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The following is the passage from Spencer:
THE PHILOSOPHY OF SPACE AND TIME (1864)
(North American Review, July 1864, Vol. xcix. No. 204.5, pp. 64-116)
Francis Ellingwood Abbot
EDITORIAL NOTE (by Joseph Ransdell): In the process of transcription from the original, many errors occurred in optical character recognition, and the errors in the footnotes in French, Latin, and German are yet to be corrected. If your scholarly research requires immediate use of the footnoted material and accuracy is essential you can retrieve a copy of the entire North American Review of 1864 in an Adobe (pdf) document of some 600+ pages using the URL below. The URL takes you to the document which contains the whole year of the NAR magazine (1864), volume number 99, with the present article running from p. 64 to 116./64/
In the higher metaphysics no ideas are more worthy of critical examination than the ideas of Space and Time, or yield richer rewards to a careful and patient analysis. Ramifying in all directions, they connect themselves with almost every important philosophical inquiry. Before proving the subjectivity of matter and motion, Idealism must first prove the subjectivity of Space and Time; while the doctrine of their objectivity is the impregnable fortress of Realism. By establishing the a posteriori origin of these ideas, Empiricism accomplishes its derivation of all knowledge from experience; while, by demonstrating their origin a priori, transcendentalism proves the existence of non-empirical elements in intelligence. By reducing them to the category of mere generalizations from physical phenomena, Materialism takes a vast stride towards /65/ the identification of mind with matter; while by showing the impossibility of such a reduction, and profoundly discriminating between general and necessary truths, immaterialism grounds the generic distinction between physical and mental processes on an indestructible basis. By elucidating the characteristics of these ideas, the doctrine of the relativity of human knowledge is shown to be greatly abused when so construed as to exclude all absolute elements from intelligence, and is restored to a rational interpretation. By the same elucidation, the origin, nature, and extent of the cognition of infinity are exhibited without the necessity of self-contradiction. And, lastly, an analytic and synthetic exposition of these ideas will be found to conduce greatly to a thorough criticism of philosophical systems, and to furnish a truer criterion of their value in some respects than can otherwise be obtained. Yet, vast as is the importance of an exhaustive and systematic doctrine of Space and Time, it still remains a philosophical desideratum; although its disjecta membra might perhaps be gathered from the labors of the past, we are aware of no treatment of the problem conducted upon rigorously scientific principles, combining at once breadth of survey and depth of insight with a methodical development. The following monograph is presented in the hope that it may partially supply this want, and that, however imperfect, it may serve at least as the hint and suggestion of a more thorough analysis.
PART I. ANTITHESIS OF SPACE AND EXTENSION.
1. Before discussing the problem of Space itself, it is necessary to show the essential difference between Space and Extension. The speculations of metaphysics demand the utmost precision of language; and perhaps no single inaccuracy of nomenclature in the whole range of the science implies greater confusion of thought, or has led to more important errors in philosophy, than the use of these two words as convertible terms. Space and Extension, so far from being identical, are in all respects to be contrasted; and although it will be necessary, in distinguishing them, to assume by anticipation some of the results of the succeeding analyses, the intelligibility of the analyses themselves will depend in large measure upon the /66/ previous establishment of the distinction. We will, therefore, proceed at once to its elucidation.
2. Extension is the continuity of matter in Space, as Protension (to borrow a convenient term from Hamilton) is the continuity of existence in Time. There can be no extension unless there is something extended, and there can be no protension unless there is something protended. They cannot be cognized except in necessary connection with some object to which they belong; in other words, they are simply qualities or attributes. The limitations of extension in objects produce shape (figure, form) and bulk; the limitations of pretension (duration) produce succession. When extended and protended objects are presented to us, they are immediately perceived; but this is not all. Objects are not only extended and protended, but they must be extended in Space, and protended in Time; when immediately cognized, therefore, they must be cognized in necessary relation to the conditions of their existence, namely, Space and Time. Space and Time as such, however, are not sensuously perceptible in phenomena, and cannot be derived from them, but are perceived only under the special determinations of extension and protension. By the perceptions of extension and protension, therefore, or, rather, by the perception of objects as continuous in Space and Time, the ideas of Space and Time are developed in the mind as necessary logical conditions of these perceptions; that is, as the absolute and necessary correlates of extension and protension.
It is not necessary to unravel the tangled operations of nascent intelligence, which, on account of the silence of memory and the impossibility of observation, are hidden in hopeless obscurity; nor to frame ingenious hypotheses incapable of verification. Such attempts are an utter waste of labor, and from the nature of the case must forever be barren of results. It is sufficient to analyze the processes of matured intelligence, note their genetic connection, and deduce from these data what must be the fixed laws of thought. When Mr. Spencer claims that what are necessities of thought to the adult are not necessities of thought to the infant, and that adult thought differs in nature from infantile thought (Principles of Psychology, p. 254) or when /67/ he maintains the possibility of a state of consciousness in which the idea of Space is not present (Ibid., p. 231); he deserts the path of rigorous philosophical method to roam without a guide in the fields of wild conjecture. Conditioned and condition, Extension and Space, are necessarily known in an indivisible cognitive act. To quote the admirable language of M. Cousin, the idea of Space in the logical order (l'ordre logique) precedes that of body, but in the chronological order (l'ordre chronologique) succeeds it;* although in absolute strictness the two ideas should be regarded as simultaneously arising in consciousness. Extension, therefore, which is always an attribute of body, presupposes Space as its conditio sine qua non. So likewise of Pretension and Time. Now the ideas of Space and Time possess characteristics not pertaining to Extension and Protension; and it is indispensable to our purpose to determine precisely what these characteristics are. For the sake of brevity, we shall confine ourselves in this undertaking to Space alone, since the general characteristics of the two ideas of Space and Time are identical. But before beginning the analysis of Space itself, it is necessary to develop more in detail the antithesis between Space and Extension.
* Histoire de la Philosophie du XVIII Siecle. The expression alone is original with Cousin; for the idea, see Kant, Kritik der reinen Vernunft, Werke, Vol. II. p. 360, footnote, et passim; Aristotle, Categ., ch. 12, and Metaph. VIII. 8.
3. Extension differs from Space in six important respects. Regarded psychologically, as subjective ingredients of human knowledge, they differ in complexity, logical function, and the faculties to which their cognition must be referred. Regarded ontologically, as objective existences, they differ in their nature, the sources from which their cognition is derived, and the specific character of this cognition. These two departments of the distinctions shade into each other, but the general division is advisable. That our distinction between Space and Extension may be more thoroughly understood and more vividly realized in thought, let it be borne in mind that Space would be unaffected by the annihilation or creation of a universe. Matter may displace other matter, but not Space; for the displacement of Space would be the displacement of place itself. /68/ Matter may move in Space; but inasmuch as its extension or continuity must move with it, while the space it at any one moment occupies cannot move at all, matter can simply leave one position in Space to occupy another. Extension is mobile, Space is immobile; the absence, presence, or motion of extended matter leaves Space unchanged and unchangeable.
4. Extension is a simple and undecomposable notion, while Space is, as we shall see hereafter, the indissoluble synthesis of three distinct elements.
The continuity of matter must be regarded as ultimately absolute, and cannot be explained away by the hypothesis of the repetition of an infinite number of points. If these points are themselves extended, each point must occupy a certain portion of space and be absolutely continuous in that space. If they are not extended, their aggregate sum cannot constitute an extended body; and the perception of extended body is illusory. Unless, therefore, we embrace Idealism, we must admit the existence of an absolute continuity in matter as one of its primary qualities, irreducible to lower terms, and therefore existing in our thought as a simple idea. And even the adoption of Idealism will not release us from the necessity of admitting the subjective reality of this simple idea. The triplicity of the idea of pure Space will be shown hereafter.
5. Extension is always, and Space never, a quality; in other words, Extension is always a predicate, Space always a subject.
By this distinction it is not meant that the quality of extendedness or continuity cannot be abstracted like all other qualities, and made the grammatical subject of a proposition; for if that were the case, the very sentence in which the distinction is stated would be its own refutation. But it is meant that Extension, being a mere quality of matter, exists in and through matter; and that, if matter were annihilated, Extension would be annihilated also; whereas Space, not being a quality or attribute, depends for its existence on no other object, and would not be involved in the annihilation of matter. Extension and matter are reciprocally conditioned as subject and attribute, and both are conditioned on Space. Extension must always be predicated of matter, and, except as a predicated attribute, has no existence in thought, but Space can be predicated of nothing.
6. Extension is cognized through the senses and the sensuous imagination, while Space is cognized only through the non-sensuous reason or the faculty of pure intellection.
This distinction is more fruitful of results than the preceding, and demands a much fuller explication. It is not yet necessary to state the grounds of the conclusion that Extension is immediately cognized both by visual and tactual perception; for present purposes it is sufficient to assume the truth of this position, and concentrate our attention on the thesis that the sensuous imagination is conversant with Extension alone, while Space is known by a very different faculty. It seems best in this place to notice the dangerous ambiguity of the words conceive, conception, conceivable, and inconceivable, which refer now to the act of the sensuous imagination in reproducing or re-combining the data of sensuous perception, and now to the act of comprehension by the higher mental faculties. Innumerable fallacies have crept into the arguments of the best thinkers from the unwary employment of these vague and shifting expressions; and we take occasion to mention that, in all the succeeding pages, we shall adhere strictly in our own use of the words to the former signification.
The sensuous imagination is that faculty of the mind which creates a mental representation of objects which have previously been presented to sensuous perception, or by means of which new syntheses of these previous perceptions are formed under the superintendence of the Elaborative Faculty. Sense perceives only qualities of matter, secondary, secundo-primary, or primary. Whenever we see, we see extended colors; and whenever we move, being always subjected to atmospheric pressure, we move under conditions of greater or less material resistance. Absolute vacuity, or empty Space, is never presented either to visual or tactual experience; and even if it were, no perception of it would be possible, inasmuch as it implies the utter absence of matter and material qualities, and no perception would be possible in the absence of all objects of perception. It must be admitted that there is no sensuous perception of pure Space. It follows, therefore, that the sensuous imagination, which can never transcend the data of sensuous perception, forms no conception of absolute vacuity or pure /70/ Space. It has been recorded by medical observers, that, in cases of loss of sight, if the eye alone is disorganized, the power of imaginative vision remains unimpaired; whereas, if the optic nerves and thalami are likewise disorganized, this power is lost. This fact, taken in connection with other considerations, has led Sir William Hamilton to the conclusion that "imagination employs the organs of sense in the representations of sensible objects." (Lectures on Metaphysics, pp. 461, 462, Amer. ed.). Going a step farther in the same direction, we deduce an important corollary; namely, The sensuous imagination can mentally reproduce no object without at the same time reproducing the physical conditions of the perception of that object. These conditions, in the case of sight, are light and extension (color and shape); in the case of touch, resistance and extension (solidity and continuity). Hence, in imagining an object of sight, we must imagine it under modifications of color and shape. Conversely, whenever we mentally represent color and shape, we imagine what has been, or, if existent, might have been an object of sight; and this representation is necessarily a conception of the sensuous imagination. The bearing of these remarks will be at once obvious. Extension is, and Space is not, an object of sensuous perception; consequently, extension alone can be an object of representation under modifications of shape and color, while pure Space, whether as finite or infinite, is absolutely excluded from such representation. Neither being perceived in phenomena, nor conceived under phenomenal conditions, Space is known only as the necessary condition of phenomena in the world of matter. That is to say, Extension is cognized only through the senses and the sensuous imagination, while pure Space is cognized only through the non-sensuous reason or intellect.
This simple distinction at once explains why infinite Space is "inconceivable." We can only represent to the mind by the imagination color and extension, which, be it remembered, are only qualities of matter, perceived and conceived under conditions of limitation; no conception whatsoever of pure Space is possible, however much the assertion may contradict the /71/ ordinary notions on this point. To confirm the foregoing conclusion, we will cite the unconscious testimony of two witnesses, of antagonistic schools in philosophy, who yet evince perfect coincidence of opinion on the question at issue. The one is Sir William Hamilton, who holds the transcendental doctrine, that the idea of Space originates a priori; the other is Mr. Herbert Spencer, who holds the doctrine of empiricism, that the idea of Space originates a posteriori; both equally confound Space with Extension, which accounts for their singular coincidence of opinion. (Extension is only another name for Space." (Hamilton's Lectures on Metaphysics, p. 346.) "Extension and Space are convertible terms." (Spencer's First Principles, p. 48.)) We will first quote from Hamilton:So far, in fact, is the doctrine which divorces the perceptions of color and extension from being true, that we cannot even represent extension to the mind, except as colored. "In this act [I. e. imagination] I can easily annihilate all corporeal existence,I can imagine empty space. But there are two attributes of which I cannot divest it, that is, shape and color. For example : extension is only presented to sense under some modification of color, and even imagination cannot represent extension except as colored. We may view it in fantasy as black or white, as translucent or opaque; but represent it we cannot, except either under some positive variety of light, or under the negation of light, which is darkness. But psychologically considered, blackness or darkness is as much a color, that is, a positive sensation, as whiteness or redness; and thus we cannot imagine to ourselves aught ex tended, not even space itself, out of relation to color." "But if we do our utmost to realize this notion of infinite extension by a positive act of imagination, how do we proceed? Why, we think out from a centre, and endeavor to carry the circumference of the sphere to infinity. But by no effort of imagination can we accomplish this; and as we cannot do it at once by one infinite act, it would require an eternity of successive finite efforts, an endless series of imaginings beyond imaginings to equalize the thought with its object. The very attempt is contradictory."* /72/* Lecture. on Metaphysics., p. 387. Contrast the profounder thought of Leibnitz, Spinoza, and Malebranche. "Le vrai infini a larigueur n'est que dans I'absolu, qui est anterieur a toute composition et n'est point forme par 1'addition des parties." (Leibnitz, Nouv. Essais, Liv. II. Ch. XVII. 1.) And again : "Le ve'ritable infini ne se trouve point dans un tout compose' de parties." (Reflex. sur 1'Essai de Locke, Opp. Phil., ed. Erdmann, p. 138.) " Quantitatem infinitam [esse] mensurabilem et ex partibus firritis conflari supponunt; quare ex absurdis, quse inde sequuntur, nihil aliud concluderepossunt, quam quod quantitas infinita non sit mensurabilis et quod ex partibus finiris conflari non possit. Atque hoc idem est, quod nos supra jam demonstravimus." (Spinoza, Eth., Pars I. Prop. XV. Schol.) " Car le firri, quelque grand qu'il soit, appliqud ou repete tant qu'on voudra, ne peut jamais %aler 1'infini." (Malebranche, Entretiens sur la Me'taphysique, Premier Entr., $ viii.) So Locke, Essay on Human Understanding, II. 17, $11.
" It [i. e. the experience-hypothesis] accounts for a certain peculiarity in our conceptions of Space, which the Kantian hypothesis does not account for; this peculiarity being, that every conception of Space which can be formed by a single mental act, is limited to such portion of Space as we can have experience of at one time. Let any one attempt to form an idea of the whole surrounding sphere of Space simultaneously, and he will find it impossible to do so. When standing upright, he can very well conceive the hemisphere of Space extending in front of him; but he cannot in the same act of thought include the hemisphere of Space that is behind. On watching his mind, he will perceive that, in thinking of the Space that is behind, he becomes momentarily unconscious of the Space that is in front. If, to get rid of perturbing circumstances, he mentally abolishes the earth and all objects, and supposes himself in an infinite void, he will still find that the infinity at any moment occupying his imagination is the infinity extending on one side of him, and never the infinity on both sides. Now the Kantian hypothesis not only leaves this fact unaccounted for, but is at variance with it; for if Space be a form of thought, our conception of it should be simple, total, uniform, and altogether unrelated to external perception. Whereas, the experience- hypothesis not only accounts for it, but involves it, as an inevitable corollary; for if all knowledge is from without, the conception which we can by one act form of Space cannot exceed the perception which one act can give us of it. To the first theory the fact is an obstacle; to the second, it is a confirmation." /73/No one who carefully observes his own mind can question the perfect accuracy of these psychological results. If we identify Space and Extension, it must be confessed that our conception of Space is hampered and embarrassed by all these difficulties; though it may well be wondered how a logical mind can refuse to accept the empiricism which is the natural result of the identification. Spencer is perfectly right in the conclusions he deduces from these premises, and Hamilton is strangely inconsequent. But the distinction between Space and Extension shows on the one hand that our so-called conception of Space is only the conception of Extension, and on the other that the idea of Space must exist a priori as the necessary condition of this very conception. Try the experiment; try to represent to the mind Space by itself, and you will discover that you can succeed only in representing some extended color, blue, gray, black, or some other. But how palpably evident it is that an extended color is not, as Hail ton calls it, "empty space"! Our only conception of empty space is, according to Hamilton, the conception of a colored something which is indefinitely extended; yet color and extension are only qualities of matter, however sublimated or attenuated it may be. Consequently we have only conceived matter indefinitely extended and colored. But every extended object must be contained in Space; the more we expand our conception of Extension, therefore, the farther we are from embracing the all of Space. The irresistible impulse of which we are conscious to dilate our conception arises from the logical necessities of the case, and is referable only to a faculty which transcends the imagination and reveals these logical necessities. The indefinite is the only infinite of the sensuous imagination. Not even by a series of additions which should exhaust eternity itself should we, as Hamilton supposes, approximate the conception of an infinite Space; for Space must still subsist as the eternal condition of our conception. It would be as impossible to overtake the idea of Space by continuing the accretions of imagination through eternity, as it would be to /74/ overtake our own shadow by endlessly chasing it. Our inability to represent Space by the sensuous imagination, however, is no argument either for or against its reality or its infinity.
7. Extension is a congeries of infinitesimal units, Space is an infinite unit.
That a cubic foot of iron consists of two halves, that each half consists of two fourths, each fourth of two eighths, &c., is a necessary truth; and of such division there can be no arbitrary termination. But the partition in this case is merely nominal, not real; and it extends so far only as the mind follows out the process in imagination. Since, however, the mind can never complete an infinite series, such divisibility cannot properly be called infinite, but only indefinite. Matter, there fore, is not infinitely divisible in thought. Neither can an infinite division of matter be possible in fact. For, inasmuch as mere division cannot annihilate anything, it could never reduce extended matter to nonentity; and concrete units must still subsist as the condition of all discrete quantity. In other words, matter, while existent at all, must exist as an aggregation of extended parts. The same conclusion follows directly from the consideration that an interminable series, being by its nature aggregated, can never be infinite, since infinity is contradictory of aggregation or number. It having been shown (Sec. 4) that absolute continuities must be assumed as the elements of composite continuities, extended units of matter actually indivisible by existent forces must be concluded to exist. That greater forces might continue the division is not to be rashly denied; but to suppose that any force whatever could continue it far enough to destroy all ultimate units of extension, is to fall into a metaphysical absurdity. Nothing short of the annihilation of matter would abolish the necessity of the existence of infinitesimal extended atoms; their inconceivability by imagination has no bearing on the question.
The continuity of matter, as empirically known by us, is never absolute. Experiment has yet found no limit to the divisibility of matter. On the one hand, the universal compressibility of matter disproves the absolute character of its extension as perceived by us, by proving the existence of pores or vacua; on the other hand, its impenetrability or ultimate /75/ incompressibility proves the existence of ultimate resistant atoms, inasmuch as an infinity of non-resistances could never constitute resistance. From the two starting-points of solidity and extension, therefore, the two primary qualities of matter, we equally arrive at the same necessity of the existence of infinitesimal extended and resistant atoms or units. The infinite unity of Space, though apparently a self-contradiction, will be demonstrated hereafter.
8. Extension is presented to cognition a posteriori, Space a priori.
The space-predicates of body are extension (dimension, continuity, &c.), magnitude (bulk, size, &c.), figure (shape, form, &c.), position (place, location, situation, &c.), distance, and direction. It is believed that all others are either synonymous with these, or reducible to them. Of the six, extension is primary or absolute; the rest are secondary or relative. Magnitude being quantity of extension, as determined by reference to some special limited extension assumed as the unit of linear, superficial, or cubic measure, and figure being the mutual relations of the surfaces of limited magnitudes, magnitude and figure are plainly nothing but relations among the limits of extension. Direction being the relative bearing of two positions, and distance being the length of a right line conceived to exist between them and measured by a linear unit, are nothing but relations of position. Position itself is empirically known only as a relation among objects; and although every single object must have absolute position in the sense of occupying a certain part of Space and no other, still even this is a relation between the object and pure Space; a relation, moreover, which, from the nature of one correlate, Space, is inconceivable. But absolute continuity, which in an ultimate unit of matter is limited, and in Space is unlimited, although in both cases equally inconceivable by the imagination, is psychologically a simple idea, and does not of itself involve any other idea. It is true, every minimum of matter, being limited in continuity, must of necessity have surfaces or extremities among which exist relations of magnitude, form, position, distance, and direction; but these predicates are involved, not in the continuity itself, but only in its limitation, as is /76/ evidenced by the fact that the continuity of Space does not involve them. Moreover, there being actually no such thing as absolute rest in Space, although Space itself is in absolute rest, every unit of matter is forever changing its position with reference to Space, while its continuity remains unchanged, another proof that continuity or extension is the only absolute space-predicate of matter. It may be observed, by the way, that the mobility of Extension, as thus contrasted with the immobility of Space, would alone necessitate a distinction between the two.
Extension and its modifications, magnitude, form, position, distance, and direction, being thus the space-predicates of matter, it remains to determine which of them are immediately perceived by the senses, and also by which of the senses we perceive them.
Extended matter is the sole object of immediate sensuous perception; except as extended, matter is imperceptible. By whatever senses, therefore, we immediately cognize matter, we immediately cognize its extension. Science and philosophy teach us that this extension cannot, except in the ultimate units of matter, be absolute; but the breaks of continuity in densely compacted substances, as a polished metallic surface, for instance, are imperceptible on account of the imperfection of the senses. The real discontinuity, therefore, is apparent continuity; and, considering the question from the psychological point of view, we do perceive matter as strictly continuous. It is indubitable that all the senses are modifications of touch (Hamilton, Lect. on Metaph., p. 374); but, setting aside the obscure problem of the perceptibility of extension by hearing, smell, and taste, and the relative pro portions of sensation and perception in the activity of those senses, we will confine ourselves to the cases of sight and touch proper.
Now the five modifications of extension above described are all relations among the limits of extension; and inasmuch as relations cannot possibly be objects of sensuous perception, but only of a higher faculty, it follows that extension alone, and not its modifications, is immediately cognized by sense. /77/
Whether these relations can in any way be cognized immediately, or only by a process of inference, it is unnecessary here to inquire, and the problem may be postponed to a future occasion; suffice it to say, that if we really know the objective relations of things, there must be some faculty of pure and immediate cognition of relations. The subject of the present inquiry has been narrowed down to this : Do we obtain immediate knowledge of extension through touch, through sight, or through both ? And as all who admit any really immediate knowledge of matter whatever admit also the cognoscibility of extension by touch, and as we do not here propose to refute Idealism, we may still further narrow the inquiry as follows: Is extension visually perceptible?
Berkeley argues that color, which is universally admitted to be the only immediate object of vision, is also universally admitted not to be "without the mind"; and that, therefore, since the extension of color must exist where the color itself is, the extension also must be in the mind. (New Theory of Vision, xliii.) This argument stands or falls with the general theory of Idealism. As to the question whether the perceptions of extension and color mutually involve each other, he professes himself unable to decide. f Sir William Hamilton replies to all scepticism on this point by the argument that, inasmuch as we perceive two different colors in juxtaposition, and inasmuch as their mutual limitation affords a breadthless line of demarcation, the perception of linear extension is given in that of the colors. Against this argument, which we regard as conclusive, Mr. Spencer retorts as follows: " Superficial extension cannot be conceived except as the attribute of something separate from consciousness, something belonging, not to the mind, but to an object out of the mind. That is to say, it implies the idea of outness, or, in other words, the idea of distance. Hence, as it is admitted that distance is knowable only through experiences of motion, it follows that visible extension is knowable only through such experiences." (Ibid., cxxx. Condillac is more dogmatic, and gives up the objectivity of Extension. Traite' des Systemes, (Euvres Completes, Tome II. pp. 136, 139. Lectures on Metaphysics, pp. 383, 385. Principles of Psychology, p. 220)
The reason assigned by Berkeley for holding that distance is not visually perceptible, is this: "For distance being a line directed endwise to the eye, it projects only one point in the fund of the eye, which point remains invariably the same, whether the distance be longer or shorter." (New Theory of Vision, ii.) This reason must be admitted to be valid, so far as relative distance is concerned, that is, the greater or less distance between the perceived color and the eye; but is it equally valid as regards merely the externality of the color? One of two things is certain: in the primary visual act, the color must either seem to be in contact with the eye, or not to be in contact with it. Now, if the latter is the case, the idea of indefinite distance is indissolubly associated with the perception of the color. But if the former is the case, then the visual perception is on a par with any other tactual perception; and, it being admitted that tactual perception in general involves the idea of outness, it must equally be admitted that the visual perception involves the idea of outness. Moreover, without framing any hypothesis as to the how or the why, it is certain that the perception of objective extension is indissolubly linked with the subjective sensation consequent upon contact between an object and a sensitive surface; and, inasmuch as more or less of the retinal surface is affected whenever an object is seen, and inasmuch as it is admitted that sight is reducible to touch, it follows that, if extension is tactually perceptible, it is also visually perceptible. These considerations not only set aside Spencer's attempted refutation of Hamilton's argument, but strengthen the conclusion of the latter, that the perception of extension is included in the perception of color. Another argument, equally strong, and tending to the same conclusion, is this. No object is perceptible to sight unless it subtends in the eye a visual angle of a certain magnitude (minimum visibile); and the object is thus, as it were, the base of a triangle whose sides are formed by the rays reflected from its extremities. Now unless the base of this triangle possesses a perceptible continuity, or extension, or length, it will not subtend an angle of the requisite magnitude, and will therefore fall below the limit of /79/ the minimum visibile, and become altogether imperceptible. The perceptibility of the object, therefore, depends absolutely on the perceptibility of its extension or continuity.
The cognition of Extension, being thus an essential element of visual and tactual perception, evidently originates a posteriori, that is, is derived from experience alone. But pure Space, not being contained in sensuous data, either as a perceptible object or as an attribute of perceptible objects, must be known in some other way than by experience. Since the only possible sources of knowledge are the ego and the non-ego, and since the idea of Space is not derivable from the non-ego, it follows that it must be derived from the ego, that is, a priori. How the idea is shown, notwithstanding its subjective origin, to correspond necessarily with an objective fact, will appear below.
9. Extension is mathematically, and Space metaphysically, cognized.
Mathematicians are not agreed as to the precise object of mathematical cognition. According to Price, "the subject- matter of geometry is space, abstracted from all consideration of the space which we occupy, and in which we are; and the science consists in the development of this idea of space. (Treatise on Infinitesimal Calculus, Vol. III. p. 8.) According to Montferrier, mathematics in general is " the science of quantities "; and, defining extension as " the limited space occupied by an object," he defines geometry as the " science of extension." * Wronski defines mathematics as the " science of the laws of space and time." ** Comte defines it as having for its object the "indirect measurement of magnitudes," or "the determination of magnitudes by means of each other, according to the exact relations existing among them."*** According to Descartes, " order and measurement" are the objects of mathematics.**** Hamilton uses the following explicit language: "Without inquiring into the reality of existences, and without borrowing from or attributing to them anything, arithmetic, the science of discrete quantity, creates its numbers, and geometry, the science of continuous quantity, creates its figures; and both operate upon these their objects in absolute independence of all external actuality. The two mathematical sciences are dependent for their several objects only on the notion of time and the notion of space, notions under which alone matter can be conceived as possible, for all matter supposes space, and all matter is moved in space and in time. But to the notions of space and time the existence or non-existence of matter is indifferent; indifferent consequently to geometry and arithmetic, so long at least as they remain in the lofty regions of pure speculation, and do not descend to the practical application of their principles. If matter had no existence, nay, if space and time existed only in our minds, mathematics would still be true; but their truth would be of a purely formal and ideal character, would furnish us with no knowledge of objective realities."***** /80/
* "Les mathcmatiques, considere'es dans leur ensemble on comme formant tine seule science, peuventetre de'finies : la science des quantites" (Encyclopedic Mathe matiqne, Tome I. p. xiii.) "L'objet general de cette science est, comme nous avons dit (Tome I. p. 14), 1'espace particulier occupe par un objet pbysique dans 1'espace absolu et sans limites qui s'etend en tous sens autour de nous. L'espace limite occupe par un objet se nomme I'&endue de cet objet/' (Ibid., Tome IV. p. 3.)
** "La science des lois du temps et de 1'espace." (Ibid., Tome I. p. xiv.)
*** "Nous sommes done parvenu maintenant a definir avec exactitude la science mathematique, en lui assignant pour but, la mesure indirecte des grandeurs, et disant qu'on s'y propose constamment de determiner les grandeurs les unes par les autres, d'apres les relations pre'cises qui existent entr'elles " (Cours de Philosophic Positive, Tome I. p. 129.)
**** "Ilia omnia tantum, in quibus ordo vel mensura examinatur, ad mathesin referri." (Dictionnaire des Sciences Philosophiques, s. v. Mathe'matiques.)
***** Lectures on Logic, p. 380. Compare the passages there referred to, Esser's System der Logik, p. 355, and Fries's System der Logik, p. 409.
A definition somewhat more comprehensive than any of the foregoing seems to be needed. All mathematical symbols whatsoever represent quantities or their relations, and all syntheses of these symbols into equations and formulae, from the simplest up to the most complicated and abstract, still represent relations among quantities. We will therefore hazard the following definition: Mathematics is the Science of Quantities and Quantitative Relations. It branches into the three grand divisions of Algorithmy, the science of Number; Geometry, the science of Extension; and Phoronomy, the science of Motion.* /81/
* " En effet, il nous suffit de faire observer que la quantite est dite pe'ometrique, lorsqu'elle se rapporte au plus ou moins d'espace occupe par un objet : on la nomine alors Y&endue de cet objet; et qu'elle est dite numerique, lorsqu'elle se rapporte au plus ou moins des parties dont 1'objet est compose : on la nomme alors le nombre de cet objet. Le nombre et Vetendue forment done deux objets generaux des mathematiques qui se divisent ainsi en deux branches fondamentales, en deux sciences distinctes : la science des nombres ou Valgorithmie, et la science de I'&endue on la geometric. Outre ces deux branches fondamentales, il en existe ne'cessairement une troisieme, dans laquelle le temps et 1'espace concourent pour former une nouvelle espece de quantite; cette quantite, diffe'rente de Ye'tendue et du nombre, mais partici pant de Tune et de 1'autre, est le mouvement. Nous avons done encore, comme branche fondamentale des mathematiques, la science du mouvement ou la Phorono- mie" (Encyclope'die Mathematique, Tome I. p. xiv.)
It is geometry alone, with its subdivisions of elementary geometry, trigonometry, descriptive geometry, and analytical geometry, which is concerned in our present discussion. The problem is briefly this, Is pure Space, Extension in the sense of limited spaces, or Extension in the sense of the continuity of body, the object-matter of the science? And inasmuch as quantities and their relations are confessedly the sole objects of mathematical cognition, the problem is still further reduced to this, Does pure Space, whether as unlimited or as limited, come under the category of quantity?
The word quantity is defined as " that which is susceptible of augmentation or diminution." * But every geometrical quantity or magnitude susceptible of augmentation or diminution must possess definite limits, since it is the mutual recession or approximation of its limits which constitutes its augmentation or diminution. Space, therefore, as an illimitable totality, cannot be called an infinite quantity without a contradiction in terms. It avails nothing to reply that this term cannot be self- contradictory, because the higher mathematics habitually employ it in the calculus, and attain true results by its use; for the word infinite in mathematical usage denotes inconceivable or immeasurable greatness, not absolute negation of limitation in all directions. The expressions "different orders of /82/ infinites," "infinity squared," etc., must not blind us to the fact that the real infinite can never be a term in an actual quantitative relation.
* Par quantite, on entend ordinairement tout ce qui est susceptible d' 'augmentation ou de diminution, un objet materiel, par exemple, tel qu'un monceau de sable, ne nous apparait, en faisant abstraction de sa nature physique, que comme un aggregat de parties, c'est-a-dire comme une quantite plus ou moins grande, susceptible d'etre augmented en ajoutant de nouvelles parties, ou d'etre diminue'e en retranchant quel- ques unes de celles-ci que la composent. En outre, la portion d'espace que ce monceau occupe est elle-meme une quantite', qui peut croitre ou decroitre selon qu'on I ajoute ou qu'on retranche des parties composantes, ou encore selon que ces parties Isont plus ou moins pressees les unes centre les autres." (Ibid. p. xiii.) VOL. XCIX. NO. 204. 6
But, admitting that the term quantity is applicable solely to the finite, and that therefore Space as an infinite whole is excluded from the category of quantity, it is nevertheless con tended by nearly all mathematicians that finite spaces, or the portions of space occupied by determinate magnitudes, are mathematically cognized. Our citations from Montferrier have shown that, defining extension as the " limited space occupied by an object," he regards that extension as the object of geo metrical cognition. It is unnecessary to mention other mathematicians who hold the same opinion; but they are numerous. Our argument, therefore, must concentrate itself on this point, and show if possible the untenability of the opinion in question.
Recurring to the definition of quantity above cited, and bearing in mind that a geometrical quantity exists solely in and through its definite limitation, it will be evident that such an expression as " limited space " is self-contradictory. Not to insist on the fact that we invariably regard Space as illimitable, and that what is true of the whole is equally true of its so-called parts, it will be sufficient to say here that no arbitrary limitation can create any break in the continuity of Space, and to refer for a fuller consideration of this point to the psycho logical analysis given below. For it will, of course, be replied, that by a " limited " space is only meant an " enclosed " space, and that, since arbitrary limits do necessarily enclose more or less space, our criticism of the definition is a mere verbal quibble.
But, considering the nature of geometry itself, it will be manifest that, while elementary geometry begins with figures, analytical geometry, inasmuch as all its algebraic formulae are interpretable into points, lines, and surfaces, ends with figures; in a word, that all geometrical ratiocination is based on construction. From the definition of a triangle up to the generation of the hyperbolic paraboloid, the entire science depends on the figuration of points, lines, and surfaces, either objectively on paper, or subjectively in imagination. Consequently, since surfaces as the limits of solids, lines as the limits of surfaces, /83/ and points as the limits of lines, are the ultimate elements of all construction, it will be necessary to consider what the figuration of points, lines, and surfaces involves. Now, at the very outset, we find a striking want of harmony between the mathematical definitions of these ultimate elements, and their actual realization on paper or in imagination. A point is position without extension, a line is length without breadth, a surface is length and breadth without thickness; but, as such, they are, according to the admission of geometers themselves, neither presentable to perception nor representable by imagination. Actual limits must either be perceived or conceived, as the necessary data of geometrical reasoning; and from this it follows that geometrical operations and investigations are absolutely impossible without the aid of sensuous symbols. The dependence of geometry as a science, therefore, upon the faculties of perception and sensuous imagination, is seen to be absolute and unconditional; and inasmuch as these faculties are conversant with Extension only, and not with Space, it follows that the object-matter of geometry is, not Space, but Extension.
Against this it will be argued, that all these symbols are imperfect and inexact; that geometrical formulae, being absolutely exact, cannot be true of the symbols to which they but qualifiedly apply, but must be true only of the things symbolized, namely, points, lines, and surfaces, as geometrically defined; and that, therefore, since mathematical points, lines, and surfaces are not found in Extension, whether perceptively or imaginatively cognized, they must be properties of pure Space. Consequently it will be concluded that, though employing sensuous symbols in its processes, geometry reveals actual relations among the parts of Space, which must, therefore, be its object-matter.
The reply to this counter-argument is as follows. Geometrical quantities, that is, points, lines, and surfaces, and the quantitative relations existing among them, are the sole objects of geometrical cognition; intuitions of points, lines, and surfaces, and equations or general formulae expressing their relations, constitute the science of geometry. The existence of the relations is necessarily conditioned on the existence of the /84/ quantities; and, since we admit without reservation that the sensuous symbols employed are indeed inexact representations of the real objects of geometrical cognition, it is necessary to define our conceptions of these objects, that is, to determine what actually existent points, lines, and surfaces are, in their objective reality. Now on this point there can be but one opinion. A given solid cube being presented in Space, its surfaces are manifestly nothing apart from the cube itself, but are simply the terminations of its extension in various directions; the only existences are Space and the cube, the general surface not constituting a third independent existence. Similarly, the bounding lines are merely the terminations of the surfaces, and the points, or vertices of the angles, are merely the terminations of the bounding lines. But the relations among these surfaces, lines, and points are unaffected by the physical nature of the cube; and whether the cube consist of iron, or wood, or any other substance whatsoever, so long as the amount of its extension remains unchanged, the geometrical relations remain unchanged. But if it be possible to suppose the cube absolutely annihilated, the bounding surfaces, lines, and points, not being distinct from the body bounded, are likewise annihilated. They are simply limits, and are therefore absolutely conditioned on the object limited; and all mathematical formulae, expressions, or equations, representing their mutual relations, become objectively valueless on the annihilation of the object. The mathematical definitions are strictly correct; the point really has no extension, the line no breadth, the surface no thickness, and the reason is, that they exist only as the limits or terminations of extended matter. For this very same reason, they are inconceivable by themselves; and all sensuous symbols are necessarily inexact, without thereby affecting the absolute exactitude of mathematical formulae, which are not based on these negative conceptions of limitation, but on positive conceptions of the extension they limit. But without positively perceiving or conceiving the object limited, mathematical cognition is impossible; hence mathematical equations and formulas possess objective validity so long only as the objects exist, and subjective validity so long only as their sensuous representations exist. The only possible escape from /85/ this conclusion is to suppose that the annihilated cube leaves an impression of its form as a residuum in pure Space, which in some inexplicable way becomes an object of mathematical cognition; than which no supposition could be more absurd. The most abstract conception of a cube in pure Space is that of a delicate framework of edges, supposed to remain after the rest of the cube has been removed; and the absence of the interior portion has led to the false notion that this framework of edges is a conception of pure Space. If we try to retain a conception of the cube, however, after abolishing this residual framework, the illusion becomes manifest; and nothing is then clearer than that the representation of form, independent of material images, is a psychological impossibility. Points, lines, and surfaces, existing both in nature and in thought only as the limitations of objects, are conditioned on the existence of matter; and since they are only modifications or limitations of Extension, it follows that Extension is the sole object of geometry.
But still it may be retorted, that, granting all this, it remains true that limited spaces are quantities, and therefore geometrically cognizable. Let a cubic yard of matter be sensuously conceived. From this cube of matter let all the interior be supposed to be removed and not replaced by other matter, (a supposition which involves no self-contradiction,) leaving only six superficial laminae of absolutely inappreciable thinness. It must be admitted that they include pure Space, unless we would fall into the absurdity of supposing a part of the universe possible in which Space does not exist. Now although these laminae do not limit Space in the sense of breaking its continuity, they do limit it in the sense that the Space included by them does not extend beyond them; hence the pure Space enclosed is limited, in so far as segregated or distinguished from the rest of Space by definite limits. But since these limits differentiate a portion of Space from infinite Space by exactly mensurable dimensions, which are mathematically cognized, we may say in absolute strictness that the cubic yard contains twenty-seven cubic feet of pure Space mathematically cognized as a given quantity.
To this we reply as follows. The simple fact that it makes /86/ absolutely no difference whether we conceive the laminae as enclosing matter or as enclosing pure Space, proves demonstratively that the attention is entirely abstracted from the interior of the cube, and entirely concentrated on the sensuously-conceived surfaces; in a word, that the object of our mathematical cognition is, not the contents of the laminae, but the laminae themselves as mutually related. Three edges of the cube, meeting at the same solid angle, are measured by reference to a certain unit of extension, namely, a foot; each contains three units of extension, and their product is twenty-seven units of extension. Now how is it that these twenty-seven units of extension are transmuted into twenty-seven units of Space ? Does the multiplication of a quantity change its nature ? Or is it not rather self-evident that every product must be of the same nature as its factors ? Extension, whether of one, two, or three dimensions, is still Extension, and not Space. The twenty-seven cubic feet, therefore, are not to be taken as the measure of the Space enclosed, but simply of the quantity of extension possible within the bounding laminae, which with their mutual relations are the real objects of geometrical cognition. No philosophical mathematician of the present day defines an angle as the quantity of Space enclosed by its sides; and the day is at hand when no philosophical mathematician will so define geometrical solids. We by no means deny, but rather affirm, that these solids do enclose pure Space; what we deny is, that this enclosed Space is the object of mathematical cognition, rather than its enclosing limits and their mutual relations.
In fine, that there can be no mathematical analysis of pure Space follows directly from the fact that it is absolutely continuous, illimitable, and indivisible; for that which has neither parts nor limits can sustain no mathematical relations with itself. That is, no mathematical cognition is possible of an absolute vacuity, wherein no points, lines, and surfaces are conceived to exist; for without these there could be no relations of number and quantity, which would have no basis of existence where things to be numbered and measured were absolutely absent. An infinite unit can bear no numerical or quantitative relation with itself, being secluded by its very /87/ nature from all mathematical categories. Only when specialized or determined in extension, which it underlies and renders possible, does it admit of computation or measurement. It cannot even be said that we know pure Space as possessing three dimensions; for Altitude, Latitude, and Longitude are cogitable only through the mental representation of three lines intersecting mutually at right angles, which representation, being possible through sensuous symbolism alone, is a representation, not of Space, but of Extension. The very word dimension implies mensurability, a predicate inapplicable to pure Space. If we could frame non-sensuous conceptions of dimension, we could cognize Space as tridimensional; as it is, however, we can cognize it, as such, solely in and through Extension.
We cannot close this section without briefly referring to a possible perversion of our doctrine. It may be said that, if the annihilation of matter would involve the annihilation of all mathematical relations, then mathematics becomes purely contingent in its character, and is degraded from the rank of an absolute science to that of the physical sciences. We cannot here discuss a question which goes down deep into the very foundation of the universe. Suffice it to say, that mathematical law is the form of matter, or that in accordance with which alone matter can exist. In a word, mathematical law is a part of the ultimate Nature of Things, on which we evermore abut, uncreated, eternal, absolutely unconditioned; yet, if all things were abolished, there would be no Nature of Things except as the law of their possibility.* This shows that mathematics could not exist as a science, if matter had never existed; and yet that matter depends absolutely on mathematics, and not reciprocally.
* Compare the striking phrase of Baggesen, Philosophischer Nachlass, VoL IL p. 150 : " Die Trias von Zeit, Raum, und Zahl ist, in ihrer nothwendigen ursprunglichen Uebereinstimmung, die Grundform der Schopfung, und somit in ihrem Unterschiede das ewige Gesetz des Weltalls." /88/
The metaphysical nature of the cognition of pure Space will presently be developed, and need not be here anticipated.
10. The antithesis of Space and Extension may be summed up as follows:
A. Subjective Distinctions:
Extension is a simple idea, Space a complex idea.
Extension is always a predicate, Space always a subject.
Extension is known by sense and imagination, Space by pure intellect.
B. Objective Distinctions:
Extension is a congeries of infinitesimal units, Space an infinite unit.
Extension is known a posteriori, Space a priori.
Extension is an object of mathematical science, Space an object of metaphysical science.
11. Before leaving this department of our subject and entering upon the discussion of Space itself, it may be well to present a synoptical table of the possible consequences of identifying and distinguishing the ideas of Space and Extension. /89/
Space and Extension Identified:
1. Space Infinite: matter must likewise be infinite. Cartesian and Spinozistic tenet.
2. Space Finite: matter must likewise be finite. Paralogism; see below, Psychological Analysis.
* "Non si puo fuggir il vacuo, ponendo il mondo finite." Giordano Bruno, Letter to Michel di Castelnovo, prefixed to De l'Infinito Universo e Mondi (ed. Wagner).
3. Space Ideal: matter must likewise be ideal. Logical result of the Kantian system.
Space and Extension Distinguished:
4. Space Infinite, Extension Infinite : the distinction vanishes. This case coincides with No. 1.
5. Space Finite, Extension Finite : ditto. This case coincides with No. 2.
6. Space Ideal, Extension Ideal : the ideality of one involves that of the other. But the distinction itself disproves the ideality.
7. Space Finite, Extension Infinite : a contradictory and unmeaning proposition.
8. Space Infinite, Extension Finite : matter must be finite.
For the sake of completeness, we have here exhibited all the possible suppositions: but of the eight, only the 1st, 3d, and 8th have, so far as we know, ever been defended. Either an infinite Space is permeated with an infinite matter, or else an infinite Space contains a finite matter. In the former case, we adopt the absolute plenum of Descartes; it then becomes as legitimate to say that matter contains Space as that Space contains matter, the ground of distinction between Space and Extension is lost, the dogma that Extension constitutes the interior essence of matter, and the theory of vortices, follow by a natural transition, and the most indefensible parts of Cartesianism are logically necessitated. In the latter case, innumerable absurdities are avoided, and the instinctive affirmations of " common sense " are justified. The only third possibility is the mutual ideality of Space and Extension. Berkeley makes the ideality of extension prove the ideality of matter; (Principles of Human Knowledge, Part I) and Kant, while postulating a pseudo-reality of matter, substantially tends to the same doctrine. ("1st Raum und Zeit mit meiner blossen Subjectivitat gegeben, so hat ja der Idealist recht so hat ja Berkeley recht." Baggesen, Philosophischer Nachlass, Vol. I. p. 180.) As Descartes ignored force (resistance, solidity) as a primary quality of matter, assumed that the essence of matter is extension, and thereby fell into the vagaries above pointed out, so Leibnitz, ignoring extension and assuming that the essence of matter is force, fell into the opposite extreme of teaching that we know matter only as an aggregation of forces, a doctrine which is still held by many who are less conversant with philosophy than with physical science. In this doctrine, however, which Kant adopted, ("Die Substanz im Kaume kennen wir nur durch Krafte." Werke, Vol. IL p. 218.) may be detected the germ of his idealistic theory of Space and Time. Inasmuch as all these extravagances are the result of a defective analysis of Space and a neglect of the distinction between it and Extension, the importance of a correct analysis is evident, and the impropriety of using Space and Extension as synonyms is equally evident.* Empiricism is impossible except through the confusion, intentional or unintentional, of these two terms; for the admission of Space as absolute in itself, and yet at the same time as an object of human cognition, is fatal to the hypothesis that all knowledge is derived from empirical sources. It will appear hereafter how complete is the revolution accomplished by a rigorous adherence to this distinction, in all philosophical systems which, like those of Descartes, Kant, Trendelenburg, Hamilton, Spencer, and others, are based on the fusion of the two ideas.
* James Mill, who identifies the two notions, and makes Space the synonyms of infinite Extension, is consistent with himself and strictly logical when he reduces /90/ Space to a purely empirical notion, and considers it as the abstraction of the quality of extendedness from the concrete extended body. (Analysis of the Human Mind, Chap. XIV. $ 4.) Spencer, who makes the same identification, also teaches the empirical genesis of the idea of Space : " Our conceptions of Time and Space are generated, as other abstracts are generated from other concretes : the only difference being, that the organization of experiences has, in these cases, been going on throughout the entire evolution of intelligence." (First Principles, p. 230.) Compare his Principles of Psychology, pp. 53, 231, 233- 243, 267, 297. So also Stewart, Dissertation, &c., Works, Vol. I. p. 595, Hamilton's edit.; and Samuel Bailey, Letters on the Philosophy of the Human Mind, p. 222, et seq.
PART II. ANALYSIS OF PURE SPACE.
12. The problem of pure Space is threefold, and divides itself into three co-ordinate departments,* as follows:
(1) Psychological Problem. To determine the elements of the idea of Space as a subjective phenomenon.
(2) Logical Problem. To determine the genesis of the idea of Space as an a priori form of sensibility, as an a posteriori generalization, or as the absolute correlation of a necessity of thought with a necessity of things.
(3) Ontological Problem . To determine the nature of Space as an objective existence.
* This general division of the problem was suggested by an admirable article by M. Benard, published in the Dictionnaire des Sciences Philosophiques, s. v. Espace. In the Precis de Philosophie, pp. 119, 121, by the same author, we have found opinions more nearly approximating our own conclusions than those of any other writer.
13. It was stated as the first distinction between Space and Extension, that, while the latter is a simple notion, the former is the indissoluble synthesis of three distinct elements. Now all /91/ these constituent elements of the idea of Space are negative notions. It is erroneous to suppose that a negative notion is the negation of all thought, and conveys absolutely no knowledge; the contrary is easily demonstrable. If, however, as will be proved, we know Space only by negative characteristics, this simple consideration will sufficiently explain why we can never represent it, either as finite or infinite, by the sensuous imagination.
14. The first of the three notions is that of receptivity. Matter is contained in Space, Space contains matter. Extended matter is the only object of perception and of the sensuous imagination; but the existence of Space is the necessary condition of the possibility of Extension. In other words, matter could not be continuous if Space did not exist; yet Space is not given as an object of positive cognition by any faculty. Every act of the pure intellect is the intuition of a relation between terms; and, the proposition, "Matter is contained in Space," being the formula of the receptivity of Space, the terms of this relation are Matter and Space, and the relation itself is that of inclusion. Now in all instances in which the relation of inclusion is positively cognized, the two terms are alike in kind, as when we say that the pitcher contains the water, or the apple contains its seeds; here both terms are extended matter. But in the above formula the two terms are unlike in kind, and the second is simply a negation of the first. It is true that the formula is not precisely interpretable into "Matter is contained in Not-matter" for time, mind, &c. equally come under the category Not-matter, which shows that the terms Space and Not-matter are not strictly convertible. As to its receptivity, Space may be defined, "that which is not matter, but which renders matter possible"; or, more briefly, " the immaterial condition of matter "; or, again, "the absence of matter plus its possibility." But these definitions do more than simply define Space as one term in the formula, " Matter is contained in Space"; for they also state the relation between the terms. So far forth as a single term only, analysis can reach no result beyond this, that Space or absolute vacuity is the negation of matter; we cannot go further and say that this not-matter, as distinguished from other not-matters, /92/ is the necessary condition of Extension, without stating the relation between this term and the other term, matter. This very fact proves the absolute necessity which characterizes the psychological relation between Space and Matter; neither term can be thought independently of the other, and the correlation is therefore unconditioned and indissoluble. Moreover, the receptivity of Space is cognized, not by the sensuous imagination, but by the faculty of pure intellection, or the non-sensuous reason: Space cannot be imagined as containing, but only Extension as contained. When imagination pictures Extension, reason postulates an underlying and including Space; annihilate Extension, and imagination is paralyzed. Space does not exist at all to the sensuous imagination, which is con versant only with that which exists in Space. But Reason asserts that, while Space contains and underlies Extension, it is not the Extension contained, beyond which she is silent. The receptivity of Space, therefore, as an idea of the reason, is simply the negation of matter; and this is the primary and intuitional element of the triple synthesis, on which the other two depend as logical necessities.
15. The second of the three notions is that of unity. All division of Space into parts is arbitrary and seeming only. Extension, so far as perceivable and conceivable, is indefinitely divisible and separable; Space must be indivisible. For suppose it divided into parts, the parts must either be separated or not separated. If they are not separated, Space is still continuous, and the division is illusory, like that of the earth's surface by parallels and meridians, which serve for convenience of reckoning, but indicate no actual partition. But if they are separated, they must either be separated by matter or by vacuity; whereas Space on the one hand is presupposed by matter, and on the other hand is itself, in the absence of matter, vacuity. Consequently Space must be divided from itself by Space, or, in other words, is not divided at all. Hence we see it involves a logical contradiction not to predicate unity of Space. Now human intelligence can form both an imaginative conception and a rational idea of unity, radically diverse in their characteristics because the products of radically diverse mental faculties. Positive unity is the /93/ attribute of a complete, bounded whole, limited in the very fact of its completeness, which can be conceived and grasped as one by the imagination; it necessarily involves, on the one hand, complexity of parts, and, on the other, finitude or limitation; it is the unity of matter which we perceive by the senses, or conceive by the sensuous imagination. Negative unity is the negation of plurality and complexity of parts, and is strictly synonymous with simplicity or indivisibility; it involves necessarily neither finitude nor infinitude, limitation nor illimitation, and is perfectly compatible with either; it cannot be pictured or grasped by the sensuous imagination, and is a purely rational idea; it is the unity which is possessed by mind, and is in fact only the condensation of the proposition, " There are no parts." The imagination, from its nature, can represent only the positive unity of Extension, which is unavoidably conceived as a limited aggregate of parts; whereas the reason, cognizing the negative unity of Space as simply the attribute of indivisibility, discovers no contradiction between the unity and the infinity of Space.
In regard to the unity of Space but two propositions are possible, namely :
1. Space is many.
2. Space is not many, i. e. one.
The propositions are contradictory and mutually exclusive. Now the first, affirming the plurality or divisibility of Space, has just been shown to involve in its very statement a logical absurdity; and is therefore set aside. But the removal of one contradictory necessitates the positing of the other; the second proposition, therefore, which, although utterly inconceivable by the sensuous imagination, is not a contradiction in itself, is necessarily established. The negative unity of the reason, which is seen to be simply the negation of plurality, is the second notion of the triple synthesis.
16. The third notion is that of infinity. Extension, being only a quality of matter, must be limited; for matter, how ever vastly extended, must still be extended in Space, and is consequently limited by the space beyond itself. Extension, being included, is ipso facto limited by the including Space. All limitation of Space, however, is purely arbitrary, and /94/ non-inclusive of that which is to be included. For suppose it limited, it must be limited either by matter or by vacuity; but Space is presupposed by matter, and is itself, in the absence of matter, vacuity : consequently Space must be limited by Space, or, in other words, is not limited at all, but is uniformly continuous. Hence we see it involves a logical contradiction and absurdity to predicate finitude of Space. But here again it must be noticed that the infinity of Space is not the infinity of the sensuous imagination, which is merely the indefinite expansion of Extension. The putative infinity of the imagination is simply the absence of limitation, whereas the cognized infinity of the reason is the impossibility of limitation. The imagination wearies itself in fruitless essays to represent an infinite extension, and gives over its attempt from sheer exhaustion; set a boundary in thought, and, goaded on by reason, which asserts Space still beyond, the jaded pinions of imagination flutter hopelessly onward, until, like Noah's dove, she flies back in awe to her abandoned home. The imagination cannot compass infinity; but the reason can.
In regard to the limitation of Space but two propositions are possible, namely :
1. Space is limited.
2. Space is unlimited.
These propositions are contradictory and mutually exclusive. Now the first, affirming the limitation of Space, has just been shown to involve a manifest absurdity, and is therefore set aside. But this necessitates the positing of the second, which, although utterly inconceivable by the sensuous imagination, is not contradictory in itself. The infinity of the reason, therefore, which is seen to be simply the absolute negation of limitation is the third notion of the triple synthesis.*
*"Non, Ariste, 1'esprit ne voit pas une etendue infinie, en ce sens que sa pensee ou sa perfection e'gale une e'tendue infinie. Si cela etait, il la comprendrait, et il serait infini lui-meme Mais 1'esprit voit actuellement que son objet immediat est infini : il voit actuellement que 1'e'teudue intelligible est infinie. Et ce n'est pas, comme vous le pensez, parcequ'il n'en voit pas le bout; car si cela etait, il pourrait espeVer de le trouver, ou de moins il pourrait douter si elle en a, ou si elle n'en a point; mais parcequ'il voit clairement qu'elle n'en a point." Malebranche, Entretiens sur la Me'taphysique, Prem. Entr. ix.
17. We have thus completed the psychological analysis of /95/ the idea of Space as it exists in the human mind. The conclusiveness of the arguments which demonstrate the unity and infinity of Space can only be impugned by impugning the validity of the reductio ad absurdum. The general result may be summed up as follows. We know Space only by negative characteristics, and these are cognized, not by the sensuous imagination, but by the non-sensuous reason. Our idea of it is a triple synthesis, the constituent elements of which are three negative notions, receptivity, unity, and infinity; the first is the negation of matter, the second the negation of divisibility, and the third the negation of limitation. To some one of these three, every other possible predicate of Space may be reduced; hence the analysis is exhaustive. For instance, to the receptivity of Space may be referred its penetrability, incorporeity, incorruptibility, &c.; to its unity may be referred its simplicity, uniformity, &c.; and to its infinity may be reduced its immutability, immobility, &c. We have not included necessity in the synthesis, inasmuch as it attaches to the idea of Space as the logical condition of the possibility of Extension, and to each of the three notions as the constituent elements of this necessary idea; but it is not a distinct element in the synthesis.*
* "1. L'idce de corps est une idee contingente et relative, tandis quo 1'idee d'espace est une idee necessaire et absolue; 2. 1'idee de corps implique 1'idee de limite, et 1'idee d'espace implique 1'absence de toute limite; 3. enfin 1'idee de corps est une representation sensible, et 1'idee d'espace est une conception pure et toute rationnelle." (Cousin, Hist. de la Phil, du XVIII* Siecle, Vol. II. p. 138.)
For Cousin's purpose, the refutation of Locke, this professedly imperfect contrast of Space and Matter is sufficient; but as an analysis of the idea of Space it is scientifically defective and erroneous. Not to mention the omission of receptivity and unity, the necessity and rational origin of the idea are co-ordinated with one of its elements, infinity. The necessity of the idea, psychologically considered, is simply the modality of the proposition, "Space exists "; or, in other words, the energy of its mental affirmation. The origin of the idea is a question extraneous to that of the determination of its constituent elements.
18. The theories hitherto advanced respecting the genesis of the idea of Space fall into two main classes, that of Transcendentalism and that of Empiricism; the former deriving the idea a priori from the ego, the latter deriving it a posteriori from the non-ego. Of the former class, Kant is the originator /96/ and principal champion; of the latter class, which is the more numerous of the two, the latest and perhaps most ingenious representative is Herbert Spencer. We shall confine ourselves to the criticism of their arguments. Kant's argument is long, abstruse, and marked with the obscurity of his style and terminology; but we shall endeavor to present it in as clear, forcible, and condensed a form as possible. To avoid repeated references, we refer once for all to the Introduction and Part First of the Critique of Pure Reason.
19. Knowledge is of three kinds. Knowledge a posteriori, or empirical knowledge, is that derived exclusively from experience; as, for instance, the proposition, " Snow is white." Pure knowledge a priori is that which is independent of experience, not of this or that experience only, but absolutely of all experience; as, for instance, "Two lines cannot enclose space." Impure knowledge a priori is that which contains both a priori and a posteriori elements; for instance, the proposition, "Every change has a cause," contains the a priori principle of causality and the a posteriori conception of change, which can only be derived from experience. The criterion of knowledge a priori is necessity and universality; if a proposition involves the idea of strict necessity, and absolute, not inductive universality, it is a proposition a priori. Mathematics affords an example how far we may carry pure a priori knowledge independently of all experience.
Judgments are of two kinds. Analytical judgments are those which analyze or evolve distinctly in the predicate what is contained obscurely in the subject, without adding anything to the conception of the subject; for example, "All bodies are extended," a proposition in which the predicate is extended "is involved in the very conception of the subject. Synthetical judgments are those which by means of the predicate add to the conception of the subject a new and augmentative conception; for example, the proposition, "All bodies are heavy," adds the conception of weight to that of the subject, which does not involve it.
Now judgments a posteriori are always synthetical, inasmuch as reflection only, not experience, can analyze a conception into its elements. But judgments a priori may be either analytical, /97/ as in the decomposition of a given conception independently of experience, or synthetical, as in the judgment, "Every change has a cause." Here the conception cause is augmentative, and not contained in the conception change; yet the judgment possesses the distinctive marks of strict necessity and universality, and is therefore a priori. It is an indubitable fact, that synthetical judgments a priori constitute the fundamental principles of all mathematical science; for instance, the axiom, " A straight line between two points is the shortest," is plainly synthetical, inasmuch as the conception straight is qualitative merely, while that of shortest is quantitative. Again, the proposition, 7 + 5 = 12, is synthetical, since the conception of the union of 7 and 5 does not contain the conception of their arithmetical sum: this will be still more evident by the substitution of larger numbers. The science of physics likewise lays down synthetical judgments a priori as principles; as, "In all changes of the material world, the quantity of matter remains unchanged"; and, "In all communication of motion, action and reaction must always be equal"; truths which transcend possible experience. It is thus evident that the most stable of human sciences can have no validity or worth whatsoever, unless we admit the legitimacy of synthetical judgments a priori.
Thus is developed the universal problem of pure reason, namely, "How are synthetical judgments a priori possible ?" For look at the indubitable facts. We find the human intellect positing certain propositions, not only as actually true, but as necessarily and universally true; and these propositions are not mere evolutions and analyses of subjective conceptions, to which necessity and universality may well attach, but synthetical assertions laid down as objective laws which govern the universe. Experience cannot be the origin of these propositions, for experience, whether external or internal, can certify only what is, never what must be. How, then, can reason, in dependently of and antecedently to all experience of external objects, enact absolutely necessary and universal laws regarding it? The stability and certitude of human knowledge demand imperatively a full answer to this question; and its discovery is one great object of the Critique of Pure Reason. /98/ The solution of this problem, to be intelligible, must be prefaced by a few definitions.
The only manner in which we derive immediate knowledge from objects is by means of an intuition (Anschauung)* An intuition can take place only so far as we are affected by objects; and our receptivity of impressions from objects is called the sensibility ( Sinnlichkeit) , which embraces an external and an internal sense. The sensibility alone furnishes intuitions, which, being thought (gedacht) by the understanding (Verstand), furnish conceptions (Begriffe). When an object is presented to the sensibility, that which corresponds to the sensation is called the matter of the phenomenon; while that which causes that the various constituents (Mannigfaltige) of the phenomenon can be arranged under certain relations, the mould, as it were, in which they are run, is called its form. But that in which our sensations are merely arranged, and by which they are susceptible of assuming a certain form, is not itself sensation.** The matter of phenomena is given a posteriori; but their form lies in the mind a priori, and can be regarded apart from sensation. Now the intuition of the shifting and contingent matter of phenomena is empirical intuition; but the intuition of their constant and necessary form is pure intuition, that is, wholly independent of sensation and experience. /99/
* It is all-important that the peculiar use of this word, which plays so distinguished a part in Kant's system, should be correctly understood. An intuition is the direct contemplation of an individual object, whether of sense or of thought; a looking at the object, either by the external or the internal sense, and hence the direct product of the sensibility as affected by the object. It includes perceptions by the outward senses, representations by the sensuous imagination, and all representations of objects of thought in general as distinguished from the relations of these objects. "The understanding was defined above only negatively, as a non-sensuous faculty of cognition. Now, independently of sensibility, we cannot possibly have any intuition; consequently the understanding is no faculty of intuition." (Krit. d. r. Vern., p. 69.) Hamilton gives the following as one of five senses of the word: "3. The knowledge which we can adequately represent in imagination, in contradistinction to the ' symbolical ' knowledge which we cannot image, but only think or conceive, through and under a sign or word. (Hence, probably, Kant's application of the term to the forms of the sensibility, the imaginations of Time and Space, in contrast to the forms or categories of the Understanding.)" Reid's Works, Note A, sect. 5, p. 759. See Mansel, Proleg. Log., p. 22, note.
** The vagueness of meaning here is irremediable, and, to quote the fine phrase of Montbeillard, should be ascribed to " 1'obscurite toujours inherente a I'erreur." De 1'Ethique de Spinoza, p. 13.
Thus, abstracting from the intuition of a body all its empirical attributes, we find extension and shape left behind as the residuum of the intuition; and these, therefore, belong to pure in tuition. So much by way of preface and definition of terms.
Space cannot be a conception derived from outward experience. For the first act of intelligence, namely, the distinction between what is within and what is without me, is possible only through a prior representation of Space, as the foundation of this cognition. Without this antecedent representation, I could not distinguish between the different parts of Space, nor discern the difference between within and without. It is evident, therefore, that, instead of the conception of Space being derived from external experience, external experience itself is only possible through this very conception.
Geometry determines the properties of Space synthetically, and yet a priori. Now the source of all synthetical propositions is intuition; mere analysis of conceptions can furnish only analytical propositions. Consequently the origin of geometrical principles must be intuition. But all geometrical principles possess the characteristics of necessity and universality, and therefore cannot be derived from experience. Consequently, the origin of such principles must be, not merely intuition, but pure intuition a priori, wholly independent of all experience. Now how can the mind, anterior to all perception of external objects, possess a priori knowledge of their relations, and lay down absolutely necessary and universal laws regarding them? Such a fact (and that it is a fact the existence of geometry as a science demonstrates) can be possible only on the supposition that the mind itself creates its geometrical objects in an intuition a priori, cognizes a priori their necessary relations, and then imposes these cognized relations on outward objects in the very act of perception.* In /100/ other words, Space, whose properties geometry can determine antecedently to all objective experience of them, can be only a subjective and regulative intuition, the form of all phenomena of the external sense, the subjective condition of the sensibility in accordance with which alone external intuition is possible. The sensibility is thus, as it were, a vase which imparts its own form to whatever liquid is poured into it, while the liquid possesses no determinate form in and of itself.
* To show that Kant is not here misrepresented, we cite the following : " Ihr seht Euch genothigt, zur Anschauung Eure Zuflucht zu nehmen, wie es die Geome tric auch jederzeit thut. Ihr gebt Euch also einen Gegenstand in der Anschauung; von welcher Art aber ist diese, ist es eine reine Anschauung a priori oder eine em- pirische ? Ware das Letzte, so konnte niemals ein allgemein gultiger, noch weniger ein apodiktischer Satz daraus werden; denn Erfahrung kann dergleichen niemals liefern. Ihr miisst also Euren Gegenstand a priori in der Anschauung geben, und auf diesen Euren synthetischen Satz griinden." Krit. d. r. Vernunft, p. 53.
20. Such, in a very condensed form, rearranged and stripped as far as possible of obscurity and technicalities, is the powerful argument for the ideality of Space devised by the most subtle analyst that ever lived. Since his day, the majority of philosophers have accepted as an established fact the a priori origin of the idea of Space, even while rejecting the doctrine of its exclusive subjectivity. In regard to Kant's argument, Hamilton expresses the following opinion : " The analysis of Kant, independently of all that has been done by other philosophers, has placed this truth [that Space is a fundamental law of thought] beyond the possibility of doubt, to all who understand the meaning and conditions of the problem." (Lectures on Metaphysics, p. 346.) On the same page, however, he claims that Space is known both a priori and a posteriori: and this mature position must be regarded as essentially modifying his earlier complete acceptance of Kant's doctrine.* But Kant's argument professes to be demonstrative; and his point is, not merely that Space is a form of intuition, but that it cannot possibly be anything else. To admit the conclusiveness of the argument, and yet reject the conclusion, is strange enough. If Kant's reasoning is sound, then Space does not objectively exist; but if Space does objectively exist, then Kant's reasoning is unsound. It is treason to intelligence to admit that what is false in fact can be true in logic. There must somewhere be fallacy in the process, if there is error in the result. Notwithstanding Cousin's attempted refutation in his "Philosophies de Kant," he fails to point out the flaws in Kant's argument, and contents himself with an appeal to common sense, and with general criticisms /101/ of the whole system. For the credit of philosophy, we hope that some sufficient refutation of this generally rejected conclusion exists, although we have not met it. Meantime we offer the following.
* Discussions on Philosophy, pp. 23, 35, 37. His disciple, Manse, cannot decide whether Space is subjective or not (Prolegomena Logica, p. 82).
21. Against this theory we have two objections to bring forward, either of which, if substantiated, must subvert it. The first is, that from a false premise at the outset a thread of fallacy runs through the whole; this false premise is the assumed possibility of pure knowledge a priori, and is based on the neglect of the broad distinction between Space and Extension, upon which we have already laid so much stress. The fact that Hamilton himself is guilty of this same neglect to a much greater degree, accounts for his not detecting it in another. The second is, that, admitting the possibility of synthetical judgments a priori, Kant does not account for their necessity and universality, as he professes to do, but merely carries the question one step further back, and leaves it unanswered at last.
22. All knowledge is relative; and the antithesis of a thinking subject and an object thought is the general law of its relativity. It is possible only in accordance with certain regulative special laws of thought. These laws, which are truly a priori, and do not depend on experience, are merely the modes of activity to which the mind, from the very nature of intelligence, is necessarily self-determined on the presentation of objects of cognition. Anterior to the presentation of objects, they exist only potentially, as the capacity for knowledge, and are in themselves nothing until actualized as the modes of individual cognitive acts. In thought, the laws and the acts they regulate are separable; in fact, they are inseparable. The mind cannot create objects of its own cognition; that were to suppose a pure, spontaneous mental activity, which would violate the law of the relativity of knowledge. All objects must be presented to it, and hence all objects of knowledge whatsoever are, mediately or immediately, empirical. What the object contributes to the act of knowledge, therefore, is a posteriori; what the mind contributes is a priori. It is thus evident that all knowledge without exception must be impure, that is, must involve both a priori and a posteriori elements; and the division of it into three kinds, as /102/ a posteriori, pure a priori, and impure a priori, is consequently deceptive. The laws of knowledge are a priori, and absolutely in dependent of experience; but knowledge itself, being from its very nature the knowledge of objects and of their relations, is not possible until the presentation of objects, and is consequently so far dependent on experience. Laws are known only in phenomena; phenomena are known only according to laws; hence every act of knowledge involves both an object of the act and laws which regulate the act. Let us take, for example, the proposition, " Snow is white," which appears to be, and according to Kant is, purely empirical. An indivisible object, white snow, is empirically presented to cognition; and according to the a priori laws of intelligence the indivisible object is separated in thought into substance (snow) and attribute (white), that is, into subject and predicate. Previous to the presentation, the law which determined this separation had only potential existence; subsequently, it existed only as the manner of knowing the object, the mode of the act of knowledge, although by abstraction it may itself become an object of knowledge. Again, the proposition, " A straight line between two points is the shortest,'* is claimed by Kant as purely a priori, and independent of experience; but points and lines are objects of knowledge, and experience alone can furnish them. Mathematics, which is brought forward by Kant as a " brilliant example " of pure knowledge a priori, has been shown (Sec. 9) to be wholly dependent upon empirical conceptions; number, quantity, unity, plurality, points, lines, surfaces, solids, etc., are conceptions derived from experience, yet mathematics would be impossible without them. Hence we must conclude, that knowledge purely a priori and knowledge purely a posteriori are alike impossible.
23. It is evident, therefore, that we must slightly modify the question which Kant proposes as the "universal problem of pure reason." It should no longer read, " How are synthetical judgments a priori possible?" for they are not possible; but "How can any synthetical judgment be absolutely necessary and universal?" which is the real fact Kant means to explain. Induction from experience can warrant only conclusions of comparative universality, the legitimate extent of the /103/ conclusion being regulated, ceteris paribus, by the number of instances embraced in the induction. But we find geometry and other sciences collecting from a few instances, often from only one, conclusions which are claimed to be coextensive with the universe, and necessary in the nature of things. Since experience cannot be the source of such judgments, what is their source ? The important error in Kant's answer to this question is the result of his slight error in stating it. His reply is this. Absolutely prior to and independent of experience, we possess a pure a priori intuition of Space in itself, unmixed with any empirical elements; therefore we can determine the relations and properties of the parts of Space irrespectively of external objects; therefore these relations and properties are determined solely by the nature of our own minds, and are necessarily imposed on all perceived objects as the very conditions of their perception; therefore the judgments we pass concerning Space a priori will apply necessarily and universally to all objects a posteriori; and on no other theory than this of the ideal nature of Space as the mere form of the sensibility, can such necessity and universality be possibly accounted for. Now nothing can be clearer than that by the "pure intuition of Space a priori" Kant means the mental image we form of empty Space; but we have shown conclusively (Sec. 6), that of empty Space, whether finite or infinite, no imaginative conception is possible, and that our putative imagination of Space is really that of Extension merely. It follows inevitably, that, inasmuch as Extension is only a quality of matter, and all knowledge of matter is empirical in its origin, such a thing as a "pure intuition of Space a priori " is an utter impossibility. It only remains to show that this is the fair and true interpretation of Kant's phrase, which is easily done by quoting his own illustrations of its meaning: "We cannot cogitate a geometrical line without drawing it in thought, nor a circle without describing it, nor represent the three dimensions of Space without drawing three lines from the same point perpendicular to one another. We cannot even cogitate time, unless, in drawing a straight line (which is to serve as the external figurative representation of time), we fix our attention on the act of the synthesis of the manifold, whereby," &c. (Kritik d. r. Vern., p. 748.) And again, more plainly still: "Thus, if I take away from our representation of a body all that the understanding thinks as belonging to it, as substance, force, divisibility, &c., and also whatever belongs to sensation, as impenetrability, hardness, color, &c., yet there is still something left us from this empirical intuition, namely, Extension (Ausdehnung) and Shape (Gestalt). These belong to pure intuition, which exists a priori in the mind, as a mere form of the sensibility, and without any real object of the senses or any sensation." (p. 32) Even if some may be found still disposed to vindicate the imaginability of Space, surely no one will be hardy enough to assert that shape, which is merely the quality of Extension as limited in an object, can be other than an empirical representation of the sensuous imagination. Extension with its modification, shape, being only a primary quality of matter, and hence cognized only a posteriori, Kant's pure intuition a priori reduces itself to an empirical intuition a posteriori. Geometry, which according to Kant "determines the properties of Space synthetically, yet a priori" is not the science of Space, but of Extension; the properties of Extension are mathematical, and furnish positive conceptions of the sensuous imagination, while the properties of Space (receptivity, unity, and infinity) are metaphysical, and furnish only negative conceptions of the non- sensuous reason. (Kritik d. r. Vern., p. 32. ) (See 9.) The mathematical definitions of a point as position without extension, a line as length without breadth, <fcc., are confessedly impossible to be imagined; and geometry is consequently obliged to represent these notions by sensuous symbols. Thus it is clear that Kant's idealistic theory of Space has its root in the confusion of Space with Extension; and a fatal flaw in the argument is detected.
24. The second objection is still more sweeping. The problem is this : How shall we account for the necessity and universality of synthetic judgments a priori? We admit unconditionally, that mere induction from experience can never possess these characteristics. We will likewise admit, for the sake of argument, that synthetical judgments purely a priori and pure intuitions a priori are possible, and actually exist in human intelligence. But notwithstanding these admissions, we maintain that the problem is not solved by the offered solution. /105/
For an illustration, let the following theorem be taken: "The sum of the three angles of any triangle is equal to two right angles." We draw or conceive a particular triangle for the sake of the demonstration; and, having gone through a series of successive intuitions, we arrive at last at the demonstrated truth of the theorem in this particular instance. This is the utmost that experience will warrant us in concluding. But this conclusion is instantaneously and irresistibly extended to all possible triangles, although such an extension of it is objectively inadmissible without being fully accounted for. The only possible explanation, according to Kant, is to hold that the mind determines this property of triangles in an intuition of Space a priori; that this pure intuition, being the form of all empirical intuition, necessarily imposes its own laws upon all our intuitions of objects a posteriori; and that thus the theorem is proved strictly necessary and absolutely universal. But this explanation is no explanation at all. It is impossible, says Kant, to derive a universal conclusion from a solitary instance in intuition a posteriori; we must derive it from intuition a priori. But if a conclusion from a single instance in empirical intuition can possess only limited validity, how can a conclusion from a single instance in pure intuition possess unlimited validity? In either case, the universal is deduced from the particular; what is the difference in the two cases ? It does not follow that the theorem is true of all triangles possible to pure intuition, simply because it is true of one, unless it equally follows that the theorem is true of all triangles possible to empirical intuition, because found true of one triangle. In fact, in the effort to escape experience outwardly, Kant unwittingly admits experience inwardly into his postulated intuition a priori; but experience is experience still, whether out ward or inward, and can never of itself yield a universal and necessary principle. Without some further hypothesis, the problem is left precisely where it was taken up; a form of pure intuition itself must be postulated on exactly the same grounds as a form of empirical intuition; then a form of this form, and so on ad infinitum. In reality, Kant did not foresee this difficulty, and makes no provision in his theory against it; yet it renders the subjectivity of Space a purely gratuitous /106/ assumption, contradicting the voice of mankind, yet explaining nothing. If we come no nearer to a solution of the proposed problem by taking one of an endless series of steps, the law of parsimony suggests the propriety of omitting that one, thus leaving the solid conviction of "common sense" unassailed. It would be foreign to our present purpose to go further in search of an answer to the really great problem of the origin of necessary and universal judgments; by setting aside an illusory answer to it, it remains still an open question, as before. In fine, we have found, first, that Kant's famous hypothesis in regard to Space and Time, resting on the possibility of pure knowledge and pure intuition a priori, is founded on the confusion of Space and Extension; and, secondly, we have found that, even admitting the possibility of such knowledge, the hypothesis fails to solve the problem it was devised to solve.
25. Turning now to the empiricistic view, we will briefly mention the leading points of Mr. Spencer's argument. He professes to fuse the transcendental and empiricistic hypotheses, and reduce them to harmony and mutual consistency. (Principles of Psychology, p. 23, foot-note; also p. 577.) But inasmuch as experience is still regarded as the sole source of the idea of Space, although in a new and modified sense, and inasmuch as the essence of the a priori theory is the irreducibility of this idea to the category of empirical cognitions, his attempted reconciliation is evidently merely a partisan advocacy of one of the theories to be reconciled. In regard to the immediate perceptibility of extension, it is impossible to make out a consistent doctrine from Mr. Spencer's statements. On the one hand, he says: " Without doubt, by the adult human consciousness all tactile resistances are unconditionally known as coexistent with some extension; and all tactile extensions are unconditionally known as coexistent with some resistance." (pp. 208, 209) And again: "The two terms of the relation, extension and resistance, cannot be cognized in absolutely the same state of consciousness. The apparently incessant presentation of both is really a rapid alternation, an alternation so rapid as to produce the effect of continuity."(p. 514.) Whence, waiving all criticism of the apparent contradiction, that a coexistence can /107/ be a sequence, it is fair to infer that, at least during an inappreciable time, extension is immediately cognized. On the other hand, many passages seem to state more explicitly that extension is not immediately cognizable: We know extension only through a combination of resistances: we know resistance immediately by itself." (p. 266; comp. pp. 191, 218, 224.) "Reduced to its lowest terms, then, extension is knowable as some series of states of consciousness." (p. 298) But while thus denying that Extension is an object of immediate perception, and while identifying Extension with Space, ( First Principles, p. 48) he still holds to the empirical genesis of the idea of Space,* and denies that it is a necessity of thought per se, (Principles of Psychology, p. 231) or can be known by us as an absolute existence. First Principles, p. 231. His theory of the origination of the idea is peculiar, and we will present it in his own words: " Body and Space being distinguished as resistant extension and non-resistant extension, it is impossible to treat of extension in any of its modes without virtually treating of them both." (Principles of Psychology, p. 230.) Fundamentally, Space and co existence are two sides of the same cognition. On the one hand, Space cannot be thought of without coexistent positions being thought of; on the other hand, coexistence cannot be thought of without at least two points in space being thought of. A relation of coexistence implies two somethings that coexist."(p. 243) All modes of extension are resolvable into relations of coexistent positions. Space is known to us as an infinitude of coexistent positions that do not resist; Body, as a congeries of coexistent positions that do resist." (p. 297) It was pointed out that our inability to banish from our minds the idea of Space was readily to be accounted for on the experience hypothesis : seeing that, if Space be a universal form of the non-ego, it must produce some corresponding universal form in the ego, a form which, as being the constant element of all impressions presented in experience, and therefore of all impressions represented in thought, is independent of every /108/ particular impression; and consequently remains when every particular impression is banished. ( Principles of Psychology, p. 231; compare pp. 53, 233, 577-583.) For, joined with this hypothesis [the development hypothesis], the simple universal law that the cohesion of psychical states is proportionate to the frequency with which they have followed one another in experience, requires but to be supplemented by the law that habitual psychical successions entail some hereditary tendency to such successions, which, under persistent conditions, will become cumulative in generation after generation, to supply an explanation of all psychological phenomena; and, among others, of the so-called 'forms of thought.' If there are certain relations which are experienced by all organisms whatever, relations which are experienced every instant of their waking lives, relations which are experienced along with every other experience, relations which consist of extremely simple elements, relations which are absolutely constant, absolutely universal, there will be gradually established in the organism answering relations that are absolutely constant, absolutely universal. Such relations we have in those of Space and Time." (Ibid, 578, 579) "Corresponding to absolute external relations, there are developed in the nervous system absolute internal relations, relations that are developed before birth; that are antecedent to, and independent of, individual experiences; and that are automatically established along with the very first cognitions I hold that these pre-established internal relations, though independent of the experiences of the individual, are not independent of experiences in general; but that they have been established by the accumulated experiences of preceding organisms." (p. 583)
* Principles of Psychology, pp. 233-243, 267; First Principles, pp. 229, 230.
26. On this ill-jointed theory we will make but two criticisms, and those brief ones.
In the first place, Mr. Spencer violates the distinction between Space and Extension both in word and in thought. He violates it in asserting that Space and Extension are "convertible terms." He violates it in holding that Body and Space are merely "modes of extension," whereas Extension is a mode, or rather attribute, of Body, and Space is a mode of nothing. /109/ He violates it in holding that, while the relation of coexistence implies "some things that coexist," Space is known as an "infinitude of coexistent positions"; thereby confounding the unity of Space with the plurality of Extension. And, in fine, he violates it in holding that "relations of Space" are empirically cognized; whereas relations of Extension are the objects of empirical cognition, and Space, having no relations with itself, because without parts, is only known a priori as the condition of Extension.
In the second place, even ignoring this confusion of Space with Extension and the numerous errors traceable to it, it is absurd to suppose that the non-ego could impress on the ego its universal conditions or forms, except in a universal experience. Experience of only an infinitesimal fraction of the non-ego, which is all that is possible to an individual, will warrant no conclusions transcending the exact limits of that experience; whatever conclusions of intelligence do transcend these limits must be derived from other than empirical sources. Even the generalization of the actual experiences themselves presupposes a power not to be derived from them; much more, then, will they fail to account for principles necessarily ex tended beyond all possible experience, so as to include the totality of the universe. To suppose that the extension of such principles is accounted for by the law of hereditary transmission, or legitimated by the accumulation of experiences in successive organisms, is singularly irrational. Does this hypothesis account for the necessity of the law of contradiction, as well as for the necessity of Space ? It cannot rationally be denied that there are absolute elements in thought per se, that there is an absolute and necessary nature of intelligence; and to hold the doctrine of the relativity of human knowledge in such a way as to deny the existence of these absolute elements, is deliberately to wink out of sight half the data of the great problem offered for our solution. The only way to be consistent in empiricism is absolutely to deny the possibility of the absolute, whether in thought or in existence; and such consistency is, on the very face of it, glaring inconsistency.
27. What, then, is our own answer to the logical problem of pure Space? Is Space a Law of Thought, a Law of Things, or both a Law of Thought and a Law of Things? /110/
That Space is a law of thought has been shown above, in the Psychological Analysis. Whether we can regard it also as a law of things, depends entirely on the theory of perception we adopt. Admitting, as we do, Hamilton's theory of direct and immediate perception of the external world, but one reply is possible. For the nexus between Space and Extension is absolute and unconditioned; if one is subjective or objective only, the other is subjective or objective only. If I do not immediately cognize Extension as an objective reality, there is no ground whatever for assuming the objectivity of Space. But if I do immediately perceive extended matter exterior to my own percipient mind, then Extension exists objectively; and if Extension exists objectively, Space, its conditio sine qua non, also exists objectively. The original and central datum of consciousness, namely, the distinction of self from not-self, be comes thus the demonstration of Space as an objective reality. It is unnecessary here to canvass the arguments for admitting this indivisible duality of consciousness; the labors of Hamilton in this field render the investigation superfluous, and have earned for him an imperishable wreath of laurel.
28. A succinct historical summary of the most striking answers heretofore given in modern philosophy to the inquiry concerning the ontological nature of pure Space, will properly precede the answer to be deduced from the conclusions we have already reached.
Descartes conceived the extension of matter to be its interior essence,* and was consequently compelled to accept the Platonic identification of matter and Space.** The inevitable and logical corollaries from this theory are the infinity of matter and the impossibility of vacua; the latter Descartes admitted unconditionally,*** while the hesitation with which he accepted /111/ the former does more honor to his candor than to his philosophical consistency.**** The same doctrine is likewise expounded by Spinoza, and apparently adopted without modification.*****
* Je conçois son etendue [i. e. l'etendue de la matiere] ou la propriete qu'elle a d'occuper de l'espace, non point comme un accident, mais corarae sa vraie forme et son essence." Le Monde, Chap. VI. )
** La meme etendue en longueur, largeur, et profondeur qui constitue 1'espace constituele corps." Princ. de la Phil., Seconde Partie, 10. )
*** "Pour ce qui est du vide, au sens que les philosophes prennent ce mot, a savoir pour un espace ou il n'y a point de substance, il est Evident qu'il n'y a point d'espace en 1'univers qui soil tel, parceque," &c. Ibid., 16.
**** "Nous saurons aussi que ce monde, ou la mature etendue qui compose 1'univers, n'a point de bornes, parceque," &c. (Ibid., 21.) But this must be modified by the following extract from a letter to Henry More, dated 1649 : " Ne regardez point comme une modestie affectee, mais commeune sage precaution a mon avis lorsque je dis qu'il y a certaines choses plutot indefinies qu'infinies Pour le reste, comme 1'etendue du monde, le nombre des parties divisibles de la matiere, et autres choses semblables, j'avoue ingenuement que je ne sais point si elles sont absolument infinies ou non." Likewise by the following extract from (Euvres ine'dites de Des cartes, published for the first time in 1859 by M. Foucher de Carcil (Tom. I. p. 66): " En disant qu'il est inde'fini, dit-il, nous ne nions pas que peut-etre dans la rdalite ii ne soit fini; mais nous nions seulement qu'une intelligence comme la notre puisse comprendre qu'il ait des bornes ou des extremites quelconques." That is to say, if our intelligence is a trustworthy interpreter of the reality of things, Descartes held strictly to the infinity of matter.
***** "Spatium ab extensione non nisi ratione distinguimus, sive in re non differt." (Princip Phil. Cartes., Pars II. Def. VI.) " Corporis sive materiae nattira in sola ex tensione consistit Spatium et corpus in re non difFerunt." (Ibid., Prop. II. et Coroll.) " Qui autem durationem et tempus ante res creatas imaginantur, eodem prsejudicio laborant ac illi, que extra materiam spatium fingunt, ut per se satis est manifestum." (Cogitata Metaphysica, Pars II. Cap. X. 5.) In these works we have no criterion by which to separate Spinoza's own opinions from those of Des cartes, whose doctrine he professes to explain; yet no other theory will consist with Spinoza's reduction of all the attributes of the One Substance to two, Thought and Extension. Compare also Epist. IV. Opera, Vol. II. p. 151 : *' Si una pars ma- teriee annihilaretur, simul etiam tota extensio evanesceret."
Leibnitz saw the glaring untenability of the Cartesian hypothesis, and strenuously opposed it, on the ground that, if extension were the essence of matter, it should account for all the essential qualities of matter; whereas the force and natural inertia of material substance are not explicable by extension. He says : "Beside extension, there must be a subject which, is extended, that is to say, a substance to which it pertains to be repeated or continued. For extension signifies only a repetition or continuous multiplicity of that which is extended, a plurality, continuity, and coexistence of parts; and consequently it does not suffice to explain the essence of the substance extended or repeated, since the notion of the substance is anterior to that of its repetition." (Journal des Savans, 18 Juin, 1691; and 5 Janvier,) 1693 This is excellently said; and is indisputably true of extension, so far as it is a multiplicity of continuities which are ultimately absolute; but by failing to observe the distinction between Space and Extension, Leibnitz passed, by an easy transition, to the theory that Space itself is nothing apart from material bodies, of which it is the mutual relation, an "order of coexistences, as time is an order of successions." * He reasons thus: "Nevertheless, although it is true that, in conceiving body, we conceive something beside Space, it does not follow that there are two extensions, that of Space and that of body. When we conceive several things at once, we conceive something beside number, to wit, the things numbered; and yet there are not two multitudes, the one abstract, that of number, and the other concrete, that of the things numbered. So we should not imagine two extensions, the one abstract, that of Space, and the other concrete, that of body." (Nouveaux Essais, Livre II. Chap. IV. 4.) The annihilation of material bodies, by this theory, would involve the annihilation of Space itself, their mutual relation of course depending on their existence; and the idea of Space, being degraded from the rank of unconditioned to that of conditioned ideas, is no longer necessary, but contingent. This fact has made the theory of Leibnitz a favorite one with the sensationalists, Condillac, for instance. **
* Correspondence of Clarke and Leibnitz, in 1715-6, on Principles of Natural Philosophy and Religion : Leibnitz's Third Paper, $ 4.
** "Dans la realite' des choses, 1'e'tendue n'est done que 1'ordre qui est entre leg monades et les aggre'gats." And in a foofc-note : " C'est-lk ce qu'entend Leib nitz, quand il dit que 1'e'tendue n'est que 1'ordre des co-existans." (Traite des Sys- temes, CEuvres Completes, Tome II. p. 139.) Bailly, in his Eloge de Leibnitz, espouses the same theory. Compare D'Alembert, Melanges, Tome V. xvi. : " S'il n'y avoit point de corps et de succession, 1'espace et le temps seroient possi bles, mais ils n'existeroient pas." Even Calderwood (Philosophy of the Infinite, 2d ed., p. 334) gives up the infinity of Space, and (p. 331) resolves Space into the "relative position" of bodies : in his first edition he held more philosophical views.
In opposition to this view, Dr. Samuel Clarke advanced an other equally indefensible. Space, being evidently independent of the objects it contains, and not contingent, must be some thing more than their mere order or mutual relation; it must be an actual existence. But every existence is necessarily either a substance or an attribute; and, as Space cannot be a substance, it must be an attribute. Since Space, however, is infinite, and since only an Infinite Being can possess infinite attributes, Space must be the immensity of God himself. This argument was probably suggested by Clarke's friend, Sir Isaac Newton, at least in its germ.* Dr. Isaac Barrow, the learned preceptor of Newton, was more cautious in his language concerning Space, and can hardly be considered responsible for the opinion of his pupil.** The weaknesses of Clarke's theory, however, being manifold and obvious, engaged him in much controversy. In 1713, a correspondence with Bishop Butler (then a young man) on Necessary Existence developed Clarke's famous argument a priori for the necessity of a God, founded on the necessity of Space, his attribute; in this contest, victory inclined to the side of the younger combatant. In 1715 and 1716, a joust took place between Clarke and Leibnitz in defence of their respective theories, and each knight succeeded in unhorsing his antagonist. The dispute was prematurely closed by the death of Leibnitz; but the attack was soon resumed by minor assailants. The Pantheistic tendencies of Clarke's doctrine have been well exposed by a recent French writer: /113/
* "Deus durat semper et adest ubique, et existendo semper et ubique durationem et spatinm, seternitatem et infinitatem, constituit." (Principia Mathematica, Schol. Gen.) For some distinctions, mainly judicious, between Space Absolute and Rela tive, compare Definitio VIII., Schol.
** " Dicerem primo spatium revera dari, distinctum a magnitudine; hoc est, illo nomine designari quid, ei conceptum respondere, fundatum in re, alium a conceptu, magnitudinis, ac ita quidem ut ubi non existit magnitudo, quamvis ea non existeret omnino, spatium nihilominus extiturum. Dicerem secundo, spatium non esse quid actu existens, actuque diversum a rebus quantis, nedum ut habeat dimensiones aliquas sibi proprias, a magnitudinis dimensionibus actu separatas Spatium nihii est aliud quam pura puta potentia, mera capacitas, ponibilitas, aut (vocabulis istis veniam) interponibilitas magnitudinis alicujus." Mathematical Works, Whew- ell's ed., p. 158.
*** Th.-Henri Martin, Examen d'un Problems de Theodicee, 1859, p. 28.
Since Clarke's day, few notable ontological theories in regard to Space have, so far as we know, been brought forward. Following the wise example of Locke, who modestly confesses his ignorance (Essay on Human Understanding, II. 13, 15.), the Scotch school and the kindred French school, as well as Cousin and Comte,* have refrained from positively /114/ dogmatizing on the subject. It would have been discreet in the German philosophers to have observed a similar reticence. Schelling denied Space, "Pure being, with the negation of all activity," as he defined Time, "Pure activity, with the negation of all being"; definitions possessing some poetical beauty, but destitute of any precise meaning; while Hegel styled it the "first determination of the Idea (i. e. Absolute Being) in the world of nature." (See Dictionnaire des Sciences Philosophiques, s. v. Espace.) Trendelenburg (Logische Untersuchungen, Vol. I. pp. 131-232 (ed. 1862).) deduces the idea of Space from the idea of Motion, arguing that the pure a priori intuitions of figures in Space are impossible except through a process of mental construction, and that this mental construction or description necessarily involves Motion. By citations from the Critique of Pure Reason, he shows that Kant, on his own theory, ought to have made Motion the condition of all pure intuitions. Not content, however, with his psychological deduction, which seems plausible if we admit his identification of Space with Extension, and sensuous imagination with pure thought, he also holds the strange ontological theory that Space, objectively considered, is "extended and created" by the motion of physical forces.** In the Philosophical Remains of the poet Jens Baggesen, published in 1863, appears an ingenious theory of Space and Time, which tempts us to apply the doctrine of metempsychosis to philosophical hypotheses, inasmuch as it is a revivification of Clarke's speculation, though in an entirely new body. We commend it to the attention of those who are fond of juggling with the words Existence, Nothing, Being, and Becoming. (Philos. Nachlass, Vol. II. pp. 145, 146.) /115/ Baggesen's rhapsody recalls to mind the fine illustration of Hume: "Men of bright fancies may, in this respect, be compared to those angels whom the Scripture represents as covering their eyes with their wings."
* In no part of his works does Comte more conclusively prove his need of discipline at the hands of his despised "metaphysicians," than in his account of the empirical origination of the idea of Space, of which this is a sample: "Quant a la nature physique de cet espace indefini, nous devons spontane'ment nous le repre- senter, pour plus de facilite, com me analogue au milieu effectif dans lequel nous vivons, tellement que si ce milieu etait liquide, au lieu d'etre gazeux, notre espace geometrique serait sans doute con9u aussi comme liquide." (Cours de Philosophie Positive, Tome I. p. 353.) He plainly has not soared above Extension, and unconsciously confirms the doctrine of our Sec. 9 above.
** "Wie sich die Vorstellung den Raum erst dehnen und schaffen muss, so dehnen und schaffen ihn ausser uns ewige Krafte." (Ibid. p. 218.) To a limited extent, E. V. Neale accepts Trendelenburg's psychological theory (Analogy of Thought and Nature, 1863, pp. 28, 29) : "As Trendelenburg has shown, all attempts to explain the thought of Space made by the profoundcst thinkers, either imply the thought of motion, or fall into absurdity The thought of Space is no sooner formed, than it distinguishes itself into two opposite thoughts, that of centre and circumference; which imply, while they deny each other. 7 ' Such a "thought of Space" is clearly a sensuous image of Extension. Morell, in his Introduction to Mental Philosophy, 1862, pp. 115-148, adopts Trendelenburg'a theory more completely, and makes no distinction between thought and imagination.
29. The true solution of the ontological problem of pure Space is involved in the solutions we have obtained of the psychological and logical problems. It has been shown that Space is objectively real, and it has likewise been shown that we cognize it only by negative characteristics. Hence it is. evident that no theory in regard to the positive nature of Space per se is possible, which does not transcend the limits of human knowledge. All ontological theories, therefore, apart from special refutation, are refuted en masse by this simple consideration. The whole of the little knowledge we can derive from these negative characteristics may be summed up briefly in the following definition:
Space is the infinite and indivisible Receptacle of Matter
By drawing a distinction between Pretension and Time, analogous, with some slight modifications, to that drawn between Extension and Space, the idea of Time is likewise resolvable into the notions of receptivity, unity, and infinity; with this difference, that the receptivity of Time is not the negation of matter, but of existences. In fact, eternity is the synonym of pure Time.* Hence we define Time as follows:
Time is the infinite and indivisible Receptacle of Existences.
* L'immensité ou 1'unité de 1'espace, 1'éternitée ou 1'unité de temps (Cousin, Hist, de la Phil, du XVII Siëcle, Introd., p. 121.) Compare Royer-Collard, in Jouffroy's Reid, Vol. III. p. 434: "La duree se perd dans 1'eternite, comme Tespace dans rimmensite'." The negativity of our cognitions of Space and Time forbids any dogmatic exposition of their mutual relations; but we cannot forbear citing a passage from Reid, expressed with remarkable dignity and vigor: "All limited duration is comprehended in Time, and all limited extension in Space. These, in their capacious womb, contain all finite existences, but are contained by none. Created things have their particular place in Space, and their particular place in Time; but Time is everywhere, and Space at all times. They embrace each other, and have that mysterious union which the schoolmen conceive between soul and body, the whole of each is in every part of the other." (Intellectual Powers of Man, Ess. III. Chap. II.) This passage is translated almost word for word by Royer-Collard, without acknowledgment. See Jouffroy's Reid, Vol. IV. p. 441. /116/
30. It will be seen that these definitions express no more than is contained in the instinctive judgments of mankind; and we rejoice that our analysis may thus justly claim the general approbation. For it is unphilosophic in the last degree to despise those naturae judicia, or ultimate beliefs, for which men commonly can assign no reason, yet which are fortunately too deeply rooted to be shaken by reasoning. Common sense is the well at the bottom of which lies Truth. It is the high and true function of philosophy to "convert alêthês doxa into epistêmê, the right opinion into scienceto clarify and elucidate the thought that lies crude in the universal understanding of the race, rather than to rear gigantic superstructures for vanity to dwell in.
One word as to the general result of our critique, and we will close. As Transcendentalism, starting from the a priori cognition of Space, denies the a posteriori cognition of Extension; so Empiricism, starting from the a posteriori cognition of Extension, denies the a priori cognition of Space. Each repudiates a truth possessed by the other, and grounds its thesis on the identification of Space and Extension. The only means of their common refutation is the establishment of a profound distinction between Space and Extension, and a rigorous adhesion to it. It then becomes apparent that, though distinguished by antithetical characteristics, and opposed as two terms of a relation, Space and Extension are united by an absolute and necessary nexus; in a word, that the cognition of one is possible only through that of the other. Extension is known only as contained in Space; Space is known only as containing Extension. But inasmuch as Space is cognized solely a priori, and Extension solely a posteriori, the recognition by philosophy of their absolute and necessary correlation becomes a bridge whereby the chasm between the subjective and the objective may be spanned, and whereby Thought may be brought face to face with Existence. From a profoundly true philosophy of Space, therefore, much light will be thrown on the fundamental problem of all philosophy, the validity of human knowledge. In a second paper we design to apply some of the foregoing conclusions to Hamilton's Law of the Conditioned, and to the cognition of the Infinite in general.
END OF: F. A. Abbot's "The Philosophy of Space and Time"
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