism in Mr. Peirce’s world-scheme as a totality. For have we not Synechism and Tychism as well? With the latter Mr. Peirce does not deal in the paper now under consideration. He must, however, be credited, or debited, with it, as held in reserve. For our present purpose it is not necessary to examine Tychism in detail. Its alleged existence is sufficient. For, and here let the significance of what follows be noted, in Mr. Peirce’s view, as opposed to determinism, Tychism exists as a principle. It is, otherwise it could not be expounded as operative. But it also exists as an idea, first, it may be, in our author’s mind, and subsequently in the minds of his disciples. Thus it falls into the synechistic province: “As an idea it can only be affected by an idea, by anything but an idea it cannot be affected at all.” (“The Law of Mind,” p. 557.) Yet to affirm Tychism thus impotent, because unaffectible, outside the synechistic sphere, is to contradict Mr. Peirce’s conclusions, for if Tychism is nothing outside the ideal realm, it is altogether inside it. Hence Synechism is everything practically, and Tychism nothing. But that Mr. Peirce will not have. He has a two-fold Tychism, that is the fact; actual and operative on the one hand, ideal on the other. And this is dualism confessed.

Mr. Peirce’s method is quite fertile in duplication of the subjective entity. His latest paper, “Man’s Glassy Essence,” (The Monist, Vol. III, No. 1,) contains some typical instances.

“Viewing a thing from the outside, considering its relations of action and reaction with other things, it appears as matter. Viewing it from the inside, looking at its immediate character as feeling, it appears as consciousness.” (P. 20.)

This is the strictly empirical view. And it may be possibly defended with the contention that all problems, to be duly examined, must, in the first place, be viewed from that standpoint. But it must be plainly manifest to any unprejudiced thinker that, even granted a total cosmical problem made up of separate problems of an individual nature, the same method of solving the sum cannot be employed which is used in solving its constituents. In the above instance, considering matter in its totality, and consciousness in its totality, what is left to view them indifferently from “outside,” or “inside”? Plainly nothing. Still more transparent an example is the following:

“The consciousness of a habit involves a general idea. In each action of that habit certain atoms get thrown out of their orbit, and replaced by others. Upon all the different occasions it is different atoms that are thrown off, but they are analogous from a physical point of view, and there is an inward sense of their being analogous. Every time one of the associated feelings recurs, there is a more or less vague sense that there are others, that it has a general character, and of about what this general character is.” (P. 20.)

This is part of the answer to the query: How do general ideas appear in the molecular theory of protoplasm? Now, without discussing the value of this rationale, as affecting Mr. Peirce’s own theories, it is not difficult to see what its acceptance would “lead to.” Certain atoms of a molecule get thrown out and are replaced by others. This happens repeatedly. On different occasions different atoms come and go. Yet they are “analogous,” and there is “an inward sense” of this. Upon whose shoulders is the burden of proving the analogy placed, or of experiencing it even? With whom or what is there “an inward sense”? Perhaps it is better not to answer otherwise than to say that if this faculty be not present in the ever changing molecule to begin with, it cannot be logically reached by any process of multiplying it.

THE MONISTIC SOLUTION.

Monism, as a unitary system of the universe, does not necessarily commend itself to acceptance simply as monism. To say, this is dualism, therefore it cannot be a correct rationale of the universe, since the only true one must be monistic, is to start with an unphilosophical prepossession. The true solution may be two-fold, or it may be manifold. But it is not too much to say, perhaps, on the other hand, that, even as causes may not be multiplied without necessity, even so phenomena must not logically be divided into independent groupings without sufficient reason given. Preference should be accorded to a monistic, rather than to a dualistic, system, not on the ground alone of the simplicity of the former, but on the ground that a theory which has one explanation for one set of phenomena and another explanation for a second set, must first demonstrate that a unitary conception of the universe is, at least, improbable, otherwise it will always be hinted that the dualism in question has not gone deep enough to find a synthetic bond wherewith to unite the apparently diverse. Mr. Peirce, throughout his article on Synechism, constantly touches, despite his latent dualism, the margin of a truth so great as to merit the title of transcendent. As often he misses it. And his concluding words are, in this connection, almost wistful: “The facts that stand before our face and eyes and stare us in the face, are far from being, in all cases, the ones most easily discerned. That has been remarked from time immemorial.” (P. 559.) But though thus “remarked,” the maxim has, as immemorially, been neglected in practice. To none can this remark be more fitly applied than to the excogitator of Synechism, himself seeing that, having arrived at the point of asserting that “there must be something like the feelings in the excitants themselves,” he does not see that the excitant and the feeling are one and the same; and that there is no second or third term in the cosmical equation.

Does this seem “extravagant”? If so, the reply must be not that it is the only escape from an otherwise inexplicable difficulty, but that there is really no difficulty at all. What Mr. Peirce’s own Synechism “leads to” is that the past, the present, and the to-come, alike of matter and idea, are not reconciled by “time and its flow,” or even by the logic of infinitesimals, subtle though that may be, but that the contents of each and all, with all their apparently infinite variety, resolve into a consistent unity.

THE “MISSING LINK.”

Pushed to a logical conclusion, the excitants and the feelings owe their apparent variety to their assigned position in a series, the correspondence or relation between them being only another link in the self-same chain. Vulgar realism never fathoms this explanation. It always harps upon the one string that idealism, and more especially idealistic monism, fails to account for variety or difference; forgetting, or rather never seeing, that difference or variety which is its essence, is only one more added perception on the same plane with ordinary perceptions; so that given a, b, c, d,—sundry perceptions,—their essential variety may be stated as e. Or this may be stated numerically; variety, as a whole, being nothing more than the sum of differences, which is always something other than the terms differentiated, but always on the same level with them—the difference between any continuous number, above unity, and another number being a third number, which is different from either. Variety in numbers cannot be expressed otherwise than numerically. So, in the last recess, the variety of colors is only colorable, of tones audible, and so on. The “vibrations of inconceivable complexity” which, according to Mr. Peirce, “excite sight and hearing,” can be approximately stated numerically, so that the difference between red and, say, yellow, is a number corresponding to another color, which may be orange or not; it being part of the present scientific theory of light that any specific number of ethereal undulations happening between the colors of the ocular spectrum, corresponds to a possible color, although the retinal expanse may be insensible to these particular rates of tremor. To Mr. Peirce it may appear “extravagant,” but the difference between any two colors and tones is another color, another tone; just as the difference between any two numbers is a third number. This is the logical outcome of his own Synechism; this, in part, is what it “leads to.”

TIME AND ITS “FLOW” RATIONALISED.

Excitants and feelings being unified, and the element of variety, hitherto supposed to be the exclusive copyright of vulgar realism, shown to be nothing but another term added to the series, or, numerically, a concurrent series—so that should a, b, c, d ... be a series, the variety of the series may be expressed as e, or the individual differences as f, g, h ...—it only needs an examination of what Mr. Peirce terms “time and its flow,” to render his system a completely monistic one, and this although true monism is much more than the negation of determinism, synechistically expressed.

In Mr. Peirce’s article under examination, “The Law of Mind,” the notation of infinitesimals, which forms the keystone of Synechism, is only introduced after a lament over the incapacity, or unworkableness rather, of finite time, when the duration of consciousness is involved. If finite time is to come in as a factor—“an idea once past [in the sense of an event in an individual consciousness] is gone forever, and any supposed recurrence of it is another idea” (p. 534). And the problem which Mr. Peirce sets himself to solve is how in effect to bring back this past idea—not vicariously—but in all its pristine freshness, into the now-time. This is sought to be accomplished by the explanation that the past idea is “not wholly past, it is only going, less past than any assignable past date”—and so on through the intricacies of Mr. Peirce’s infinitesimal theory, into which we need not enter at present. But the statement of the, supposed, difficulty which finite time presents in this connection,—the past idea really past and gone, and the recurrence of it another idea,—if put in a slightly different form, hints a solution, in continuity with the foregoing pages, without the aid of the infinitesimal at all. That an idea is once past and gone, any occurrence, or recurrence, of this idea, is another idea.⁠[73]

But, in the meantime, let us see what Mr. Peirce has to say regarding “time and its flow”:

“One of the most marked features about the law of mind is, that it makes time to have a definite direction of flow from past to future. The relation of past to future is, in reference to the law of mind, different from the relation of future to past. This makes one of the great contrasts between the law of mind and the law of physical force, where there is no more distinction between the two opposite directions in time than between moving northward, and moving southward” (p. 546).

This for once is not very clear. It is difficult to see how “the law of physical force” can be spoken of as “in time,” to the exclusion of mind; not easy, also, to understand the distinction further insisted upon. But the intention is evident, viz., to perpetuate, if not to originate, a cosmical duality. Time, it would seem, marches indifferently in at least two directions, though it is not very clear how this is accomplished. And then the old fiction follows, that “Time, as the universal form of change, cannot exist unless there is something to undergo change, etc.” (p. 547.)

The same notation suits in this case as in the foregoing. Time is only another term in the series. If a, b, c, d be a series, e is the variety, f the whole time involved, and g the individual intervals. Of course all this is not a simple series, it is an infinitely complicated one; the above arrangement is only intended to show that difference, variety, time, etc., are no mysterious entities pervading events, acting as their “form” or carrying them in their “flow,” but simply percepts, or concepts, on a level with others.

This is not patent on the surface, it may be. Time has the appearance of a current in which events float. But this is an illusion dispelled by examination. Events cannot be submerged in time. Time cannot be the vehicle of events. It is impossible to conceive time as existing simultaneously with an event. It always follows it. What to Mr. Peirce appears as a “flow,” arises from the foregoing. Take events, percepts, or concepts, as a hypothetical series, a, b, c, d ... and their times as a´, b´, c´, d´ ... the first series contains the event per se, or as happening; the “time when” is contained in the second series, practically inseparable from the first, but the time when necessarily follows—consequently if the first be a, the second must be, at least a. But no concept or percept is abstract, except the concept time itself, which, being unconnected, seems anywhere, and, like its fellow-abstract space, is spread out, to us, tri-dimensionally, as past, present, and to come. And, as in space the position is simply spectral,⁠[74] a question of perspective or adjustment, so, in time, the timal series is adjusted to the substantive idea. But this two-fold spectral succession breeds by comparative intensity (which is another complex series) the sense of a flow, where there is none, but only the idea of a flowing, which is another matter. Thus, the so-called “veil of the future” is no more a veil than it is a brickbat. It is simply the indeterminateness of an unconnected adjective—as if one should say, white—and the query arises, What is it that is white? When the noun is supplied you have something definite. Just so, when the future lapses into the present.

Thus there is never anything without, at least, these three additions: first, variety or difference; second, time; third, relation, spatial or otherwise. These are all terms in a series, or set of concurrent series. Nothing can be, practically, isolated, for everything runs in a series. But this is a much broader theory of continuity than that which Synechism affords.⁠[75] All apparent perplexities vanish. The difficulty no longer exists that to perceive a series we must hold it, as it were, in solution. Since other than series nothing is. Hence the cosmos is an illimitable series or complex of series. But inasmuch as the timal element (as also the spatial) occurs through the series having time-term and space-term resident within it, all difficulty in apprehending it as a series vanishes. The impracticability, if any, would be in viewing any term as isolated.

THE RESULT RE TYCHISM.

What a flood of light does such a system shed indirectly upon Tychism, since the controversy between the latter and determinism mainly hinges upon the “must be,” the imperative, as it were, of the series! It has been very ably pointed out by Dr. Carus in his article re Mr. Peirce’s “Onslaught on the Doctrine of Necessity” (The Monist, Vol. II, No. 4, pp. 573-4.) that the formula adopted by Mr. Peirce in his Tychism, “chance is first, law is second, the tendency of habits is third,” involves its author in the admission of a law in a system professing to be, in its inception at all events, chanceful and lawless. Mr. Peirce’s “Synechism” professes to be the law of mind. Parenthetically, however, it may be remarked, that the distinction as to law, and lawlessness or “chance,” narrows itself to the plane of one term more or less in a series, or even to less than that subordinate place. For, although, for convenience sake, and for facility of contrast, we have followed Mr. Peirce’s figure of a series, to show more clearly also to what his theory leads, it is nevertheless plain, that time and its accompanying relations being placed on their proper level, that of integral percepts and concepts, the figure of a series is simply a matter of convenience of arrangement. Certainly as the “time when” is necessarily annexed to every percept and concept the timal element may be said to follow, not to precede, its fellow-term. Really, however, they may be said to be simultaneous, since the timal refinements of finite, infinite, past, present, and future are each of them contained in a percept of its own.

EXTERNALITY A SERIAL TERM.

But if the timal element be independent as a separate percept, the spatial as another, and so on, it follows that, although the terms of the series may, as it were, run, though we cannot conceive them separated, or as, in practice, otherwise than as continuous in their flow, still, theoretically, a series or complex of series it is, and a series may be interrupted at any term. Thus externality itself being a spatial relation, is but one term more, non-essential in theory, to the term preceding. So that when the Neo-Kantians speak of the “constitution of the objective” it ought to be added that it is not only the content of the objective which is thus constituted by consciousness, but that externality, all that goes to make up what is termed “out-sidedness,” is constituted by consciousness also.

THE NOW-TIME.

“The present is half past, and half to come,” (p. 546) like the color of a curved boundary line on a particolored surface; i. e. “betwixt and between” the two. It is here that the theory of Synechism shows its chief defect. Up to this stage we have been dealing with ideas, feelings, a, b, c, d ... successively passing through a point of consciousness e. And the infinitesimal notation suits the required process fairly well. It is complicated enough, but it is ingenious, and at least plausible. Nothing up to this stage would lead us to suppose that any additional element was to be imported into the rationale which Mr. Peirce presents. As we have seen, finite time would not serve his purpose. By however minute a finite interval have a, b, c or d passed the point e, all chance of their recovery is hopeless. Well, we have recourse to infinitesimals, and find (to put it popularly, and not in Mr. Peirce’s technical terms) that a past the point of consciousness by an infinitesimal interval heralds b. So that e is simultaneously confronted with the disappearing form of the first and the appearing form of the second, and the same with b, and c, in turn, and so on. Thus the present, in the sense of ideas successively passing through consciousness, is half a and half b, then half b and half c, this infinitesimal gradation ultimately ensuring the presence of the whole series in the last “moment.”

But this will not avail with the concept time itself as distinguished from timed succession. That these two are separate with Mr. Peirce it is impossible to doubt. He says, e. g., “Time with its continuity logically involves some other kind of continuity than its own,” (p. 547) and speaks of “time and its flow,” and of “time as the universal form of change.” And it is confusing, to say the least, when we are shifted without warning from what is practically the perceptual to the conceptual region. Granted the ideas, the feelings, or what not, “gliding almost imperceptibly” (as did the late Mr. Bardell to another sphere) past the central point of consciousness, yet not wholly past, only going, less past than any assignable past date, granted this, the assertion is not consequently warranted that time itself, the present, as time, not as involving the succession of ideas, is “half past and half to come.” The ideas, the feelings, of which Mr. Peirce writes, successively pass through the stage of being thus half past and half to come, but that is by no means the same thing as saying that the present is half past, half to come, as Synechism avers. With our theory, as presented in the foregoing pages there is indeed no such difficulty, but Mr. Peirce, on the other hand, has elected to stand by infinitesimally measuring time, as applied to ideas etc., as separate from conceptual time, and must take the consequences of his decision. He says the present, not the present idea.

Now, in the concept time as a whole, in its entire range, a definite point may be selected—to the exclusion of other points—a point having position but not extension, as the present. Is it, then,—the present,—half past, half to come, as a timed idea is? Certainly not. There is nothing of the flow of a series in it. Further, this selection of the “now,” as a point, does not interfere with its permanence. “Nowness” may persist. And the moment it partook, even infinitesimally, of the character of the past or of the future, it would cease to be the present. In the case of a series of ideas in time the difficulty is to get them all in present solution, as it were, without detriment to their evident continuity, but the definition of the present as a point in time presents no such difficulty. The conditions are quite distinct. Yet regarding this time point—the present—Mr. Peirce assures us that it is “half past, half to come,” which is just that of which it is the precise negation, if words are to have any meaning.

Again, Mr. Peirce’s rationale shows, upon the face of it, that there is (1) finitely divisible time and, (2) time divided infinitesimally, for what finite time could not do, in that it had limitations, the infinitesimal notation readily accomplishes. In its ulterior consequences, this is somewhat unfortunate for Synechism, inasmuch as the consciousness of ideas in continuity being confined to the infinitesimal theory, where, it may be asked, is the place, in consciousness, for the succession of finite intervals? Consciousness must be practically doubled, so to speak, if it is to hold both of these together. This is what comes of making one’s world-scheme hang upon a mathematical subtlety—the subtlety in question partaking as a rule, more or less of the nature of an escape from the difficulties of the vulgar notation, the vulgar notation remains to be reckoned with, and both have to be credited to consciousness. As an instance of this take the following from Mr. Peirce’s late article,⁠[76] “Man’s Glassy Essence”—p. 15:

“In order that a sub-molecule of food may be thoroughly and firmly assimilated into a broken molecule of protoplasm, it is necessary not only that it should have precisely the right chemical composition, but also that it should be at precisely the right spot at the right time and should be moving in precisely the right direction with precisely the right velocity. If all these conditions are not fulfilled, ... it will be in special danger of being thrown out again” (The italics are not in the original).

Now here is a “time when” which can be exactly specified in accordance with the conditions. Certain results follow unless it is kept to. This is what Mr. Peirce would doubtless consider as a timed physical event, part and parcel of the regularity of matter, and yet an event which, in its own time and way, goes to account for both feeling and habit-taking—capable, therefore, of being stated in terms of finite time, as happening at a given instant, and neither before nor after it. But when this same molecule is, by virtue of keeping its appointment punctually, safely installed in feeling protoplasm, the succession of ideas, or feelings, of which, as subject, it is capable, obeys another rule—a given instant obtains no longer; it is the moment which is everything⁠[77]—a moment half its predecessor, half its successor. Even granted the function of the infinitesimal, this looks very much like a reduction to absurdity. For, if the above mentioned timed coalescence of the sub-molecule with the broken molecule were also a matter of subjective feeling, passed as process through a consciousness, the conclusion follows that the juncture of the molecules happens at two different times! There is no escape from this. Given the instant in the one case, the moment in the other, these two cannot possibly be the same point in time. The moment partakes, however insensibly, of the preceding and succeeding stages, the instant does not. Hence they are not the same but different times.

OTHERNESS.

The foregoing has a distinct bearing upon the question of “other selves” of which Mr. Peirce writes as follows:

“The recognition by one person of another’s personality takes place by means to some extent identical with the means by which he is conscious of his own personality. The idea of the second personality, which is as much as to say that second personality itself, enters within the field of direct consciousness of the first person, and is as immediately perceived as his ego, though less strongly. At the same time, the opposition between the two persons is perceived, so that the externality of the second is recognised.” (“The Law of Mind,” p. 558.)

This is the scheme of “otherness” which, in the case of the Neo-Kantians, particularly the French section, represented by M. Pillon, M. Renouvier, and others, has proved such a snare. To these thinkers, (as indeed to the late Prof. T. H. Green, of Oxford, though in a less degree,) the so-called external world lies in “other” thinking subjects—in “foreign centres of representations.” The free-trade doctrine has verily penetrated to the philosophic region—the wholesale admission of foreign wares to the detriment of home products. Why should I place the content of that so-called external world, which, external or internal, is my very own inalienably, in a centre of representation other than my own, thus making my cognition of it rest entirely upon the “ejective” plane? It is only when I discover, as I must sooner or later, that there is nothing in the report of an “outsider” (or in any number of them) beyond what I credit him or her with in my own consciousness; and that the outsider is on the same plane as other objects, it is only then that the mystification is cleared up. I do not cognise, or recognise, the external at second-hand. The “note” of otherness is simply another term more or less in the cosmical series.

It is, however, not only with the familiar “other selves” of ordinary life that we are confronted in Synechism. In the creed of animism

and Mr. Peirce speaks of “spiritual influences” (p. 559) as having at least no hindrance presented to them by his doctrine. But he has some other shadowy personalities at command, which, it must be confessed, are well calculated to give us pause. “There should be something like⁠[78] personal consciousness in bodies of men who are in intimate and intensely sympathetic communion.... None of us can fully realise which the minds of corporations are.... But the law of mind clearly points to the existence of such personalities.” It is probably true that the “minds of corporations,” must ever present an insoluble riddle of perversity to the suburban dweller, vexed with the mockery of paving and lighting. But we need not linger over this speculation, for there are other shades behind.

“If such a fact is capable of being made out anywhere it should be in the Church.... Surely a personality ought to have developed in that Church, in that ‘bride of Christ,’ as they call it.” (“Man’s Glassy Essence,” pp. 21-22.)

A PERSONAL CREATOR.

Bearing our ecclesiastical divisions in mind, it is difficult to conceive the unity of a “corporate personality” of this kind, but, to let that pass, it may be remarked that, when any one begins to imagine that there are others in the universe besides himself, he is not, as a rule, content with two or three companions of his solitude. They come in battalions. Thus, behind the other selves, corporate personalities and spiritual influences of Synechism, there looms a transcendent personality. “A genuine evolutionary philosophy,” we are told, “... is so far from being antagonistic to the idea of a personal Creator, that it is really inseparable from that idea.” And a philosophy of pseudo-evolutionism is “hostile to all hopes of personal relations to God.” (“The Law of Mind,” p. 557.)

Mr. Peirce thus assigns to his first cause a place in the continuum of ideas, and says that if there is a personal God we must have a direct perception of that person and “indeed be in personal communication with him.” The difficulty, he admits, is that if this be so, how is it possible that the existence of this being should ever have been doubted by anybody. And the only answer he can at present make is, that “facts that stand before our face and eyes, and stare us in the face, are far from being in all cases the ones most easily discerned. That,” he adds, “has been remarked from time immemorial.” (“The Law of Mind,” pp. 558-559.)

One of the ablest of living philosophical writers, Professor Veitch, of Glasgow University, puts it somewhat similarly, though with his own realistic coloring, when he says:

“God, if at all, must rise above the line of finite regress; He cannot be a cause in that; He cannot be a cause dependent on another cause; He must be somewhere, or at some point, in the line of an otherwise endless scientific regress, there, above it, yet related to it, and in it; otherwise He is nothing for us.” (“Knowing and Being,” p. 320.)

The parallelism is worth noting. Those views embody what has been the contention of the present writer throughout this paper, with this most notable difference: that no term of a series may thus transcend the series, or be other than on a level with the other terms, being itself only a term, a link, in the series itself. And with this falls forever the idea of a cause uncaused.

Yet am I not in the series? For all that is in the series is mine every percept, every concept; so that, “extravagant” as it may appear, it is I who am the series. In other words, the ego is the universe-synthesis, and the universe-synthesis the ego.

Is Mr. Peirce prepared to take the consequences of that which his Synechism leads to?

G. M. McCrie.

FOOTNOTES:

THE FOURTH DIMENSION. MATHEMATICAL AND SPIRITUALISTIC.

INTRODUCTORY.

The tendency to generalise long ago led mathematicians to extend the notion of three-dimensional space, which is the space of sensible representation, and to define aggregates of points, or spaces, of more than three dimensions, with the view of employing these definitions as useful means of investigation. They had no idea of requiring people to imagine four-dimensional things and worlds, and they were even still less remote from requiring of them to believe in the real existence of a four-dimensioned space. In the hands of mathematicians this extension of the notion of space was a mere means devised for the discovery and expression, by shorter and more convenient ways, of truths applicable to common geometry and to algebra operating with more than three unknown quantities. At this stage, however, the spiritualists came in, and coolly took possession of this private property of the mathematicians. They were in great perplexity as to where they should put the spirits of the dead. To give them a place in the world accessible to our senses was not exactly practicable. They were compelled, therefore, to look around after some terra incognita, which should oppose to the spirit of research inborn in humanity an insuperable barrier. The residence of the spirits had to be a place inaccessible to our senses and full of mystery to the mind. This property the four-dimensioned space of the mathematicians possessed. With an intellectual perversity which science has no idea of, these spiritualists boldly asserted, first, that the whole world was so situated in a four-dimensioned space as a plane might be situated in the space familiar to us, secondly, that the spirits of the dead lived in such a four-dimensioned space, thirdly, that these spirits could accordingly act upon the world and, consequently, upon the human beings resident in it, exactly as we three-dimensioned creatures can produce effects upon things that are two-dimensional; for example, such effects as that produced when we shatter a lamina of ice, and so influence some possibly existing two-dimensioned ice-world.

Since spiritualism, under the leadership of the Leipsic Professor Zöllner, thus proclaimed the existence of a four-dimensioned space, this notion, which the mathematicians are thoroughly master of,—for in all their operations with it, though they have forsaken the path of actual representability, they have never left that of the truth,—this notion has also passed into the heads of lay persons who have used it as a catchword, ordinarily without having any clear idea of what they or any one else mean by it. To clear up such ideas and to correct the wrong impressions of cultured people who have not a technical mathematical training, is the purpose of the following pages. A similar elucidation was aimed at in the tracts which Schlegel (Riemann, Berlin, 1888) and Cranz (Virchow-Holtzendorff’s Sammlung, Nos. 112 and 113) have published on the so-called fourth dimension. Both treatises possess indubitable merits, but their methods of presentation are in many respects too concise to give a lay mind any profound comprehension of the subject. The author, accordingly, has been able to add to the reflections which these excellent treatises offer, a great deal that appears to him necessary for a thorough explanation in the minds of non-mathematicians of the notion of the fourth dimension.

I. THE CONCEPT OF DIMENSION.

Many text-books of stereometry begin with the words: “Every body has three dimensions, length, breadth, and thickness.” If we should ask the author of a book of this description to tell us the length, breadth, and thickness of an apple, of a sponge, or of a cloud of tobacco smoke, he would be somewhat perplexed and would probably say, that the definition in question referred to something different. A cubical box, or some similar structure, whose angles are all right angles and whose bounding surfaces are consequently all rectangles is the only body of which it can at all be unmistakably asserted that there are three principal directions distinguishable in it, of which any one can be called the length, any other the breadth, and any third the thickness. We thus see that the notions of length, breadth, and thickness are not sufficiently clear and universal to enable us to derive from them any idea of what is meant when it is said that every body possesses three dimensions, or that the space of the world is three-dimensional.

This distinction may be made sharper and more evident by the following considerations: We have, let us suppose, a straight line on which a point is situated, and the problem is proposed to determine the position of the point on the line in an unequivocal manner. The simplest way to solve this is, to state how far the point is removed in the one or the other direction from some given fixed point; just as in a thermometer the position of the surface of the mercury is given by a statement of its distance in the direction of cold or heat from a predetermined fixed point—the point of freezing water. To state, therefore, the position of a point on a straight line, the sole datum necessary is a single number, for beforehand we have fixed upon some standard line, like the centimetre, and some definite point to which we give the value zero, and have also previously decided in what direction from the zero-point, points must be situated whose position is expressed by positive numbers, and also in what direction those must lie whose position is expressed by negative numbers. This last mentioned fact, that a single number is sufficient to determine the place of a point in a straight line, is the real reason why we attribute to the straight line or to any part of it a single dimension.

More generally, we call every totality or system, of infinitely numerous things, one-dimensional, in which one number is all that is requisite to determine and distinguish any particular one of these things amidst the entire totality. Thus, time is one-dimensional. We, as inhabitants of the earth, have naturally chosen as our unit of time, the period of the rotation of the earth about its axis, namely, the day, or a definite portion of a day. The zero-point of time is regarded in Christian countries as the year of the birth of Christ, and the positive direction of time is the time subsequent to the birth of Christ. These data fixed, all that is necessary to establish and distinguish any definite point of time amid the infinite totality of all the points of time, is a single number. Of course this number need not be a whole number, but may be made up of the sum of a whole number and a fraction in whose numerator and denominator we may have numbers as great as we please. We may, therefore, also say that the totality of all conceivable numerical magnitudes, or of only such as are greater than one definite number and smaller than some other definite number, is one-dimensional.

We shall add here a few additional examples of one-dimensioned magnitudes presented by geometry. First, the circumference of a circle is a one-dimensional magnitude, as is every curved line, whether it returns into itself or not. Further, the totality of all equilateral triangles which stand on the same base is one-dimensional, or the totality of all circles that can be described through two fixed points. Also, the totality of all conceivable cubes will be seen to be one-dimensional, provided they are distinguished, not with respect to position, but with respect to magnitude.

In conformity with the fundamental ideas by which we define the notion of a one-dimensional manifoldness, it will be seen that the attribute two-dimensional must be applied to all totalities of things in which two numbers are necessary (and sufficient) to distinguish any determinate individual thing amid the totality. The simplest two-dimensioned complex which we know of is the plane. To determine accurately the position of a point in a plane, the simplest way is to take two axes at right angles to each other, that is, fixed straight lines, and then to specify the distances by which the point in question is removed from each of these axes.

This method of determining the position of a point in a plane suggested to the celebrated philosopher and mathematician Descartes the fundamental idea of analytical geometry, a branch of mathematics in which by the simple artifice of ascribing to every point in a plane two numerical values, determined by its distances from the two axes above referred to, planimetrical considerations are transformed into algebraical. So, too, all kinds of curves that graphically represent the dependence of things on time, make use of the fact that the totality of the points in a plane is two-dimensional. For example, to represent in a graphical form the increase of the population of a city, we take a horizontal axis to represent the time, and a perpendicular one to represent the numbers which are the measures of the population. Any two lines, then, whose lengths practical considerations determine, are taken as the unit of time, which we may say is a year, and as the unit of population, which we will say is one thousand. Some definite year, say 1850, is fixed upon as the zero point. Then, from all the equally distant points on the horizontal axis, which points stand for the years, we proceed in directions parallel to the other axis, that is, in the perpendicular direction, just so much upwards as the numbers which stand for the population of that year require. The terminal points so reached, or the curve which runs through these terminal points, will then present a graphic picture of the rates of increase of the population of the town in the different years. The rectangular axes of Descartes are employed in a similar way for the construction of barometer curves, which specify for the different localities of a country the amount of variation of the atmospheric pressure during any period of time. Immediately next to the plane the surface of the earth will be recognised as a two-dimensional aggregate of points. In this case geographical latitude and longitude supply the two numbers that are requisite accurately to determine the position of a point. Also, the totality of all the possible straight lines that can be drawn through any point in space is two-dimensional, as we shall best understand if we picture to ourselves a plane which is cut in a point by each of these straight lines and then remember that by such a construction every point on the plane will belong to some one line and, vice versa, a line to every point, whence it follows that the totality of all the straight lines which pass through the point assigned are of the same dimensions as the totality of the points of the imagined plane.

The question might be asked, In what way and to what extent in this case is the specification of two numbers requisite and sufficient to determine amid all the rays which pass through the specified point a definite individual ray? To get a clear idea of the problem here involved, let us imagine the ray produced far into the heavens, where some quite definite point will correspond to it. Now, the position of a point in the heavens depends, as does the position of a point on all spherical surfaces, on two numbers. In the heavens these two numbers are ordinarily supplied by the two angles called altitude, or the distance above the plane of the horizon, and azimuth, or the angular distance between the circle on which the altitude is measured and the meridian of the observer. It will be seen thus that the totality of all the luminous rays that an eye, conceived as a point, can receive from the outer world is two-dimensional, and also that a luminous point emits a two-dimensional group of luminous rays. It will also be observed, in connection with this example, that the two-dimensional totality of all the rays that can be drawn through a point in space is something different from the totality of the rays that pass through a point but are required to lie in a given plane. Such a group of objects as the last-named one, is a one-dimensional totality.

Now that we have sufficiently discussed the attributes that are characteristic of one and two-dimensional aggregates, we may, without any further investigation of the subject, propose the following definition, that, generally, an n-dimensional totality of infinitely numerous things is such, with respect to which the specification of n numbers is necessary and sufficient to indicate a definite individual amid the totality of all the infinitely numerous individuals of the group.

Accordingly, the point-aggregate made up of the world-space which we inhabit, is a three-dimensional totality. To get true bearings in this space and to define any determinate point in it, we have therefore to lay through any point which we take as our zero-point three axes at right angles to each other, one running from right to left, one backwards and forwards, and one upwards and downwards. We then join each two of these axes by a plane and are enabled thus to specify the position of every point in space by the three perpendicular distances by which the point in question is removed in a positive or negative sense from these three planes. It is customary to denote the numbers which are the measures of these three distances by x, y, and z, the positive x, positive y, and positive z ordinarily being reckoned in the right hand, the forward, and the upward directions from the origin. If now, with direct reference to this fundamental axial system, any particular specification of x, y, and z be made, there will, by such an operation, be cut out and isolated from the three-dimensional manifoldness of all the points of space a totality of less dimensions. If, for example, z is equal to seven units or measures, this is equivalent to a statement that only the two-dimensional totality of the points is meant, which constitute the plane that can be laid at right angles to the upward-passing z-axis at a distance of seven measures from the zero-point. Consequently, every imaginable equation between x, y, and z isolates and defines a two-dimensional aggregate of points. If two different equations obtain between x, y, and z, two such two-dimensional totalities will be isolated from among all the points of space. But as these last must have some one-dimensional totality in common, we may say that the co-existence of two equations between x, y, and z defines a one-dimensional totality of points, that is to say a straight line, a line curved in a plane, or even, perhaps, one curved in space. It is evident from this that the introduction of the three axes of reference forms a bridge between the theory of space and the theory of equations involving three variable quantities, x, y, z. The reason that the theory of space cannot thus be brought into connection with algebra in general, that is, with the theory of indefinitely numerous equations, but only with the algebra of three quantities, x, y, z, is simply to be sought in the fact that space, as we picture it, can only have three dimensions.

We have now only to supply a few additional examples of n-dimensional totalities. All particles of air are four-dimensional in magnitude when in addition to their position in space we also consider the variable densities which they assume, as they are expressed by the different heights of the barometer in the different parts of the atmosphere. Similarly, all conceivable spheres in space are four-dimensional magnitudes, for their centres form a three-dimensional point-aggregate, and around each centre there may be additionally conceived a one-dimensional totality of spheres, the radii of which can be expressed by every numerical magnitude from zero to infinity. Further, if we imagine a measuring stick of invariable length to assume every conceivable position in space, the positions so obtained will constitute a five-dimensional aggregate. For, in the first place, one of the extremities of the measuring stick may be conceived to assume a position at every point of space, and this determines for one extremity alone of the stick a three-dimensional totality of positions; and secondly, as we have seen above, there proceeds from every such position of this extremity a two-dimensional totality of directions, and by conceiving the measuring-stick to be placed lengthwise in every one of these directions we shall obtain all the conceivable positions which the second extremity can assume, and consequently, the dimensions must be 3 plus 2 or 5. Finally, to find out how many dimensions the totality of all the possible positions of a square, invariable in magnitude, possesses, we first give one of its corners all conceivable positions in space, and we thus obtain three dimensions. One definite point in space now being fixed for the position of one corner of the square, we imagine drawn through this point all possible lines, and on each we lay off the length of the side of the square and thus obtain two additional dimensions. Through the point obtained for the position of the second corner of the square we must now conceive all the possible directions drawn that are perpendicular to the line thus fixed, and we must lay off once more on each of these directions the side of the square. By this last determination the dimensions are only increased by one, for only one one-dimensional totality of perpendicular directions is possible to one straight line in one of its points. Three corners of the square are now fixed and therewith the position of the fourth also is uniquely determined. Accordingly, the totality of all equal squares which only differ from one another by their position in space, constitutes a manifoldness of six dimensions.

II. THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL POINT-AGGREGATES, PERMISSIBLE.

In the preceding section it was shown that we can conceive not only of manifoldnesses of one, two, and three dimensions, but also of manifoldnesses of any number of dimensions. But it was at the same time indicated that our world-space, that is, the totality of all conceivable points that differ only in respect of position, cannot in agreement with our notions of things possess more than three dimensions. But the question now arises, whether, if the progress of science tends in such a direction, it is permissible to extend the notion of space by the introduction of point-aggregates of more than three dimensions, and to engage in the study of the properties of such creations, although we know that notwithstanding the fact that we may conceptually establish and explore such aggregates of points, yet we cannot picture to ourselves these creations as we do the spatial magnitudes which surround us, that is, the regular three-dimensional aggregates of points.

To show the reader clearly that this question must be answered in the affirmative, that the extension of our notion of space is permissible, although it leads to things which we cannot perceive by our senses, I may call the reader’s attention to the fact that in arithmetic we are accustomed from our youth upwards to extensions of ideas, which, accurately viewed, as little admit of graphic conception as a four-dimensional space, that is, a point-aggregate of four dimensions. By his senses man first reaches only the idea of whole numbers—the results of counting. The observation of primitive peoples⁠[79] and of children clearly proves that the essential decisive factors of counting are these three: First, we abstract, in the counting of things, completely from the individual and characteristic attributes of these things, that is, we consider them as homogeneous. Second, we associate individually with the things which we count other homogeneous things. These other things are even now, among uncivilised peoples, the ten fingers of the two hands. They may, however, be simple strokes, or, as in the case of dice and dominoes, black points on a white background. Third, we substitute for the result of this association some concise symbol or word; for example, the Romans substituted for three things counted, three strokes placed side by side, namely: III; but for greater numbers of things they employed abbreviated signs. The Aztecs, the original inhabitants of Mexico, had time enough, it seems, to express all the numbers up to nineteen by equal circles placed side by side. They had abbreviated signs only for the numbers 20, 400, 8000, and so forth. In speaking, some one same sound might be associated with the things counted; but this method of counting is nowadays employed only by clocks: the languages of men since prehistoric times have fashioned concise words for the results of the association in question. From the notion of number, thus fixed as the result of counting, man reached the notion of the addition of two numbers, and thence the notion that is the inverse of the last process, the notion of subtraction. But at this point it clearly appears that not every problem which may be propounded is soluble; for there is no number which can express the result of the subtraction of a number from one which is equally large or from one which is smaller than itself. The primary school pupil who says that 8 from 5 “won’t go” is perfectly right from his point of view. For there really does not exist any result of counting which added to eight will give five.

If humanity had abided by this point of view and had rested content with the opinion that the problem “5 minus 8” is not solvable, the science of arithmetic would never have received its full development, and humanity would not have advanced as far in civilisation as it has. Fortunately, men said to themselves at this crisis: “If 5 minus 8 won’t go, we’ll make it go; if 5 minus 8 does not possess an intelligible meaning, we will simply give it one.” As a fact, things which have not a meaning always afford men a pleasing opportunity of investing them with one. The question is, then, what significance is the problem “5 minus 8” to be invested with?

The most natural and, therefore, the most advantageous solution undoubtedly is to abide by the original notion of subtraction as the inverse of addition, and to make the significance of 5 minus 8 such, that for 5 minus 8 plus 8 we shall get our original minuend 5. By such a method all the rules of computation which apply to real differences will also hold good for unreal differences, such as 5 minus 8. But it then clearly appears that all forms expressive of differences in which the number that stands before the minus direction is less by an equal amount than that which follows it may be regarded as equal; so that the simplest course seems to be to introduce as the common characteristic of all equal differential forms of this description a common sign, which will indicate at the same time the difference of the two numbers thus associated. Thus it came about, that for 5 minus 8, as well as for every differential form which can be regarded as equal thereto the sign “-3” was introduced. But in calling differential forms of this description numbers, the notion of number was extended and a new domain was opened up, namely, the domain of negative numbers.

In the further development of the science of arithmetic, through the operation of division viewed as the inverse of multiplication, a second extension of the idea of number was reached, namely, the notion of fractional numbers as the outcome of divisions that had led to numbers hitherto undefined. We find, thus, that the science of arithmetic throughout its whole development has strictly adhered to the principle of conformity and consistency and has invested every association of two numbers, which before had no significance, by the introduction of new numbers, with a real significance, such that similar operations in conformity with exactly the same rules could be performed with the new numbers, viewed as the results of this association, as with the numbers which were before known and perfectly defined. Thus the science proceeded further on its way and reached the notions of irrational, imaginary, and complex numbers.

The point in all this, which the reader must carefully note, is, that all the numbers of arithmetic, with the exception of the positive whole numbers, are artificial products of human thought, invented to make the language of arithmetic more flexible, and to accelerate the progress of science. All these numbers lack the attributes of representability.

No man in the world can picture to himself “minus three trees.” It is possible, of course, to know that when three trees of a garden have been cut down and carried away, that three are missing, and by substituting for “missing” the inverse notion of “added,” we may say, perhaps, that “minus three trees” are added. But this is quite different from the feat of imagining a negative number of trees. We can only picture to ourselves a number of trees that results from actual counting, that is, a positive whole number. Yet, notwithstanding all this, people had not the slightest hesitation in extending the notion of number. Exactly so must it be permitted us in geometry to extend the notion of space, even though such an extension can only be mentally defined and can never be brought within the range of human powers of representation.

In mathematics, in fact, the extension of any notion is admissible, provided such extension does not lead to contradictions with itself or with results which are well established. Whether such extensions are necessary, justifiable, or important for the advancement of science is a different question. It must be admitted, therefore, that the mathematician is justified in the extension of the notion of space as a point-aggregate of three dimensions, and in the introduction of space or point-aggregates of more than three dimensions, and in the employment of them as means of research. Other sciences also operate with things which they do not know exist, and which, though they are sufficiently defined, cannot be perceived by our senses. For example, the physicist employs the ether as a means of investigation, though he can have no sensory knowledge of it. The ether is nothing more than a means which enables us to comprehend mechanically the effects known as action at a distance and to bring them within the range of a common point of view. Without the assumption of a material which penetrates everything, and by means of whose undulations impulses are transmitted to the remotest parts of space, the phenomena of light, of heat, of gravitation, and of electricity would be a jumble of isolated and unconnected mysteries. The assumption of an ether, however, comprises in a systematic scheme all these isolated events, facilitates our mental control of the phenomena of nature, and enables us to produce these phenomena at will. But it must not be forgotten in such reflections that the ether itself is even a greater problem for man, and that the ether-hypothesis does not solve the difficulties of phenomena, but only puts them in a unitary conceptual shape. Notwithstanding all this, physicists have never had the least hesitation in employing the ether as a means of investigation. And as little do reasons exist why the mathematicians should hesitate to investigate the properties of a four-dimensioned point-aggregate, with the view of acquiring thus a convenient means of research.

III. THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT-AGGREGATES OF SERVICE TO RESEARCH.

From the concession that the mathematician has the right to define and investigate the properties of point-aggregates of more than three dimensions, it does not necessarily follow that the introduction of an idea of this description is of value to science. Thus, for example, in arithmetic, the introduction of operations which spring from involution, as involution and its two inverse operations proceed from multiplication, is undoubtedly permitted. Just as for “a times a times a” we write the abbreviated symbol “a³,” (which we read, a to the third power,) and investigate in detail the operation of involution thus defined, so we might also introduce some shorthand symbol for “a to the aᵗʰ power to the aᵗʰ power” and thus reach an operation of the fourth degree, which would regard a as a passive number and the number 3, or any higher number, as the active number, that is, as the number which indicates how often a is taken as the base of a power whose exponent may be a, or “a to the aᵗʰ,” or “a to the aᵗʰ to the aᵗʰ power.”