Negation and Direction

Lloyd, Alfred H. “Negation and Direction.” The Philosophical Review 25, no. 3 (1916): 383.

THE directional value of negation seems to me a not unprofitable subject for discussion at this time. This is perhaps only to say that I think I have something worth submitting on the subject, but in any case, practically and logically negation Isa very common attitude or motive in experience, at the present time being very much in evidence, and its value, in particular its relation to direction, is a subject of real interest. Also, whatever may be said for the substance or the manner of the discussion that follows, the subject is certainly one that may be chosen for the present occasion, when special honor to Professor Royce is intended. Any subject, however, seriously undertaken, would have Royce's approval.

Anarchy, agnosticism, irrationalism and many other cults or attitudes in negation not all of them, socialism, for example, or liberalism or naturalism, bearing names of negative form are surely among the signs of the time, and accordingly give more than a mere formal or abstractly logical interest to the problem of direction. Now, moreover, as at any time, there are many who from thoughtlessness or superficiality wonder how there can be any real direction in negation, their opinion being that negations can lead nowhere or at least nowhere in particular or nowhere pertinently. Not only have popular notions taken this discouraging view, but also even expert theories have often failed to recognize clearly and appreciate fully the part that negation has or may have in direction. So, I would discuss the question; and my thesis is just this: Logically and practically negation can never be merely and absolutely negative, as often assumed; on the contrary, in general it does and must lead somewhere and, what is more, in a pertinent and orderly way. Indeed, there can be no real negation without direction, and even this: direction can be significant in the life of anything positive only through negation.

In support and explanation of this thesis I begin with certain very simple and familiar principles. Thus, for the first of these, whenever there is definite assertion or 'position,' then is also, in tendency if not in fulfilment, generalization, and the outcome of generalization is always negation, transcendence of the positive. The idea here is manifestly Hegelian, but apart from its Hegelianism in real life propagandism, imperialism, all forms of what in general I may call monarchism or monism or even monomania, reputable or abnormal, show both how inseparable position and generalization are and how negation or opposition results inevitably from their union. With such origin, however, negation cannot escape a certain inheritance from its parents, position, and generalization.

Secondly, then, among the familiar principles referred to, nothing positive may be negated or say transcended by reason of its generalization, without assertion, open or implied, of the principle, the general principle, of that for which in particular form the positive thing negated has been standing. Thus, you cannot honestly proclaim someone an impostor without ascribing actual significance to that which he has claimed to represent. There must be thrones, if there be pretenders; truth in things if there be lies or liars about them. Again, to deny the letter of some creed is to assert the spirit and even anarchy is really a call for a new regime. A metaphysical nihilism, declaring there is no reality, can be only a disguised or indirect realism, being nihilistic only relative to some passing notion of reality. So, to recur to the biological figure and to enlarge upon it a bit, although the negative may not or apparently may not inherit the formally manifest traits or characters of the positive, although really or apparently it may not inherit any of these quite intact, at least it must inherit the general principle, the basal radical life or nature of the positive; showing, if never the exact formal structure, the essential function.

Nor can negation, thirdly, he said to inherit only the general principle of the positive which it negates. Can any negative ever be free from the formal context, from the positive conditions, of its origin? Logically a negative, even if seemingly superlatively negative, must still always be relative or relational. However negative, whatever else be true of it, it must be morality and immorality, although very distinct from each other, both cases of general morality. Opposition is possible only between things alike. Anarchy attacking organization must nevertheless adopt organization. Your very worst enemy can indeed fight with you only as he adapts himself to your nature and methods. It is only matter that may not penetrate matter. Even infinity can be only another finite. This necessary affinity of context or at least formal identity of negative and positive, this relativity of the negative, is a thing much too often forgotten or, if remembered, too little appreciated. In this, as in its many other aspects, the negative is so peculiarly elusive. So easily one has regard only to the obtrusive side of its nature. Yet how deeply and subtly the child enjoys willing not to touch, taste, handle or otherwise disturb or molest the forbidden jam! Justin his negative attitude and his filial cultivation of it lies, hidden perhaps but very much alive, a sweetly persisting jam context. Logically, I say, and with not less truth practically the context of any positive must persist in the negative. What were negative terms, impure or untied or apathetic, with only the prefix?

But, fourthly, and not so simply, now that the negative, born of position and generalization, has been shown to inherit both the general principle and the specific context, described above also as the manifest traits or characters, of the positive, there is some danger that the negative itself will be taken for a mere shell, an empty fiction, quite lacking in real meaning and effect; in other words, that, inheriting so much, it will seem to offer nothing really new; and this danger must be quickly removed ,although the sheer absurdity of such a conclusion might be counted on to take care of it. Thanks to nothing less than that origin in the meeting of position and generalization, negation can never be idle or empty. It does inherit the general principle or function of the positive, but it retains this only as freed from the positive parental expression for some new expression. Again, it does inherit the context or particular form of the positive, but this it does not and cannot remain intact or unchanged in value. The formal context of the positive does indeed persist in the negative, or for the negative, but not as something final and intrinsic; it persists only as something having meaning, as something real or valid only mediately, not any longer immediately. Negation, as our story has it, shows the positive neither wholly denied nor of course not this merely reasserted, but made means instead of end, this change having in point of fact a radical character not easily exaggerated. The end to which the positive becomes only means lies of course in the comprehensive general principle or function which the negation has freed from its identification with the positive. So is there truly great difference between real negation and ' absolute 'negation so-called, the latter being as idle or abstract or formal as 'absolute. ‘Real negation is relative, and its rise in experience must always show the two things already pointed out: (I) the liberated principle as end or meaning, and (2) mediation in the sense of change from immediate reality or value to only mediate reality or value of the positive thing negated. 'Absolute' negation can at best give only another case, perhaps a last or limiting case, of the positive; real negation quite transcends the positive by making it not opposed, but mediate.

It seems worth-while to add here that, viewing negation from any one of those three standpoints which were so closely associated above, from the contextual affinity of negative and positive, or from the necessity of opposition being in kind as well as from the negative's relativity, one must always find mediation of the positive as incident to the negation. As to either the affinity or the opposition, what two mutually opposed things have in common obviously can be only medium, a 'medium of exchange ‘perhaps, a common weapon or instrument, the always necessary common ground on which distinct differences meet. Just by dint of the difference, the opposition, the negation, it simply cannot be immediate any longer. But the relativity of the negatives of most direct interest here. So, to return to that, not only are all real negatives relative, but also in all relativism there is negation, perhaps often disguised, however poorly, never really hidden; and relativism, as is a commonplace at least in history if not formally in logic, has never been without its notable associate, utilitarianism. The relative, besides being the questioned if not denied, the mistrusted if not opposed, has been also the merely useful or mediate and so, I add, the forerunner of some real change.

But, fifthly, now to reach an important conclusion from what has so far been presented, negation, having such origin and such inheritance, brings difference or change of a sort which I think can best be described as dimensional. Real negation means, it implies and induces dimensional difference; this being, as I conceive it, neither difference in mere degree nor absolute difference in kind, but a true tertium quid. The term dimensional or dimension is of course borrowed from mathematics, and borrowed by a layman in mathematics at that, but some intimation of its meaning even as used here should be evident from its source, although many may regard the term in this place as quite too metaphorical to be profitable. Also, as must be conceded, philosophy has need in general of guarding herself against too much mathematicalism, professional or lay. The meaning here, however, is the chief thing, be the term wisely borrowed or not, and in the present intention a dimensional difference or change is one which, although qualitative, although really negative of something, although in kind, is still both congruous with and dependent upon, not directly but mediately dependent upon, that from which the different thing is said to be different. Otherwise put, anything become only mediate is dimensionally inferior to that to which it is mediate, the latter being dimensionally greater. This may, then, be a bold use of the term dimension, suggesting as plainly it does that dimensional difference is intimately related to the distinction between means and end, Buti think I can at least make out a plausible case.

The whole question of dimensions is of course not just one of length, breadth, and thickness, nor of the rectangular relation that these may have to each other. Like other things, dimensions are always of wider and deeper principle than any given case of them can ever adequately exemplify. Simply any given case must be relative to some particular situation. I submit, then, as already suggested, that in general in dimensional difference or change there is involved the distinction between means and end, that where you have the distinction you have dimensional difference and where you have dimensional difference you have the distinction. As to the objection, which is quite likely to be raised, that dimensions coexist, whereas in the distinction between means and end there is always an implication of movement or action, the former thus being special and the latter temporal, I would simply say that dimensional variation may very properly be viewed genetically and that in any given instance coexisting dimensions, like those of ordinary three-dimensional space, may only (I) represent certain accomplished adjustments or mediations and yet also (2) just by their structurally determined region be mediative of some activity in time that realizes new dimensional variation of the mediating region. Ordinary space’s three dimensions only bound or define a region that formally or structurally is what it is relatively to such established adaptations of varying but functionally related factors as accord with the possibility of locomotion or change of place, or even with the possibility only of a certain type of locomotion or change of place. Locomotion, in other words, is so much a matter, if physical or objective, of mere mechanical routine or, if subjective, of free habit or second nature, that its sphere or region, its space, appears quite staid, seeming static in character and coexistent and eternal in all its parts component or dimensional. But this staid character, or rigidity, is relative to the freedom of the locomotion or to the perfect adjustments which the freedom shows. The space of the locomotion, itself three-dimensioned, may still be only mediate to something different, dimensionally different.

In most general terms, if one view any dimensional variation genetically and so in accordance with the distinction between means and end, the new dimension, say the n + 1st dimension, instead of being just one more statically and numerically, as if its ordinate place and character had no distinct value, as if with the advance there was no enhancement and progression of meaning, even no advance in quality, must always be a mark of something to which the lower dimensional field, that is, the w-field, has become only means or medium. Indeed, is not the mediation, here suggested, to be detected in the familiar functional relation that maintains between dimensions, even when viewed quite statically or existentially? Any two, if functions of each other, are in the relation of means to end, so to speak, reciprocally; the dynamic character of the relation being only hidden in the poise, the established balance, of the function and being indeed only truly dynamic because of the reciprocity. A function so accurately established as to be reciprocal or reversible, like the functions of coexisting variants or dimensions, is the very basis of a freely active force. Furthermore, if an established function thus shows reciprocity in the relation between means and end and so also gives evidence of a freely active force, one needs only to look in order to see that the situation thus comprising at once a rigid system, the established region of the function, and a liberated force, the movement within the system, must be potential with something else, with something different, to which the situation itself is become only mediate. Ordinary space, the rigid sphere of free locomotion, may be mediative of activities dimensionally much more complex.

To ordinary space and locomotion, I shall have occasion to refer hereafter. Here, besides pointing out that any dimensional difference or variation may be viewed genetically and under the relation of means and end, a variation in dimensions, n, n + In + 2, etc., showing a progressive mediation, I would sargassos perhaps now quite unnecessarily that any dimensional variation must involve more than a quantitative change. In other words, a manifoldly dimensional field or region can be, or contain, no mere homogeneous mass, but must involve heterogeneity, its dimensions making only a systematic distribution of qualitatively different factors. Thus, very simply put, a four dimensioned field varied by a fifth dimension involves a difference that is not like in kind to the ordinary quantitative difference between four and five. In the former case there is a structural change; in the latter, only a change quite within given structural conditions. In ordinary mathematical terms, which cannot wholly conceal the facts, the former involves ratios and multiplication for a new dimension is always a multiplier, a constant factor; but the latter involves mere quanta or masses and addition. Logically the context of multiplication is very different, qualitatively different, from that of addition; as different as ratio from mass. Multiplication may be, as we used to be taught, merely a short method of addition, but this does not preclude its being a different kind of thing. A dimensions difference, then, is not a quantitative difference; or, if a quantitative difference, is its own kind of quantitative difference, unlike that of mere aggregation.

At risk of offending with much repetition, in any dimension change, n being what you please and the n+1st dimension being multiplier of the w-dimensioned field or structure, the change reduces the w-field from an aggregate of mass-values to a system of ratio-values; and ratios, as was said, certainly do give a different context from that supplied by mere masses. Also, as showing another phase of the change, from the standpoint of the n+1st dimension there is realized a peculiar superiority to the merely quantitative conditions or limitations of the n-field. Fifteen, for example, as a whole, is a distinct sort of a whole, a whole of a higher kind, when the multiplicand of some number, its multiplier, as compared with fifteen as a whole simply increased by some addition. As a multiplicand it is a functional whole, mediated whole, an integral system of ratios or related parts become the medium of something formally different, and in this character of the system there is that peculiar superiority. In the difference between length and area or between area and solidity there is to be seen the change of context and quality above referred to, for the lower region as well as the mediation of that region, length being only mediate to area, area to solidity. But of course, the idea is not confined to such commonplace geometry. In any dimensional difference the lower field, becoming mediate changes from one kind of whole to another kind of whole, from an aggregate whole to a relational whole. Dimensional variation, to sum up, and mediation and heterogeneity go together; and, lest we forget, in each of these there is evident a negation of something or, say, evidence of what it is to negate something. Any given region, negated, mediates something different.

Furthermore, still to consider the nature of dimensions and to court still the favor or at least the patience of mathematics, there is an incident of dimensional variation, of any change from to + 1 dimensions, that should have close attention; since, as seems to me, it throws important light on the meaning of negation and of the various accompaniments of negation which have been pointed out here. Thus, to begin with, the n + 1st dimension must always stand, of course for something formally different from the mediating lower field, the n-field, but also for something essentially possible to, or potential in, that field. Such an essential in distinction from formal or mechanical? potentiality might be concluded from the functional relation that the new dimension must have to the mediating field; and the conclusion itself suggests an interesting, if bold, question. Can it be that dimensional variation is closely akin to the difference between mechanism and organism, the mechanical and the organic? Certainly, the organic, always depending on the mediation of some mechanism, must be something essentially potential in the mechanical although at the same time itself notice the negative non-mechanical. But such bold speculation aside, so much being said of the potentiality of the new dimension in the mediating field, it remains to be added that much of the present story, at least a very important chapter of it, is to be found in the infinity of an infinite series and especially in the last term which must represent a possibility of the series, but being infinite, not a formal possibility.

Infinity, however negative, is always the infinity of something simple circumstance not infrequently overlooked; it can no more be free from the context of some finite than any negative can be free from the context of its positive. But, furthermore, nothing infinite can ever be duly accounted for as merely the largest or the smallest possible, for the infinity of a thing must be more than just the supreme variation in number or size, and not to see it as more is to fail to give the negative in its full significance. Also, it is to belie the last term by treating it as a formal possibility. Thus, a so-called infinite term, a last term or limit, of a series cannot possibly be a term of the series and also last or infinite; a difficulty that is quite too old and familiar to Needmore than mention. Simply by its negative the infinity adds something besides maximum or minimum size to the finite. Any series of terms must obviously be more than just the terms of the series and at least a part, an important part, of the meaning of the infinite term must somehow be that by which the series is more, the very infinity even affecting a certain abstraction of the positive finite terms of the series and revealing and asserting, apart from these terms and their formal character, something essential to the series, general to all the terms, and formally different. The 'last term,' for example, of the simple series:1, ½, ¼, 1/8… is describable in various ways, every one of them having some regard to this peculiarity. It is hardly the half of anything, since at infinity there would be nothing left to halve, but halving itself as a principle, a function, at last free from any

particular application and so implying all possible applications. It is, then, as implying all possible applications, not so much a term of the series as the series’ unity or law that has been exemplified in every term. If you would call it zero, you must remember that it is a contextual zero. Is there any other kind of zero? Moreover, in so far as it is zero, not only is abstraction made of all the positive formal terms, but also the abstraction is no sooner accomplished than these very positive terms return to the series in a new character, all of them having the nature of the infinite term. In short, they return to the series as a constituting system rather than a series, as 'relations' rather than 'things, ‘ratios rather than quanta. So does the infinite term show itself like any other negative, to be born of position and generalization, to have inherited the principle or function of the positive, and to have rendered the positive only a mediating system. With regard to the mediation, it remains to be said that by the change from series to system, which logically the infinite term completes, the formal series is, so to speak, taken up aufgehoben? into a region dimensionally enhanced, mediation and dimensional variation being inseparable. Thus, again, the logical value of the last or infinite term includes a dimensional change for the field within which the finite terms have their manifest form; and the term itself, so valued, appears indeed as a true tertium quid

between difference in degree the term as essentially although not formally a term of the series and difference in kind the term as standing for something in and of the series but formally different. Possibly the same story is told at least as plainly in the following way: An infinite series, whatever its positive manifest form, must always be expressive of a functional so different from a structural unity; of such a unity between two formally different things, making some w-field mediate to then + 1st dimension or taking an n field up into an n + 1-field.

Parenthetically I venture to remark that the real logic of mathematics is commonly hidden in the very abstraction, the extremely formal character, of mathematics. In a world of purely formal relationships, the real things related are made as invisible as ghosts. Graphical representations, therefore, are bound to be of great value, since in some measure they bring to view important logical implications that otherwise would be quite hidden. It has often puzzled me, for example, that one could ever get the sum of an infinite series; for, however formally correct the calculated sum might be, there has still seemed tube something not accounted for; but, perhaps only in my layman’s folly or my super angelic aggressiveness, I think that I sea simple way out of the difficulty. The sum of the series: 1, ½, ¼,…is, of course, 2; and, formally, one need go no farther; but, to go farther, if the series be formally in an n-field, the 2, if really ‘satisfying' the infinity and what this brings to the series, must be in an n + 1-field; in other words, 2 as a product rather than 2 as a sum; graphically, 2 as an area rather than a length or as a solid rather than an area, depending on the value of n. To the formal mathematician such a distinction will doubtless seem too subtle or altogether empty and futile; but whatever be its value or lack of value in the mere technique of mathematics, it strikes me as highly important in the real logic of mathematics. I do not know if I have succeeded in conveying my meaning. Yet what I would say is that the case of infinity affords a specially interesting illustration of the negative as involving dimensional difference or change. If any given finite, or structure, be of dimensions, then its real in distinction from its formal or only hypothetical infinite must have + n dimensions. Infinite space, for example, is not just formally like but bigger than the biggest possible finite space; for (a) a finite space is always formally a definite and specific thing, in other words a thing of n dimensions, of given structure and originality, (b) it is made infinite only serially or gradually, that is only by some persisting function, operation or principle, the series’ unity or law, such as bisection, uniform addition, parallelism, or any regular variation you please, and (c) as infinite, far from being the series’ biggest, or smallest, possible case or term, it reveals something true of all the terms, informal or supernormal to them and virtually transforming of the series as a whole, making the positive series only mediate to something formally different. A finite space, then, may not become infinite and remain formally intact. As an n-space it must remain always finite, formally unchanged; infinite, it is an n + 1-space; its very infinity being so rounded up or brought to earth. As an n +1i-space, although infinite relative to the w-space mediating it, although positively manifesting and incarnating the infinity of that space, it is itself quite earthy for being, within its own higher region, capable of indefinite finite expressions.

Have I at last lost myself and perhaps others in the maze of mathematics? I make no apology. Also, my story, although not yet finished, is approaching its last chapter and there may be relief in that. Before leaving the field of mathematics, however, or rather the field of the real logic of mathematics, there is a conclusion from much that has now been said which, although possibly somewhat aside, I cannot pass without mention. Thus, in just a word or two, with every instance of mediation or dimensional variation there must be a change in the meaning or value of what it is to be the part of a +whole or, as of special interest, of what it is to have position or location. A given structure or region having n dimensions and having become a mediating system to an n + 1st dimension, every part or position in the new region must have a value comprehensive of the whole of the mediating n-dimensioned structure; every part or position must be intensive with the complete extension of the lower region and so must be said to have the freedom of that region, transcending the limitations of partiality or particular position in it so appears in a new way that peculiar superiority which was claimed above for the new region in any case of mediation over the old, any part or position in the new being as if all parts or as if everywhere in the old; and this fact of superiority, suggestive even of the infinite for which, if I may paraphrase the words of the Psalmist, all finite places are as one place, opens up most interesting reflections on the whole problem of location or participation. In any valuation of part or place one must first know, let me say, the dimensional coefficient. To speak in the familiar symbols, an n + 1-here is an n-everywhere.

Now, to recapitulate, negation has been seen to be sprung from position and generalization, inheriting from its allied forebears at once the freed principle of the thing posited and the thing posited itself as a mediated whole. Negation, furthermore, having such inheritance, brought difference, but dimensional difference, which I venture to speak of now as the difference of change by mediation. Thus, there is mechanical change, variation under conditions of uniformity and commensurability, infect routine or accumulation rather than real change. There is, again, absolute change, the rationalistic change of an old-time theology, change by causation ab extra or production ex nihilo; difference, then, by a complete dualism or pluralism; not real change. And there is change or difference by mediation; the change of dimensional difference, discoverable, as has now been submitted, even in the dimensional variations of space; real change; and, as may be added here, change that has real direction. So much have we seen. But for appreciation of what has been found there is need of other than mathematical illustrations of dimensional difference, useful as these have been; for, as should be remembered, in its rise here the idea of such difference was as general as negation. In the variation of an -field by an n + 1st, dimension masses were seen to give way to ratios, component parts to relational parts; in other words, certain assumed absolutes became relative; and relativism succeeding absolutism is certainly no mere mathematical phenomenon, being quite atypical of the worlds of psychology and sociology. In these worlds, too, the relative, as was indeed remarked, is also the mediate, the useful, quite as truly as in the world of mathematics. To explain a little, relativism in general, when supplanting absolutism, always means the passing of certain positive standards, or 'measures.'

Once treated as final and absolute, these become only 'relative' and with the change the pertaining whole, perhaps the organized life of some people, in which they have maintained, becomes in the course of history a mediated whole, losing at once its isolation and a certain inner discreteness or separation of its parts that has made it more social aggregate than a unity. In short the relativism shows a social system come or coming into free, open use; its various component institutes changing from things of direct interest, each with its own cherished and intrinsic value, to so many related conventions or utilities; from distinct institutes, each quick with its own human life and passion, to mere instruments generally and freely in use because become conventional and humanly dead; for relativism and utilitarianism, I say again, are of the same day and generation. It may be only a coincidence, but it seems a coincidence well worth some reflection, that among the ancient Greeks mathematics came to a consciousness of the difference between mass and ratio, leading eventually to Euclid's mingling that strange and incongruous book of proportions with the other books of his geometry, at about the same time that brought the relativistic and utilitarian dictum of Protagoras: Man is the measure of all things. The mass-unit and the definite or positive standard or measure of any kind were thus dethroned contemporaneously; all such measures being henceforth only ‘relative'; there being no longer any supposed commensurability of men or things. But Euclid's book of proportions aside, in what I have said above of relativism and of a social system coming into use, becoming mediated, I shall seem to some to be blinding myself to an important fact. Relativism, they will insist, by discrediting traditional positive standards and institutions really brings individualism and serious social disorganization instead of 'social system in free use.' This fact or rather this notion I deny. Such a way of putting the case rests on a misunderstanding of individualism, of the disorganization so often seen and of a social system becoming mediated whole or come into real use. Thus, the assertive individual, always more the cosmopolitan than the provincial patriot, has at last found the various elements of the organized life around him only so many adjustable parts of a useful whole, which just in being used becomes a real unity or system, and their results, relatively to the traditional sanctity and conservative integrity of things, seeming instability and disorganization. Yet the 'disorganization' is only an incident of the use or mediation, very much as a law's loss of rigor is an incident of its application in real practice. There must ever be 'disorganization,' when mere use or only mediate interest succeeds immediate devotion, when the sacred turns secular. Deeply, however, nothing is so organized as secularization. Moreover, the individual is at once a truer and bigger whole, a more comprehensive and a more complex unity, than the local system or order which, thanks to the possible adjustments of parts, he has been able to make only the means to his now cosmopolitan life. I might add, too, that the individual is always something of a foreigner and, thanks tohis cosmopolitanism, is never without his invitation to what is foreign. But the foreign, by the very negation in it, is only an influence sure to bring to completion the relativism and utilitarianism in the life of a people. Invading foreigners, although like bulls in a china-shop, are so much freer even than the cosmopolitans at home really to use what they find, unhampered as they are by any lingering emotional associations. Just here, then, that is, in the negation which the new or foreign realizes, lies what must give special significance to the analogy, here suggested, between sociological and mathematical relativism and the several incidents, mediation and dimensional difference, which relativism implies. Thus, with the negation from the foreign there must come into life real difference, but a difference however bold or seemingly crude it may be to describe it so that is no more or no less radical than a dimensional variation. It is a dimensional variation under the here adopted definition; for the new life that is brought about is always dependent upon the mediation of the old regime. Such mediation, moreover, is more radical than revolution, which commonly brings only opposition and succession in kind; it is as radical as evolution; as the change from the ancient civilizations to the Christian or from the medieval regime to the modern. In history, as in logic, change by mediation is the only real change.

Further illustration of what is meant here by mediation and dimensional difference, by change through mediation, as coming from negation, is afforded by the various circumstances that always underlie a movement for democracy. This case of democracy, I may say, was the subject of a paper published recently under the title: "The Duplicity of Democracy, or Democratic Equality and the Principle of Relativity." In any democratic call for equality, a call that, however seemingly abstract and general in its terms, is always in effect relative to some particular social and political context, that is, to some established type of social organization, there is to be seen, in the first place, something positive, so far as also general, becoming only mediate. In both the generalization and the mediation, moreover, one can see negation, the democratic attack upon the positive, that is, conflict of democracy with some aristocracy and its peculiar privilege; and the result of all is that special privilege turns into general opportunity, the institutes of the aristocracy becoming the general instruments of a democracy. Democracy, then, at least under present definitions, may be spoken of as a dimensional variation of aristocracy. Also, unlike manner, constitutionalism, for which the given law is only means to an end, is a dimensional variation of political absolutism; induction, of deduction; rationalism and mathematicalism, of a positive and dogmatic legalism; industrialism, of militarism; and, not to make a longer list, Protestantism, or liberalism generally in religion, of a religion of authority. In all of these ‘dimensional variations,' as in the case of democracy, something positive has been at once generalized and negated and so has been made mediate to something new, to something radically different.

"Is he diagramming history," I imagine someone asking at this point, "for the entertainment of mankind? Would he draw the life of one period of history in n dimensions and of a thickening period in n+1? How humorously profound! So, to illustrate his story is truly delightful, although possibly more delightful than true." Let a dimensional history amuse, if it must, or may. Of course, the intended meaning is the important thing, and the meaning is, again, that the significant changes of history are changes by mediation, the later thing, the new, being what it is only by the free mediation of the old or passing thing; only with apologies for the worn refrain by one-time institutes becoming the instruments of human life. Can the new, if new, if different, if negative of the old, ever escape the context and mediation of the old?

But, waiving further illustration, whether from mathematics or from history, and resuming the recounting of general principles and their story, I turn at last, sixthly, to the simple conclusion, virtually stated already, that in dimensional difference, consequent upon negation, in change by mediation, lies the direction which I would claim as not less practically than logically belonging to negation, even to such negatives as were mentioned at the beginning of this paper, anarchism, atheism, agnosticism, irrationalism, and many others, including those bearing names not negative in form. In such change by mediation there is real direction; for, in the first place, it is real change, and, in the second place, the change is always, so to speak, mindful of a context. A dimensional difference might even be defined as a difference mindful of a context; and certainly, significant direction must depend on such mindfulness. Here, then, this paper might come to an end, for in essential principles its course is run, its story told. It is, however, a poor story that awakens no afterthought; and so, to save my tale at least from the appearance of poverty, a few reflections, some with a view to meeting possible criticisms, some perhaps of a lived-happily-ever-after character, are appended.

In human experience as worked out socially, as developed in social organization, where social classes exist under the conditions of division of labor, specialism and all sorts of isolated cults and interests, one may often have difficulty in detecting the conditions and results of negation here asserted. Like other attitudes of organized and more or less isolated groups, negation may often seem to be assumed and maintained absolutely and unqualifiedly, that is, just for its own sake, and an apparently aimless and directionless violence accordingly may quite obscure everything else. But in the logic of human experience, one needs to remember that no attitude or cult of a group, no defined class interest, positive or negative in character, should ever be taken by itself. No such interest ever represents the experience whole. Clear as this is, it is often overlooked. From the standpoint of wholeness, then, of the essential unity of human experience, I think that the social expression of human experience, always more disruptive and analytical than the personal or individual, can afford no real case against the idea that logically or practically negation leads somewhere, having real direction by the dimensional difference, the mediation of the positive, which it brings.

As to the idea being practical as well as logical, it has certainly had its place in psychology and biology which at least seem to deal more directly with what is actual than logic. In these fields the fact of mediation is evident in the part taken in all theories of adaptation by the distinction between the structural and the functional, the formal and the vital, the mechanical and the organic, or even, recalling Professor Dewey's valuable contribution to psychological ethics, impulse and will, will being in his phrase the 'mediation of impulse,' and impulse being the response of given structure to something external and different. Biologically or psychologically, as well as sociologically, the logic of negation and mediation, of positive structures becoming mediate, has no lack of illustrations. Structures become mediate, as they confront alien, negative conditions; as the self, identified with them, adapts itself to a not self or external environment.

But somebody says here that it is not the doctrine of mediation but the application of the term dimension which gives him pause. Logically and sociologically and psychologically and biologically there seems to be a case for change by mediation, but to make the phrase, 'dimensional difference,' cover all such changes is fantastic and to get behind a definition of one's own is quite too arbitrary to be accepted without some protest. So, must I, the offender, return once more to the scene of my offense. Replying to the critic of my admittedly very comprehensive dimensionless shall get out from behind my definition and suggest: (a) that dimensional difference, like most if not all other things, is bound to be, as has in fact been said here already, more in principle than in its usual acceptance or application; (b) that psychologically even the dimensional values of ordinary space are acquired by processes of adaptation and mediation; (c) that ordinary space, whether regarded psychologically or mathematically, is ordinary and three-dimensional only by virtue of an abstraction, which, however warranted, needs now to be recognized as arbitrary and so misleading; and (d) that ordinary and three-dimensional space itself is or may be in reality, that is, when seen without constraint of any arbitrary abstraction, a space of many more than three dimensions and so of a much more complex adaptation or a much larger or fuller mediation than the abstract standpoint referred to can possibly disclose.

(a) The first suggestion needs no explanation.

(b) The second will hardly be disputed.

(c) The third, on its psychological side, obviously has reference

to the fact that space has been for the most part and under the prevailing habit of mind regarded and explained only as a region or medium of bodily locomotion. This fact has already been a matter of some discussion here, but it may be enlarged upon. Thus, the old theory of space perception by association of visual sensations, local signs and muscular sensations was certainly relative to the notion of the self or subject, whether in its whole-body or in distinct parts, like the moving legs, the gesturing arms, or the adapting eyes, as locomotive; and, so far as I can appreciate, later theories or later variations of this theory have not really freed themselves from such an isolation of the locomotive self. Relatively to the locomotive, self space may be so and so; perhaps, thanks to the bilateral structure, the erect position, and the free mobility, three-dimensional, possessing height and width and perspective or depth; but one must always remember the relativity and the abstract standpoint determining it. Unreality locomotion is very far indeed from exhausting the nature or meaning of the self's special activity, even of its activity 'in space,' and the space itself in which the activity takes place cannot therefore be merely, so to speak, a room to move about in, while one does, thanks to more abstraction, a lot of non-speciating’s. The self's so-called non-special activities are quite as truly of space as in space. The higher human activities in general may have their space in quo, but also, in a sense that may not yet be apparent, they must have their space ob. quern. My Latin, I think, is correct.

(d) To turn to the fourth suggestion, which is only complementary to the third, space, even ordinary space, really must be deep with the values of activities far more complex in their adaptive or mediative character than those of mere locomotion and must be accordingly differentiated with many more dimensions or terms of functional relation than the three which locomotion seems to require or which express the natural space or extension of only so much of human activity or, objectively, of mere change of place. Admittedly it is a very artificial way of showing the meaning here only to point out that, dimensions being multipliers or terms of functional relation, the space which to the conventional locomotive view seems only three-dimensional may really be reproduced or multiplied into itself indefinitely, a fourth dimension if the structural basis of variation be rectangular being formally but not actually identical with the first, a fifth with the second, and so on; but, however artificial this notion, in form doubtless more mathematical than psychological, it is well to recognize and appreciate once more that the ordinary special world really does mediate far more than just the locomotive activities and so that the actual dimensions of this world's space cannot possibly stop at three mathematically or psychologically. True, the psychology of the fourth, fifth, sixth . . . twentieth dimension has yet to be worked out and would, if ever undertaken, be sure to meet great difficulties; but, humorously or seriously, psychology has not been in the habit of stopping at difficulties. I wonder if it may not be said that psychologically, as well as mathematically, the free and orderly motion of a body in space expresses or realizes how shall I put what I would say? a field or space which is dimensionally, or functionally, superior to that merely mediating space in which the motion as motion seems to take place. Does not the very freedom and order of such motion imply a functional relation, a dimension, not formally manifest in the special structure either of the body itself or of the merely containing space? The mathematician may duly account for such a complication by his device of 'powers,' one or more of the dimensions being squared or cubed, and yet fail to realize that the motion he has so described expresses more than the space it seems to be in, expressing a space ob. quern that dimensionally transcends the space in quo, but the mathematician's blindness should not set the limit to the evidence. Wherever there is orderly activity within a given structural system, there is realized a space ob. quern dimensionally superior to the space in quo of the activity. Think, furthermore, of the dimensional variation or complexity that must be realized when a freely moving body freely uses or functions with a freely moving body; as, for example, when a human being makes use of a tool or in any way expresses himself or his 'thought ‘through any mobile medium. The situation so presented is truly a complex one, probably far too complex for successful analysis; but the pertinent fact about it is that the activity especially mediated and that the space even of its mediation must lie quite within the ordinary three-dimensional space of mere locomotion.

The same abstraction which has hidden from view the deeper values of ordinary space has also hidden the meanings, the living and immanent meanings, of location and of the external world, the world sensuously perceived in space. Of location or position something has been said already. To know the value of any location one must know the dimensional coefficient. Aston the perceived world, to all human activities save those of locomotion, this has been indeed an external world. Primarily only a world of motor-signs, landmarks, tactile values, and such conditions of locomotion, it has had no direct and intimate parting of the super-locomotive activities. Man's nature, in short, has been divided. He has been a creature of physical or sensuous activities and of non-physical or non-sensuous activities. His consciousness has depended on the distinct faculties of ordinarily special sense-perception and non-special and non-sensuous thought. It is true, of course, that in recent times these divisions have been losing character, or animus, but the illusion of Themis not wholly dispelled. Yet, even as the space in which men move and act is in reality indefinitely dimensional, being unlimitedly potential with what may be called mediative power, being a space, whose resources are, so to speak, only very slightly exploited by locomotion, the perceived world in space can be no mere world, as ordinarily understood, of 'external perception.'

It is itself always a world of thought, and not less sensuous especially for being a world of thought. The very essence of thoughts mediation. Thought and the life it accompanies and directs, instead of being non-special, really comprise only deeper variations, fuller mediations, of that same space, itself a realization of thought, the ordinary three-dimensional space, in which men move and perceive sensuously and 'externally.' Perhaps, if the much abused impractical, abstruse thinker, walking the streets of life, had his head less in the air, realizing that his thinking could mean only added dimensions for the space of his walking, in other words, a larger and deeper mediation of the world and the life around him, his thinking would be less abstruse and impractical and he, at the next crossing, less in danger of being run over.

An important conclusion from the foregoing view of the external world is plainly this. The distinction between external and internal cannot be at any time a fixed one, single in application; it must be, on the contrary, not indeed an unreal distinction, but moving or functional; always an incident, specific as to the application and content of the terms, of every mediation. Always the mediating field is 'external,' but its meaning, that which it mediates, is internal; yet internal in a relative and at the same time somewhat special sense. Thus the 'internal' meanings so, or is said to be so, for being different in kind from the form or structure of the medium. Meaning must always be thus different in kind and accordingly, although mediated by the given structure, cannot be formally identified with this but, relative to it, must seem hidden, mysterious, not placeable anywhere, 'internal.' So internal, however, it at the same time comprehends a greater sphere than that in which it has no determined place. In the adopted language of this article, while internal or without position relative to the space in quo, the n-space, it has its place and part in the space ob. quern, the n + 1 – space, any one of whose here’s, as should be remembered, is an n-everywhere. How absurd one's language sometimes gets! But for illustration of this account of the external and the internal I would call attention to certain facts of history, which is only human experience written large. In history, in social evolution, as we have seen, it is the destiny of the institutional to become instrumental, of the immediate to turn only mediate, and, further, with this change there always arises a vigorous assertive individualism. In other words, the personal members of society acquire a life to self, a life by reservation, an inner life, to which the institutional order of the time becomes only the external means or medium. Such an inner life, however, is so only relatively. The individual feeling and asserting it is also very much a man of the world, the outside world, another world, in feeling and volition being universal, cosmopolitan, natural, identifying himself even with things quite alien to the old order and retaining the old order only for its use to his new and different life. Historically the man of deep 'inner' life at any time has himself lived and helped others to live at once in a larger 'outer' world and in a different world different for the mediation, different by the 'dimensional variation.'

So, as to my offense of dimensionless, far from needing to get behind my own definition of the term, dimension, which may have seemed an arbitrary definition, in my use of the term cannot have made any serious departure from anything essential to ordinary usage.

Finally, in a very simple sum of the whole story, which has been told here, real negation does possess directional value; it brings, not change by mechanical variation nor change by causation ab extra or ex nihilo, but change by mediation; real change; change, as may now be added, that strikes in as well as out, developing inner life as well as larger and different spheres of life; change, again, that really leads somewhere, just because always mindful of a context, always using instead of just opposing what it changes and always being at once inward and outward.

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