The Mystery of Space

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ric Space Differentiated-The Finity of Space-Difference Between the Purely Formal and the Ac

o-called fourth dimension is an aspect thereof. It is now deemed fitting that we should enter into a more detai

al. In its application to space there is involved a great deal of confusion because of the inferential character of its definition. For instance, commonly we measure a body in spa

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f the position of the point b. In this way geometers have determined that our space is tridimensional; but it is obvious that this conclusion is based not upon any examination of space itself but upon the measurement of bodies in space. Upon this view it is seen that conclusions based upon such a procedure render our notion of the extension of bodies in space identical with the notion of spatial extensity. In other words, we take bodies in space and by examining their characteristics and properties arrive at an alleged apodeictic judgment of space. It is by means of this conventional norm of geometric knowledge that various other spaces, notably the one-, two-, four-and n-space, have been devised. It would appear that if some more absolute standard of measurement or definition of space were adopted the confusion which now clings to the conception of dimension could be

mensionality is very aptly described

A space being given, its dimensionality is not therewith determined, but depends upon the will of the investigator who by a proper choice of

determining the dimension and essential nature of real space. Its significance for philosophy lies in the need of a logical, rigidly exclusive and absolutely peculiar standard of space definition.

ly drawn. So that when reference is made to either there will be no doubt as to which is meant. And then, too, conceptual space is no space at all,

rd-stick, the kilometer, by which it is measured and apportioned. It is logically impossible to predicate the same conclusion for both of them. That is, to do so causes a profound fracture of the fundamental norms of logic. Such conclusions being thus illeg

ment so far removed from this primary congruence as the non-Euclidean system is should exhibit more obvious signs of correspondence? But the advocates of the n-dimensionality of space have illatively asserted the identity of space and its dimensions. Accordingly, there is not recognized any

hat if, as is commonly done, dimension be made to mean direction of extent, there would be an unlimited number of directions of extent and they would all be perceptible. But this is really another fundamental fault. Non-Euclideans have stretched

e have, by adding a fourth or n-dimensions, involved ourselves into a more complex situation than before? For by postulating a fourth dimension either we have created a new world whose dimensions are four in number or we have expl

arth in which space extends in four primary directions and in which four co?rdinates are necessary for its determination or we are driven to the other horn of the dilemma where we are brought face to face with the conclusion that the three perceptual space dimensions have in common a hitherto unknown property or extension in virtue of which it may be viewed

ons as there are directions. Primarily, there are six directions of space and an unlimited number of subsidiary directions. On this view it is not necessary to invent a new domain of space if the object be merely to discover and utilize a greater number of dimensio

e definition leaves out of account the idea that space is at the same time a direction or collection of directions. The term extension is generic and when applied to space means extension in all

s there are systems of space-measurement and the latter may be limitless. But if the totality of spaces are to be viewed as one space then we shall have one space with an indefinite number of dimensions; also an indefinite number of space measurements which woul

n for identifying it with these norms. Here we have the source of all error in mathematical conclusions about the nature of space; because all such conclusions are based not upon the intrinsic nature of space, but upon artificial forms which we choose to impose upon it for our own convenience. But it should be remembered that the irregularities which we note are not in space itself but inhere in the forms which we use. For t

gain the question as to just what are the limits of mathematical certainty and necessity and just how far we shall depend upon the validity of mathematics to determine for us absolutely certain conclusions about the nature of space. In view of the uncertainty noted, are we justified in following too closely the mathematic lead even in matters of logic, to say nothing o

f knowledge. It is perhaps well that we should expunge the word absolute from our vocabularies. It is really a misnomer and has no m

ange its qualities as it evolves. All knowledge is also relative and what is true of one state may not be true of another; what is true of one life may not be true of another life; the limitations of one degree of knowledge may not have any bearings upon another degree. The norms of one will not satisfy the conditions of another stage of manifestation. It is always within limits that the criterion of knowledge will be found to satisfy a given set of conditions. Hence within certain limits mathematical conclusions will maintain their validity. Error is committe

f sectarian fanatics. To them it is quite clear there may be two straight lines through

tioned by Carus, that he can with evident lack of mental perturbation proclaim that two straight lines, noncoincident with each other, may pass through a point and yet be parallel to a third line. But this is a mere trifle, a bagatelle, to the many other infractions of which he is guilty. The wonder is that he is able to secure such obsequious ac

mit, unending in extent and eternally existent. Riemann, whose thought life found its most convenient mode of expression by means of pure mathematics, was the first in the history of human thought to surmise that space is not infinite but limited even though unbounded. But his conception has been much vitiated on account of its entanglement with an idealized construction by which space is regarded as a thing to be manipulated and generated by act of thought. Were it not for this his conception would indeed mark the beginning of a new era in psychogenesis. As it

ing to take a new view of space in general. The departure is especial

The trend of Physics, say with such a representative as Ostwald, is to make things the expression of force; the constitutio

not an empty void. They know that the atmosphere fills a great deal of space. They also have extended their conception in this direction to include the ether and occultism goes further and postulates four kinds of ether-the chemical

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convenient for it; but everything makes its own spaciousness by its own defensi

r own space but the appearance of that in which no physical object is makes room for itself by its own dynamic for

f. And this very force-appearance, constituting space, is that which makes it perceivable. For instance, let us say the space that exists between the earth and the moon, is not really empty nor does it have an existence prior to itself, but is a mode of dynamic appearance which is the cause of its own existence. Its dynamic character makes it to appear p

transcends the comprehension of finite minds or processes. It is finite because it is in manifestation. Everything that is in manifestation is finite. The infinite is not in manifestation. Infinity has to be l

opolar with wrong; good is copolar with evil; night with day; manifestation with non-manifestation; truth with falsity; infinity with finity and so on, throughout the whole gamut of the pairs of opposites. What is the logical inference? Spac

come dangerously near to imagining it. The distance from excogitating upon, discussing and describing the properties of four-space to imagining it is not so great after all. Truly it is difficult indeed, it seems, to be able to describe a thing yet not be able to imagine or make a mental image of it. There is an evident fallacy here. Either the description of four-space

gether, in their manifestation of action and interaction, in their dynamic appearance, if you please, they constitute space. If these were removed with all that their existence implies there would result a condition of spacelessness i

ctual. The universe does not exist as pure form, neither does space. As purely formal constructions of the intellect these can have no perceptible existence. The phenomenal or sensible may not be judged by exactly the same standard as the formal. The phenomenal or sensible represents things as they appear to the senses, or, so far as the actualized universe is concerned, as they really are. The formal represents things as they are made to appear by the mind. It cannot be actualized. It may be said that the purely formal is the limit of e

licable. Transfinity is much better and more accurate. Space is transfinite because its scope is gr

; but was compelled, by his own limitations, to make an idealized construction in which he could actualize his conception. And for real, dynamic space, he substituted his ideal construction and proc

which is truly symbolic of the finity of the universe and the infinity of the archetypal. Just as a square or a series of polygonal figures inscribed in a circle cannot be made to coincide exactly with the circle so cannot the actual be made to coincide with the ideal. The circumference of the circle is the unapproachable limit of inscribed squares. If it were possible so to multiply squares thus inscribed that a figure coincident with the circumf

g element it is still more difficult to understand the connotation. But even granting this view, are we not compelled to recognize the dynamism of space as a necessary inference? Another definition which Carus gives is that space is a "pure form of extension." If it be granted that space is a pure form of extension we should have to conclude that it has no actual existence; for pure form does not exist except as an idealized

s, or sense experience. All knowledge therefore whose origin can be traced to the senses is knowledge a posteriori. Now, knowledge a priori should be just the opposite of this. It should indicate such knowledge as that which does not have its origin in the senses, or which is not dependent upon the ordinary avenues of sense-experience. Abstract thought is as truly experience as smelling, seeing or hearing. It is by traversing its scope of motility that the mind finds out what the norms of logic are. It could not remain quiescent and discover them. It has to be active, examining, co

stitute the principle of apriority in the mind. Aprioriness is a principle of mind partaking of the very nature and essence of mind. It is the very mainspring of mentality. Perception and conception are processes which the mind performs intuitively. The mind perceives and conceives because it is impossible for the normal mind to do otherwise. We take a view upon a given question; we assume certain

sential principle of mind. It should not be identified

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produce a field of abstraction in which we can construct purely formal combination, such as numbers, or the ideas of types and species. Thus we create a world of pure thought which has the adv

ich would be placed both the process of abstraction and its results. It is therefore exceedingly difficult to understand why so cursory a consi

al; the one the knower and the other the known. Thus, the only reason that the formal is found to be answerable to the a priori at all is due to the fact that it is construable only upon the basis of the a priori. But being so is not sufficient warrant for its identification with the a priori. The formal merely represents the totality of possibilities in the universe as viewed by the mind; but as

diverse tendences, qualities and characteristics. Apriorily, it acts in unison with the eternal purpose of life and the universal mind

tural processes, and every activity connected therewith belong to the category of the a priori. Cells multiply, divide, build up and tear down tissues and they do it intuitively. Most certainly these functions are perform

tution of things and of the universe. And if so, it is right and without illusion. It is also evident that whatever notion a posteriori the intellect may entertain with respect to these questions is unavoidably l

which is found to exist in the mind as an eternal aspect of its nature? Does the mind intuitively measure its contents or its operations by the empirical standard of space-measurement known as dimension? Is the attitude of the mind towards the objectively real one of discrimination a priori as to the direction or dimension in which a percept may originate? In other words, does the mind habitually and intuitively refer its data to a system of co?rdinates for final determination? There is no other answer but that the mind makes no such reference and is dependent upon no kind of

and appropriate to devise an additional ordinate or dimension to satisfy the necessities of its more complex ramifications into the nature of things and to determine their greatly increased space-relations. It may be even possible for the mind to function normally in a space of four dimensions. But this would simply be a new adjustment, not a change in the essential nature of mind. It would be like the series of adjustments to environments which man has made in the onward movement of civilization. There has been no serious change in the manhood per se of man. That has remained t